Michael W. Mahoney

ICLR Conference 2025 Conference Paper

A Statistical Framework for Ranking LLM-based Chatbots

• Siavash Ameli
• Siyuan Zhuang
• Ion Stoica
• Michael W. Mahoney

Large language models (LLMs) have transformed natural language processing, with frameworks like Chatbot Arena providing pioneering platforms for evaluating these models. By facilitating millions of pairwise comparisons based on human judgments, Chatbot Arena has become a cornerstone in LLM evaluation, offering rich datasets for ranking models in open-ended conversational tasks. Building upon this foundation, we propose a statistical framework that incorporates key advancements to address specific challenges in pairwise comparison analysis. First, we introduce a factored tie model that enhances the ability to handle ties—an integral aspect of human-judged comparisons—significantly improving the model's fit to observed data. Second, we extend the framework to model covariance between competitors, enabling deeper insights into performance relationships and facilitating intuitive groupings into performance tiers. Third, we resolve optimization challenges arising from parameter non-uniqueness by introducing novel constraints, ensuring stable and interpretable parameter estimation. Through rigorous evaluation and extensive experimentation, our framework demonstrates substantial improvements over existing methods in modeling pairwise comparison data. To support reproducibility and practical adoption, we release leaderbot, an open-source Python package implementing our models and analyses.

ICML Conference 2025 Conference Paper

Determinant Estimation under Memory Constraints and Neural Scaling Laws

• Siavash Ameli
• Chris van der Heide
• Liam Hodgkinson
• Fred Roosta
• Michael W. Mahoney

Calculating or accurately estimating log-determinants of large positive definite matrices is of fundamental importance in many machine learning tasks. While its cubic computational complexity can already be prohibitive, in modern applications, even storing the matrices themselves can pose a memory bottleneck. To address this, we derive a novel hierarchical algorithm based on block-wise computation of the LDL decomposition for large-scale log-determinant calculation in memory-constrained settings. In extreme cases where matrices are highly ill-conditioned, accurately computing the full matrix itself may be infeasible. This is particularly relevant when considering kernel matrices at scale, including the empirical Neural Tangent Kernel (NTK) of neural networks trained on large datasets. Under the assumption of neural scaling laws in the test error, we show that the ratio of pseudo-determinants satisfies a power-law relationship, allowing us to derive corresponding scaling laws. This enables accurate estimation of NTK log-determinants from a tiny fraction of the full dataset; in our experiments, this results in a $\sim$100, 000$\times$ speedup with improved accuracy over competing approximations. Using these techniques, we successfully estimate log-determinants for dense matrices of extreme sizes, which were previously deemed intractable and inaccessible due to their enormous scale and computational demands.

TMLR Journal 2025 Journal Article

Elucidating the Design Choice of Probability Paths in Flow Matching for Forecasting

• Soon Hoe Lim
• Yijin Wang
• Annan Yu
• Emma Hart
• Michael W. Mahoney
• Sherry Li
• N. Benjamin Erichson

Flow matching has recently emerged as a powerful paradigm for generative modeling and has been extended to probabilistic time series forecasting. However, the impact of the specific choice of probability path model on forecasting performance, particularly for high-dimensional spatio-temporal dynamics, remains under-explored. In this work, we demonstrate that forecasting spatio-temporal data with flow matching is highly sensitive to the selection of the probability path model. Motivated by this insight, we propose a novel probability path model designed to improve forecasting performance. Our empirical results across various dynamical system benchmarks show that our model achieves faster convergence during training and improved predictive performance compared to existing probability path models. Importantly, our approach is efficient during inference, requiring only a few sampling steps. This makes our proposed model practical for real-world applications and opens new avenues for probabilistic forecasting.

ICML Conference 2025 Conference Paper

Enhancing Foundation Models for Time Series Forecasting via Wavelet-based Tokenization

• Luca Masserano
• Abdul Fatir Ansari
• Boran Han
• Xiyuan Zhang
• Christos Faloutsos
• Michael W. Mahoney
• Andrew Gordon Wilson
• Youngsuk Park

How to best develop foundational models for time series forecasting remains an important open question. Tokenization is a crucial consideration in this effort: what is an effective discrete vocabulary for a real-valued sequential input? To address this question, we develop WaveToken, a wavelet-based tokenizer that allows models to learn complex representations directly in the space of time-localized frequencies. Our method first scales and decomposes the input time series, then thresholds and quantizes the wavelet coefficients, and finally pre-trains an autoregressive model to forecast coefficients for the forecast horizon. By decomposing coarse and fine structures in the inputs, wavelets provide an eloquent and compact language for time series forecasting that simplifies learning. Empirical results on a comprehensive benchmark, including 42 datasets for both in-domain and zero-shot settings, show that WaveToken: i) performs on par or better than recently proposed foundation models for forecasting while using a much smaller vocabulary (1024 tokens), and is competitive with modern deep learning models trained specifically on each dataset; ii) exhibits superior generalization capabilities, achieving the best average rank across all datasets for three complementary metrics; and iii) easily captures complex temporal patterns of practical relevance that are challenging for other recent pre-trained models, including trends, sparse spikes, and non-stationary time series with varying frequencies evolving over time.

ICML Conference 2025 Conference Paper

Fundamental Bias in Inverting Random Sampling Matrices with Application to Sub-sampled Newton

• Chengmei Niu
• Zhenyu Liao 0001
• Zenan Ling
• Michael W. Mahoney

A substantial body of work in machine learning (ML) and randomized numerical linear algebra (RandNLA) has exploited various sorts of random sketching methodologies, including random sampling and random projection, with much of the analysis using Johnson–Lindenstrauss and subspace embedding techniques. Recent studies have identified the issue of inversion bias – the phenomenon that inverses of random sketches are not unbiased, despite the unbiasedness of the sketches themselves. This bias presents challenges for the use of random sketches in various ML pipelines, such as fast stochastic optimization, scalable statistical estimators, and distributed optimization. In the context of random projection, the inversion bias can be easily corrected for dense Gaussian projections (which are, however, too expensive for many applications). Recent work has shown how the inversion bias can be corrected for sparse sub-gaussian projections. In this paper, we show how the inversion bias can be corrected for random sampling methods, both uniform and non-uniform leverage-based, as well as for structured random projections, including those based on the Hadamard transform. Using these results, we establish problem-independent local convergence rates for sub-sampled Newton methods.

ICLR Conference 2025 Conference Paper

Gradient-Free Generation for Hard-Constrained Systems

• Chaoran Cheng
• Boran Han
• Danielle C. Maddix
• Abdul Fatir Ansari
• Andrew Stuart
• Michael W. Mahoney
• Yuyang Wang 0001

Generative models that satisfy hard constraints are critical in many scientific and engineering applications, where physical laws or system requirements must be strictly respected. Many existing constrained generative models, especially those developed for computer vision, rely heavily on gradient information, which is often sparse or computationally expensive in some fields, e.g., partial differential equations (PDEs). In this work, we introduce a novel framework for adapting pre-trained, unconstrained flow-matching models to satisfy constraints exactly in a zero-shot manner without requiring expensive gradient computations or fine-tuning. Our framework, *ECI sampling*, alternates between extrapolation (E), correction (C), and interpolation (I) stages during each iterative sampling step of flow matching sampling to ensure accurate integration of constraint information while preserving the validity of the generation. We demonstrate the effectiveness of our approach across various PDE systems, showing that ECI-guided generation strictly adheres to physical constraints and accurately captures complex distribution shifts induced by these constraints. Empirical results demonstrate that our framework consistently outperforms baseline approaches in various zero-shot constrained generation tasks and also achieves competitive results in the regression tasks without additional fine-tuning.

ICLR Conference 2025 Conference Paper

HOPE for a Robust Parameterization of Long-memory State Space Models

• Annan Yu
• Michael W. Mahoney
• N. Benjamin Erichson

State-space models (SSMs) that utilize linear, time-invariant (LTI) systems are known for their effectiveness in learning long sequences. To achieve state-of-the-art performance, an SSM often needs a specifically designed initialization, and the training of state matrices is on a logarithmic scale with a very small learning rate. To understand these choices from a unified perspective, we view SSMs through the lens of Hankel operator theory. Building upon it, we develop a new parameterization scheme, called HOPE, for LTI systems that utilizes Markov parameters within Hankel operators. Our approach helps improve the initialization and training stability, leading to a more robust parameterization. We efficiently implement these innovations by nonuniformly sampling the transfer functions of LTI systems, and they require fewer parameters compared to canonical SSMs. When benchmarked against HiPPO-initialized models such as S4 and S4D, an SSM parameterized by Hankel operators demonstrates improved performance on Long-Range Arena (LRA) tasks. Moreover, our new parameterization endows the SSM with non-decaying memory within a fixed time window, which is empirically corroborated by a sequential CIFAR-10 task with padded noise.

ICLR Conference 2025 Conference Paper

Mitigating Memorization in Language Models

• Mansi Sakarvadia
• Aswathy Ajith
• Arham Mushtaq Khan
• Nathaniel C. Hudson
• Caleb Geniesse
• Kyle Chard
• Yaoqing Yang 0002
• Ian T. Foster

Language models (LMs) can “memorize” information, i.e., encode training data in their weights in such a way that inference-time queries can lead to verbatim regurgitation of that data. This ability to extract training data can be problematic, for example, when data are private or sensitive. In this work, we investigate methods to mitigate memorization: three regularizer-based, three fine-tuning-based, and eleven machine unlearning-based methods, with five of the latter being new methods that we introduce. We also introduce TinyMem, a suite of small, computationally-efficient LMs for the rapid development and evaluation of memorization-mitigation methods. We demonstrate that the mitigation methods that we develop using TinyMem can successfully be applied to production-grade LMs, and we determine via experiment that: regularizer-based mitigation methods are slow and ineffective at curbing memorization; fine-tuning-based methods are effective at curbing memorization, but overly expensive, especially for retaining higher accuracies; and unlearning-based methods are faster and more effective, allowing for the precise localization and removal of memorized information from LM weights prior to inference. We show, in particular, that our proposed unlearning method BalancedSubnet outperforms other mitigation methods at removing memorized information while preserving performance on target tasks.

ICML Conference 2025 Conference Paper

Models of Heavy-Tailed Mechanistic Universality

• Liam Hodgkinson
• Zhichao Wang
• Michael W. Mahoney

Recent theoretical and empirical successes in deep learning, including the celebrated neural scaling laws, are punctuated by the observation that many objects of interest tend to exhibit some form of heavy-tailed or power law behavior. In particular, the prevalence of heavy-tailed spectral densities in Jacobians, Hessians, and weight matrices has led to the introduction of the concept of heavy-tailed mechanistic universality (HT-MU). Multiple lines of empirical evidence suggest a robust correlation between heavy-tailed metrics and model performance, indicating that HT-MU may be a fundamental aspect of deep learning efficacy. Here, we propose a general family of random matrix models—the high-temperature Marchenko-Pastur (HTMP) ensemble —to explore attributes that give rise to heavy-tailed behavior in trained neural networks. Under this model, spectral densities with power laws on (upper and lower) tails arise through a combination of three independent factors (complex correlation structures in the data; reduced temperatures during training; and reduced eigenvector entropy), appearing as an implicit bias in the model structure, and they can be controlled with an "eigenvalue repulsion” parameter. Implications of our model on other appearances of heavy tails, including neural scaling laws, optimizer trajectories, and the five-plus-one phases of neural network training, are discussed.

ICML Conference 2025 Conference Paper

QuantSpec: Self-Speculative Decoding with Hierarchical Quantized KV Cache

• Rishabh Tiwari
• Haocheng Xi
• Aditya Tomar
• Coleman Richard Charles Hooper
• Sehoon Kim 0001
• Maxwell Horton
• Mahyar Najibi
• Michael W. Mahoney

Large Language Models (LLMs) are increasingly being deployed on edge devices for long-context settings, creating a growing need for fast and efficient long-context inference. In these scenarios, the Key-Value (KV) cache is the primary bottleneck in terms of both GPU memory and latency, as the full KV cache must be loaded for each decoding step. While speculative decoding is a widely accepted technique to accelerate autoregressive decoding, existing methods often struggle to achieve significant speedups due to inefficient KV cache optimization strategies and result in low acceptance rates. To address these challenges, we propose a novel self-speculative decoding framework, QuantSpec, where the draft model shares the architecture of the target model but employs a hierarchical 4-bit quantized KV cache and 4-bit quantized weights for acceleration. QuantSpec maintains high acceptance rates ($>$90%) and reliably provides consistent end-to-end speedups upto $\sim2. 5\times$, outperforming other self-speculative decoding methods that use sparse KV cache for long-context LLM inference. QuantSpec also reduces the memory requirements by $\sim 1. 3\times$ compared to these alternatives.

ICLR Conference 2025 Conference Paper

Tuning Frequency Bias of State Space Models

• Annan Yu
• Dongwei Lyu
• Soon Hoe Lim
• Michael W. Mahoney
• N. Benjamin Erichson

State space models (SSMs) leverage linear, time-invariant (LTI) systems to effectively learn sequences with long-range dependencies. By analyzing the transfer functions of LTI systems, we find that SSMs exhibit an implicit bias toward capturing low-frequency components more effectively than high-frequency ones. This behavior aligns with the broader notion of frequency bias in deep learning model training. We show that the initialization of an SSM assigns it an innate frequency bias and that training the model in a conventional way does not alter this bias. Based on our theory, we propose two mechanisms to tune frequency bias: either by scaling the initialization to tune the inborn frequency bias; or by applying a Sobolev-norm-based filter to adjust the sensitivity of the gradients to high-frequency inputs, which allows us to change the frequency bias via training. Using an image-denoising task, we empirically show that we can strengthen, weaken, or even reverse the frequency bias using both mechanisms. By tuning the frequency bias, we can also improve SSMs' performance on learning long-range sequences, averaging an $88.26\\%$ accuracy on the Long-Range Arena (LRA) benchmark tasks.

TMLR Journal 2024 Journal Article

$\clubsuit$ CLOVER $\clubsuit$: Probabilistic Forecasting with Coherent Learning Objective Reparameterization

• Kin G. Olivares
• Geoffrey Négiar
• Ruijun Ma
• Oinam Nganba Meetei
• Mengfei Cao
• Michael W. Mahoney

Obtaining accurate probabilistic forecasts is an operational challenge in many applications, such as energy management, climate forecasting, supply chain planning, and resource allocation. Many of these applications present a natural hierarchical structure over the forecasted quantities; and forecasting systems that adhere to this hierarchical structure are said to be coherent. Furthermore, operational planning benefits from the accuracy at all levels of the aggregation hierarchy. However, building accurate and coherent forecasting systems is challenging: classic multivariate time series tools and neural network methods are still being adapted for this purpose. In this paper, we augment an MQForecaster neural network architecture with a modified multivariate Gaussian factor model that achieves coherence by construction. The factor model samples can be differentiated with respect to the model parameters, allowing optimization on arbitrary differentiable learning objectives that align with the forecasting system's goals, including quantile loss and the scaled Continuous Ranked Probability Score (CRPS). We call our method the Coherent Learning Objective Reparametrization Neural Network (CLOVER). In comparison to state-of-the-art coherent forecasting methods, CLOVER achieves significant improvements in scaled CRPS forecast accuracy, with average gains of 15%, as measured on six publicly-available datasets.

NeurIPS Conference 2024 Conference Paper

AlphaPruning: Using Heavy-Tailed Self Regularization Theory for Improved Layer-wise Pruning of Large Language Models

• Haiquan Lu
• Yefan Zhou
• Shiwei Liu
• Zhangyang Wang
• Michael W. Mahoney
• Yaoqing Yang

Recent work on pruning large language models (LLMs) has shown that one can eliminate a large number of parameters without compromising performance, making pruning a promising strategy to reduce LLM model size. Existing LLM pruning strategies typically assign uniform pruning ratios across layers, limiting overall pruning ability; and recent work on layerwise pruning of LLMs is often based on heuristics that can easily lead to suboptimal performance. In this paper, we leverage Heavy-Tailed Self-Regularization (HT-SR) Theory, in particular the shape of empirical spectral densities (ESDs) of weight matrices, to design improved layerwise pruning ratios for LLMs. Our analysis reveals a wide variability in how well-trained, and thus relatedly how prunable, different layers of an LLM are. Based on this, we propose AlphaPruning, which uses shape metrics to allocate layerwise sparsity ratios in a more theoretically-principled manner. AlphaPruning can be used in conjunction with multiple existing LLM pruning methods. Our empirical results show that AlphaPruning prunes LLaMA-7B to 80% sparsity while maintaining reasonable perplexity, marking a first in the literature on LLMs.

