Arrow Research search

Author name cluster

Surya Ganguli

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

48 papers
2 author rows

Possible papers

48

NeurIPS Conference 2025 Conference Paper

Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks

  • Daniel Kunin
  • Giovanni Luca Marchetti
  • Feng Chen
  • Dhruva Karkada
  • James Simon
  • Michael Deweese
  • Surya Ganguli
  • Nina Miolane

What features neural networks learn, and how, remains an open question. In this paper, we introduce Alternating Gradient Flows (AGF), an algorithmic framework that describes the dynamics of feature learning in two-layer networks trained from small initialization. Prior works have shown that gradient flow in this regime exhibits a staircase-like loss curve, alternating between plateaus where neurons slowly align to useful directions and sharp drops where neurons rapidly grow in norm. AGF approximates this behavior as an alternating two-step process: maximizing a utility function over dormant neurons and minimizing a cost function over active ones. AGF begins with all neurons dormant, corresponding to an initialization at the origin. At each iteration, a dormant neuron activates, triggering the acquisition of a feature and a drop in the loss. AGF quantifies the order, timing, and magnitude of these drops, matching experiments across several commonly studied architectures. We show that AGF unifies and extends existing saddle-to-saddle analyses in fully connected linear networks and attention-only linear transformers, where the learned features are singular modes and principal components, respectively. In diagonal linear networks, we prove AGF converges to gradient flow in the limit of vanishing initialization. Applying AGF to quadratic networks trained to perform modular addition, we give the first complete characterization of the training dynamics, revealing that networks learn Fourier features in decreasing order of coefficient magnitude. Altogether, AGF offers a promising step towards understanding feature learning in neural networks.

PDF Details

ICML Conference 2025 Conference Paper

An analytic theory of creativity in convolutional diffusion models

  • Mason Kamb
  • Surya Ganguli

We obtain an analytic, interpretable and predictive theory of creativity in convolutional diffusion models. Indeed, score-matching diffusion models can generate highly original images that lie far from their training data. However, optimal score-matching theory suggests that these models should only be able to produce memorized training examples. To reconcile this theory-experiment gap, we identify two simple inductive biases, locality and equivariance, that: (1) induce a form of combinatorial creativity by preventing optimal score-matching; (2) result in fully analytic, completely mechanistically interpretable, local score (LS) and equivariant local score (ELS) machines that, (3) after calibrating a single time-dependent hyperparameter can quantitatively predict the outputs of trained convolution only diffusion models (like ResNets and UNets) with high accuracy (median $r^2$ of $0. 95, 0. 94, 0. 94, 0. 96$ for our top model on CIFAR10, FashionMNIST, MNIST, and CelebA). Our model reveals a locally consistent patch mosaic mechanism of creativity, in which diffusion models create exponentially many novel images by mixing and matching different local training set patches at different scales and image locations. Our theory also partially predicts the outputs of pre-trained self-attention enabled UNets (median $r^2 \sim 0. 77$ on CIFAR10), revealing an intriguing role for attention in carving out semantic coherence from local patch mosaics.

Details

ICML Conference 2025 Conference Paper

Features are fate: a theory of transfer learning in high-dimensional regression

  • Javan Tahir
  • Surya Ganguli
  • Grant M. Rotskoff

With the emergence of large-scale pre-trained neural networks, methods to adapt such "foundation" models to data-limited downstream tasks have become a necessity. Fine-tuning, preference optimization, and transfer learning have all been successfully employed for these purposes when the target task closely resembles the source task, but a precise theoretical understanding of “task similarity” is still lacking. We adopt a feature-centric viewpoint on transfer learning and establish a number of theoretical results that demonstrate that when the target task is well represented by the feature space of the pre-trained model, transfer learning outperforms training from scratch. We study deep linear networks as a minimal model of transfer learning in which we can analytically characterize the transferability phase diagram as a function of the target dataset size and the feature space overlap. For this model, we establish rigorously that when the feature space overlap between the source and target tasks is sufficiently strong, both linear transfer and fine-tuning improve performance, especially in the low data limit. These results build on an emerging understanding of feature learning dynamics in deep linear networks, and we demonstrate numerically that the rigorous results we derive for the linear case also apply to nonlinear networks.

Details

NeurIPS Conference 2025 Conference Paper

Rethinking Fine-Tuning when Scaling Test-Time Compute: Limiting Confidence Improves Mathematical Reasoning

  • Feng Chen
  • Allan Raventós
  • Nan Cheng
  • Surya Ganguli
  • Shaul Druckmann

Recent progress in large language models (LLMs) highlights the power of scaling test-time compute to achieve strong performance on complex tasks, such as mathematical reasoning and code generation. This raises a critical question: how should model training be modified to optimize performance under a subsequent test-time compute strategy and budget? To explore this, we focus on pass@N, a simple test-time strategy that searches for a correct answer in N independent samples. We show, surprisingly, that training with cross-entropy (CE) can be misaligned with pass@N in that pass@N accuracy decreases with longer CE training. We explain the origins of this misalignment in terms of model overconfidence induced by CE, and experimentally verify our prediction of overconfidence as an impediment to scaling test-time compute via pass@N. Furthermore we suggest a principled, modified training loss that is better aligned to pass@N by limiting model confidence and rescuing pass@N test performance. Our algorithm demonstrates improved mathematical reasoning on MATH and MiniF2F benchmarks under several scenarios: (1) providing answers to math questions both with and without Chain-of-Thought reasoning traces; and (2) proving theorems by searching over proof trees of varying shapes. Overall our work underscores the importance of co-designing two traditionally separate phases of LLM development: training-time protocols and test-time search and reasoning strategies.

PDF Details

NeurIPS Conference 2024 Conference Paper

Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning

  • Daniel Kunin
  • Allan Raventós
  • Clémentine Dominé
  • Feng Chen
  • David Klindt
  • Andrew Saxe
  • Surya Ganguli

While the impressive performance of modern neural networks is often attributed to their capacity to efficiently extract task-relevant features from data, the mechanisms underlying this rich feature learning regime remain elusive, with much of our theoretical understanding stemming from the opposing lazy regime. In this work, we derive exact solutions to a minimal model that transitions between lazy and rich learning, precisely elucidating how unbalanced layer-specific initialization variances and learning rates determine the degree of feature learning. Our analysis reveals that they conspire to influence the learning regime through a set of conserved quantities that constrain and modify the geometry of learning trajectories in parameter and function space. We extend our analysis to more complex linear models with multiple neurons, outputs, and layers and to shallow nonlinear networks with piecewise linear activation functions. In linear networks, rapid feature learning only occurs from balanced initializations, where all layers learn at similar speeds. While in nonlinear networks, unbalanced initializations that promote faster learning in earlier layers can accelerate rich learning. Through a series of experiments, we provide evidence that this unbalanced rich regime drives feature learning in deep finite-width networks, promotes interpretability of early layers in CNNs, reduces the sample complexity of learning hierarchical data, and decreases the time to grokking in modular arithmetic. Our theory motivates further exploration of unbalanced initializations to enhance efficient feature learning.

PDF Details DOI

ICLR Conference 2023 Conference Paper

Disentanglement with Biological Constraints: A Theory of Functional Cell Types

  • James C. R. Whittington
  • Will Dorrell
  • Surya Ganguli
  • Timothy Edward John Behrens

Neurons in the brain are often finely tuned for specific task variables. Moreover, such disentangled representations are highly sought after in machine learning. Here we mathematically prove that simple biological constraints on neurons, namely nonnegativity and energy efficiency in both activity and weights, promote such sought after disentangled representations by enforcing neurons to become selective for single factors of task variation. We demonstrate these constraints lead to disentanglement in a variety of tasks and architectures, including variational autoencoders. We also use this theory to explain why the brain partitions its cells into distinct cell types such as grid and object-vector cells, and also explain when the brain instead entangles representations in response to entangled task factors. Overall, this work provides a mathematical understanding of why single neurons in the brain often represent single human-interpretable factors, and steps towards an understanding task structure shapes the structure of brain representation.

