Author name cluster

Aurélien Lucchi

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.

17 papers

1 author row

ICLR Conference 2025 Conference Paper

Adaptive Methods through the Lens of SDEs: Theoretical Insights on the Role of Noise

Enea Monzio Compagnoni
Tianlin Liu
Rustem Islamov
Frank Norbert Proske
Antonio Orvieto
Aurélien Lucchi

Despite the vast empirical evidence supporting the efficacy of adaptive optimization methods in deep learning, their theoretical understanding is far from complete. This work introduces novel SDEs for commonly used adaptive optimizers: SignSGD, RMSprop(W), and Adam(W). These SDEs offer a quantitatively accurate description of these optimizers and help illuminate an intricate relationship between adaptivity, gradient noise, and curvature. Our novel analysis of SignSGD highlights a noteworthy and precise contrast to SGD in terms of convergence speed, stationary distribution, and robustness to heavy-tail noise. We extend this analysis to AdamW and RMSpropW, for which we observe that the role of noise is much more complex. Crucially, we support our theoretical analysis with experimental evidence by verifying our insights: this includes numerically integrating our SDEs using Euler-Maruyama discretization on various neural network architectures such as MLPs, CNNs, ResNets, and Transformers. Our SDEs accurately track the behavior of the respective optimizers, especially when compared to previous SDEs derived for Adam and RMSprop. We believe our approach can provide valuable insights into best training practices and novel scaling rules.

ICML Conference 2024 Conference Paper

Characterizing Overfitting in Kernel Ridgeless Regression Through the Eigenspectrum

Tin Sum Cheng
Aurélien Lucchi
Anastasis Kratsios
David Belius

We derive new bounds for the condition number of kernel matrices, which we then use to enhance existing non-asymptotic test error bounds for kernel ridgeless regression in the over-parameterized regime for a fixed input dimension. For kernels with polynomial spectral decay, we recover the bound from previous work; for exponential decay, our bound is non-trivial and novel. Our conclusion is two-fold: (i) kernel regressors whose eigenspectrum decays polynomially must generalize well, even in the presence of noisy labeled training data; these models exhibit so-called tempered overfitting; (ii) if the eigenspectrum of any kernel ridge regressor decays exponentially, then it generalizes poorly, i. e. , it exhibits catastrophic overfitting. This adds to the available characterization of kernel ridge regressors exhibiting benign overfitting as the extremal case where the eigenspectrum of the kernel decays sub-polynomially. Our analysis combines new random matrix theory (RMT) techniques with recent tools in the kernel ridge regression (KRR) literature.

ICML Conference 2024 Conference Paper

Initial Guessing Bias: How Untrained Networks Favor Some Classes

Emanuele Francazi
Aurélien Lucchi
Marco Baity-Jesi

Understanding and controlling biasing effects in neural networks is crucial for ensuring accurate and fair model performance. In the context of classification problems, we provide a theoretical analysis demonstrating that the structure of a deep neural network (DNN) can condition the model to assign all predictions to the same class, even before the beginning of training, and in the absence of explicit biases. We prove that, besides dataset properties, the presence of this phenomenon, which we call Initial Guessing Bias (IGB), is influenced by model choices including dataset preprocessing methods, and architectural decisions, such as activation functions, max-pooling layers, and network depth. Our analysis of IGB provides information for architecture selection and model initialization. We also highlight theoretical consequences, such as the breakdown of node-permutation symmetry, the violation of self-averaging and the non-trivial effects that depth has on the phenomenon.

ICML Conference 2023 Conference Paper

A Theoretical Analysis of the Learning Dynamics under Class Imbalance

Emanuele Francazi
Marco Baity-Jesi
Aurélien Lucchi

Data imbalance is a common problem in machine learning that can have a critical effect on the performance of a model. Various solutions exist but their impact on the convergence of the learning dynamics is not understood. Here, we elucidate the significant negative impact of data imbalance on learning, showing that the learning curves for minority and majority classes follow sub-optimal trajectories when training with a gradient-based optimizer. This slowdown is related to the imbalance ratio and can be traced back to a competition between the optimization of different classes. Our main contribution is the analysis of the convergence of full-batch (GD) and stochastic gradient descent (SGD), and of variants that renormalize the contribution of each per-class gradient. We find that GD is not guaranteed to decrease the loss for each class but that this problem can be addressed by performing a per-class normalization of the gradient. With SGD, class imbalance has an additional effect on the direction of the gradients: the minority class suffers from a higher directional noise, which reduces the effectiveness of the per-class gradient normalization. Our findings not only allow us to understand the potential and limitations of strategies involving the per-class gradients, but also the reason for the effectiveness of previously used solutions for class imbalancesuch as oversampling.