ICML Conference 2024 Conference Paper

An LLM Compiler for Parallel Function Calling

• Sehoon Kim 0001
• Suhong Moon
• Ryan Tabrizi
• Nicholas Lee
• Michael W. Mahoney
• Kurt Keutzer
• Amir Gholami

The reasoning capabilities of the recent LLMs enable them to execute external function calls to overcome their inherent limitations, such as knowledge cutoffs, poor arithmetic skills, or lack of access to private data. This development has allowed LLMs to select and coordinate multiple functions based on the context to tackle more complex problems. However, current methods for function calling often require sequential reasoning and acting for each function which can result in high latency, cost, and sometimes inaccurate behavior. To address this, we introduce LLMCompiler, which executes functions in parallel to efficiently orchestrate multiple function calls. Drawing inspiration from the principles of classical compilers, LLMCompiler enables parallel function calling with three components: (i) a Function Calling Planner, formulating execution plans for function calling; (ii) a Task Fetching Unit, dispatching function calling tasks; and (iii) an Executor, executing these tasks in parallel. LLMCompiler automatically generates an optimized orchestration for the function calls and can be used with both open-source and closed-source models. We have benchmarked LLMCompiler on a range of tasks with different patterns of function calling. We observe consistent latency speedup of up to $3. 7 \times$, cost savings of up to $6. 7 \times$, and accuracy improvement of up to $\sim 9 %$ compared to ReAct. Our code is available at https: //github. com/SqueezeAILab/LLMCompiler.

TMLR Journal 2024 Journal Article

Chronos: Learning the Language of Time Series

• Abdul Fatir Ansari
• Lorenzo Stella
• Ali Caner Turkmen
• Xiyuan Zhang
• Pedro Mercado
• Huibin Shen
• Oleksandr Shchur
• Syama Sundar Rangapuram

We introduce Chronos, a simple yet effective framework for pretrained probabilistic time series models. Chronos tokenizes time series values using scaling and quantization into a fixed vocabulary and trains existing transformer-based language model architectures on these tokenized time series via the cross-entropy loss. We pretrained Chronos models based on the T5 family (ranging from 20M to 710M parameters) on a large collection of publicly available datasets, complemented by a synthetic dataset that we generated via Gaussian processes to improve generalization. In a comprehensive benchmark consisting of 42 datasets, and comprising both classical local models and deep learning methods, we show that Chronos models: (a) significantly outperform other methods on datasets that were part of the training corpus; and (b) have comparable and occasionally superior zero-shot performance on new datasets, relative to methods that were trained specifically on them. Our results demonstrate that Chronos models can leverage time series data from diverse domains to improve zero-shot accuracy on unseen forecasting tasks, positioning pretrained models as a viable tool to greatly simplify forecasting pipelines.

NeurIPS Conference 2024 Conference Paper

Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning

• Wuyang Chen
• Jialin Song
• Pu Ren
• Shashank Subramanian
• Dmitriy Morozov
• Michael W. Mahoney

Recent years have witnessed the promise of coupling machine learning methods and physical domain-specific insights for solving scientific problems based on partial differential equations (PDEs). However, being data-intensive, these methods still require a large amount of PDE data. This reintroduces the need for expensive numerical PDE solutions, partially undermining the original goal of avoiding these expensive simulations. In this work, seeking data efficiency, we design unsupervised pretraining for PDE operator learning. To reduce the need for training data with heavy simulation costs, we mine unlabeled PDE data without simulated solutions, and we pretrain neural operators with physics-inspired reconstruction-based proxy tasks. To improve out-of-distribution performance, we further assist neural operators in flexibly leveraging a similarity-based method that learns in-context examples, without incurring extra training costs or designs. Extensive empirical evaluations on a diverse set of PDEs demonstrate that our method is highly data-efficient, more generalizable, and even outperforms conventional vision-pretrained models. We provide our code at https: //github. com/delta-lab-ai/data efficient nopt.

ICLR Conference 2024 Conference Paper

Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs

• Ilan Naiman
• N. Benjamin Erichson
• Pu Ren
• Michael W. Mahoney
• Omri Azencot

Generating realistic time series data is important for many engineering and scientific applications. Existing work tackles this problem using generative adversarial networks (GANs). However, GANs are unstable during training, and they can suffer from mode collapse. While variational autoencoders (VAEs) are known to be more robust to the these issues, they are (surprisingly) less considered for time series generation. In this work, we introduce Koopman VAE (KoVAE), a new generative framework that is based on a novel design for the model prior, and that can be optimized for either regular and irregular training data. Inspired by Koopman theory, we represent the latent conditional prior dynamics using a linear map. Our approach enhances generative modeling with two desired features: (i) incorporating domain knowledge can be achieved by leveraging spectral tools that prescribe constraints on the eigenvalues of the linear map; and (ii) studying the qualitative behavior and stability of the system can be performed using tools from dynamical systems theory. Our results show that KoVAE outperforms state-of-the-art GAN and VAE methods across several challenging synthetic and real-world time series generation benchmarks. Whether trained on regular or irregular data, KoVAE generates time series that improve both discriminative and predictive metrics. We also present visual evidence suggesting that KoVAE learns probability density functions that better approximate the empirical ground truth distribution.

NeurIPS Conference 2024 Conference Paper

How many classifiers do we need?

• Hyunsuk Kim
• Liam Hodgkinson
• Ryan Theisen
• Michael W. Mahoney

As performance gains through scaling data and/or model size experience diminishing returns, it is becoming increasingly popular to turn to ensembling, where the predictions of multiple models are combined to improve accuracy. In this paper, we provide a detailed analysis of how the disagreement and the polarization (a notion we introduce and define in this paper) among classifiers relate to the performance gain achieved by aggregating individual classifiers, for majority vote strategies in classification tasks. We address these questions in the following ways. (1) An upper bound for polarization is derived, and we propose what we call a neural polarization law: most interpolating neural network models are 4/3-polarized. Our empirical results not only support this conjecture but also show that polarization is nearly constant for a dataset, regardless of hyperparameters or architectures of classifiers. (2) The error rate of the majority vote classifier is considered under restricted entropy conditions, and we present a tight upper bound that indicates that the disagreement is linearly correlated with the error rate, and that the slope is linear in the polarization. (3) We prove results for the asymptotic behavior of the disagreement in terms of the number of classifiers, which we show can help in predicting the performance for a larger number of classifiers from that of a smaller number. Our theoretical findings are supported by empirical results on several image classification tasks with various types of neural networks.

NeurIPS Conference 2024 Conference Paper

KVQuant: Towards 10 Million Context Length LLM Inference with KV Cache Quantization

• Coleman Hooper
• Sehoon Kim
• Hiva Mohammadzadeh
• Michael W. Mahoney
• Yakun S. Shao
• Kurt Keutzer
• Amir Gholami

LLMs are seeing growing use for applications which require large context windows, and with these large context windows KV cache activations surface as the dominant contributor to memory consumption during inference. Quantization is a promising approach for compressing KV cache activations; however, existing solutions fail to represent activations accurately in sub-4-bit precision. Our work, KVQuant, facilitates low precision KV cache quantization by incorporating several novel methods: (i) Per-Channel Key Quantization, where we adjust the dimension along which we quantize the Key activations to better match the distribution; (ii) Pre-RoPE Key Quantization, where we quantize Key activations before the rotary positional embedding to mitigate its impact on quantization; (iii) Non-Uniform KV Cache Quantization, where we derive per-layer sensitivity-weighted non-uniform datatypes that better represent the distributions; and (iv) Per-Vector Dense-and-Sparse Quantization, where we isolate outliers separately for each vector to minimize skews in quantization ranges. By applying our method to the LLaMA, Llama-2, Llama-3, and Mistral models, we achieve < 0. 1 perplexity degradation with 3-bit quantization on both Wikitext-2 and C4, outperforming existing approaches. Our method enables serving LLaMA-7B with a context length of up to 1 million on a single A100-80GB GPU and up to 10 million on an 8-GPU system. We develop custom CUDA kernels for KVQuant, showing that we can achieve up to ~1. 7x speedups, compared to baseline fp16 matrix-vector multiplications, for the LLaMA-7B model.

ICLR Conference 2024 Conference Paper

Robustifying State-space Models for Long Sequences via Approximate Diagonalization

• Annan Yu
• Arnur Nigmetov
• Dmitriy Morozov
• Michael W. Mahoney
• N. Benjamin Erichson

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable ``perturb-then-diagonalize'' (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

NeurIPS Conference 2024 Conference Paper

Sharpness-diversity tradeoff: improving flat ensembles with SharpBalance

• Haiquan Lu
• Xiaotian Liu
• Yefan Zhou
• Qunli Li
• Kurt Keutzer
• Michael W. Mahoney
• Yujun Yan
• Huanrui Yang

Recent studies on deep ensembles have identified the sharpness of the local minima of individual learners and the diversity of the ensemble members as key factors in improving test-time performance. Building on this, our study investigates the interplay between sharpness and diversity within deep ensembles, illustrating their crucial role in robust generalization to both in-distribution (ID) and out-of-distribution (OOD) data. We discover a trade-off between sharpness and diversity: minimizing the sharpness in the loss landscape tends to diminish the diversity of individual members within the ensemble, adversely affecting the ensemble's improvement. The trade-off is justified through our rigorous theoretical analysis and verified empirically through extensive experiments. To address the issue of reduced diversity, we introduce SharpBalance, a novel training approach that balances sharpness and diversity within ensembles. Theoretically, we show that our training strategy achieves a better sharpness-diversity trade-off. Empirically, we conducted comprehensive evaluations in various data sets (CIFAR-10, CIFAR-100, TinyImageNet) and showed that SharpBalance not only effectively improves the sharpness-diversity trade-off but also significantly improves ensemble performance in ID and OOD scenarios.

ICML Conference 2024 Conference Paper

SqueezeLLM: Dense-and-Sparse Quantization

• Sehoon Kim 0001
• Coleman Richard Charles Hooper
• Amir Gholami
• Zhen Dong 0003
• Xiuyu Li
• Sheng Shen 0001
• Michael W. Mahoney
• Kurt Keutzer

Generative Large Language Models (LLMs) have demonstrated remarkable results for a wide range of tasks. However, deploying these models for inference has been a significant challenge due to their unprecedented resource requirements. This has forced existing deployment frameworks to use multi-GPU inference pipelines, which are often complex and costly, or to use smaller and less performant models. In this work, we demonstrate that the main bottleneck for generative inference with LLMs is memory bandwidth, rather than compute, specifically for single batch inference. While quantization has emerged as a promising solution by representing weights with reduced precision, previous efforts have often resulted in notable performance degradation. To address this, we introduce SqueezeLLM, a post-training quantization framework that not only enables lossless compression to ultra-low precisions of up to 3-bit, but also achieves higher quantization performance under the same memory constraint. Our framework incorporates two novel ideas: (i) sensitivity-based non-uniform quantization, which searches for the optimal bit precision assignment based on second-order information; and (ii) the Dense-and-Sparse decomposition that stores outliers and sensitive weight values in an efficient sparse format. When applied to the LLaMA models, our 3-bit quantization significantly reduces the perplexity gap from the FP16 baseline by up to 2. 1x as compared to the state-of-the-art methods with the same memory requirement. Furthermore, when deployed on an A6000 GPU, our quantized models achieve up to 2. 3x speedup compared to the baseline. Our code is available at https: //github. com/SqueezeAILab/SqueezeLLM.

ICML Conference 2024 Conference Paper

Towards Scalable and Versatile Weight Space Learning

• Konstantin Schürholt
• Michael W. Mahoney
• Damian Borth

Learning representations of well-trained neural network models holds the promise to provide an understanding of the inner workings of those models. However, previous work has either faced limitations when processing larger networks or was task-specific to either discriminative or generative tasks. This paper introduces the SANE approach to weight-space learning. SANE overcomes previous limitations by learning task-agnostic representations of neural networks that are scalable to larger models of varying architectures and that show capabilities beyond a single task. Our method extends the idea of hyper-representations towards sequential processing of subsets of neural network weights, thus allowing one to embed larger neural networks as a set of tokens into the learned representation space. SANE reveals global model information from layer-wise embeddings, and it can sequentially generate unseen neural network models, which was unattainable with previous hyper-representation learning methods. Extensive empirical evaluation demonstrates that SANE matches or exceeds state-of-the-art performance on several weight representation learning benchmarks, particularly in initialization for new tasks and larger ResNet architectures.

ICML Conference 2024 Conference Paper

Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs

• S. Chandra Mouli
• Danielle C. Maddix
• Shima Alizadeh
• Gaurav Gupta
• Andrew Stuart
• Michael W. Mahoney
• Yuyang Wang 0001

Existing work in scientific machine learning (SciML) has shown that data-driven learning of solution operators can provide a fast approximate alternative to classical numerical partial differential equation (PDE) solvers. Of these, Neural Operators (NOs) have emerged as particularly promising. We observe that several uncertainty quantification (UQ) methods for NOs fail for test inputs that are even moderately out-of-domain (OOD), even when the model approximates the solution well for in-domain tasks. To address this limitation, we show that ensembling several NOs can identify high-error regions and provide good uncertainty estimates that are well-correlated with prediction errors. Based on this, we propose a cost-effective alternative, DiverseNO, that mimics the properties of the ensemble by encouraging diverse predictions from its multiple heads in the last feed-forward layer. We then introduce Operator-ProbConserv, a method that uses these well-calibrated UQ estimates within the ProbConserv framework to update the model. Our empirical results show that Operator-ProbConserv enhances OOD model performance for a variety of challenging PDE problems and satisfies physical constraints such as conservation laws.

NeurIPS Conference 2023 Conference Paper

A Heavy-Tailed Algebra for Probabilistic Programming

• Feynman T. Liang
• Liam Hodgkinson
• Michael W. Mahoney

Despite the successes of probabilistic models based on passing noise through neural networks, recent work has identified that such methods often fail to capture tail behavior accurately---unless the tails of the base distribution are appropriately calibrated. To overcome this deficiency, we propose a systematic approach for analyzing the tails of random variables, and we illustrate how this approach can be used during the static analysis (before drawing samples) pass of a probabilistic programming language (PPL) compiler. To characterize how the tails change under various operations, we develop an algebra which acts on a three-parameter family of tail asymptotics and which is based on the generalized Gamma distribution. Our algebraic operations are closed under addition and multiplication; they are capable of distinguishing sub-Gaussians with differing scales; and they handle ratios sufficiently well to reproduce the tails of most important statistical distributions directly from their definitions. Our empirical results confirm that inference algorithms that leverage our heavy-tailed algebra attain superior performance across a number of density modeling and variational inference (VI) tasks.

ICML Conference 2023 Conference Paper

A Three-regime Model of Network Pruning

• Yefan Zhou
• Yaoqing Yang 0002
• Arin Chang
• Michael W. Mahoney

Recent work has highlighted the complex influence training hyperparameters, e. g. , the number of training epochs, can have on the prunability of machine learning models. Perhaps surprisingly, a systematic approach to predict precisely how adjusting a specific hyperparameter will affect prunability remains elusive. To address this gap, we introduce a phenomenological model grounded in the statistical mechanics of learning. Our approach uses temperature-like and load-like parameters to model the impact of neural network (NN) training hyperparameters on pruning performance. A key empirical result we identify is a sharp transition phenomenon: depending on the value of a load-like parameter in the pruned model, increasing the value of a temperature-like parameter in the pre-pruned model may either enhance or impair subsequent pruning performance. Based on this transition, we build a three-regime model by taxonomizing the global structure of the pruned NN loss landscape. Our model reveals that the dichotomous effect of high temperature is associated with transitions between distinct types of global structures in the post-pruned model. Based on our results, we present three case-studies: 1) determining whether to increase or decrease a hyperparameter for improved pruning; 2) selecting the best model to prune from a family of models; and 3) tuning the hyperparameter of the Sharpness Aware Minimization method for better pruning performance.

ECAI Conference 2023 Conference Paper

Adaptive Self-Supervision Algorithms for Physics-Informed Neural Networks

• Shashank Subramanian
• Robert M. Kirby
• Michael W. Mahoney
• Amir Gholami

Physics-informed neural networks (PINNs) incorporate physical knowledge from the problem domain as a soft constraint on the loss function, but recent work has shown that this can lead to optimization difficulties. Here, we study the impact of the location of the collocation points on the trainability of these models. We find that the vanilla PINN performance can be significantly boosted by adapting the location of the collocation points as training proceeds. Specifically, we propose a novel adaptive collocation scheme which progressively allocates more collocation points (without increasing their total number) to areas where the model is making higher errors (based on the gradient of the loss function in the domain). This, coupled with a judicious restarting of the training during any optimization stalls (by simply resampling the collocation points in order to adjust the loss landscape) leads to better estimates for the prediction error. We present results for several problems, including a 2D Poisson and diffusion-advection system with different forcing functions. We find that training vanilla PINNs for these problems can result in up to 70% prediction error in the solution, especially in the regime of low collocation points. In contrast, our adaptive schemes can achieve up to an order of magnitude smaller error, with similar computational complexity as the baseline. Furthermore, we find that the adaptive methods consistently perform on-par or slightly better than vanilla PINN method, even for large collocation point regimes. The code for all the experiments has been open sourced.