Details

TMLR Journal 2023 Journal Article

Holistic Evaluation of Language Models

  • Percy Liang
  • Rishi Bommasani
  • Tony Lee
  • Dimitris Tsipras
  • Dilara Soylu
  • Michihiro Yasunaga
  • Yian Zhang
  • Deepak Narayanan

Language models (LMs) are becoming the foundation for almost all major language technologies, but their capabilities, limitations, and risks are not well understood. We present Holistic Evaluation of Language Models (HELM) to improve the transparency of language models. First, we taxonomize the vast space of potential scenarios (i.e. use cases) and metrics (i.e. desiderata) that are of interest for LMs. Then we select a broad subset based on coverage and feasibility, noting what’s missing or underrepresented (e.g. question answering for neglected English dialects, metrics for trustworthiness). Second, we adopt a multi-metric approach: We measure 7 metrics (accuracy, calibration, robustness, fairness, bias, toxicity, and efficiency) for each of 16 core scenarios to the extent possible (87.5% of the time), ensuring that metrics beyond accuracy don’t fall to the wayside, and that trade-offs across models and metrics are clearly exposed. We also perform 7 targeted evaluations, based on 26 targeted scenarios, to more deeply analyze specific aspects (e.g. knowledge, reasoning, memorization/copyright, disinformation). Third, we conduct a large-scale evaluation of 30 prominent language models (spanning open, limited-access, and closed models) on all 42 scenarios, including 21 scenarios that were not previously used in mainstream LM evaluation. Prior to HELM, models on average were evaluated on just 17.9% of the core HELM scenarios, with some prominent models not sharing a single scenario in common. We improve this to 96.0%: now all 30 models have been densely benchmarked on a set of core scenarios and metrics under standardized conditions. Our evaluation surfaces 25 top-level findings concerning the interplay between different scenarios, metrics, and models. For full transparency, we release all raw model prompts and completions publicly for further analysis, as well as a general modular toolkit for easily adding new scenarios, models, metrics, and prompting strategies. We intend for HELM to be a living benchmark for the community, continuously updated with new scenarios, metrics, and models.

PDF Details

NeurIPS Conference 2023 Conference Paper

Information Geometry of the Retinal Representation Manifold

  • Xuehao Ding
  • Dongsoo Lee
  • Joshua Melander
  • George Sivulka
  • Surya Ganguli
  • Stephen Baccus

The ability for the brain to discriminate among visual stimuli is constrained by their retinal representations. Previous studies of visual discriminability have been limited to either low-dimensional artificial stimuli or pure theoretical considerations without a realistic encoding model. Here we propose a novel framework for understanding stimulus discriminability achieved by retinal representations of naturalistic stimuli with the method of information geometry. To model the joint probability distribution of neural responses conditioned on the stimulus, we created a stochastic encoding model of a population of salamander retinal ganglion cells based on a three-layer convolutional neural network model. This model not only accurately captured the mean response to natural scenes but also a variety of second-order statistics. With the model and the proposed theory, we computed the Fisher information metric over stimuli to study the most discriminable stimulus directions. We found that the most discriminable stimulus varied substantially across stimuli, allowing an examination of the relationship between the most discriminable stimulus and the current stimulus. By examining responses generated by the most discriminable stimuli we further found that the most discriminative response mode is often aligned with the most stochastic mode. This finding carries the important implication that under natural scenes, retinal noise correlations are information-limiting rather than increasing information transmission as has been previously speculated. We additionally observed that sensitivity saturates less in the population than for single cells and that as a function of firing rate, Fisher information varies less than sensitivity. We conclude that under natural scenes, population coding benefits from complementary coding and helps to equalize the information carried by different firing rates, which may facilitate decoding of the stimulus under principles of information maximization.

PDF Details

NeurIPS Conference 2023 Conference Paper

Pretraining task diversity and the emergence of non-Bayesian in-context learning for regression

  • Allan Raventós
  • Mansheej Paul
  • Feng Chen
  • Surya Ganguli

Pretrained transformers exhibit the remarkable ability of in-context learning (ICL): they can learn tasks from just a few examples provided in the prompt without updating any weights. This raises a foundational question: can ICL solve fundamentally new tasks that are very different from those seen during pretraining? To probe this question, we examine ICL’s performance on linear regression while varying the diversity of tasks in the pretraining dataset. We empirically demonstrate a task diversity threshold for the emergence of ICL. Below this threshold, the pretrained transformer cannot solve unseen regression tasks, instead behaving like a Bayesian estimator with the non-diverse pretraining task distribution as the prior. Beyond this threshold, the transformer significantly outperforms this estimator; its behavior aligns with that of ridge regression, corresponding to a Gaussian prior over all tasks, including those not seen during pretraining. Thus, when pretrained on data with task diversity greater than the threshold, transformers can optimally solve fundamentally new tasks in-context. Importantly, this capability hinges on it deviating from the Bayes optimal estimator with the pretraining distribution as the prior. This study also explores the effect of regularization, model capacity and task structure and underscores, in a concrete example, the critical role of task diversity, alongside data and model scale, in the emergence of ICL.

PDF Details

NeurIPS Conference 2023 Conference Paper

Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks

  • Feng Chen
  • Daniel Kunin
  • Atsushi Yamamura
  • Surya Ganguli

In this work, we reveal a strong implicit bias of stochastic gradient descent (SGD) that drives overly expressive networks to much simpler subnetworks, thereby dramatically reducing the number of independent parameters, and improving generalization. To reveal this bias, we identify invariant sets, or subsets of parameter space that remain unmodified by SGD. We focus on two classes of invariant sets that correspond to simpler (sparse or low-rank) subnetworks and commonly appear in modern architectures. Our analysis uncovers that SGD exhibits a property of stochastic attractivity towards these simpler invariant sets. We establish a sufficient condition for stochastic attractivity based on a competition between the loss landscape's curvature around the invariant set and the noise introduced by stochastic gradients. Remarkably, we find that an increased level of noise strengthens attractivity, leading to the emergence of attractive invariant sets associated with saddle-points or local maxima of the train loss. We observe empirically the existence of attractive invariant sets in trained deep neural networks, implying that SGD dynamics often collapses to simple subnetworks with either vanishing or redundant neurons. We further demonstrate how this simplifying process of stochastic collapse benefits generalization in a linear teacher-student framework. Finally, through this analysis, we mechanistically explain why early training with large learning rates for extended periods benefits subsequent generalization.

PDF Details

ICLR Conference 2023 Conference Paper

The Asymmetric Maximum Margin Bias of Quasi-Homogeneous Neural Networks

  • Daniel Kunin
  • Atsushi Yamamura
  • Chao Ma
  • Surya Ganguli

In this work, we explore the maximum-margin bias of quasi-homogeneous neural networks trained with gradient flow on an exponential loss and past a point of separability. We introduce the class of quasi-homogeneous models, which is expressive enough to describe nearly all neural networks with homogeneous activations, even those with biases, residual connections, and normalization layers, while structured enough to enable geometric analysis of its gradient dynamics. Using this analysis, we generalize the existing results of maximum-margin bias for homogeneous networks to this richer class of models. We find that gradient flow implicitly favors a subset of the parameters, unlike in the case of a homogeneous model where all parameters are treated equally. We demonstrate through simple examples how this strong favoritism toward minimizing an asymmetric norm can degrade the robustness of quasi-homogeneous models. On the other hand, we conjecture that this norm-minimization discards, when possible, unnecessary higher-order parameters, reducing the model to a sparser parameterization. Lastly, by applying our theorem to sufficiently expressive neural networks with normalization layers, we reveal a universal mechanism behind the empirical phenomenon of Neural Collapse.

Details

ICLR Conference 2023 Conference Paper

Unmasking the Lottery Ticket Hypothesis: What's Encoded in a Winning Ticket's Mask?