ICML Conference 2023 Conference Paper

An SDE for Modeling SAM: Theory and Insights

Enea Monzio Compagnoni
Luca Biggio
Antonio Orvieto
Frank Norbert Proske
Hans Kersting
Aurélien Lucchi

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones – by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

ICML Conference 2022 Conference Paper

Anticorrelated Noise Injection for Improved Generalization

Antonio Orvieto
Hans Kersting
Frank Norbert Proske
Francis R. Bach
Aurélien Lucchi

Injecting artificial noise into gradient descent (GD) is commonly employed to improve the performance of machine learning models. Usually, uncorrelated noise is used in such perturbed gradient descent (PGD) methods. It is, however, not known if this is optimal or whether other types of noise could provide better generalization performance. In this paper, we zoom in on the problem of correlating the perturbations of consecutive PGD steps. We consider a variety of objective functions for which we find that GD with anticorrelated perturbations ("Anti-PGD") generalizes significantly better than GD and standard (uncorrelated) PGD. To support these experimental findings, we also derive a theoretical analysis that demonstrates that Anti-PGD moves to wider minima, while GD and PGD remain stuck in suboptimal regions or even diverge. This new connection between anticorrelated noise and generalization opens the field to novel ways to exploit noise for training machine learning models.

ICLR Conference 2022 Conference Paper

Generalization Through the Lens of Leave-One-Out Error

Gregor Bachmann
Thomas Hofmann 0001
Aurélien Lucchi

Despite the tremendous empirical success of deep learning models to solve various learning tasks, our theoretical understanding of their generalization ability is very limited. Classical generalization bounds based on tools such as the VC dimension or Rademacher complexity, are so far unsuitable for deep models and it is doubtful that these techniques can yield tight bounds even in the most idealistic settings~\citep{nagarajan2019uniform}. In this work, we instead revisit the concept of leave-one-out (LOO) error to measure the generalization ability of deep models in the so-called kernel regime. While popular in statistics, the LOO error has been largely overlooked in the context of deep learning. By building upon the recently established connection between neural networks and kernel learning, we leverage the closed-form expression for the leave-one-out error, giving us access to an efficient proxy for the test error. We show both theoretically and empirically that the leave-one-out error is capable of capturing various phenomena in generalization theory, such as double descent, random labels or transfer learning. Our work therefore demonstrates that the leave-one-out error provides a tractable way to estimate the generalization ability of deep neural networks in the kernel regime, opening the door to potential, new research directions in the field of generalization.

ICLR Conference 2022 Conference Paper

Phenomenology of Double Descent in Finite-Width Neural Networks

Sidak Pal Singh
Aurélien Lucchi
Thomas Hofmann 0001
Bernhard Schölkopf

`Double descent' delineates the generalization behaviour of models depending on the regime they belong to: under- or over-parameterized. The current theoretical understanding behind the occurrence of this phenomenon is primarily based on linear and kernel regression models --- with informal parallels to neural networks via the Neural Tangent Kernel. Therefore such analyses do not adequately capture the mechanisms behind double descent in finite-width neural networks, as well as, disregard crucial components --- such as the choice of the loss function. We address these shortcomings by leveraging influence functions in order to derive suitable expressions of the population loss and its lower bound, while imposing minimal assumptions on the form of the parametric model. Our derived bounds bear an intimate connection with the spectrum of the Hessian at the optimum, and importantly, exhibit a double descent behaviour at the interpolation threshold. Building on our analysis, we further investigate how the loss function affects double descent --- and thus uncover interesting properties of neural networks and their Hessian spectra near the interpolation threshold.

ICML Conference 2021 Conference Paper

Neural Symbolic Regression that scales

Luca Biggio
Tommaso Bendinelli
Alexander Neitz
Aurélien Lucchi
Giambattista Parascandolo

Symbolic equations are at the core of scientific discovery. The task of discovering the underlying equation from a set of input-output pairs is called symbolic regression. Traditionally, symbolic regression methods use hand-designed strategies that do not improve with experience. In this paper, we introduce the first symbolic regression method that leverages large scale pre-training. We procedurally generate an unbounded set of equations, and simultaneously pre-train a Transformer to predict the symbolic equation from a corresponding set of input-output-pairs. At test time, we query the model on a new set of points and use its output to guide the search for the equation. We show empirically that this approach can re-discover a set of well-known physical equations, and that it improves over time with more data and compute.

ICML Conference 2020 Conference Paper

An Accelerated DFO Algorithm for Finite-sum Convex Functions

Yuwen Chen
Antonio Orvieto
Aurélien Lucchi

Derivative-free optimization (DFO) has recently gained a lot of momentum in machine learning, spawning interest in the community to design faster methods for problems where gradients are not accessible. While some attention has been given to the concept of acceleration in the DFO literature, existing stochastic algorithms for objective functions with a finite-sum structure have not been shown theoretically to achieve an accelerated rate of convergence. Algorithms that use acceleration in such a setting are prone to instabilities, making it difficult to reach convergence. In this work, we exploit the finite-sum structure of the objective in order to design a variance-reduced DFO algorithm that provably yields acceleration. We prove rates of convergence for both smooth convex and strongly-convex finite-sum objective functions. Finally, we validate our theoretical results empirically on several tasks and datasets.