ICML Conference 2023 Conference Paper

Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching

• Ilgee Hong
• Sen Na
• Michael W. Mahoney
• Mladen Kolar

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

ICLR Conference 2023 Conference Paper

Gradient Gating for Deep Multi-Rate Learning on Graphs

• T. Konstantin Rusch
• Ben Chamberlain 0001
• Michael W. Mahoney
• Michael M. Bronstein
• Siddhartha Mishra

We present Gradient Gating (G$^2$), a novel framework for improving the performance of Graph Neural Networks (GNNs). Our framework is based on gating the output of GNN layers with a mechanism for multi-rate flow of message passing information across nodes of the underlying graph. Local gradients are harnessed to further modulate message passing updates. Our framework flexibly allows one to use any basic GNN layer as a wrapper around which the multi-rate gradient gating mechanism is built. We rigorously prove that G$^2$ alleviates the oversmoothing problem and allows the design of deep GNNs. Empirical results are presented to demonstrate that the proposed framework achieves state-of-the-art performance on a variety of graph learning tasks, including on large-scale heterophilic graphs.

ICLR Conference 2023 Conference Paper

Learning differentiable solvers for systems with hard constraints

• Geoffrey Négiar
• Michael W. Mahoney
• Aditi S. Krishnapriyan

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our method leverages differentiable optimization and the implicit function theorem to effectively enforce physical constraints. Inspired by dictionary learning, our model learns a family of functions, each of which defines a mapping from PDE parameters to PDE solutions. At inference time, the model finds an optimal linear combination of the functions in the learned family by solving a PDE-constrained optimization problem. Our method provides continuous solutions over the domain of interest that accurately satisfy desired physical constraints. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.

ICML Conference 2023 Conference Paper

Learning Physical Models that Can Respect Conservation Laws

• Derek Hansen
• Danielle C. Maddix
• Shima Alizadeh
• Gaurav Gupta
• Michael W. Mahoney

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy” PDE operators (e. g. , elliptic and parabolic), with less emphasis on relatively “hard” PDE operators (e. g. , hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticity. In each case, it achieves superior predictive performance on downstream tasks.

ICML Conference 2023 Conference Paper

Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes

• Liam Hodgkinson
• Chris van der Heide
• Fred Roosta
• Michael W. Mahoney

Despite their importance for assessing reliability of predictions, uncertainty quantification (UQ) measures in machine learning models have only recently begun to be rigorously characterized. One prominent issue is the curse of dimensionality: it is commonly believed that the marginal likelihood should be reminiscent of cross-validation metrics and both should deteriorate with larger input dimensions. However, we prove that by tuning hyperparameters to maximize marginal likelihood (the empirical Bayes procedure), performance, as measured by the marginal likelihood, improves monotonically with the input dimension. On the other hand, cross-validation metrics exhibit qualitatively different behavior that is characteristic of double descent. Cold posteriors, which have recently attracted interest due to their improved performance in certain settings, appear to exacerbate these phenomena. We verify empirically that our results hold for real data, beyond our considered assumptions, and we explore consequences involving synthetic covariates.

NeurIPS Conference 2023 Conference Paper

Speculative Decoding with Big Little Decoder

• Sehoon Kim
• Karttikeya Mangalam
• Suhong Moon
• Jitendra Malik
• Michael W. Mahoney
• Amir Gholami
• Kurt Keutzer

The recent emergence of Large Language Models based on the Transformer architecture has enabled dramatic advancements in the field of Natural Language Processing. However, these models have long inference latency, which limits their deployment and makes them prohibitively expensive for various real-time applications. The inference latency is further exacerbated by autoregressive generative tasks, as models need to run iteratively to generate tokens sequentially without leveraging token-level parallelization. To address this, we propose Big Little Decoder (BiLD), a framework that can improve inference efficiency and latency for a wide range of text generation applications. The BiLD framework contains two models with different sizes that collaboratively generate text. The small model runs autoregressively to generate text with a low inference cost, and the large model is only invoked occasionally to refine the small model’s inaccurate predictions in a non-autoregressive manner. To coordinate the small and large models, BiLD introduces two simple yet effective policies: (1) the fallback policy that determines when to hand control over to the large model; and (2) the rollback policy that determines when the large model needs to correct the small model's inaccurate predictions. To evaluate our framework across different tasks and models, we apply BiLD to various text generation scenarios encompassing machine translation on IWSLT 2017 De-En and WMT 2014 De-En, and summarization on XSUM and CNN/DailyMail. On an NVIDIA T4 GPU, our framework achieves a speedup of up to 2. 12x speedup with minimal generation quality degradation. Furthermore, our framework is fully plug-and-play and can be applied without any modifications in the training process or model architecture. Our code is open-sourced.

NeurIPS Conference 2023 Conference Paper

Temperature Balancing, Layer-wise Weight Analysis, and Neural Network Training

• Yefan Zhou
• Tianyu Pang
• Keqin Liu
• Charles Martin
• Michael W. Mahoney
• Yaoqing Yang

Regularization in modern machine learning is crucial, and it can take various forms in algorithmic design: training set, model family, error function, regularization terms, and optimizations. In particular, the learning rate, which can be interpreted as a temperature-like parameter within the statistical mechanics of learning, plays a crucial role in neural network training. Indeed, many widely adopted training strategies basically just define the decay of the learning rate over time. This process can be interpreted as decreasing a temperature, using either a global learning rate (for the entire model) or a learning rate that varies for each parameter. This paper proposes TempBalance, a straightforward yet effective layer-wise learning rate method. TempBalance is based on Heavy-Tailed Self-Regularization (HT-SR) Theory, an approach which characterizes the implicit self-regularization of different layers in trained models. We demonstrate the efficacy of using HT-SR-motivated metrics to guide the scheduling and balancing of temperature across all network layers during model training, resulting in improved performance during testing. We implement TempBalance on CIFAR10, CIFAR100, SVHN, and TinyImageNet datasets using ResNets, VGGs and WideResNets with various depths and widths. Our results show that TempBalance significantly outperforms ordinary SGD and carefully-tuned spectral norm regularization. We also show that TempBalance outperforms a number of state-of-the-art optimizers and learning rate schedulers.

NeurIPS Conference 2023 Conference Paper

Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior

• Shashank Subramanian
• Peter Harrington
• Kurt Keutzer
• Wahid Bhimji
• Dmitriy Morozov
• Michael W. Mahoney
• Amir Gholami

Pre-trained machine learning (ML) models have shown great performance for awide range of applications, in particular in natural language processing (NLP)and computer vision (CV). Here, we study how pre-training could be used forscientific machine learning (SciML) applications, specifically in the context oftransfer learning. We study the transfer behavior of these models as (i) the pretrainedmodel size is scaled, (ii) the downstream training dataset size is scaled, (iii) the physics parameters are systematically pushed out of distribution, and (iv)how a single model pre-trained on a mixture of different physics problems canbe adapted to various downstream applications. We find that—when fine-tunedappropriately—transfer learning can help reach desired accuracy levels with ordersof magnitude fewer downstream examples (across different tasks that can even beout-of-distribution) than training from scratch, with consistent behaviour across awide range of downstream examples. We also find that fine-tuning these modelsyields more performance gains as model size increases, compared to training fromscratch on new downstream tasks. These results hold for a broad range of PDElearning tasks. All in all, our results demonstrate the potential of the “pre-train andfine-tune” paradigm for SciML problems, demonstrating a path towards buildingSciML foundation models. Our code is available as open-source.

NeurIPS Conference 2023 Conference Paper

When are ensembles really effective?

• Ryan Theisen
• Hyunsuk Kim
• Yaoqing Yang
• Liam Hodgkinson
• Michael W. Mahoney

Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious. We study, both theoretically and empirically, the fundamental question of when ensembling yields significant performance improvements in classification tasks. Theoretically, we prove new results relating the \emph{ensemble improvement rate} (a measure of how much ensembling decreases the error rate versus a single model, on a relative scale) to the \emph{disagreement-error ratio}. We show that ensembling improves performance significantly whenever the disagreement rate is large relative to the average error rate; and that, conversely, one classifier is often enough whenever the disagreement rate is low relative to the average error rate. On the way to proving these results, we derive, under a mild condition called \emph{competence}, improved upper and lower bounds on the average test error rate of the majority vote classifier. To complement this theory, we study ensembling empirically in a variety of settings, verifying the predictions made by our theory, and identifying practical scenarios where ensembling does and does not result in large performance improvements. Perhaps most notably, we demonstrate a distinct difference in behavior between interpolating models (popular in current practice) and non-interpolating models (such as tree-based methods, where ensembling is popular), demonstrating that ensembling helps considerably more in the latter case than in the former.

NeurIPS Conference 2022 Conference Paper

A Fast Post-Training Pruning Framework for Transformers

• Woosuk Kwon
• Sehoon Kim
• Michael W. Mahoney
• Joseph Hassoun
• Kurt Keutzer
• Amir Gholami

Pruning is an effective way to reduce the huge inference cost of Transformer models. However, prior work on pruning Transformers requires retraining the models. This can add high training cost and high complexity to model deployment, making it difficult to use in many practical situations. To address this, we propose a fast post-training pruning framework for Transformers that does not require any retraining. Given a resource constraint and a sample dataset, our framework automatically prunes the Transformer model using structured sparsity methods. To retain high accuracy without retraining, we introduce three novel techniques: (i) a lightweight mask search algorithm that finds which heads and filters to prune based on the Fisher information; (ii) mask rearrangement that complements the search algorithm; and (iii) mask tuning that reconstructs the output activations for each layer. We apply our method to BERT-base and DistilBERT, and we evaluate its effectiveness on GLUE and SQuAD benchmarks. Our framework achieves up to 2. 0x reduction in FLOPs and 1. 56x speedup in inference latency, while maintaining < 1% loss in accuracy. Importantly, our framework prunes Transformers in less than 3 minutes on a single GPU, which is over two orders of magnitude faster than existing pruning approaches that retrain the models.

JMLR Journal 2022 Journal Article

Asymptotic Analysis of Sampling Estimators for Randomized Numerical Linear Algebra Algorithms

• Ping Ma
• Yongkai Chen
• Xinlian Zhang
• Xin Xing
• Jingyi Ma
• Michael W. Mahoney

The statistical analysis of Randomized Numerical Linear Algebra (RandNLA) algorithms within the past few years has mostly focused on their performance as point estimators. However, this is insufficient for conducting statistical inference, e.g., constructing confidence intervals and hypothesis testing, since the distribution of the estimator is lacking. In this article, we develop an asymptotic analysis to derive the distribution of RandNLA sampling estimators for the least-squares problem. In particular, we derive the asymptotic distribution of a general sampling estimator with arbitrary sampling probabilities in a fixed design setting. The analysis is conducted in two complementary settings, i.e., when the objective of interest is to approximate the full sample estimator, and when it is to infer the underlying ground truth model parameters. For each setting, we show that the sampling estimator is asymptotically normally distributed under mild regularity conditions. Moreover, the sampling estimator is asymptotically unbiased in both settings. Based on our asymptotic analysis, we use two criteria, the Asymptotic Mean Squared Error (AMSE) and the Expected Asymptotic Mean Squared Error (EAMSE), to identify optimal sampling probabilities. Several of these optimal sampling probability distributions are new to the literature, e.g., the root leverage sampling estimator and the predictor length sampling estimator. Our theoretical results clarify the role of leverage in the sampling process, and our empirical results demonstrate improvements over existing methods. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

ICML Conference 2022 Conference Paper

AutoIP: A United Framework to Integrate Physics into Gaussian Processes

• Da Long
• Zheng Wang 0042
• Aditi S. Krishnapriyan
• Robert M. Kirby
• Shandian Zhe
• Michael W. Mahoney

Physical modeling is critical for many modern science and engineering applications. From a data science or machine learning perspective, where more domain-agnostic, data-driven models are pervasive, physical knowledge {—} often expressed as differential equations {—} is valuable in that it is complementary to data, and it can potentially help overcome issues such as data sparsity, noise, and inaccuracy. In this work, we propose a simple, yet powerful and general framework {—} AutoIP, for Automatically Incorporating Physics {—} that can integrate all kinds of differential equations into Gaussian Processes (GPs) to enhance prediction accuracy and uncertainty quantification. These equations can be linear or nonlinear, spatial, temporal, or spatio-temporal, complete or incomplete with unknown source terms, and so on. Based on kernel differentiation, we construct a GP prior to sample the values of the target function, equation related derivatives, and latent source functions, which are all jointly from a multivariate Gaussian distribution. The sampled values are fed to two likelihoods: one to fit the observations, and the other to conform to the equation. We use the whitening method to evade the strong dependency between the sampled function values and kernel parameters, and we develop a stochastic variational learning algorithm. AutoIP shows improvement upon vanilla GPs in both simulation and several real-world applications, even using rough, incomplete equations.

ICLR Conference 2022 Conference Paper

Doubly Adaptive Scaled Algorithm for Machine Learning Using Second-Order Information

• Majid Jahani
• Sergey Rusakov 0001
• Zheng Shi
• Peter Richtárik
• Michael W. Mahoney
• Martin Takác 0001

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyper-parameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

ICML Conference 2022 Conference Paper

Fat-Tailed Variational Inference with Anisotropic Tail Adaptive Flows

• Feynman T. Liang
• Michael W. Mahoney
• Liam Hodgkinson

While fat-tailed densities commonly arise as posterior and marginal distributions in robust models and scale mixtures, they present a problematic scenario when Gaussian-based variational inference fails to accurately capture tail decay. We first improve previous theory on tails of Lipschitz flows by quantifying how they affect the rate of tail decay and expanding the theory to non-Lipschitz polynomial flows. Next, we develop an alternative theory for multivariate tail parameters which is sensitive to tail-anisotropy. In doing so, we unveil a fundamental problem which plagues many existing flow-based methods: they can only model tail-isotropic distributions (i. e. , distributions having the same tail parameter in every direction). To mitigate this and enable modeling of tail-anisotropic targets, we propose anisotropic tail-adaptive flows (ATAF). Experimental results confirm ATAF on both synthetic and real-world targets is competitive with prior work while also exhibiting appropriate tail-anisotropy.

ICML Conference 2022 Conference Paper

GACT: Activation Compressed Training for Generic Network Architectures

• Xiaoxuan Liu
• Lianmin Zheng
• Dequan Wang
• Yukuo Cen
• Weize Chen
• Xu Han 0007
• Jianfei Chen 0001
• Zhiyuan Liu 0001

Training large neural network (NN) models requires extensive memory resources, and Activation Compression Training (ACT) is a promising approach to reduce training memory footprint. This paper presents GACT, an ACT framework to support a broad range of machine learning tasks for generic NN architectures with limited domain knowledge. By analyzing a linearized version of ACT’s approximate gradient, we prove the convergence of GACT without prior knowledge on operator type or model architecture. To make training stable, we propose an algorithm that decides the compression ratio for each tensor by estimating its impact on the gradient at run time. We implement GACT as a PyTorch library that readily applies to any NN architecture. GACT reduces the activation memory for convolutional NNs, transformers, and graph NNs by up to 8. 1x, enabling training with a 4. 2x to 24. 7x larger batch size, with negligible accuracy loss.

ICML Conference 2022 Conference Paper

Generalization Bounds using Lower Tail Exponents in Stochastic Optimizers

• Liam Hodgkinson
• Umut Simsekli
• Rajiv Khanna
• Michael W. Mahoney

Despite the ubiquitous use of stochastic optimization algorithms in machine learning, the precise impact of these algorithms and their dynamics on generalization performance in realistic non-convex settings is still poorly understood. While recent work has revealed connections between generalization and heavy-tailed behavior in stochastic optimization, they mainly relied on continuous-time approximations; and a rigorous treatment for the original discrete-time iterations is yet to be performed. To bridge this gap, we present novel bounds linking generalization to the lower tail exponent of the transition kernel associated with the optimizer around a local minimum, in both discrete- and continuous-time settings. To achieve this, we first prove a data- and algorithm-dependent generalization bound in terms of the celebrated Fernique-Talagrand functional applied to the trajectory of the optimizer. Then, we specialize this result by exploiting the Markovian structure of stochastic optimizers, and derive bounds in terms of their (data-dependent) transition kernels. We support our theory with empirical results from a variety of neural networks, showing correlations between generalization error and lower tail exponents.