  • Mansheej Paul
  • Feng Chen 0046
  • Brett W. Larsen
  • Jonathan Frankle
  • Surya Ganguli
  • Gintare Karolina Dziugaite

As neural networks get larger and costlier, it is important to find sparse networks that require less compute and memory but can be trained to the same accuracy as the full network (i.e. matching). Iterative magnitude pruning (IMP) is a state of the art algorithm that can find such highly sparse matching subnetworks, known as winning tickets. IMP iterates through cycles of training, pruning a fraction of smallest magnitude weights, rewinding unpruned weights back to an early training point, and repeating. Despite its simplicity, the principles underlying when and how IMP finds winning tickets remain elusive. In particular, what useful information does an IMP mask found at the end of training convey to a rewound network near the beginning of training? How does SGD allow the network to extract this information? And why is iterative pruning needed, i.e. why can't we prune to very high sparsities in one shot? We investigate these questions through the lens of the geometry of the error landscape. First, we find that—at higher sparsities—pairs of pruned networks at successive pruning iterations are connected by a linear path with zero error barrier if and only if they are matching. This indicates that masks found at the end of training convey to the rewind point the identity of an axial subspace that intersects a desired linearly connected mode of a matching sublevel set. Second, we show SGD can exploit this information due to a strong form of robustness: it can return to this mode despite strong perturbations early in training. Third, we show how the flatness of the error landscape at the end of training limits the fraction of weights that can be pruned at each iteration of IMP. This analysis yields a new quantitative link between IMP performance and the Hessian eigenspectrum. Finally, we show that the role of retraining in IMP is to find a network with new small weights to prune. Overall, these results make progress toward demystifying the existence of winning tickets by revealing the fundamental role of error landscape geometry in the algorithms used to find them.

Details

NeurIPS Conference 2022 Conference Paper

Beyond neural scaling laws: beating power law scaling via data pruning

  • Ben Sorscher
  • Robert Geirhos
  • Shashank Shekhar
  • Surya Ganguli
  • Ari Morcos

Widely observed neural scaling laws, in which error falls off as a power of the training set size, model size, or both, have driven substantial performance improvements in deep learning. However, these improvements through scaling alone require considerable costs in compute and energy. Here we focus on the scaling of error with dataset size and show how in theory we can break beyond power law scaling and potentially even reduce it to exponential scaling instead if we have access to a high-quality data pruning metric that ranks the order in which training examples should be discarded to achieve any pruned dataset size. We then test this improved scaling prediction with pruned dataset size empirically, and indeed observe better than power law scaling in practice on ResNets trained on CIFAR-10, SVHN, and ImageNet. Next, given the importance of finding high-quality pruning metrics, we perform the first large-scale benchmarking study of ten different data pruning metrics on ImageNet. We find most existing high performing metrics scale poorly to ImageNet, while the best are computationally intensive and require labels for every image. We therefore developed a new simple, cheap and scalable self-supervised pruning metric that demonstrates comparable performance to the best supervised metrics. Overall, our work suggests that the discovery of good data-pruning metrics may provide a viable path forward to substantially improved neural scaling laws, thereby reducing the resource costs of modern deep learning.

PDF Details

ICLR Conference 2022 Conference Paper

How many degrees of freedom do we need to train deep networks: a loss landscape perspective

  • Brett W. Larsen
  • Stanislav Fort
  • Nic Becker
  • Surya Ganguli

A variety of recent works, spanning pruning, lottery tickets, and training within random subspaces, have shown that deep neural networks can be trained using far fewer degrees of freedom than the total number of parameters. We analyze this phenomenon for random subspaces by first examining the success probability of hitting a training loss sublevel set when training within a random subspace of a given training dimensionality. We find a sharp phase transition in the success probability from $0$ to $1$ as the training dimension surpasses a threshold. This threshold training dimension increases as the desired final loss decreases, but decreases as the initial loss decreases. We then theoretically explain the origin of this phase transition, and its dependence on initialization and final desired loss, in terms of properties of the high-dimensional geometry of the loss landscape. In particular, we show via Gordon's escape theorem, that the training dimension plus the Gaussian width of the desired loss sublevel set, projected onto a unit sphere surrounding the initialization, must exceed the total number of parameters for the success probability to be large. In several architectures and datasets, we measure the threshold training dimension as a function of initialization and demonstrate that it is a small fraction of the total parameters, implying by our theory that successful training with so few dimensions is possible precisely because the Gaussian width of low loss sublevel sets is very large. Moreover, we compare this threshold training dimension to more sophisticated ways of reducing training degrees of freedom, including lottery tickets as well as a new, analogous method: lottery subspaces.

Details

NeurIPS Conference 2022 Conference Paper

Lottery Tickets on a Data Diet: Finding Initializations with Sparse Trainable Networks

  • Mansheej Paul
  • Brett Larsen
  • Surya Ganguli
  • Jonathan Frankle
  • Gintare Karolina Dziugaite

A striking observation about iterative magnitude pruning (IMP; Frankle et al. 2020) is that—after just a few hundred steps of dense training—the method can find a sparse sub-network that can be trained to the same accuracy as the dense network. However, the same does not hold at step 0, i. e. random initialization. In this work, we seek to understand how this early phase of pre-training leads to a good initialization for IMP both through the lens of the data distribution and the loss landscape geometry. Empirically we observe that, holding the number of pre-training iterations constant, training on a small fraction of (randomly chosen) data suffices to obtain an equally good initialization for IMP. We additionally observe that by pre-training only on "easy" training data, we can decrease the number of steps necessary to find a good initialization for IMP compared to training on the full dataset or a randomly chosen subset. Finally, we identify novel properties of the loss landscape of dense networks that are predictive of IMP performance, showing in particular that more examples being linearly mode connected in the dense network correlates well with good initializations for IMP. Combined, these results provide new insight into the role played by the early phase training in IMP.

PDF Details

ICLR Conference 2022 Conference Paper

MetaMorph: Learning Universal Controllers with Transformers

  • Agrim Gupta
  • Linxi Fan
  • Surya Ganguli
  • Li Fei-Fei 0001

Multiple domains like vision, natural language, and audio are witnessing tremendous progress by leveraging Transformers for large scale pre-training followed by task specific fine tuning. In contrast, in robotics we primarily train a single robot for a single task. However, modular robot systems now allow for the flexible combination of general-purpose building blocks into task optimized morphologies. However, given the exponentially large number of possible robot morphologies, training a controller for each new design is impractical. In this work, we propose MetaMorph, a Transformer based approach to learn a universal controller over a modular robot design space. MetaMorph is based on the insight that robot morphology is just another modality on which we can condition the output of a Transformer. Through extensive experiments we demonstrate that large scale pre-training on a variety of robot morphologies results in policies with combinatorial generalization capabilities, including zero shot generalization to unseen robot morphologies. We further demonstrate that our pre-trained policy can be used for sample-efficient transfer to completely new robot morphologies and tasks.

Details

ICML Conference 2021 Conference Paper

A theory of high dimensional regression with arbitrary correlations between input features and target functions: sample complexity, multiple descent curves and a hierarchy of phase transitions

  • Gabriel Mel
  • Surya Ganguli

The performance of neural networks depends on precise relationships between four distinct ingredients: the architecture, the loss function, the statistical structure of inputs, and the ground truth target function. Much theoretical work has focused on understanding the role of the first two ingredients under highly simplified models of random uncorrelated data and target functions. In contrast, performance likely relies on a conspiracy between the statistical structure of the input distribution and the structure of the function to be learned. To understand this better we revisit ridge regression in high dimensions, which corresponds to an exceedingly simple architecture and loss function, but we analyze its performance under arbitrary correlations between input features and the target function. We find a rich mathematical structure that includes: (1) a dramatic reduction in sample complexity when the target function aligns with data anisotropy; (2) the existence of multiple descent curves; (3) a sequence of phase transitions in the performance, loss landscape, and optimal regularization as a function of the amount of data that explains the first two effects.