ICLR Conference 2022 Conference Paper

Long Expressive Memory for Sequence Modeling

• T. Konstantin Rusch
• Siddhartha Mishra
• N. Benjamin Erichson
• Michael W. Mahoney

We propose a novel method called Long Expressive Memory (LEM) for learning long-term sequential dependencies. LEM is gradient-based, it can efficiently process sequential tasks with very long-term dependencies, and it is sufficiently expressive to be able to learn complicated input-output maps. To derive LEM, we consider a system of multiscale ordinary differential equations, as well as a suitable time-discretization of this system. For LEM, we derive rigorous bounds to show the mitigation of the exploding and vanishing gradients problem, a well-known challenge for gradient-based recurrent sequential learning methods. We also prove that LEM can approximate a large class of dynamical systems to high accuracy. Our empirical results, ranging from image and time-series classification through dynamical systems prediction to speech recognition and language modeling, demonstrate that LEM outperforms state-of-the-art recurrent neural networks, gated recurrent units, and long short-term memory models.

JMLR Journal 2022 Journal Article

LSAR: Efficient Leverage Score Sampling Algorithm for the Analysis of Big Time Series Data

• Ali Eshragh
• Fred Roosta
• Asef Nazari
• Michael W. Mahoney

We apply methods from randomized numerical linear algebra (RandNLA) to develop improved algorithms for the analysis of large-scale time series data. We first develop a new fast algorithm to estimate the leverage scores of an autoregressive (AR) model in big data regimes. We show that the accuracy of approximations lies within $(1+\mathcal{O}({\varepsilon}))$ of the true leverage scores with high probability. These theoretical results are subsequently exploited to develop an efficient algorithm, called LSAR, for fitting an appropriate AR model to big time series data. Our proposed algorithm is guaranteed, with high probability, to find the maximum likelihood estimates of the parameters of the underlying true AR model and has a worst case running time that significantly improves those of the state-of-the-art alternatives in big data regimes. Empirical results on large-scale synthetic as well as real data highly support the theoretical results and reveal the efficacy of this new approach. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

ICML Conference 2022 Conference Paper

Neurotoxin: Durable Backdoors in Federated Learning

• Zhengming Zhang 0001
• Ashwinee Panda
• Linyue Song
• Yaoqing Yang 0002
• Michael W. Mahoney
• Prateek Mittal
• Kannan Ramchandran
• Joseph E. Gonzalez

Federated learning (FL) systems have an inherent vulnerability to adversarial backdoor attacks during training due to their decentralized nature. The goal of the attacker is to implant backdoors in the learned model with poisoned updates such that at test time, the model’s outputs can be fixed to a given target for certain inputs (e. g. , if a user types “people from New York” into a mobile keyboard app that uses a backdoored next word prediction model, the model will autocomplete their sentence to “people in New York are rude”). Prior work has shown that backdoors can be inserted in FL, but these backdoors are not durable: they do not remain in the model after the attacker stops uploading poisoned updates because training continues, and in production FL systems an inserted backdoor may not survive until deployment. We propose Neurotoxin, a simple one-line backdoor attack that functions by attacking parameters that are changed less in magnitude during training. We conduct an exhaustive evaluation across ten natural language processing and computer vision tasks and find that we can double the durability of state of the art backdoors by adding a single line with Neurotoxin.

ICLR Conference 2022 Conference Paper

Noisy Feature Mixup

• Soon Hoe Lim
• N. Benjamin Erichson
• Francisco Utrera
• Winnie Xu
• Michael W. Mahoney

We introduce Noisy Feature Mixup (NFM), an inexpensive yet effective method for data augmentation that combines the best of interpolation based training and noise injection schemes. Rather than training with convex combinations of pairs of examples and their labels, we use noise-perturbed convex combinations of pairs of data points in both input and feature space. This method includes mixup and manifold mixup as special cases, but it has additional advantages, including better smoothing of decision boundaries and enabling improved model robustness. We provide theory to understand this as well as the implicit regularization effects of NFM. Our theory is supported by empirical results, demonstrating the advantage of NFM, as compared to mixup and manifold mixup. We show that residual networks and vision transformers trained with NFM have favorable trade-offs between predictive accuracy on clean data and robustness with respect to various types of data perturbation across a range of computer vision benchmark datasets.

NeurIPS Conference 2022 Conference Paper

Squeezeformer: An Efficient Transformer for Automatic Speech Recognition

• Sehoon Kim
• Amir Gholami
• Albert Shaw
• Nicholas Lee
• Karttikeya Mangalam
• Jitendra Malik
• Michael W. Mahoney
• Kurt Keutzer

The recently proposed Conformer model has become the de facto backbone model for various downstream speech tasks based on its hybrid attention-convolution architecture that captures both local and global features. However, through a series of systematic studies, we find that the Conformer architecture’s design choices are not optimal. After re-examining the design choices for both the macro and micro-architecture of Conformer, we propose Squeezeformer which consistently outperforms the state-of-the-art ASR models under the same training schemes. In particular, for the macro-architecture, Squeezeformer incorporates (i) the Temporal U-Net structure which reduces the cost of the multi-head attention modules on long sequences, and (ii) a simpler block structure of multi-head attention or convolution modules followed up by feed-forward module instead of the Macaron structure proposed in Conformer. Furthermore, for the micro-architecture, Squeezeformer (i) simplifies the activations in the convolutional block, (ii) removes redundant Layer Normalization operations, and (iii) incorporates an efficient depthwise down-sampling layer to efficiently sub-sample the input signal. Squeezeformer achieves state-of-the-art results of 7. 5%, 6. 5%, and 6. 0% word-error-rate (WER) on LibriSpeech test-other without external language models, which are 3. 1%, 1. 4%, and 0. 6% better than Conformer-CTC with the same number of FLOPs. Our code is open-sourced and available online.

ICML Conference 2021 Conference Paper

ActNN: Reducing Training Memory Footprint via 2-Bit Activation Compressed Training

• Jianfei Chen 0001
• Lianmin Zheng
• Zhewei Yao
• Dequan Wang
• Ion Stoica
• Michael W. Mahoney
• Joseph E. Gonzalez

The increasing size of neural network models has been critical for improvements in their accuracy, but device memory is not growing at the same rate. This creates fundamental challenges for training neural networks within limited memory environments. In this work, we propose ActNN, a memory-efficient training framework that stores randomly quantized activations for back propagation. We prove the convergence of ActNN for general network architectures, and we characterize the impact of quantization on the convergence via an exact expression for the gradient variance. Using our theory, we propose novel mixed-precision quantization strategies that exploit the activation’s heterogeneity across feature dimensions, samples, and layers. These techniques can be readily applied to existing dynamic graph frameworks, such as PyTorch, simply by substituting the layers. We evaluate ActNN on mainstream computer vision models for classification, detection, and segmentation tasks. On all these tasks, ActNN compresses the activation to 2 bits on average, with negligible accuracy loss. ActNN reduces the memory footprint of the activation by 12x, and it enables training with a 6. 6x to 14x larger batch size.

ICLR Conference 2021 Conference Paper

Adversarially-Trained Deep Nets Transfer Better: Illustration on Image Classification

• Francisco Utrera
• Evan Kravitz
• N. Benjamin Erichson
• Rajiv Khanna
• Michael W. Mahoney

Transfer learning has emerged as a powerful methodology for adapting pre-trained deep neural networks on image recognition tasks to new domains. This process consists of taking a neural network pre-trained on a large feature-rich source dataset, freezing the early layers that encode essential generic image properties, and then fine-tuning the last few layers in order to capture specific information related to the target situation. This approach is particularly useful when only limited or weakly labeled data are available for the new task. In this work, we demonstrate that adversarially-trained models transfer better than non-adversarially-trained models, especially if only limited data are available for the new domain task. Further, we observe that adversarial training biases the learnt representations to retaining shapes, as opposed to textures, which impacts the transferability of the source models. Finally, through the lens of influence functions, we discover that transferred adversarially-trained models contain more human-identifiable semantic information, which explains -- at least partly -- why adversarially-trained models transfer better.

NeurIPS Conference 2021 Conference Paper

Characterizing possible failure modes in physics-informed neural networks

• Aditi Krishnapriyan
• Amir Gholami
• Shandian Zhe
• Robert Kirby
• Michael W. Mahoney

Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. The typical approach is to incorporate physical domain knowledge as soft constraints on an empirical loss function and use existing machine learning methodologies to train the model. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena for even slightly more complex problems. In particular, we analyze several distinct situations of widespread physical interest, including learning differential equations with convection, reaction, and diffusion operators. We provide evidence that the soft regularization in PINNs, which involves PDE-based differential operators, can introduce a number of subtle problems, including making the problem more ill-conditioned. Importantly, we show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize. We then describe two promising solutions to address these failure modes. The first approach is to use curriculum regularization, where the PINN's loss term starts from a simple PDE regularization, and becomes progressively more complex as the NN gets trained. The second approach is to pose the problem as a sequence-to-sequence learning task, rather than learning to predict the entire space-time at once. Extensive testing shows that we can achieve up to 1-2 orders of magnitude lower error with these methods as compared to regular PINN training.

UAI Conference 2021 Conference Paper

Geometric rates of convergence for kernel-based sampling algorithms

• Rajiv Khanna
• Liam Hodgkinson
• Michael W. Mahoney

The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geometric rate of convergence for target measures that are nearly atomic. Furthermore, we show these algorithms perform comparably to the theoretical best possible sampling algorithm under the maximum mean discrepancy. An analysis is also conducted in a distributed setting. Our theoretical developments are supported by empirical observations on simulated data as well as a real world application.

ICML Conference 2021 Conference Paper

HAWQ-V3: Dyadic Neural Network Quantization

• Zhewei Yao
• Zhen Dong 0003
• Zhangcheng Zheng
• Amir Gholami
• Jiali Yu
• Eric Tan
• Leyuan Wang
• Qijing Huang 0001

Current low-precision quantization algorithms often have the hidden cost of conversion back and forth from floating point to quantized integer values. This hidden cost limits the latency improvement realized by quantizing Neural Networks. To address this, we present HAWQ-V3, a novel mixed-precision integer-only quantization framework. The contributions of HAWQ-V3 are the following: (i) An integer-only inference where the entire computational graph is performed only with integer multiplication, addition, and bit shifting, without any floating point operations or even integer division; (ii) A novel hardware-aware mixed-precision quantization method where the bit-precision is calculated by solving an integer linear programming problem that balances the trade-off between model perturbation and other constraints, e. g. , memory footprint and latency; (iii) Direct hardware deployment and open source contribution for 4-bit uniform/mixed-precision quantization in TVM, achieving an average speed up of 1. 45x for uniform 4-bit, as compared to uniform 8-bit for ResNet50 on T4 GPUs; and (iv) extensive evaluation of the proposed methods on ResNet18/50 and InceptionV3, for various model compression levels with/without mixed precision. For ResNet50, our INT8 quantization achieves an accuracy of 77. 58%, which is 2. 68% higher than prior integer-only work, and our mixed-precision INT4/8 quantization can reduce INT8 latency by 23% and still achieve 76. 73% accuracy. Our framework and the TVM implementation have been open sourced (HAWQ, 2020).

NeurIPS Conference 2021 Conference Paper

Hessian Eigenspectra of More Realistic Nonlinear Models

• Zhenyu Liao
• Michael W. Mahoney

Given an optimization problem, the Hessian matrix and its eigenspectrum can be used in many ways, ranging from designing more efficient second-order algorithms to performing model analysis and regression diagnostics. When nonlinear models and non-convex problems are considered, strong simplifying assumptions are often made to make Hessian spectral analysis more tractable. This leads to the question of how relevant the conclusions of such analyses are for realistic nonlinear models. In this paper, we exploit tools from random matrix theory to make a precise characterization of the Hessian eigenspectra for a broad family of nonlinear models that extends the classical generalized linear models, without relying on strong simplifying assumptions used previously. We show that, depending on the data properties, the nonlinear response model, and the loss function, the Hessian can have qualitatively different spectral behaviors: of bounded or unbounded support, with single- or multi-bulk, and with isolated eigenvalues on the left- or right-hand side of the main eigenvalue bulk. By focusing on such a simple but nontrivial model, our analysis takes a step forward to unveil the theoretical origin of many visually striking features observed in more realistic machine learning models.

ICML Conference 2021 Conference Paper

I-BERT: Integer-only BERT Quantization

• Sehoon Kim 0001
• Amir Gholami
• Zhewei Yao
• Michael W. Mahoney
• Kurt Keutzer

Transformer based models, like BERT and RoBERTa, have achieved state-of-the-art results in many Natural Language Processing tasks. However, their memory footprint, inference latency, and power consumption are prohibitive efficient inference at the edge, and even at the data center. While quantization can be a viable solution for this, previous work on quantizing Transformer based models use floating-point arithmetic during inference, which cannot efficiently utilize integer-only logical units such as the recent Turing Tensor Cores, or traditional integer-only ARM processors. In this work, we propose I-BERT, a novel quantization scheme for Transformer based models that quantizes the entire inference with integer-only arithmetic. Based on lightweight integer-only approximation methods for nonlinear operations, e. g. , GELU, Softmax, and Layer Normalization, I-BERT performs an end-to-end integer-only BERT inference without any floating point calculation. We evaluate our approach on GLUE downstream tasks using RoBERTa-Base/Large. We show that for both cases, I-BERT achieves similar (and slightly higher) accuracy as compared to the full-precision baseline. Furthermore, our preliminary implementation of I-BERT shows a speedup of 2. 4- 4. 0x for INT8 inference on a T4 GPU system as compared to FP32 inference. The framework has been developed in PyTorch and has been open-sourced.

JMLR Journal 2021 Journal Article

Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning

• Charles H. Martin
• Michael W. Mahoney

Random Matrix Theory (RMT) is applied to analyze the weight matrices of Deep Neural Networks (DNNs), including both production quality, pre-trained models such as AlexNet and Inception, and smaller models trained from scratch, such as LeNet5 and a miniature-AlexNet. Empirical and theoretical results clearly indicate that the DNN training process itself implicitly implements a form of Self-Regularization, implicitly sculpting a more regularized energy or penalty landscape. In particular, the empirical spectral density (ESD) of DNN layer matrices displays signatures of traditionally-regularized statistical models, even in the absence of exogenously specifying traditional forms of explicit regularization, such as Dropout or Weight Norm constraints. Building on relatively recent results in RMT, most notably its extension to Universality classes of Heavy-Tailed matrices, and applying them to these empirical results, we develop a theory to identify 5+1 Phases of Training, corresponding to increasing amounts of Implicit Self-Regularization. These phases can be observed during the training process as well as in the final learned DNNs. For smaller and/or older DNNs, this Implicit Self-Regularization is like traditional Tikhonov regularization, in that there is a “size scale” separating signal from noise. For state-of-the-art DNNs, however, we identify a novel form of Heavy-Tailed Self-Regularization, similar to the self-organization seen in the statistical physics of disordered systems (such as classical models of actual neural activity). This results from correlations arising at all size scales, which for DNNs arises implicitly due to the training process itself. This implicit Self-Regularization can depend strongly on the many knobs of the training process. In particular, we demonstrate that we can cause a small model to exhibit all 5+1 phases of training simply by changing the batch size. Our results suggest that large, well-trained DNN architectures should exhibit Heavy-Tailed Self-Regularization, and we discuss the theoretical and practical implications of this. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

IJCAI Conference 2021 Conference Paper

Improved Guarantees and a Multiple-descent Curve for Column Subset Selection and the Nystrom Method (Extended Abstract)

• Michał Dereziński
• Rajiv Khanna
• Michael W. Mahoney

The Column Subset Selection Problem (CSSP) and the Nystrom method are among the leading tools for constructing interpretable low-rank approximations of large datasets by selecting a small but representative set of features or instances. A fundamental question in this area is: what is the cost of this interpretability, i. e. , how well can a data subset of size k compete with the best rank k approximation? We develop techniques which exploit spectral properties of the data matrix to obtain improved approximation guarantees which go beyond the standard worst-case analysis. Our approach leads to significantly better bounds for datasets with known rates of singular value decay, e. g. , polynomial or exponential decay. Our analysis also reveals an intriguing phenomenon: the cost of interpretability as a function of k may exhibit multiple peaks and valleys, which we call a multiple-descent curve. A lower bound we establish shows that this behavior is not an artifact of our analysis, but rather it is an inherent property of the CSSP and Nystrom tasks. Finally, using the example of a radial basis function (RBF) kernel, we show that both our improved bounds and the multiple-descent curve can be observed on real datasets simply by varying the RBF parameter.

JMLR Journal 2021 Journal Article

Limit theorems for out-of-sample extensions of the adjacency and Laplacian spectral embeddings

• Keith D. Levin
• Fred Roosta
• Minh Tang
• Michael W. Mahoney
• Carey E. Priebe

Graph embeddings, a class of dimensionality reduction techniques designed for relational data, have proven useful in exploring and modeling network structure. Most dimensionality reduction methods allow out-of-sample extensions, by which an embedding can be applied to observations not present in the training set. Applied to graphs, the out-of-sample extension problem concerns how to compute the embedding of a vertex that is added to the graph after an embedding has already been computed. In this paper, we consider the out-of-sample extension problem for two graph embedding procedures: the adjacency spectral embedding and the Laplacian spectral embedding. In both cases, we prove that when the underlying graph is generated according to a latent space model called the random dot product graph, which includes the popular stochastic block model as a special case, an out-of-sample extension based on a least-squares objective obeys a central limit theorem. In addition, we prove a concentration inequality for the out-of-sample extension of the adjacency spectral embedding based on a maximum-likelihood objective. Our results also yield a convenient framework in which to analyze trade-offs between estimation accuracy and computational expenses, which we explore briefly. Finally, we explore the performance of these out-of-sample extensions as applied to both simulated and real-world data. We observe significant computational savings with minimal losses to the quality of the learned embeddings, in keeping with our theoretical results. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

ICLR Conference 2021 Conference Paper

Lipschitz Recurrent Neural Networks

• N. Benjamin Erichson
• Omri Azencot
• Alejandro F. Queiruga
• Liam Hodgkinson
• Michael W. Mahoney

Viewing recurrent neural networks (RNNs) as continuous-time dynamical systems, we propose a recurrent unit that describes the hidden state's evolution with two parts: a well-understood linear component plus a Lipschitz nonlinearity. This particular functional form facilitates stability analysis of the long-term behavior of the recurrent unit using tools from nonlinear systems theory. In turn, this enables architectural design decisions before experimentation. Sufficient conditions for global stability of the recurrent unit are obtained, motivating a novel scheme for constructing hidden-to-hidden matrices. Our experiments demonstrate that the Lipschitz RNN can outperform existing recurrent units on a range of benchmark tasks, including computer vision, language modeling and speech prediction tasks. Finally, through Hessian-based analysis we demonstrate that our Lipschitz recurrent unit is more robust with respect to input and parameter perturbations as compared to other continuous-time RNNs.

UAI Conference 2021 Conference Paper

LocalNewton: Reducing communication rounds for distributed learning

• Vipul Gupta
• Avishek Ghosh
• Michal Derezinski
• Rajiv Khanna
• Kannan Ramchandran
• Michael W. Mahoney

To address the communication bottleneck problem in distributed optimization within a master-worker framework, we propose LocalNewton, a distributed second-order algorithm with local averaging. In LocalNewton, the worker machines update their model in every iteration by finding a suitable second-order descent direction using only the data and model stored in their own local memory. We let the workers run multiple such iterations locally and communicate the models to the master node only once every few (say $L$) iterations. LocalNewton is highly practical since it requires only one hyperparameter, the number $L$ of local iterations. We use novel matrix concentration based techniques to obtain theoretical guarantees for LocalNewton, and we validate them with detailed empirical evaluation. To enhance practicability, we devise an adaptive scheme to choose $L$, and we show that this reduces the number of local iterations in worker machines between two model synchronizations as the training proceeds, successively refining the model quality at the master. Via extensive experiments using several real-world datasets with AWS Lambda workers and an AWS EC2 master, we show that LocalNewton requires fewer than $60%$ of the communication rounds (between master and workers) and less than $40%$ of the end-to-end running time, compared to state-of-the-art algorithms, to reach the same training loss.

ICML Conference 2021 Conference Paper

Multiplicative Noise and Heavy Tails in Stochastic Optimization

• Liam Hodgkinson
• Michael W. Mahoney

Although stochastic optimization is central to modern machine learning, the precise mechanisms underlying its success, and in particular, the precise role of the stochasticity, still remain unclear. Modeling stochastic optimization algorithms as discrete random recurrence relations, we show that multiplicative noise, as it commonly arises due to variance in local rates of convergence, results in heavy-tailed stationary behaviour in the parameters. Theoretical results are obtained characterizing this for a large class of (non-linear and even non-convex) models and optimizers (including momentum, Adam, and stochastic Newton), demonstrating that this phenomenon holds generally. We describe dependence on key factors, including step size, batch size, and data variability, all of which exhibit similar qualitative behavior to recent empirical results on state-of-the-art neural network models. Furthermore, we empirically illustrate how multiplicative noise and heavy-tailed structure improve capacity for basin hopping and exploration of non-convex loss surfaces, over commonly-considered stochastic dynamics with only additive noise and light-tailed structure.

NeurIPS Conference 2021 Conference Paper

Newton-LESS: Sparsification without Trade-offs for the Sketched Newton Update

• Michal Derezinski
• Jonathan Lacotte
• Mert Pilanci
• Michael W. Mahoney

In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a trade-off between the computational cost of sketching and the convergence rate of the optimization. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which produces unbiased estimates of the exact Newton step and offers strong problem-independent convergence guarantees. We show that the Gaussian matrix can be drastically sparsified, substantially reducing the computational cost, without affecting its convergence properties in any way. This approach, called Newton-LESS, is based on a recently introduced sketching technique: LEverage Score Sparsified (LESS) embeddings. We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings for a large class of functions. In particular, this leads to a new state-of-the-art convergence result for an iterative least squares solver. Finally, we substantially extend LESS embeddings to include uniformly sparsified random sign matrices which can be implemented efficiently and perform well in numerical experiments.

NeurIPS Conference 2021 Conference Paper

Noisy Recurrent Neural Networks

• Soon Hoe Lim
• N. Benjamin Erichson
• Liam Hodgkinson
• Michael W. Mahoney

We provide a general framework for studying recurrent neural networks (RNNs) trained by injecting noise into hidden states. Specifically, we consider RNNs that can be viewed as discretizations of stochastic differential equations driven by input data. This framework allows us to study the implicit regularization effect of general noise injection schemes by deriving an approximate explicit regularizer in the small noise regime. We find that, under reasonable assumptions, this implicit regularization promotes flatter minima; it biases towards models with more stable dynamics; and, in classification tasks, it favors models with larger classification margin. Sufficient conditions for global stability are obtained, highlighting the phenomenon of stochastic stabilization, where noise injection can improve stability during training. Our theory is supported by empirical results which demonstrate that the RNNs have improved robustness with respect to various input perturbations.

ICLR Conference 2021 Conference Paper

Sparse Quantized Spectral Clustering

• Zhenyu Liao 0001
• Romain Couillet
• Michael W. Mahoney

Given a large data matrix, sparsifying, quantizing, and/or performing other entry-wise nonlinear operations can have numerous benefits, ranging from speeding up iterative algorithms for core numerical linear algebra problems to providing nonlinear filters to design state-of-the-art neural network models. Here, we exploit tools from random matrix theory to make precise statements about how the eigenspectrum of a matrix changes under such nonlinear transformations. In particular, we show that very little change occurs in the informative eigenstructure, even under drastic sparsification/quantization, and consequently that very little downstream performance loss occurs when working with very aggressively sparsified or quantized spectral clustering problems. We illustrate how these results depend on the nonlinearity, we characterize a phase transition beyond which spectral clustering becomes possible, and we show when such nonlinear transformations can introduce spurious non-informative eigenvectors.

NeurIPS Conference 2021 Conference Paper

Stateful ODE-Nets using Basis Function Expansions

• Alejandro Queiruga
• N. Benjamin Erichson
• Liam Hodgkinson
• Michael W. Mahoney

The recently-introduced class of ordinary differential equation networks (ODE-Nets) establishes a fruitful connection between deep learning and dynamical systems. In this work, we reconsider formulations of the weights as continuous-in-depth functions using linear combinations of basis functions which enables us to leverage parameter transformations such as function projections. In turn, this view allows us to formulate a novel stateful ODE-Block that handles stateful layers. The benefits of this new ODE-Block are twofold: first, it enables incorporating meaningful continuous-in-depth batch normalization layers to achieve state-of-the-art performance; second, it enables compressing the weights through a change of basis, without retraining, while maintaining near state-of-the-art performance and reducing both inference time and memory footprint. Performance is demonstrated by applying our stateful ODE-Block to (a) image classification tasks using convolutional units and (b) sentence-tagging tasks using transformer encoder units.

JMLR Journal 2021 Journal Article

Statistical guarantees for local graph clustering

• Wooseok Ha
• Kimon Fountoulakis
• Michael W. Mahoney

Local graph clustering methods aim to find small clusters in very large graphs. These methods take as input a graph and a seed node, and they return as output a good cluster in a running time that depends on the size of the output cluster but that is independent of the size of the input graph. In this paper, we adopt a statistical perspective on local graph clustering, and we analyze the performance of the $\ell_1$-regularized PageRank method (Fountoulakis et al., 2019) for the recovery of a single target cluster, given a seed node inside the cluster. Assuming the target cluster has been generated by a random model, we present two results. In the first, we show that the optimal support of $\ell_1$-regularized PageRank recovers the full target cluster, with bounded false positives. In the second, we show that if the seed node is connected solely to the target cluster then the optimal support of $\ell_1$-regularized PageRank recovers exactly the target cluster. We also show empirically that $\ell_1$-regularized PageRank has a state-of-the-art performance on many real graphs, demonstrating the superiority of the method. From a computational perspective, we show that the solution path of $\ell_1$-regularized PageRank is monotonic. This allows for the application of the forward stagewise algorithm, which approximates the entire solution path in running time that does not depend on the size of the whole graph. Finally, we show that $\ell_1$-regularized PageRank and approximate personalized PageRank (APPR) (Andersen et al., 2006), another very popular method for local graph clustering, are equivalent in the sense that we can lower and upper bound the output of one with the output of the other. Based on this relation, we establish for APPR similar results to those we establish for $\ell_1$-regularized PageRank. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

UAI Conference 2021 Conference Paper

Stochastic continuous normalizing flows: training SDEs as ODEs

• Liam Hodgkinson
• Chris van der Heide
• Fred Roosta
• Michael W. Mahoney

We provide a general theoretical framework for stochastic continuous normalizing flows, an extension of continuous normalizing flows for density estimation of stochastic differential equations (SDEs). Using the theory of rough paths, the underlying Brownian motion is treated as a latent variable and approximated. Doing so enables the treatment of SDEs as random ordinary differential equations, which can be trained using existing techniques. For scalar loss functions, this approach naturally recovers the stochastic adjoint method of Li et al. [2020] for training neural SDEs, while supporting a more flexible class of approximations.

NeurIPS Conference 2021 Conference Paper

Taxonomizing local versus global structure in neural network loss landscapes

• Yaoqing Yang
• Liam Hodgkinson
• Ryan Theisen
• Joe Zou
• Joseph E. Gonzalez
• Kannan Ramchandran
• Michael W. Mahoney

Viewing neural network models in terms of their loss landscapes has a long history in the statistical mechanics approach to learning, and in recent years it has received attention within machine learning proper. Among other things, local metrics (such as the smoothness of the loss landscape) have been shown to correlate with global properties of the model (such as good generalization performance). Here, we perform a detailed empirical analysis of the loss landscape structure of thousands of neural network models, systematically varying learning tasks, model architectures, and/or quantity/quality of data. By considering a range of metrics that attempt to capture different aspects of the loss landscape, we demonstrate that the best test accuracy is obtained when: the loss landscape is globally well-connected; ensembles of trained models are more similar to each other; and models converge to locally smooth regions. We also show that globally poorly-connected landscapes can arise when models are small or when they are trained to lower quality data; and that, if the loss landscape is globally poorly-connected, then training to zero loss can actually lead to worse test accuracy. Our detailed empirical results shed light on phases of learning (and consequent double descent behavior), fundamental versus incidental determinants of good generalization, the role of load-like and temperature-like parameters in the learning process, different influences on the loss landscape from model and data, and the relationships between local and global metrics, all topics of recent interest.

NeurIPS Conference 2020 Conference Paper

A random matrix analysis of random Fourier features: beyond the Gaussian kernel, a precise phase transition, and the corresponding double descent

• Zhenyu Liao
• Romain Couillet
• Michael W. Mahoney

This article characterizes the exact asymptotics of random Fourier feature (RFF) regression, in the realistic setting where the number of data samples $n$, their dimension $p$, and the dimension of feature space $N$ are all large and comparable. In this regime, the random RFF Gram matrix no longer converges to the well-known limiting Gaussian kernel matrix (as it does when $N \to \infty$ alone), but it still has a tractable behavior that is captured by our analysis. This analysis also provides accurate estimates of training and test regression errors for large $n, p, N$. Based on these estimates, a precise characterization of two qualitatively different phases of learning, including the phase transition between them, is provided; and the corresponding double descent test error curve is derived from this phase transition behavior. These results do not depend on strong assumptions on the data distribution, and they perfectly match empirical results on real-world data sets.

NeurIPS Conference 2020 Conference Paper

A Statistical Framework for Low-bitwidth Training of Deep Neural Networks

• Jianfei Chen
• Yu Gai
• Zhewei Yao
• Michael W. Mahoney
• Joseph E. Gonzalez

Fully quantized training (FQT), which uses low-bitwidth hardware by quantizing the activations, weights, and gradients of a neural network model, is a promising approach to accelerate the training of deep neural networks. One major challenge with FQT is the lack of theoretical understanding, in particular of how gradient quantization impacts convergence properties. In this paper, we address this problem by presenting a statistical framework for analyzing FQT algorithms. We view the quantized gradient of FQT as a stochastic estimator of its full precision counterpart, a procedure known as quantization-aware training (QAT). We show that the FQT gradient is an unbiased estimator of the QAT gradient, and we discuss the impact of gradient quantization on its variance. Inspired by these theoretical results, we develop two novel gradient quantizers, and we show that these have smaller variance than the existing per-tensor quantizer. For training ResNet-50 on ImageNet, our 5-bit block Householder quantizer achieves only 0. 5% validation accuracy loss relative to QAT, comparable to the existing INT8 baseline.

NeurIPS Conference 2020 Conference Paper

Boundary thickness and robustness in learning models

• Yaoqing Yang
• Rajiv Khanna
• Yaodong Yu
• Amir Gholami
• Kurt Keutzer
• Joseph E. Gonzalez
• Kannan Ramchandran
• Michael W. Mahoney

Robustness of machine learning models to various adversarial and non-adversarial corruptions continues to be of interest. In this paper, we introduce the notion of the boundary thickness of a classifier, and we describe its connection with and usefulness for model robustness. Thick decision boundaries lead to improved performance, while thin decision boundaries lead to overfitting (e. g. , measured by the robust generalization gap between training and testing) and lower robustness. We show that a thicker boundary helps improve robustness against adversarial examples (e. g. , improving the robust test accuracy of adversarial training), as well as so-called out-of-distribution (OOD) transforms, and we show that many commonly-used regularization and data augmentation procedures can increase boundary thickness. On the theoretical side, we establish that maximizing boundary thickness is akin to minimizing the so-called mixup loss. Using these observations, we can show that noise-augmentation on mixup training further increases boundary thickness, thereby combating vulnerability to various forms of adversarial attacks and OOD transforms. We can also show that the performance improvement in several recent lines of work happens in conjunction with a thicker boundary.

NeurIPS Conference 2020 Conference Paper

Debiasing Distributed Second Order Optimization with Surrogate Sketching and Scaled Regularization

• Michal Derezinski
• Burak Bartan
• Mert Pilanci
• Michael W. Mahoney

In distributed second order optimization, a standard strategy is to average many local estimates, each of which is based on a small sketch or batch of the data. However, the local estimates on each machine are typically biased, relative to the full solution on all of the data, and this can limit the effectiveness of averaging. Here, we introduce a new technique for debiasing the local estimates, which leads to both theoretical and empirical improvements in the convergence rate of distributed second order methods. Our technique has two novel components: (1) modifying standard sketching techniques to obtain what we call a surrogate sketch; and (2) carefully scaling the global regularization parameter for local computations. Our surrogate sketches are based on determinantal point processes, a family of distributions for which the bias of an estimate of the inverse Hessian can be computed exactly. Based on this computation, we show that when the objective being minimized is $l_2$-regularized with parameter $\lambda$ and individual machines are each given a sketch of size $m$, then to eliminate the bias, local estimates should be computed using a shrunk regularization parameter given by $\lambda^\prime=\lambda(1-\frac{d_\lambda}{m})$, where $d_\lambda$ is the $\lambda$-effective dimension of the Hessian (or, for quadratic problems, the data matrix).