Details

NeurIPS Conference 2021 Conference Paper

Deep Learning on a Data Diet: Finding Important Examples Early in Training

  • Mansheej Paul
  • Surya Ganguli
  • Gintare Karolina Dziugaite

Recent success in deep learning has partially been driven by training increasingly overparametrized networks on ever larger datasets. It is therefore natural to ask: how much of the data is superfluous, which examples are important for generalization, and how do we find them? In this work, we make the striking observation that, in standard vision datasets, simple scores averaged over several weight initializations can be used to identify important examples very early in training. We propose two such scores—the Gradient Normed (GraNd) and the Error L2-Norm (EL2N) scores—and demonstrate their efficacy on a range of architectures and datasets by pruning significant fractions of training data without sacrificing test accuracy. In fact, using EL2N scores calculated a few epochs into training, we can prune half of the CIFAR10 training set while slightly improving test accuracy. Furthermore, for a given dataset, EL2N scores from one architecture or hyperparameter configuration generalize to other configurations. Compared to recent work that prunes data by discarding examples that are rarely forgotten over the course of training, our scores use only local information early in training. We also use our scores to detect noisy examples and study training dynamics through the lens of important examples—we investigate how the data distribution shapes the loss surface and identify subspaces of the model’s data representation that are relatively stable over training.

PDF Details

NeurIPS Conference 2021 Conference Paper

Explaining heterogeneity in medial entorhinal cortex with task-driven neural networks

  • Aran Nayebi
  • Alexander Attinger
  • Malcolm Campbell
  • Kiah Hardcastle
  • Isabel Low
  • Caitlin S Mallory
  • Gabriel Mel
  • Ben Sorscher

Medial entorhinal cortex (MEC) supports a wide range of navigational and memory related behaviors. Well-known experimental results have revealed specialized cell types in MEC --- e. g. grid, border, and head-direction cells --- whose highly stereotypical response profiles are suggestive of the role they might play in supporting MEC functionality. However, the majority of MEC neurons do not exhibit stereotypical firing patterns. How should the response profiles of these more "heterogeneous" cells be described, and how do they contribute to behavior? In this work, we took a computational approach to addressing these questions. We first performed a statistical analysis that shows that heterogeneous MEC cells are just as reliable in their response patterns as the more stereotypical cell types, suggesting that they have a coherent functional role. Next, we evaluated a spectrum of candidate models in terms of their ability to describe the response profiles of both stereotypical and heterogeneous MEC cells. We found that recently developed task-optimized neural network models are substantially better than traditional grid cell-centric models at matching most MEC neuronal response profiles --- including those of grid cells themselves --- despite not being explicitly trained for this purpose. Specific choices of network architecture (such as gated nonlinearities and an explicit intermediate place cell representation) have an important effect on the ability of the model to generalize to novel scenarios, with the best of these models closely approaching the noise ceiling of the data itself. We then performed in silico experiments on this model to address questions involving the relative functional relevance of various cell types, finding that heterogeneous cells are likely to be just as involved in downstream functional outcomes (such as path integration) as grid and border cells. Finally, inspired by recent data showing that, going beyond their spatial response selectivity, MEC cells are also responsive to non-spatial rewards, we introduce a new MEC model that performs reward-modulated path integration. We find that this unified model matches neural recordings across all variable-reward conditions. Taken together, our results point toward a conceptually principled goal-driven modeling approach for moving future experimental and computational efforts beyond overly-simplistic single-cell stereotypes.

PDF Details

ICLR Conference 2021 Conference Paper

Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics

  • Daniel Kunin
  • Javier Sagastuy-Breña
  • Surya Ganguli
  • Daniel L. K. Yamins
  • Hidenori Tanaka

Understanding the dynamics of neural network parameters during training is one of the key challenges in building a theoretical foundation for deep learning. A central obstacle is that the motion of a network in high-dimensional parameter space undergoes discrete finite steps along complex stochastic gradients derived from real-world datasets. We circumvent this obstacle through a unifying theoretical framework based on intrinsic symmetries embedded in a network's architecture that are present for any dataset. We show that any such symmetry imposes stringent geometric constraints on gradients and Hessians, leading to an associated conservation law in the continuous-time limit of stochastic gradient descent (SGD), akin to Noether's theorem in physics. We further show that finite learning rates used in practice can actually break these symmetry induced conservation laws. We apply tools from finite difference methods to derive modified gradient flow, a differential equation that better approximates the numerical trajectory taken by SGD at finite learning rates. We combine modified gradient flow with our framework of symmetries to derive exact integral expressions for the dynamics of certain parameter combinations. We empirically validate our analytic expressions for learning dynamics on VGG-16 trained on Tiny ImageNet. Overall, by exploiting symmetry, our work demonstrates that we can analytically describe the learning dynamics of various parameter combinations at finite learning rates and batch sizes for state of the art architectures trained on any dataset.

Details

ICML Conference 2021 Conference Paper

Understanding self-supervised learning dynamics without contrastive pairs

  • Yuandong Tian
  • Xinlei Chen
  • Surya Ganguli

While contrastive approaches of self-supervised learning (SSL) learn representations by minimizing the distance between two augmented views of the same data point (positive pairs) and maximizing views from different data points (negative pairs), recent \emph{non-contrastive} SSL (e. g. , BYOL and SimSiam) show remarkable performance {\it without} negative pairs, with an extra learnable predictor and a stop-gradient operation. A fundamental question rises: why they do not collapse into trivial representation? In this paper, we answer this question via a simple theoretical study and propose a novel approach, \ourmethod{}, that \emph{directly} sets the linear predictor based on the statistics of its inputs, rather than trained with gradient update. On ImageNet, it performs comparably with more complex two-layer non-linear predictors that employ BatchNorm and outperforms linear predictor by $2. 5%$ in 300-epoch training (and $5%$ in 60-epoch). \ourmethod{} is motivated by our theoretical study of the nonlinear learning dynamics of non-contrastive SSL in simple linear networks. Our study yields conceptual insights into how non-contrastive SSL methods learn, how they avoid representational collapse, and how multiple factors, like predictor networks, stop-gradients, exponential moving averages, and weight decay all come into play. Our simple theory recapitulates the results of real-world ablation studies in both STL-10 and ImageNet. Code is released\footnote{\url{https: //github. com/facebookresearch/luckmatters/tree/master/ssl}}.

Details

NeurIPS Conference 2020 Conference Paper

Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel

  • Stanislav Fort
  • Gintare Karolina Dziugaite
  • Mansheej Paul
  • Sepideh Kharaghani
  • Daniel M. Roy
  • Surya Ganguli

In suitably initialized wide networks, small learning rates transform deep neural networks (DNNs) into neural tangent kernel (NTK) machines, whose training dynamics is well-approximated by a linear weight expansion of the network at initialization. Standard training, however, diverges from its linearization in ways that are poorly understood. We study the relationship between the training dynamics of nonlinear deep networks, the geometry of the loss landscape, and the time evolution of a data-dependent NTK. We do so through a large-scale phenomenological analysis of training, synthesizing diverse measures characterizing loss landscape geometry and NTK dynamics. In multiple neural architectures and datasets, we find these diverse measures evolve in a highly correlated manner, revealing a universal picture of the deep learning process. In this picture, deep network training exhibits a highly chaotic rapid initial transient that within 2 to 3 epochs determines the final linearly connected basin of low loss containing the end point of training. During this chaotic transient, the NTK changes rapidly, learning useful features from the training data that enables it to outperform the standard initial NTK by a factor of 3 in less than 3 to 4 epochs. After this rapid chaotic transient, the NTK changes at constant velocity, and its performance matches that of full network training in 15\% to 45\% of training time. Overall, our analysis reveals a striking correlation between a diverse set of metrics over training time, governed by a rapid chaotic to stable transition in the first few epochs, that together poses challenges and opportunities for the development of more accurate theories of deep learning.