ICML Conference 2020 Conference Paper

Error Estimation for Sketched SVD via the Bootstrap

• Miles E. Lopes
• N. Benjamin Erichson
• Michael W. Mahoney

In order to compute fast approximations to the singular value decompositions (SVD) of very large matrices, randomized sketching algorithms have become a leading approach. However, a key practical difficulty of sketching an SVD is that the user does not know how far the sketched singular vectors/values are from the exact ones. Indeed, the user may be forced to rely on analytical worst-case error bounds, which may not account for the unique structure of a given problem. As a result, the lack of tools for error estimation often leads to much more computation than is really necessary. To overcome these challenges, this paper develops a fully data-driven bootstrap method that numerically estimates the actual error of sketched singular vectors/values. Furthermore, the method is computationally inexpensive, because it operates only on sketched objects, and hence it requires no extra passes over the full matrix being factored.

NeurIPS Conference 2020 Conference Paper

Exact expressions for double descent and implicit regularization via surrogate random design

• Michal Derezinski
• Feynman T. Liang
• Michael W. Mahoney

Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We provide the first exact non-asymptotic expressions for double descent of the minimum norm linear estimator. Our approach involves constructing a special determinantal point process which we call surrogate random design, to replace the standard i. i. d. design of the training sample. This surrogate design admits exact expressions for the mean squared error of the estimator while preserving the key properties of the standard design. We also establish an exact implicit regularization result for over-parameterized training samples. In particular, we show that, for the surrogate design, the implicit bias of the unregularized minimum norm estimator precisely corresponds to solving a ridge-regularized least squares problem on the population distribution. In our analysis we introduce a new mathematical tool of independent interest: the class of random matrices for which determinant commutes with expectation.

ICML Conference 2020 Conference Paper

Forecasting Sequential Data Using Consistent Koopman Autoencoders

• Omri Azencot
• N. Benjamin Erichson
• Vanessa Lin
• Michael W. Mahoney

Recurrent neural networks are widely used on time series data, yet such models often ignore the underlying physical structures in such sequences. A new class of physics-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems. In this work, we propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics. Key to our approach is a new analysis which explores the interplay between consistent dynamics and their associated Koopman operators. Our network is directly related to the derived analysis, and its computational requirements are comparable to other baselines. We evaluate our method on a wide range of high-dimensional and short-term dependent problems, and it achieves accurate estimates for significant prediction horizons, while also being robust to noise.

NeurIPS Conference 2020 Conference Paper

HAWQ-V2: Hessian Aware trace-Weighted Quantization of Neural Networks

• Zhen Dong
• Zhewei Yao
• Daiyaan Arfeen
• Amir Gholami
• Michael W. Mahoney
• Kurt Keutzer

Quantization is an effective method for reducing memory footprint and inference time of Neural Networks. However, ultra low precision quantization could lead to significant degradation in model accuracy. A promising method to address this is to perform mixed-precision quantization, where more sensitive layers are kept at higher precision. However, the search space for a mixed-precision quantization is exponential in the number of layers. Recent work has proposed a novel Hessian based framework, with the aim of reducing this exponential search space by using second-order information. While promising, this prior work has three major limitations: (i) they only use a heuristic metric based on top Hessian eigenvalue as a measure of sensitivity and do not consider the rest of the Hessian spectrum; (ii) their approach only provides relative sensitivity of different layers and therefore requires a manual selection of the mixed-precision setting; and (iii) they do not consider mixed-precision activation quantization. Here, we present HAWQ-V2 which addresses these shortcomings. For (i), we theoretically prove that the right sensitivity metric is the average Hessian trace, instead of just top Hessian eigenvalue. For (ii), we develop a Pareto frontier based method for automatic bit precision selection of different layers without any manual intervention. For (iii), we develop the first Hessian based analysis for mixed-precision activation quantization, which is very beneficial for object detection. We show that HAWQ-V2 achieves new state-of-the-art results for a wide range of tasks. In particular, we present quantization results for InceptionV3, ResNet50, and SqueezeNext, all without any manual bit selection. Furthermore, we present results for object detection on Microsoft COCO, where we achieve 2. 6 higher mAP than direct uniform quantization and 1. 6 higher mAP than the recently proposed method of FQN, with a smaller model size of 17. 9MB.

NeurIPS Conference 2020 Conference Paper

Improved guarantees and a multiple-descent curve for Column Subset Selection and the Nystrom method

• Michal Derezinski
• Rajiv Khanna
• Michael W. Mahoney

The Column Subset Selection Problem (CSSP) and the Nystrom method are among the leading tools for constructing small low-rank approximations of large datasets in machine learning and scientific computing. A fundamental question in this area is: how well can a data subset of size k compete with the best rank k approximation? We develop techniques which exploit spectral properties of the data matrix to obtain improved approximation guarantees which go beyond the standard worst-case analysis. Our approach leads to significantly better bounds for datasets with known rates of singular value decay, e. g. , polynomial or exponential decay. Our analysis also reveals an intriguing phenomenon: the approximation factor as a function of k may exhibit multiple peaks and valleys, which we call a multiple-descent curve. A lower bound we establish shows that this behavior is not an artifact of our analysis, but rather it is an inherent property of the CSSP and Nystrom tasks. Finally, using the example of a radial basis function (RBF) kernel, we show that both our improved bounds and the multiple-descent curve can be observed on real datasets simply by varying the RBF parameter.

ICML Conference 2020 Conference Paper

PowerNorm: Rethinking Batch Normalization in Transformers

• Sheng Shen 0001
• Zhewei Yao
• Amir Gholami
• Michael W. Mahoney
• Kurt Keutzer

The standard normalization method for neural network (NN) models used in Natural Language Processing (NLP) is layer normalization (LN). This is different than batch normalization (BN), which is widely-adopted in Computer Vision. The preferred use of LN in NLP is principally due to the empirical observation that a (naive/vanilla) use of BN leads to significant performance degradation for NLP tasks; however, a thorough understanding of the underlying reasons for this is not always evident. In this paper, we perform a systematic study of NLP transformer models to understand why BN has a poor performance, as compared to LN. We find that the statistics of NLP data across the batch dimension exhibit large fluctuations throughout training. This results in instability, if BN is naively implemented. To address this, we propose Power Normalization (PN), a novel normalization scheme that resolves this issue by (i) relaxing zero-mean normalization in BN, (ii) incorporating a running quadratic mean instead of per batch statistics to stabilize fluctuations, and (iii) using an approximate backpropagation for incorporating the running statistics in the forward pass. We show theoretically, under mild assumptions, that PN leads to a smaller Lipschitz constant for the loss, compared with BN. Furthermore, we prove that the approximate backpropagation scheme leads to bounded gradients. We extensively test PN for transformers on a range of NLP tasks, and we show that it significantly outperforms both LN and BN. In particular, PN outperforms LN by 0. 4/0. 6 BLEU on IWSLT14/WMT14 and 5. 6/3. 0 PPL on PTB/WikiText-103. We make our code publicly available at https: //github. com/sIncerass/powernorm.

NeurIPS Conference 2020 Conference Paper

Precise expressions for random projections: Low-rank approximation and randomized Newton

• Michal Derezinski
• Feynman T. Liang
• Zhenyu Liao
• Michael W. Mahoney

It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a low-dimensional subspace. Matrix sketching has emerged as a powerful technique for performing such dimensionality reduction very efficiently. Even though there is an extensive literature on the worst-case performance of sketching, existing guarantees are typically very different from what is observed in practice. We exploit recent developments in the spectral analysis of random matrices to develop novel techniques that provide provably accurate expressions for the expected value of random projection matrices obtained via sketching. These expressions can be used to characterize the performance of dimensionality reduction in a variety of common machine learning tasks, ranging from low-rank approximation to iterative stochastic optimization. Our results apply to several popular sketching methods, including Gaussian and Rademacher sketches, and they enable precise analysis of these methods in terms of spectral properties of the data. Empirical results show that the expressions we derive reflect the practical performance of these sketching methods, down to lower-order effects and even constant factors.

AAAI Conference 2020 Conference Paper

Q-BERT: Hessian Based Ultra Low Precision Quantization of BERT

• Sheng Shen
• Zhen Dong
• Jiayu Ye
• Linjian Ma
• Zhewei Yao
• Amir Gholami
• Michael W. Mahoney
• Kurt Keutzer

Transformer based architectures have become de-facto models used for a range of Natural Language Processing tasks. In particular, the BERT based models achieved significant accuracy gain for GLUE tasks, CoNLL-03 and SQuAD. However, BERT based models have a prohibitive memory footprint and latency. As a result, deploying BERT based models in resource constrained environments has become a challenging task. In this work, we perform an extensive analysis of fine-tuned BERT models using second order Hessian information, and we use our results to propose a novel method for quantizing BERT models to ultra low precision. In particular, we propose a new group-wise quantization scheme, and we use Hessian-based mix-precision method to compress the model further. We extensively test our proposed method on BERT downstream tasks of SST-2, MNLI, CoNLL-03, and SQuAD. We can achieve comparable performance to baseline with at most 2.3% performance degradation, even with ultra-low precision quantization down to 2 bits, corresponding up to 13× compression of the model parameters, and up to 4× compression of the embedding table as well as activations. Among all tasks, we observed the highest performance loss for BERT fine-tuned on SQuAD. By probing into the Hessian based analysis as well as visualization, we show that this is related to the fact that current training/fine-tuning strategy of BERT does not converge for SQuAD.

JMLR Journal 2019 Journal Article

A Bootstrap Method for Error Estimation in Randomized Matrix Multiplication

• Miles E. Lopes
• Shusen Wang
• Michael W. Mahoney

In recent years, randomized methods for numerical linear algebra have received growing interest as a general approach to large-scale problems. Typically, the essential ingredient of these methods is some form of randomized dimension reduction, which accelerates computations, but also creates random approximation error. In this way, the dimension reduction step encodes a tradeoff between cost and accuracy. However, the exact numerical relationship between cost and accuracy is typically unknown, and consequently, it may be difficult for the user to precisely know (1) how accurate a given solution is, or (2) how much computation is needed to achieve a given level of accuracy. In the current paper, we study randomized matrix multiplication (sketching) as a prototype setting for addressing these general problems. As a solution, we develop a bootstrap method for directly estimating the accuracy as a function of the reduced dimension (as opposed to deriving worst-case bounds on the accuracy in terms of the reduced dimension). From a computational standpoint, the proposed method does not substantially increase the cost of standard sketching methods, and this is made possible by an “extrapolation” technique. In addition, we provide both theoretical and empirical results to demonstrate the effectiveness of the proposed method. [abs] [ pdf ][ bib ] &copy JMLR 2019. ( edit, beta )

JMLR Journal 2019 Journal Article

Scalable Kernel K-Means Clustering with Nystrom Approximation: Relative-Error Bounds

• Shusen Wang
• Alex Gittens
• Michael W. Mahoney

Kernel $k$-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear $k$-means clustering algorithm. However, kernel $k$-means clustering is computationally expensive when the non-linear feature map is high-dimensional and there are many input points. Kernel approximation, e.g., the Nystrom method, has been applied in previous works to approximately solve kernel learning problems when both of the above conditions are present. This work analyzes the application of this paradigm to kernel $k$-means clustering, and shows that applying the linear $k$-means clustering algorithm to $\frac{k}{\epsilon} (1 + o(1))$ features constructed using a so-called rank-restricted Nystrom approximation results in cluster assignments that satisfy a $1 + \epsilon$ approximation ratio in terms of the kernel $k$-means cost function, relative to the guarantee provided by the same algorithm without the use of the Nystrom method. As part of the analysis, this work establishes a novel $1 + \epsilon$ relative-error trace norm guarantee for low-rank approximation using the rank-restricted Nystrom approximation. Empirical evaluations on the $8.1$ million instance MNIST8M dataset demonstrate the scalability and usefulness of kernel $k$-means clustering with Nystrom approximation. This work argues that spectral clustering using Nystrom approximation---a popular and computationally efficient, but theoretically unsound approach to non-linear clustering---should be replaced with the efficient and theoretically sound combination of kernel $k$-means clustering with Nystrom approximation. The superior performance of the latter approach is empirically verified. [abs] [ pdf ][ bib ] &copy JMLR 2019. ( edit, beta )

ICML Conference 2019 Conference Paper

Traditional and Heavy Tailed Self Regularization in Neural Network Models

• Michael W. Mahoney
• Charles H. Martin

Random Matrix Theory (RMT) is applied to analyze the weight matrices of Deep Neural Networks (DNNs), including both production quality, pre-trained models such as AlexNet and Inception, and smaller models trained from scratch, such as LeNet5 and a miniature-AlexNet. Empirical and theoretical results clearly indicate that the empirical spectral density (ESD) of DNN layer matrices displays signatures of traditionally-regularized statistical models, even in the absence of exogenously specifying traditional forms of regularization, such as Dropout or Weight Norm constraints. Building on recent results in RMT, most notably its extension to Universality classes of Heavy-Tailed matrices, we develop a theory to identify 5+1 Phases of Training, corresponding to increasing amounts of Implicit Self-Regularization. For smaller and/or older DNNs, this Implicit Self-Regularization is like traditional Tikhonov regularization, in that there is a “size scale” separating signal from noise. For state-of-the-art DNNs, however, we identify a novel form of Heavy-Tailed Self-Regularization, similar to the self-organization seen in the statistical physics of disordered systems. This implicit Self-Regularization can depend strongly on the many knobs of the training process. By exploiting the generalization gap phenomena, we demonstrate that we can cause a small model to exhibit all 5+1 phases of training simply by changing the batch size.

ICML Conference 2018 Conference Paper

Error Estimation for Randomized Least-Squares Algorithms via the Bootstrap

• Miles E. Lopes
• Shusen Wang
• Michael W. Mahoney

Over the course of the past decade, a variety of randomized algorithms have been proposed for computing approximate least-squares (LS) solutions in large-scale settings. A longstanding practical issue is that, for any given input, the user rarely knows the actual error of an approximate solution (relative to the exact solution). Likewise, it is difficult for the user to know precisely how much computation is needed to achieve the desired error tolerance. Consequently, the user often appeals to worst-case error bounds that tend to offer only qualitative guidance. As a more practical alternative, we propose a bootstrap method to compute a posteriori error estimates for randomized LS algorithms. These estimates permit the user to numerically assess the error of a given solution, and to predict how much work is needed to improve a "preliminary" solution. In addition, we provide theoretical consistency results for the method, which are the first such results in this context (to the best of our knowledge). From a practical standpoint, the method also has considerable flexibility, insofar as it can be applied to several popular sketching algorithms, as well as a variety of error metrics. Moreover, the extra step of error estimation does not add much cost to an underlying sketching algorithm. Finally, we demonstrate the effectiveness of the method with empirical results.

ICML Conference 2018 Conference Paper

Out-of-sample extension of graph adjacency spectral embedding

• Keith D. Levin
• Fred Roosta
• Michael W. Mahoney
• Carey E. Priebe

Many popular dimensionality reduction procedures have out-of-sample extensions, which allow a practitioner to apply a learned embedding to observations not seen in the initial training sample. In this work, we consider the problem of obtaining an out-of-sample extension for the adjacency spectral embedding, a procedure for embedding the vertices of a graph into Euclidean space. We present two different approaches to this problem, one based on a least-squares objective and the other based on a maximum-likelihood formulation. We show that if the graph of interest is drawn according to a certain latent position model called a random dot product graph, then both of these out-of-sample extensions estimate the true latent position of the out-of-sample vertex with the same error rate. Further, we prove a central limit theorem for the least-squares-based extension, showing that the estimate is asymptotically normal about the truth in the large-graph limit.