PDF Details

NeurIPS Conference 2020 Conference Paper

Identifying Learning Rules From Neural Network Observables

  • Aran Nayebi
  • Sanjana Srivastava
  • Surya Ganguli
  • Daniel L. Yamins

The brain modifies its synaptic strengths during learning in order to better adapt to its environment. However, the underlying plasticity rules that govern learning are unknown. Many proposals have been suggested, including Hebbian mechanisms, explicit error backpropagation, and a variety of alternatives. It is an open question as to what specific experimental measurements would need to be made to determine whether any given learning rule is operative in a real biological system. In this work, we take a "virtual experimental" approach to this problem. Simulating idealized neuroscience experiments with artificial neural networks, we generate a large-scale dataset of learning trajectories of aggregate statistics measured in a variety of neural network architectures, loss functions, learning rule hyperparameters, and parameter initializations. We then take a discriminative approach, training linear and simple non-linear classifiers to identify learning rules from features based on these observables. We show that different classes of learning rules can be separated solely on the basis of aggregate statistics of the weights, activations, or instantaneous layer-wise activity changes, and that these results generalize to limited access to the trajectory and held-out architectures and learning curricula. We identify the statistics of each observable that are most relevant for rule identification, finding that statistics from network activities across training are more robust to unit undersampling and measurement noise than those obtained from the synaptic strengths. Our results suggest that activation patterns, available from electrophysiological recordings of post-synaptic activities on the order of several hundred units, frequently measured at wider intervals over the course of learning, may provide a good basis on which to identify learning rules.

PDF Details

NeurIPS Conference 2020 Conference Paper

Predictive coding in balanced neural networks with noise, chaos and delays

  • Jonathan Kadmon
  • Jonathan Timcheck
  • Surya Ganguli

Biological neural networks face a formidable task: performing reliable computations in the face of intrinsic stochasticity in individual neurons, imprecisely specified synaptic connectivity, and nonnegligible delays in synaptic transmission. A common approach to combatting such biological heterogeneity involves averaging over large redundant networks of N neurons resulting in coding errors that decrease classically as the square root of N. Recent work demonstrated a novel mechanism whereby recurrent spiking networks could efficiently encode dynamic stimuli achieving a superclassical scaling in which coding errors decrease as 1/N. This specific mechanism involved two key ideas: predictive coding, and a tight balance, or cancellation between strong feedforward inputs and strong recurrent feedback. However, the theoretical principles governing the efficacy of balanced predictive coding and its robustness to noise, synaptic weight heterogeneity and communication delays remain poorly understood. To discover such principles, we introduce an analytically tractable model of balanced predictive coding, in which the degree of balance and the degree of weight disorder can be dissociated unlike in previous balanced network models, and we develop a mean-field theory of coding accuracy. Overall, our work provides and solves a general theoretical framework for dissecting the differential contributions neural noise, synaptic disorder, chaos, synaptic delays, and balance to the fidelity of predictive neural codes, reveals the fundamental role that balance plays in achieving superclassical scaling, and unifies previously disparate models in theoretical neuroscience.

PDF Details

NeurIPS Conference 2020 Conference Paper

Pruning neural networks without any data by iteratively conserving synaptic flow

  • Hidenori Tanaka
  • Daniel Kunin
  • Daniel L. Yamins
  • Surya Ganguli

Pruning the parameters of deep neural networks has generated intense interest due to potential savings in time, memory and energy both during training and at test time. Recent works have identified, through an expensive sequence of training and pruning cycles, the existence of winning lottery tickets or sparse trainable subnetworks at initialization. This raises a foundational question: can we identify highly sparse trainable subnetworks at initialization, without ever training, or indeed without ever looking at the data? We provide an affirmative answer to this question through theory driven algorithm design. We first mathematically formulate and experimentally verify a conservation law that explains why existing gradient-based pruning algorithms at initialization suffer from layer-collapse, the premature pruning of an entire layer rendering a network untrainable. This theory also elucidates how layer-collapse can be entirely avoided, motivating a novel pruning algorithm Iterative Synaptic Flow Pruning (SynFlow). This algorithm can be interpreted as preserving the total flow of synaptic strengths through the network at initialization subject to a sparsity constraint. Notably, this algorithm makes no reference to the training data and consistently competes with or outperforms existing state-of-the-art pruning algorithms at initialization over a range of models (VGG and ResNet), datasets (CIFAR-10/100 and Tiny ImageNet), and sparsity constraints (up to 99. 99 percent). Thus our data-agnostic pruning algorithm challenges the existing paradigm that, at initialization, data must be used to quantify which synapses are important.

PDF Details

ICML Conference 2020 Conference Paper

Two Routes to Scalable Credit Assignment without Weight Symmetry

  • Daniel Kunin
  • Aran Nayebi
  • Javier Sagastuy-Breña
  • Surya Ganguli
  • Jonathan M. Bloom
  • Daniel L. K. Yamins

The neural plausibility of backpropagation has long been disputed, primarily for its use of non-local weight transport — the biologically dubious requirement that one neuron instantaneously measure the synaptic weights of another. Until recently, attempts to create local learning rules that avoid weight transport have typically failed in the large-scale learning scenarios where backpropagation shines, e. g. ImageNet categorization with deep convolutional networks. Here, we investigate a recently proposed local learning rule that yields competitive performance with backpropagation and find that it is highly sensitive to metaparameter choices, requiring laborious tuning that does not transfer across network architecture. Our analysis indicates the underlying mathematical reason for this instability, allowing us to identify a more robust local learning rule that better transfers without metaparameter tuning. Nonetheless, we find a performance and stability gap between this local rule and backpropagation that widens with increasing model depth. We then investigate several non-local learning rules that relax the need for instantaneous weight transport into a more biologically-plausible "weight estimation" process, showing that these rules match state-of-the-art performance on deep networks and operate effectively in the presence of noisy updates. Taken together, our results suggest two routes towards the discovery of neural implementations for credit assignment without weight symmetry: further improvement of local rules so that they perform consistently across architectures and the identification of biological implementations for non-local learning mechanisms.

Details

NeurIPS Conference 2019 Conference Paper

A unified theory for the origin of grid cells through the lens of pattern formation

  • Ben Sorscher
  • Gabriel Mel
  • Surya Ganguli
  • Samuel Ocko

Grid cells in the brain fire in strikingly regular hexagonal patterns across space. There are currently two seemingly unrelated frameworks for understanding these patterns. Mechanistic models account for hexagonal firing fields as the result of pattern-forming dynamics in a recurrent neural network with hand-tuned center-surround connectivity. Normative models specify a neural architecture, a learning rule, and a navigational task, and observe that grid-like firing fields emerge due to the constraints of solving this task. Here we provide an analytic theory that unifies the two perspectives by casting the learning dynamics of neural networks trained on navigational tasks as a pattern forming dynamical system. This theory provides insight into the optimal solutions of diverse formulations of the normative task, and shows that symmetries in the representation of space correctly predict the structure of learned firing fields in trained neural networks. Further, our theory proves that a nonnegativity constraint on firing rates induces a symmetry-breaking mechanism which favors hexagonal firing fields. We extend this theory to the case of learning multiple grid maps and demonstrate that optimal solutions consist of a hierarchy of maps with increasing length scales. These results unify previous accounts of grid cell firing and provide a novel framework for predicting the learned representations of recurrent neural networks.

PDF Details

ICLR Conference 2019 Conference Paper

A Unified Theory of Early Visual Representations from Retina to Cortex through Anatomically Constrained Deep CNNs

  • Jack Lindsey
  • Samuel A. Ocko
  • Surya Ganguli
  • Stéphane Deny

The vertebrate visual system is hierarchically organized to process visual information in successive stages. Neural representations vary drastically across the first stages of visual processing: at the output of the retina, ganglion cell receptive fields (RFs) exhibit a clear antagonistic center-surround structure, whereas in the primary visual cortex (V1), typical RFs are sharply tuned to a precise orientation. There is currently no unified theory explaining these differences in representations across layers. Here, using a deep convolutional neural network trained on image recognition as a model of the visual system, we show that such differences in representation can emerge as a direct consequence of different neural resource constraints on the retinal and cortical networks, and for the first time we find a single model from which both geometries spontaneously emerge at the appropriate stages of visual processing. The key constraint is a reduced number of neurons at the retinal output, consistent with the anatomy of the optic nerve as a stringent bottleneck. Second, we find that, for simple downstream cortical networks, visual representations at the retinal output emerge as nonlinear and lossy feature detectors, whereas they emerge as linear and faithful encoders of the visual scene for more complex cortical networks. This result predicts that the retinas of small vertebrates (e.g. salamander, frog) should perform sophisticated nonlinear computations, extracting features directly relevant to behavior, whereas retinas of large animals such as primates should mostly encode the visual scene linearly and respond to a much broader range of stimuli. These predictions could reconcile the two seemingly incompatible views of the retina as either performing feature extraction or efficient coding of natural scenes, by suggesting that all vertebrates lie on a spectrum between these two objectives, depending on the degree of neural resources allocated to their visual system.

Details

NeurIPS Conference 2019 Conference Paper

From deep learning to mechanistic understanding in neuroscience: the structure of retinal prediction

  • Hidenori Tanaka
  • Aran Nayebi
  • Niru Maheswaranathan
  • Lane McIntosh
  • Stephen Baccus
  • Surya Ganguli

Recently, deep feedforward neural networks have achieved considerable success in modeling biological sensory processing, in terms of reproducing the input-output map of sensory neurons. However, such models raise profound questions about the very nature of explanation in neuroscience. Are we simply replacing one complex system (a biological circuit) with another (a deep network), without understanding either? Moreover, beyond neural representations, are the deep network's computational mechanisms for generating neural responses the same as those in the brain? Without a systematic approach to extracting and understanding computational mechanisms from deep neural network models, it can be difficult both to assess the degree of utility of deep learning approaches in neuroscience, and to extract experimentally testable hypotheses from deep networks. We develop such a systematic approach by combining dimensionality reduction and modern attribution methods for determining the relative importance of interneurons for specific visual computations. We apply this approach to deep network models of the retina, revealing a conceptual understanding of how the retina acts as a predictive feature extractor that signals deviations from expectations for diverse spatiotemporal stimuli. For each stimulus, our extracted computational mechanisms are consistent with prior scientific literature, and in one case yields a new mechanistic hypothesis. Thus overall, this work not only yields insights into the computational mechanisms underlying the striking predictive capabilities of the retina, but also places the framework of deep networks as neuroscientific models on firmer theoretical foundations, by providing a new roadmap to go beyond comparing neural representations to extracting and understand computational mechanisms.

PDF Details

NeurIPS Conference 2019 Conference Paper

Reverse engineering recurrent networks for sentiment classification reveals line attractor dynamics

  • Niru Maheswaranathan
  • Alex Williams
  • Matthew Golub
  • Surya Ganguli
  • David Sussillo

Recurrent neural networks (RNNs) are a widely used tool for modeling sequential data, yet they are often treated as inscrutable black boxes. Given a trained recurrent network, we would like to reverse engineer it--to obtain a quantitative, interpretable description of how it solves a particular task. Even for simple tasks, a detailed understanding of how recurrent networks work, or a prescription for how to develop such an understanding, remains elusive. In this work, we use tools from dynamical systems analysis to reverse engineer recurrent networks trained to perform sentiment classification, a foundational natural language processing task. Given a trained network, we find fixed points of the recurrent dynamics and linearize the nonlinear system around these fixed points. Despite their theoretical capacity to implement complex, high-dimensional computations, we find that trained networks converge to highly interpretable, low-dimensional representations. In particular, the topological structure of the fixed points and corresponding linearized dynamics reveal an approximate line attractor within the RNN, which we can use to quantitatively understand how the RNN solves the sentiment analysis task. Finally, we find this mechanism present across RNN architectures (including LSTMs, GRUs, and vanilla RNNs) trained on multiple datasets, suggesting that our findings are not unique to a particular architecture or dataset. Overall, these results demonstrate that surprisingly universal and human interpretable computations can arise across a range of recurrent networks.

PDF Details

NeurIPS Conference 2019 Conference Paper

Universality and individuality in neural dynamics across large populations of recurrent networks

  • Niru Maheswaranathan
  • Alex Williams
  • Matthew Golub
  • Surya Ganguli
  • David Sussillo

Many recent studies have employed task-based modeling with recurrent neural networks (RNNs) to infer the computational function of different brain regions. These models are often assessed by quantitatively comparing the low-dimensional neural dynamics of the model and the brain, for example using canonical correlation analysis (CCA). However, the nature of the detailed neurobiological inferences one can draw from such efforts remains elusive. For example, to what extent does training neural networks to solve simple tasks, prevalent in neuroscientific studies, uniquely determine the low-dimensional dynamics independent of neural architectures? Or alternatively, are the learned dynamics highly sensitive to different neural architectures? Knowing the answer to these questions has strong implications on whether and how to use task-based RNN modeling to understand brain dynamics. To address these foundational questions, we study populations of thousands of networks of commonly used RNN architectures trained to solve neuroscientifically motivated tasks and characterize their low-dimensional dynamics via CCA and nonlinear dynamical systems analysis. We find the geometry of the dynamics can be highly sensitive to different network architectures, and further find striking dissociations between geometric similarity as measured by CCA and network function, yielding a cautionary tale. Moreover, we find that while the geometry of neural dynamics can vary greatly across architectures, the underlying computational scaffold: the topological structure of fixed points, transitions between them, limit cycles, and linearized dynamics, often appears {\it universal} across all architectures. Overall, this analysis of universality and individuality across large populations of RNNs provides a much needed foundation for interpreting quantitative measures of dynamical similarity between RNN and brain dynamics.

PDF Details

NeurIPS Conference 2018 Conference Paper

Statistical mechanics of low-rank tensor decomposition

  • Jonathan Kadmon
  • Surya Ganguli

Often, large, high dimensional datasets collected across multiple modalities can be organized as a higher order tensor. Low-rank tensor decomposition then arises as a powerful and widely used tool to discover simple low dimensional structures underlying such data. However, we currently lack a theoretical understanding of the algorithmic behavior of low-rank tensor decompositions. We derive Bayesian approximate message passing (AMP) algorithms for recovering arbitrarily shaped low-rank tensors buried within noise, and we employ dynamic mean field theory to precisely characterize their performance. Our theory reveals the existence of phase transitions between easy, hard and impossible inference regimes, and displays an excellent match with simulations. Moreover, it reveals several qualitative surprises compared to the behavior of symmetric, cubic tensor decomposition. Finally, we compare our AMP algorithm to the most commonly used algorithm, alternating least squares (ALS), and demonstrate that AMP significantly outperforms ALS in the presence of noise.

PDF Details

NeurIPS Conference 2018 Conference Paper

Task-Driven Convolutional Recurrent Models of the Visual System

  • Aran Nayebi
  • Daniel Bear
  • Jonas Kubilius
  • Kohitij Kar
  • Surya Ganguli
  • David Sussillo
  • James DiCarlo
  • Daniel Yamins

Feed-forward convolutional neural networks (CNNs) are currently state-of-the-art for object classification tasks such as ImageNet. Further, they are quantitatively accurate models of temporally-averaged responses of neurons in the primate brain's visual system. However, biological visual systems have two ubiquitous architectural features not shared with typical CNNs: local recurrence within cortical areas, and long-range feedback from downstream areas to upstream areas. Here we explored the role of recurrence in improving classification performance. We found that standard forms of recurrence (vanilla RNNs and LSTMs) do not perform well within deep CNNs on the ImageNet task. In contrast, novel cells that incorporated two structural features, bypassing and gating, were able to boost task accuracy substantially. We extended these design principles in an automated search over thousands of model architectures, which identified novel local recurrent cells and long-range feedback connections useful for object recognition. Moreover, these task-optimized ConvRNNs matched the dynamics of neural activity in the primate visual system better than feedforward networks, suggesting a role for the brain's recurrent connections in performing difficult visual behaviors.