JMLR Journal 2018 Journal Article

Sketched Ridge Regression: Optimization Perspective, Statistical Perspective, and Model Averaging

• Shusen Wang
• Alex Gittens
• Michael W. Mahoney

We address the statistical and optimization impacts of the classical sketch and Hessian sketch used to approximately solve the Matrix Ridge Regression (MRR) problem. Prior research has quantified the effects of classical sketch on the strictly simpler least squares regression (LSR) problem. We establish that classical sketch has a similar effect upon the optimization properties of MRR as it does on those of LSR: namely, it recovers nearly optimal solutions. By contrast, Hessian sketch does not have this guarantee; instead, the approximation error is governed by a subtle interplay between the “mass” in the responses and the optimal objective value. For both types of approximation, the regularization in the sketched MRR problem results in significantly different statistical properties from those of the sketched LSR problem. In particular, there is a bias-variance trade-off in sketched MRR that is not present in sketched LSR. We provide upper and lower bounds on the bias and variance of sketched MRR; these bounds show that classical sketch significantly increases the variance, while Hessian sketch significantly increases the bias. Empirically, sketched MRR solutions can have risks that are higher by an order-of- magnitude than those of the optimal MRR solutions. We establish theoretically and empirically that model averaging greatly decreases the gap between the risks of the true and sketched solutions to the MRR problem. Thus, in parallel or distributed settings, sketching combined with model averaging is a powerful technique that quickly obtains near-optimal solutions to the MRR problem while greatly mitigating the increased statistical risk incurred by sketching. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

JMLR Journal 2018 Journal Article

Weighted SGD for $\ell_p$ Regression with Randomized Preconditioning

• Jiyan Yang
• Yin-Lam Chow
• Christopher Ré
• Michael W. Mahoney

In recent years, stochastic gradient descent (SGD) methods and randomized linear algebra (RLA) algorithms have been applied to many large-scale problems in machine learning and data analysis. SGD methods are easy to implement and applicable to a wide range of convex optimization problems. In contrast, RLA algorithms provide much stronger performance guarantees but are applicable to a narrower class of problems. We aim to bridge the gap between these two methods in solving constrained overdetermined linear regression problems---e.g., $\ell_2$ and $\ell_1$ regression problems. We propose a hybrid algorithm named PWSGD that uses RLA techniques for preconditioning and constructing an importance sampling distribution, and then performs an SGD-like iterative process with weighted sampling on the preconditioned system. By rewriting a deterministic $\ell_p$ regression problem as a stochastic optimization problem, we connect PWSGD to several existing $\ell_p$ solvers including RLA methods with algorithmic leveraging (RLA for short). We prove that PWSGD inherits faster convergence rates that only depend on the lower dimension of the linear system, while maintaining low computation complexity. Such SGD convergence rates are superior to other related SGD algorithm such as the weighted randomized Kaczmarz algorithm. Particularly, when solving $\ell_1$ regression with size $n$ by $d$, PWSGD returns an approximate solution with $\epsilon$ relative error in the objective value in $\mathcal{O}(\log n \cdot \text{nnz}(A) + \text{poly}(d)/\epsilon^2)$ time. This complexity is uniformly better than that of RLA methods in terms of both $\epsilon$ and $d$ when the problem is unconstrained. In the presence of constraints, PWSGD only has to solve a sequence of much simpler and smaller optimization problem over the same constraints. In general this is more efficient than solving the constrained subproblem required in RLA. For $\ell_2$ regression, PWSGD returns an approximate solution with $\epsilon$ relative error in the objective value and the solution vector measured in prediction norm in $\mathcal{O}(\log n \cdot \text{nnz}(A) + \text{poly}(d) \log(1/\epsilon) /\epsilon)$ time. We show that for unconstrained $\ell_2$ regression, this complexity is comparable to that of RLA and is asymptotically better over several state-of-the-art solvers in the regime where the desired accuracy $\epsilon$, high dimension $n$ and low dimension $d$ satisfy $d\geq 1/\epsilon$ and $n \geq d^2/\epsilon$. We also provide lower bounds on the coreset complexity for more general regression problems, indicating that still new ideas will be needed to extend similar RLA preconditioning ideas to weighted SGD algorithms for more general regression problems. Finally, the effectiveness of such algorithms is illustrated numerically on both synthetic and real datasets, and the results are consistent with our theoretical findings and demonstrate that PWSGD converges to a medium- precision solution, e.g., $\epsilon=10^{-3}$, more quickly. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

ICML Conference 2017 Conference Paper

Capacity Releasing Diffusion for Speed and Locality

• Di Wang 0005
• Kimon Fountoulakis
• Monika Henzinger
• Michael W. Mahoney
• Satish Rao

Diffusions and related random walk procedures are of central importance in many areas of machine learning, data analysis, and applied mathematics. Because they spread mass agnostically at each step in an iterative manner, they can sometimes spread mass “too aggressively, ” thereby failing to find the “right” clusters. We introduce a novel Capacity Releasing Diffusion (CRD) Process, which is both faster and stays more local than the classical spectral diffusion process. As an application, we use our CRD Process to develop an improved local algorithm for graph clustering. Our local graph clustering method can find local clusters in a model of clustering where one begins the CRD Process in a cluster whose vertices are connected better internally than externally by an $O(\log^2 n)$ factor, where $n$ is the number of nodes in the cluster. Thus, our CRD Process is the first local graph clustering algorithm that is not subject to the well-known quadratic Cheeger barrier. Our result requires a certain smoothness condition, which we expect to be an artifact of our analysis. Our empirical evaluation demonstrates improved results, in particular for realistic social graphs where there are moderately good—but not very good—clusters.

ICML Conference 2017 Conference Paper

Sketched Ridge Regression: Optimization Perspective, Statistical Perspective, and Model Averaging

• Shusen Wang
• Alex Gittens
• Michael W. Mahoney

We address the statistical and optimization impacts of using classical sketch versus Hessian sketch to solve approximately the Matrix Ridge Regression (MRR) problem. Prior research has considered the effects of classical sketch on least squares regression (LSR), a strictly simpler problem. We establish that classical sketch has a similar effect upon the optimization properties of MRR as it does on those of LSR—namely, it recovers nearly optimal solutions. In contrast, Hessian sketch does not have this guarantee; instead, the approximation error is governed by a subtle interplay between the “mass” in the responses and the optimal objective value. For both types of approximations, the regularization in the sketched MRR problem gives it significantly different statistical properties from the sketched LSR problem. In particular, there is a bias-variance trade-off in sketched MRR that is not present in sketched LSR. We provide upper and lower bounds on the biases and variances of sketched MRR; these establish that the variance is significantly increased when classical sketches are used, while the bias is significantly increased when using Hessian sketches. Empirically, sketched MRR solutions can have risks that are higher by an order-of-magnitude than those of the optimal MRR solutions. We establish theoretically and empirically that model averaging greatly decreases this gap. Thus, in the distributed setting, sketching combined with model averaging is a powerful technique that quickly obtains near-optimal solutions to the MRR problem while greatly mitigating the statistical risks incurred by sketching.

ICML Conference 2016 Conference Paper

A Simple and Strongly-Local Flow-Based Method for Cut Improvement

• Nate Veldt
• David F. Gleich
• Michael W. Mahoney

Many graph-based learning problems can be cast as finding a good set of vertices nearby a seed set, and a powerful methodology for these problems is based on minimum cuts and maximum flows. We introduce and analyze a new method for locally-biased graph-based learning called SimpleLocal, which finds good conductance cuts near a set of seed vertices. An important feature of our algorithm is that it is strongly-local, meaning it does not need to explore the entire graph to find cuts that are locally optimal. This method is related to other strongly-local flow-based methods, but it enables a simple implementation. We also show how it achieves localization through an implicit l1-norm penalty term. As a flow-based method, our algorithm exhibits several advantages in terms of cut optimality and accurate identification of target regions in a graph. We demonstrate the power of SimpleLocal solving segmentation problems on a 467 million edge graph based on an MRI scan.

JMLR Journal 2016 Journal Article

A Statistical Perspective on Randomized Sketching for Ordinary Least-Squares

• Garvesh Raskutti
• Michael W. Mahoney

We consider statistical as well as algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. For a LS problem with input data $(X, Y) \in \mathbb{R}^{n \times p} \times \mathbb{R}^n$, sketching algorithms use a sketching matrix, $S\in\mathbb{R}^{r \times n}$, where $r \ll n$. Then, rather than solving the LS problem using the full data $(X,Y)$, sketching algorithms solve the LS problem using only the sketched data $(SX, SY)$. Prior work has typically adopted an algorithmic perspective, in that it has made no statistical assumptions on the input $X$ and $Y$, and instead it has been assumed that the data $(X,Y)$ are fixed and worst-case (WC). Prior results show that, when using sketching matrices such as random projections and leverage-score sampling algorithms, with $p \lesssim r \ll n$, the WC error is the same as solving the original problem, up to a small constant. From a statistical perspective, we typically consider the mean-squared error performance of randomized sketching algorithms, when data $(X, Y)$ are generated according to a statistical linear model $Y = X \beta + \epsilon$, where $\epsilon$ is a noise process. In this paper, we provide a rigorous comparison of both perspectives leading to insights on how they differ. To do this, we first develop a framework for assessing, in a unified manner, algorithmic and statistical aspects of randomized sketching methods. We then consider the statistical prediction efficiency (PE) and the statistical residual efficiency (RE) of the sketched LS estimator; and we use our framework to provide upper bounds for several types of random projection and random sampling sketching algorithms. Among other results, we show that the RE can be upper bounded when $p \lesssim r \ll n$ while the PE typically requires the sample size $r$ to be substantially larger. Lower bounds developed in subsequent results show that our upper bounds on PE can not be improved. (A preliminary version of this paper appeared as Raskutti and Mahoney (2014, 2015).) [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

JMLR Journal 2016 Journal Article

Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels

• Haim Avron
• Vikas Sindhwani
• Jiyan Yang
• Michael W. Mahoney

We consider the problem of improving the efficiency of randomized Fourier feature maps to accelerate training and testing speed of kernel methods on large data sets. These approximate feature maps arise as Monte Carlo approximations to integral representations of shift-invariant kernel functions (e.g., Gaussian kernel). In this paper, we propose to use Quasi-Monte Carlo (QMC) approximations instead, where the relevant integrands are evaluated on a low-discrepancy sequence of points as opposed to random point sets as in the Monte Carlo approach. We derive a new discrepancy measure called box discrepancy based on theoretical characterizations of the integration error with respect to a given sequence. We then propose to learn QMC sequences adapted to our setting based on explicit box discrepancy minimization. Our theoretical analyses are complemented with empirical results that demonstrate the effectiveness of classical and adaptive QMC techniques for this problem. [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

JMLR Journal 2016 Journal Article

Revisiting the Nyström Method for Improved Large-scale Machine Learning

• Alex Gittens
• Michael W. Mahoney

We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods; they characterize the effects of common data preprocessing steps on the performance of these algorithms; and they point to important differences between uniform sampling and nonuniform sampling methods based on leverage scores. In addition, our empirical results illustrate that existing theory is so weak that it does not provide even a qualitative guide to practice. Thus, we complement our empirical results with a suite of worst- case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds---e.g., improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error---and they point to future directions to make these algorithms useful in even larger-scale machine learning applications. [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

SODA Conference 2016 Conference Paper

Weighted SGD for ℓ p Regression with Randomized Preconditioning

• Jiyan Yang
• Yinlam Chow
• Christopher Ré
• Michael W. Mahoney

In recent years, stochastic gradient descent (SGD) methods and randomized linear algebra (RLA) algorithms have been applied to many large-scale problems in machine learning and data analysis. SGD methods are easy to implement and applicable to a wide range of convex optimization problems. In contrast, RLA algorithms provide much stronger worst-case performance guarantees but are applicable to a narrower class of problems. We aim to bridge the gap between these two classes of methods in solving constrained overdetermined linear regression problems—e. g. , ℓ 2 and ℓ 1 regression problems. We propose a hybrid algorithm named pw SGD that uses RLA techniques for preconditioning and constructing an importance sampling distribution, and then performs an SGD-like iterative process with weighted sampling on the preconditioned system. By rewriting the ℓ p regression problem into a stochastic optimization problem, we connect pw SGD to several existing ℓ p solvers including RLA methods with algorithmic leveraging (RLA for short). We prove that pw SGD inherits faster convergence rates that only depend on the lower dimension of the linear system, while maintaining low computation complexity. Such SGD convergence rate is superior to other related SGD algorithms such as the weighted randomized Kaczmarz algorithm. Particularly, when solving ℓ 1 regression with size n by d, pw SGD returns an approximate solution with ∊ relative error on the objective value in ℴ (log n · nnz( A ) + poly( d )/∊ 2 ) time. This complexity is uniformly better than that of RLA methods in terms of both ∊ and d when the problem is unconstrained. In the presence of constraints, pw SGD only has to solve a sequence of much simpler and smaller optimization problem over the same constraints. In general this is more efficient than solving the constrained subproblem required in RLA. For ℓ 2 regression, pw SGD returns an approximate solution with ∊ relative error on the objective value and solution vector in prediction norm in ℴ (log n · nnz( A ) + poly( d ) log(1/∊)/∊) time. We show that when solving unconstrained ℓ 2 regression, this complexity is comparable to that of RLA and is asymptotically better over several state-of-the-art solvers in the regime where the desired accuracy ∊, high dimension n and low dimension d satisfy d ≥ 1/∊ and n ≥ d 2 /∊. Finally, the effectiveness of such algorithms is illustrated numerically on both synthetic and real datasets, and the results are consistent with our theoretical findings and demonstrate that pw SGD converges to a medium-precision solution, e. g. , ∊ = 10 –3, more quickly than other methods.

JMLR Journal 2015 Journal Article

A Statistical Perspective on Algorithmic Leveraging

• Ping Ma
• Michael W. Mahoney
• Bin Yu

One popular method for dealing with large-scale data sets is sampling. For example, by using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows/columns of data matrices to reduce the data size before performing computations on the subproblem. This method has been successful in improving computational efficiency of algorithms for matrix problems such as least-squares approximation, least absolute deviations approximation, and low-rank matrix approximation. Existing work has focused on algorithmic issues such as worst-case running times and numerical issues associated with providing high-quality implementations, but none of it addresses statistical aspects of this method. In this paper, we provide a simple yet effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model with a fixed number of predictors. In particular, for several versions of leverage-based sampling, we derive results for the bias and variance, both conditional and unconditional on the observed data. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms: one constructs a smaller least-squares problem with "shrinkage" leverage scores (SLEV), and the other solves a smaller and unweighted (or biased) least-squares problem (LEVUNW). A detailed empirical evaluation of existing leverage-based methods as well as these two new methods is carried out on both synthetic and real data sets. The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage- based algorithms and that the new algorithms achieve improved performance. For example, with the same computation reduction as in the original algorithmic leveraging approach, our proposed SLEV typically leads to improved biases and variances both unconditionally and conditionally (on the observed data), and our proposed LEVUNW typically yields improved unconditional biases and variances. [abs] [ pdf ][ bib ] &copy JMLR 2015. ( edit, beta )

ICML Conference 2015 Conference Paper

Statistical and Algorithmic Perspectives on Randomized Sketching for Ordinary Least-Squares

• Garvesh Raskutti
• Michael W. Mahoney

We consider statistical and algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. Prior results show that, from an \emphalgorithmic perspective, when using sketching matrices constructed from random projections and leverage-score sampling, if the number of samples r much smaller than the original sample size n, then the worst-case (WC) error is the same as solving the original problem, up to a very small relative error. From a \emphstatistical perspective, one typically considers the mean-squared error performance of randomized sketching algorithms, when data are generated according to a statistical linear model. In this paper, we provide a rigorous comparison of both perspectives leading to insights on how they differ. To do this, we first develop a framework for assessing, in a unified manner, algorithmic and statistical aspects of randomized sketching methods. We then consider the statistical prediction efficiency (PE) and the statistical residual efficiency (RE) of the sketched LS estimator; and we use our framework to provide upper bounds for several types of random projection and random sampling algorithms. Among other results, we show that the RE can be upper bounded when r is much smaller than n, while the PE typically requires the number of samples r to be substantially larger. Lower bounds developed in subsequent work show that our upper bounds on PE can not be improved.

ICML Conference 2014 Conference Paper

A Statistical Perspective on Algorithmic Leveraging

• Ping Ma 0001
• Michael W. Mahoney
• Bin Yu 0001

One popular method for dealing with large-scale data sets is sampling. Using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows/columns of data matrices to reduce the data size before performing computations on the subproblem. Existing work has focused on algorithmic issues, but none of it addresses statistical aspects of this method. Here, we provide an effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model. In particular, for several versions of leverage-based sampling, we derive results for the bias and variance, both conditional and unconditional on the observed data. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms: one constructs a smaller least-squares problem with “shrinked” leverage scores (SLEV), and the other solves a smaller and unweighted (or biased) least-squares problem (LEVUNW). The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance.