PDF Details

NeurIPS Conference 2018 Conference Paper

The emergence of multiple retinal cell types through efficient coding of natural movies

  • Samuel Ocko
  • Jack Lindsey
  • Surya Ganguli
  • Stephane Deny

One of the most striking aspects of early visual processing in the retina is the immediate parcellation of visual information into multiple parallel pathways, formed by different retinal ganglion cell types each tiling the entire visual field. Existing theories of efficient coding have been unable to account for the functional advantages of such cell-type diversity in encoding natural scenes. Here we go beyond previous theories to analyze how a simple linear retinal encoding model with different convolutional cell types efficiently encodes naturalistic spatiotemporal movies given a fixed firing rate budget. We find that optimizing the receptive fields and cell densities of two cell types makes them match the properties of the two main cell types in the primate retina, midget and parasol cells, in terms of spatial and temporal sensitivity, cell spacing, and their relative ratio. Moreover, our theory gives a precise account of how the ratio of midget to parasol cells decreases with retinal eccentricity. Also, we train a nonlinear encoding model with a rectifying nonlinearity to efficiently encode naturalistic movies, and again find emergent receptive fields resembling those of midget and parasol cells that are now further subdivided into ON and OFF types. Thus our work provides a theoretical justification, based on the efficient coding of natural movies, for the existence of the four most dominant cell types in the primate retina that together comprise 70% of all ganglion cells.

PDF Details

ICML Conference 2017 Conference Paper

Continual Learning Through Synaptic Intelligence

  • Friedemann Zenke
  • Ben Poole
  • Surya Ganguli

While deep learning has led to remarkable advances across diverse applications, it struggles in domains where the data distribution changes over the course of learning. In stark contrast, biological neural networks continually adapt to changing domains, possibly by leveraging complex molecular machinery to solve many tasks simultaneously. In this study, we introduce intelligent synapses that bring some of this biological complexity into artificial neural networks. Each synapse accumulates task relevant information over time, and exploits this information to rapidly store new memories without forgetting old ones. We evaluate our approach on continual learning of classification tasks, and show that it dramatically reduces forgetting while maintaining computational efficiency.

Details

ICML Conference 2017 Conference Paper

On the Expressive Power of Deep Neural Networks

  • Maithra Raghu
  • Ben Poole
  • Jon M. Kleinberg
  • Surya Ganguli
  • Jascha Sohl-Dickstein

We propose a new approach to the problem of neural network expressivity, which seeks to characterize how structural properties of a neural network family affect the functions it is able to compute. Our approach is based on an interrelated set of measures of expressivity, unified by the novel notion of trajectory length, which measures how the output of a network changes as the input sweeps along a one-dimensional path. Our findings show that: (1) The complexity of the computed function grows exponentially with depth (2) All weights are not equal: trained networks are more sensitive to their lower (initial) layer weights (3) Trajectory regularization is a simpler alternative to batch normalization, with the same performance.

Details

NeurIPS Conference 2017 Conference Paper

Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice

  • Jeffrey Pennington
  • Samuel Schoenholz
  • Surya Ganguli

It is well known that weight initialization in deep networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is O(1) is essential for avoiding exponentially vanishing or exploding gradients. Moreover, in deep linear networks, ensuring that all singular values of the Jacobian are concentrated near 1 can yield a dramatic additional speed-up in learning; this is a property known as dynamical isometry. However, it is unclear how to achieve dynamical isometry in nonlinear deep networks. We address this question by employing powerful tools from free probability theory to analytically compute the {\it entire} singular value distribution of a deep network's input-output Jacobian. We explore the dependence of the singular value distribution on the depth of the network, the weight initialization, and the choice of nonlinearity. Intriguingly, we find that ReLU networks are incapable of dynamical isometry. On the other hand, sigmoidal networks can achieve isometry, but only with orthogonal weight initialization. Moreover, we demonstrate empirically that deep nonlinear networks achieving dynamical isometry learn orders of magnitude faster than networks that do not. Indeed, we show that properly-initialized deep sigmoidal networks consistently outperform deep ReLU networks. Overall, our analysis reveals that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning.

PDF Details

NeurIPS Conference 2017 Conference Paper

Variational Walkback: Learning a Transition Operator as a Stochastic Recurrent Net

  • Anirudh Goyal ALIAS PARTH GOYAL
  • Nan Rosemary Ke
  • Surya Ganguli
  • Yoshua Bengio

We propose a novel method to {\it directly} learn a stochastic transition operator whose repeated application provides generated samples. Traditional undirected graphical models approach this problem indirectly by learning a Markov chain model whose stationary distribution obeys detailed balance with respect to a parameterized energy function. The energy function is then modified so the model and data distributions match, with no guarantee on the number of steps required for the Markov chain to converge. Moreover, the detailed balance condition is highly restrictive: energy based models corresponding to neural networks must have symmetric weights, unlike biological neural circuits. In contrast, we develop a method for directly learning arbitrarily parameterized transition operators capable of expressing non-equilibrium stationary distributions that violate detailed balance, thereby enabling us to learn more biologically plausible asymmetric neural networks and more general non-energy based dynamical systems. The proposed training objective, which we derive via principled variational methods, encourages the transition operator to "walk back" (prefer to revert its steps) in multi-step trajectories that start at data-points, as quickly as possible back to the original data points. We present a series of experimental results illustrating the soundness of the proposed approach, Variational Walkback (VW), on the MNIST, CIFAR-10, SVHN and CelebA datasets, demonstrating superior samples compared to earlier attempts to learn a transition operator. We also show that although each rapid training trajectory is limited to a finite but variable number of steps, our transition operator continues to generate good samples well past the length of such trajectories, thereby demonstrating the match of its non-equilibrium stationary distribution to the data distribution. Source Code: http: //github. com/anirudh9119/walkback_nips17

PDF Details

NeurIPS Conference 2016 Conference Paper

An equivalence between high dimensional Bayes optimal inference and M-estimation

  • Madhu Advani
  • Surya Ganguli

Due to the computational difficulty of performing MMSE (minimum mean squared error) inference, maximum a posteriori (MAP) is often used as a surrogate. However, the accuracy of MAP is suboptimal for high dimensional inference, where the number of model parameters is of the same order as the number of samples. In this work we demonstrate how MMSE performance is asymptotically achievable via optimization with an appropriately selected convex penalty and regularization function which are a smoothed version of the widely applied MAP algorithm. Our findings provide a new derivation and interpretation for recent optimal M-estimators discovered by El Karoui, et. al. PNAS 2013 as well as extending to non-additive noise models. We demonstrate the performance of these optimal M-estimators with numerical simulations. Overall, at the heart of our work is the revelation of a remarkable equivalence between two seemingly very different computational problems: namely that of high dimensional Bayesian integration, and high dimensional convex optimization. In essence we show that the former computationally difficult integral may be computed by solving the latter, simpler optimization problem.

PDF Details

NeurIPS Conference 2016 Conference Paper

Deep Learning Models of the Retinal Response to Natural Scenes

  • Lane McIntosh
  • Niru Maheswaranathan
  • Aran Nayebi
  • Surya Ganguli
  • Stephen Baccus

A central challenge in sensory neuroscience is to understand neural computations and circuit mechanisms that underlie the encoding of ethologically relevant, natural stimuli. In multilayered neural circuits, nonlinear processes such as synaptic transmission and spiking dynamics present a significant obstacle to the creation of accurate computational models of responses to natural stimuli. Here we demonstrate that deep convolutional neural networks (CNNs) capture retinal responses to natural scenes nearly to within the variability of a cell's response, and are markedly more accurate than linear-nonlinear (LN) models and Generalized Linear Models (GLMs). Moreover, we find two additional surprising properties of CNNs: they are less susceptible to overfitting than their LN counterparts when trained on small amounts of data, and generalize better when tested on stimuli drawn from a different distribution (e. g. between natural scenes and white noise). An examination of the learned CNNs reveals several properties. First, a richer set of feature maps is necessary for predicting the responses to natural scenes compared to white noise. Second, temporally precise responses to slowly varying inputs originate from feedforward inhibition, similar to known retinal mechanisms. Third, the injection of latent noise sources in intermediate layers enables our model to capture the sub-Poisson spiking variability observed in retinal ganglion cells. Fourth, augmenting our CNNs with recurrent lateral connections enables them to capture contrast adaptation as an emergent property of accurately describing retinal responses to natural scenes. These methods can be readily generalized to other sensory modalities and stimulus ensembles. Overall, this work demonstrates that CNNs not only accurately capture sensory circuit responses to natural scenes, but also can yield information about the circuit's internal structure and function.