ICML Conference 2014 Conference Paper

Anti-differentiating approximation algorithms: A case study with min-cuts, spectral, and flow

• David F. Gleich
• Michael W. Mahoney

We formalize and illustrate the general concept of algorithmic anti-differentiation: given an algorithmic procedure, e. g. , an approximation algorithm for which worst-case approximation guarantees are available or a heuristic that has been engineered to be practically-useful but for which a precise theoretical understanding is lacking, an algorithmic anti-derivative is a precise statement of an optimization problem that is exactly solved by that procedure. We explore this concept with a case study of approximation algorithms for finding locally-biased partitions in data graphs, demonstrating connections between min-cut objectives, a personalized version of the popular PageRank vector, and the highly effective "push" procedure for computing an approximation to personalized PageRank. We show, for example, that this latter algorithm solves (exactly, but implicitly) an l1-regularized l2-regression problem, a fact that helps to explain its excellent performance in practice. We expect that, when available, these implicit optimization problems will be critical for rationalizing and predicting the performance of many approximation algorithms on realistic data.

ICML Conference 2014 Conference Paper

Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels

• Jiyan Yang
• Vikas Sindhwani
• Haim Avron
• Michael W. Mahoney

We consider the problem of improving the efficiency of randomized Fourier feature maps to accelerate training and testing speed of kernel methods on large datasets. These approximate feature maps arise as Monte Carlo approximations to integral representations of shift-invariant kernel functions (e. g. , Gaussian kernel). In this paper, we propose to use Quasi-Monte Carlo (QMC) approximations instead where the relevant integrands are evaluated on a low-discrepancy sequence of points as opposed to random point sets as in the Monte Carlo approach. We derive a new discrepancy measure called box discrepancy based on theoretical characterizations of the integration error with respect to a given sequence. We then propose to learn QMC sequences adapted to our setting based on explicit box discrepancy minimization. Our theoretical analyses are complemented with empirical results that demonstrate the effectiveness of classical and adaptive QMC techniques for this problem.

JMLR Journal 2014 Journal Article

Semi-Supervised Eigenvectors for Large-Scale Locally-Biased Learning

• Toke J. Hansen
• Michael W. Mahoney

In many applications, one has side information, e.g., labels that are provided in a semi-supervised manner, about a specific target region of a large data set, and one wants to perform machine learning and data analysis tasks "nearby" that prespecified target region. For example, one might be interested in the clustering structure of a data graph near a prespecified "seed set" of nodes, or one might be interested in finding partitions in an image that are near a prespecified "ground truth" set of pixels. Locally-biased problems of this sort are particularly challenging for popular eigenvector-based machine learning and data analysis tools. At root, the reason is that eigenvectors are inherently global quantities, thus limiting the applicability of eigenvector-based methods in situations where one is interested in very local properties of the data. In this paper, we address this issue by providing a methodology to construct semi-supervised eigenvectors of a graph Laplacian, and we illustrate how these locally-biased eigenvectors can be used to perform locally-biased machine learning. These semi-supervised eigenvectors capture successively-orthogonalized directions of maximum variance, conditioned on being well-correlated with an input seed set of nodes that is assumed to be provided in a semi-supervised manner. We show that these semi-supervised eigenvectors can be computed quickly as the solution to a system of linear equations; and we also describe several variants of our basic method that have improved scaling properties. We provide several empirical examples demonstrating how these semi-supervised eigenvectors can be used to perform locally-biased learning; and we discuss the relationship between our results and recent machine learning algorithms that use global eigenvectors of the graph Laplacian. [abs] [ pdf ][ bib ] &copy JMLR 2014. ( edit, beta )

STOC Conference 2013 Conference Paper

Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression

• Xiangrui Meng
• Michael W. Mahoney

Low-distortion embeddings are critical building blocks for developing random sampling and random projection algorithms for common linear algebra problems. We show that, given a matrix A ∈ R n x d with n >> d and a p ∈ [1, 2), with a constant probability, we can construct a low-distortion embedding matrix Π ∈ R O(poly(d)) x n that embeds A p , the l p subspace spanned by A's columns, into (R O(poly(d)) , |~cdot~| p ); the distortion of our embeddings is only O(poly(d)), and we can compute Π A in O(nnz(A)) time, i.e., input-sparsity time. Our result generalizes the input-sparsity time l 2 subspace embedding by Clarkson and Woodruff [STOC'13]; and for completeness, we present a simpler and improved analysis of their construction for l 2 . These input-sparsity time l p embeddings are optimal, up to constants, in terms of their running time; and the improved running time propagates to applications such as (1 pm ε)-distortion l p subspace embedding and relative-error l p regression. For l 2 , we show that a (1+ε)-approximate solution to the l 2 regression problem specified by the matrix A and a vector b ∈ R n can be computed in O(nnz(A) + d 3 log(d/ε) /ε^2) time; and for l p , via a subspace-preserving sampling procedure, we show that a (1 pm ε)-distortion embedding of A p into R O(poly(d)) can be computed in O(nnz(A) ⋅ log n) time, and we also show that a (1+ε)-approximate solution to the l p regression problem min x ∈ R d |A x - b| p can be computed in O(nnz(A) ⋅ log n + poly(d) log(1/ε)/ε 2 ) time. Moreover, we can also improve the embedding dimension or equivalently the sample size to O(d 3+p/2 log(1/ε) / ε 2 ) without increasing the complexity.

ICML Conference 2013 Conference Paper

Quantile Regression for Large-scale Applications

• Jiyan Yang
• Xiangrui Meng
• Michael W. Mahoney

Quantile regression is a method to estimate the quantiles of the conditional distribution of a response variable, and as such it permits a much more accurate portrayal of the relationship between the response variable and observed covariates than methods such as Least-squares or Least Absolute Deviations regression. It can be expressed as a linear program, and interior-point methods can be used to find a solution for moderately large problems. Dealing with very large problems, \emphe. g. , involving data up to and beyond the terabyte regime, remains a challenge. Here, we present a randomized algorithm that runs in time that is nearly linear in the size of the input and that, with constant probability, computes a (1+ε) approximate solution to an arbitrary quantile regression problem. Our algorithm computes a low-distortion subspace-preserving embedding with respect to the loss function of quantile regression. Our empirical evaluation illustrates that our algorithm is competitive with the best previous work on small to medium-sized problems, and that it can be implemented in MapReduce-like environments and applied to terabyte-sized problems.

ICML Conference 2013 Conference Paper

Revisiting the Nystrom method for improved large-scale machine learning

• Alex Gittens
• Michael W. Mahoney

We reconsider randomized algorithms for the low-rank approximation of SPSD matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results consist of an empirical evaluation of the performance quality and running time of sampling and projection methods on a diverse suite of SPSD matrices. Our results highlight complementary aspects of sampling versus projection methods, and they point to differences between uniform and nonuniform sampling methods based on leverage scores. We complement our empirical results with a suite of worst-case theoretical bounds for both random sampling and random projection methods. These bounds are qualitatively superior to existing bounds— e. g. , improved additive-error bounds for spectral and Frobenius norm error and relative-error bounds for trace norm error.

ICML Conference 2013 Conference Paper

Robust Regression on MapReduce

• Xiangrui Meng
• Michael W. Mahoney

Although the MapReduce framework is now the \emphde facto standard for analyzing massive data sets, many algorithms (in particular, many iterative algorithms popular in machine learning, optimization, and linear algebra) are hard to fit into MapReduce. Consider, \emphe. g. , the \ell_p regression problem: given a matrix A ∈\mathbbR^m \times n and a vector b ∈\mathbbR^m, find a vector x^* ∈\mathbbR^n that minimizes f(x) = \|A x - b\|_p. The widely-used \ell_2 regression, \emphi. e. , linear least-squares, is known to be highly sensitive to outliers; and choosing p ∈[1, 2) can help improve robustness. In this work, we propose an efficient algorithm for solving strongly over-determined (m ≫n) robust \ell_p regression problems to moderate precision on MapReduce. Our empirical results on data up to the terabyte scale demonstrate that our algorithm is a significant improvement over traditional iterative algorithms on MapReduce for \ell_1 regression, even for a fairly small number of iterations. In addition, our proposed interior-point cutting-plane method can also be extended to solving more general convex problems on MapReduce.

SODA Conference 2013 Conference Paper

The Fast Cauchy Transform and Faster Robust Linear Regression

• Kenneth L. Clarkson
• Petros Drineas
• Malik Magdon-Ismail
• Michael W. Mahoney
• Xiangrui Meng
• David P. Woodruff

We provide fast algorithms for overconstrained ℓ p regression and related problems: for an n × d input matrix A and vector b ∊ ℝ n, in O ( nd log n ) time we reduce the problem min x ∊ ℝ d ‖ A x − b ‖ p to the same problem with input matrix A of dimension s × d and corresponding b of dimension s × 1. Here, Ã and are a coreset for the problem, consisting of sampled and rescaled rows of A and b; and s is independent of n and polynomial in d. Our results improve on the best previous algorithms when n ≫ d, for all p ∊ [1, ∞) except p = 2; in particular, they improve the O ( nd 1. 376+ ) running time of Sohler and Woodruff (STOC, 2011) for p = 1, that uses asymptotically fast matrix multiplication, and the O ( nd 5 log n ) time of Dasgupta et al. (SICOMP, 2009) for general p, that uses ellipsoidal rounding. We also provide a suite of improved results for finding well-conditioned bases via ellipsoidal rounding, illustrating tradeoffs between running time and conditioning quality, including a one-pass conditioning algorithm for general ℓ p problems. To complement this theory, we provide a detailed empirical evaluation of implementations of our algorithms for p = 1, comparing them with several related algorithms. Among other things, our empirical results clearly show that, in the asymptotic regime, the theory is a very good guide to the practical performance of these algorithms. Our algorithms use our faster constructions of well-conditioned bases for ℓ p spaces and, for p = 1, a fast subspace embedding of independent interest that we call the Fast Cauchy Transform: a matrix Π: ℝ n → ℝ O ( d log d ), found obliviously to A, that approximately preserves the ℓ 1 norms: that is, ‖ Ax ‖ 1 ≈ ‖Π A x ‖ 1, for all x, with distortion O ( d 2 + η log d ), for an arbitrarily small constant η > 0; and, moreover, ΠA can be computed in O ( nd log d ) time. The techniques underlying our Fast Cauchy Transform include fast Johnson-Lindenstrauss transforms, low-coherence matrices, and rescaling by Cauchy random variables.

JMLR Journal 2012 Journal Article

A Local Spectral Method for Graphs: With Applications to Improving Graph Partitions and Exploring Data Graphs Locally

• Michael W. Mahoney
• Lorenzo Orecchia
• Nisheeth K. Vishnoi

The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several local regions of interest, and the second eigenvector will typically fail to provide information fine-tuned to each local region. In this paper, we introduce a locally-biased analogue of the second eigenvector, and we demonstrate its usefulness at highlighting local properties of data graphs in a semi-supervised manner. To do so, we first view the second eigenvector as the solution to a constrained optimization problem, and we incorporate the local information as an additional constraint; we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a Personalized PageRank vector; and finally, as a consequence, we show that the solution can be computed in nearly-linear time. In addition, we show that this locally-biased vector can be used to compute an approximation to the best partition near an input seed set in a manner analogous to the way in which the second eigenvector of the Laplacian can be used to obtain an approximation to the best partition in the entire input graph. Such a primitive is useful for identifying and refining clusters locally, as it allows us to focus on a local region of interest in a semi-supervised manner. Finally, we provide a detailed empirical evaluation of our method by showing how it can applied to finding locally-biased sparse cuts around an input vertex seed set in social and information networks. [abs] [ pdf ][ bib ] &copy JMLR 2012. ( edit, beta )

ICML Conference 2012 Conference Paper

Fast approximation of matrix coherence and statistical leverage

• Michael W. Mahoney
• Petros Drineas
• Malik Magdon-Ismail
• David P. Woodruff

JMLR Journal 2012 Journal Article

Fast Approximation of Matrix Coherence and Statistical Leverage

• Petros Drineas
• Malik Magdon-Ismail
• Michael W. Mahoney
• David P. Woodruff

The statistical leverage scores of a matrix A are the squared row-norms of the matrix containing its (top) left singular vectors and the coherence is the largest leverage score. These quantities are of interest in recently-popular problems such as matrix completion and Nyström-based low-rank matrix approximation as well as in large-scale statistical data analysis applications more generally; moreover, they are of interest since they define the key structural nonuniformity that must be dealt with in developing fast randomized matrix algorithms. Our main result is a randomized algorithm that takes as input an arbitrary n × d matrix A, with n >> d, and that returns as output relative-error approximations to all n of the statistical leverage scores. The proposed algorithm runs (under assumptions on the precise values of n and d ) in O(n d log n) time, as opposed to the O(nd 2 ) time required by the naïve algorithm that involves computing an orthogonal basis for the range of A. Our analysis may be viewed in terms of computing a relative-error approximation to an under constrained least-squares approximation problem, or, relatedly, it may be viewed as an application of Johnson-Lindenstrauss type ideas. Several practically-important extensions of our basic result are also described, including the approximation of so-called cross-leverage scores, the extension of these ideas to matrices with n ≈ d, and the extension to streaming environments. [abs] [ pdf ][ bib ] &copy JMLR 2012. ( edit, beta )

ICML Conference 2011 Conference Paper

Implementing regularization implicitly via approximate eigenvector computation

• Michael W. Mahoney
• Lorenzo Orecchia

UAI Conference 2010 Conference Paper

Approximating Higher-Order Distances Using Random Projections

• Ping Li 0001
• Michael W. Mahoney
• Yiyuan She

We provide a simple method and relevant theoretical analysis for efficiently estimating higherorder lp distances. While the analysis mainly focuses on l4, our methodology extends naturally to p = 6, 8, 10..., (i.e., when p is even). Distance-based methods are popular in machine learning. In large-scale applications, storing, computing, and retrieving the distances can be both space and time prohibitive. Efficient algorithms exist for estimating lp distances if 0 < p ≤ 2. The task for p > 2 is known to be difficult. Our work partially fills this gap.

SODA Conference 2009 Conference Paper

An improved approximation algorithm for the column subset selection problem

• Christos Boutsidis
• Michael W. Mahoney
• Petros Drineas

We consider the problem of selecting the “best” subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O (min{ mn 2, m 2 n }) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O ( k log k ) columns according to a judiciously-chosen probability distribution that depends on information in the top- k right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic column-selection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m × k matrix containing those k columns, let P C denote the projection matrix onto the span of those columns, and let A k denote the “best” rank- k approximation to the matrix A as computed with the singular value decomposition. Then, we prove that with probability at least 0. 7. This spectral norm bound improves upon the best previously-existing result (of Gu and Eisenstat [21]) for the spectral norm version of this Column ubset election Problem. We also prove that with the same probability. This Frobenius norm bound is only a factor of worse than the best previously existing existential result and is roughly better than the best previous algorithmic result (both of Deshpande et al. [11]) for the Frobenius norm version of this Column ubset election Problem.

SODA Conference 2008 Conference Paper

Sampling algorithms and coresets for ℓ p regression

• Anirban Dasgupta 0001
• Petros Drineas
• Boulos Harb
• Ravi Kumar 0001
• Michael W. Mahoney

SODA Conference 2006 Conference Paper

Sampling algorithms for l 2 regression and applications

• Petros Drineas
• Michael W. Mahoney
• S. Muthukrishnan 0001

JMLR Journal 2005 Journal Article

On the Nystrom Method for Approximating a Gram Matrix for Improved Kernel-Based Learning

• Petros Drineas
• Michael W. Mahoney

A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O(n 3 ), where n is the number of training examples. We develop and analyze an algorithm to compute an easily-interpretable low-rank approximation to an n × n Gram matrix G such that computations of interest may be performed more rapidly. The approximation is of the form ~ G k = CW k + C T, where C is a matrix consisting of a small number c of columns of G and W k is the best rank- k approximation to W, the matrix formed by the intersection between those c columns of G and the corresponding c rows of G. An important aspect of the algorithm is the probability distribution used to randomly sample the columns; we will use a judiciously-chosen and data-dependent nonuniform probability distribution. Let ||·|| 2 and ||·|| F denote the spectral norm and the Frobenius norm, respectively, of a matrix, and let G k be the best rank- k approximation to G. We prove that by choosing O(k/ε 4 ) columns || G-CW k + C T || ξ ≤ || G-G k || ξ + ε Σ i=1 n G ii 2, both in expectation and with high probability, for both ξ = 2, F, and for all k: 0 ≤ k ≤ rank (W). This approximation can be computed using O(n) additional space and time, after making two passes over the data from external storage. The relationships between this algorithm, other related matrix decompositions, and the Nyström method from integral equation theory are discussed. [abs] [ pdf ][ bib ] &copy JMLR 2005. ( edit, beta )