PDF Details

NeurIPS Conference 2016 Conference Paper

Exponential expressivity in deep neural networks through transient chaos

  • Ben Poole
  • Subhaneil Lahiri
  • Maithra Raghu
  • Jascha Sohl-Dickstein
  • Surya Ganguli

We combine Riemannian geometry with the mean field theory of high dimensional chaos to study the nature of signal propagation in deep neural networks with random weights. Our results reveal a phase transition in the expressivity of random deep networks, with networks in the chaotic phase computing nonlinear functions whose global curvature grows exponentially with depth, but not with width. We prove that this generic class of random functions cannot be efficiently computed by any shallow network, going beyond prior work that restricts their analysis to single functions. Moreover, we formally quantify and demonstrate the long conjectured idea that deep networks can disentangle exponentially curved manifolds in input space into flat manifolds in hidden space. Our theoretical framework for analyzing the expressive power of deep networks is broadly applicable and provides a basis for quantifying previously abstract notions about the geometry of deep functions.

PDF Details

NeurIPS Conference 2015 Conference Paper

Deep Knowledge Tracing

  • Chris Piech
  • Jonathan Bassen
  • Jonathan Huang
  • Surya Ganguli
  • Mehran Sahami
  • Leonidas Guibas
  • Jascha Sohl-Dickstein

Knowledge tracing, where a machine models the knowledge of a student as they interact with coursework, is an established and significantly unsolved problem in computer supported education. In this paper we explore the benefit of using recurrent neural networks to model student learning. This family of models have important advantages over current state of the art methods in that they do not require the explicit encoding of human domain knowledge, and have a far more flexible functional form which can capture substantially more complex student interactions. We show that these neural networks outperform the current state of the art in prediction on real student data, while allowing straightforward interpretation and discovery of structure in the curriculum. These results suggest a promising new line of research for knowledge tracing.

PDF Details

ICML Conference 2015 Conference Paper

Deep Unsupervised Learning using Nonequilibrium Thermodynamics

  • Jascha Sohl-Dickstein
  • Eric A. Weiss
  • Niru Maheswaranathan
  • Surya Ganguli

A central problem in machine learning involves modeling complex data-sets using highly flexible families of probability distributions in which learning, sampling, inference, and evaluation are still analytically or computationally tractable. Here, we develop an approach that simultaneously achieves both flexibility and tractability. The essential idea, inspired by non-equilibrium statistical physics, is to systematically and slowly destroy structure in a data distribution through an iterative forward diffusion process. We then learn a reverse diffusion process that restores structure in data, yielding a highly flexible and tractable generative model of the data. This approach allows us to rapidly learn, sample from, and evaluate probabilities in deep generative models with thousands of layers or time steps, as well as to compute conditional and posterior probabilities under the learned model. We additionally release an open source reference implementation of the algorithm.

Details

ICLR Conference 2014 Conference Paper

Exact solutions to the nonlinear dynamics of learning in deep linear neural networks

  • Andrew M. Saxe
  • James L. McClelland
  • Surya Ganguli

Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics on weights that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that as the depth of a network approaches infinity, learning speed remains finite: for a special class of initial conditions on the weights, very deep networks incur only a finite delay in learning speed relative to shallow networks. We further show that, under certain conditions on the training data, unsupervised pretraining can find this special class of initial conditions, thereby providing analytical insight into the success of unsupervised pretraining in deep supervised learning tasks.

Details

ICML Conference 2014 Conference Paper

Fast large-scale optimization by unifying stochastic gradient and quasi-Newton methods

  • Jascha Sohl-Dickstein
  • Ben Poole
  • Surya Ganguli

We present an algorithm for minimizing a sum of functions that combines the computational efficiency of stochastic gradient descent (SGD) with the second order curvature information leveraged by quasi-Newton methods. We unify these disparate approaches by maintaining an independent Hessian approximation for each contributing function in the sum. We maintain computational tractability and limit memory requirements even for high dimensional optimization problems by storing and manipulating these quadratic approximations in a shared, time evolving, low dimensional subspace. This algorithm contrasts with earlier stochastic second order techniques that treat the Hessian of each contributing function as a noisy approximation to the full Hessian, rather than as a target for direct estimation. Each update step requires only a single contributing function or minibatch evaluation (as in SGD), and each step is scaled using an approximate inverse Hessian and little to no adjustment of hyperparameters is required (as is typical for quasi-Newton methods). We experimentally demonstrate improved convergence on seven diverse optimization problems. The algorithm is released as open source Python and MATLAB packages.

Details

NeurIPS Conference 2014 Conference Paper

Identifying and attacking the saddle point problem in high-dimensional non-convex optimization

  • Yann Dauphin
  • Razvan Pascanu
  • Caglar Gulcehre
  • Kyunghyun Cho
  • Surya Ganguli
  • Yoshua Bengio

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.

PDF Details

NeurIPS Conference 2013 Conference Paper

A memory frontier for complex synapses

  • Subhaneil Lahiri
  • Surya Ganguli

An incredible gulf separates theoretical models of synapses, often described solely by a single scalar value denoting the size of a postsynaptic potential, from the immense complexity of molecular signaling pathways underlying real synapses. To understand the functional contribution of such molecular complexity to learning and memory, it is essential to expand our theoretical conception of a synapse from a single scalar to an entire dynamical system with many internal molecular functional states. Moreover, theoretical considerations alone demand such an expansion; network models with scalar synapses assuming finite numbers of distinguishable synaptic strengths have strikingly limited memory capacity. This raises the fundamental question, how does synaptic complexity give rise to memory? To address this, we develop new mathematical theorems elucidating the relationship between the structural organization and memory properties of complex synapses that are themselves molecular networks. Moreover, in proving such theorems, we uncover a framework, based on first passage time theory, to impose an order on the internal states of complex synaptic models, thereby simplifying the relationship between synaptic structure and function.

PDF Details

NeurIPS Conference 2010 Conference Paper

Short-term memory in neuronal networks through dynamical compressed sensing

  • Surya Ganguli
  • Haim Sompolinsky

Recent proposals suggest that large, generic neuronal networks could store memory traces of past input sequences in their instantaneous state. Such a proposal raises important theoretical questions about the duration of these memory traces and their dependence on network size, connectivity and signal statistics. Prior work, in the case of gaussian input sequences and linear neuronal networks, shows that the duration of memory traces in a network cannot exceed the number of neurons (in units of the neuronal time constant), and that no network can out-perform an equivalent feedforward network. However a more ethologically relevant scenario is that of sparse input sequences. In this scenario, we show how linear neural networks can essentially perform compressed sensing (CS) of past inputs, thereby attaining a memory capacity that {\it exceeds} the number of neurons. This enhanced capacity is achieved by a class of ``orthogonal recurrent networks and not by feedforward networks or generic recurrent networks. We exploit techniques from the statistical physics of disordered systems to analytically compute the decay of memory traces in such networks as a function of network size, signal sparsity and integration time. Alternately, viewed purely from the perspective of CS, this work introduces a new ensemble of measurement matrices derived from dynamical systems, and provides a theoretical analysis of their asymptotic performance. "

PDF Details
v2026.09.13