ICML Conference 2020 Conference Paper

Randomized Block-Diagonal Preconditioning for Parallel Learning

Celestine Dünner
Aurélien Lucchi

We study preconditioned gradient-based optimization methods where the preconditioning matrix has block-diagonal form. Such a structural constraint comes with the advantage that the update computation can be parallelized across multiple independent tasks. Our main contribution is to demonstrate that the convergence of these methods can significantly be improved by a randomization technique which corresponds to repartitioning coordinates across tasks during the optimization procedure. We provide a theoretical analysis that accurately characterizes the expected convergence gains of repartitioning and validate our findings empirically on various traditional machine learning tasks. From an implementation perspective, block-separable models are well suited for parallelization and, when shared memory is available, randomization can be implemented on top of existing methods very efficiently to improve convergence.

UAI Conference 2019 Conference Paper

The Role of Memory in Stochastic Optimization

Antonio Orvieto
Jonas Kohler
Aurélien Lucchi

The choice of how to retain information about past gradients dramatically affects the convergence properties of state-of-the-art stochastic optimization methods, such as Heavy-ball, Nesterov’s momentum, RMSprop and Adam. Building on this observation, we use stochastic differential equations (SDEs) to explicitly study the role of memory in gradient-based algorithms. We first derive a general continuous-time model that can incorporate arbitrary types of memory, for both deterministic and stochastic settings. We provide convergence guarantees for this SDE for weakly-quasi-convex and quadratically growing functions. We then demonstrate how to discretize this SDE to get a flexible discrete-time algorithm that can implement a board spectrum of memories ranging from short- to long-term. Not only does this algorithm increase the degrees of freedom in algorithmic choice for practitioners but it also comes with better stability properties than classical momentum in the convex stochastic setting. In particular, no iterate averaging is needed for convergence. Interestingly, our analysis also provides a novel interpretation of Nesterov’s momentum as stable gradient amplification and highlights a possible reason for its unstable behavior in the (convex) stochastic setting. Furthermore, we discuss the use of long term memory for second-moment estimation in adaptive methods, such as Adam and RMSprop. Finally, we provide an extensive experimental study of the effect of different types of memory in both convex and nonconvex settings.

ICML Conference 2018 Conference Paper

A Distributed Second-Order Algorithm You Can Trust

Celestine Dünner
Aurélien Lucchi
Matilde Gargiani
Yatao An Bian
Thomas Hofmann 0001
Martin Jaggi

Due to the rapid growth of data and computational resources, distributed optimization has become an active research area in recent years. While first-order methods seem to dominate the field, second-order methods are nevertheless attractive as they potentially require fewer communication rounds to converge. However, there are significant drawbacks that impede their wide adoption, such as the computation and the communication of a large Hessian matrix. In this paper we present a new algorithm for distributed training of generalized linear models that only requires the computation of diagonal blocks of the Hessian matrix on the individual workers. To deal with this approximate information we propose an adaptive approach that - akin to trust-region methods - dynamically adapts the auxiliary model to compensate for modeling errors. We provide theoretical rates of convergence for a wide class of problems including $L_1$-regularized objectives. We also demonstrate that our approach achieves state-of-the-art results on multiple large benchmark datasets.

ICML Conference 2018 Conference Paper

Escaping Saddles with Stochastic Gradients

Hadi Daneshmand
Jonas Kohler
Aurélien Lucchi
Thomas Hofmann 0001

We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients indeed exhibit a strong component along these directions. Furthermore, we show that - contrary to the case of isotropic noise - this variance is proportional to the magnitude of the corresponding eigenvalues and not decreasing in the dimensionality. Based upon this bservation we propose a new assumption under which we show that the injection of explicit, isotropic noise usually applied to make gradient descent escape saddle points can successfully be replaced by a simple SGD step. Additionally - and under the same condition - we derive the first convergence rate for plain SGD to a second-order stationary point in a number of iterations that is independent of the problem dimension.

ICML Conference 2017 Conference Paper

Sub-sampled Cubic Regularization for Non-convex Optimization

Jonas Kohler
Aurélien Lucchi

We consider the minimization of non-convex functions that typically arise in machine learning. Specifically, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence guarantees than first- and second-order as well as classical trust region methods. However, it suffers from a high computational complexity that makes it impractical for large-scale learning. Here, we propose a novel method that uses sub-sampling to lower this computational cost. By the use of concentration inequalities we provide a sampling scheme that gives sufficiently accurate gradient and Hessian approximations to retain the strong global and local convergence guarantees of cubically regularized methods. To the best of our knowledge this is the first work that gives global convergence guarantees for a sub-sampled variant of cubic regularization on non-convex functions. Furthermore, we provide experimental results supporting our theory.

ICML Conference 2016 Conference Paper

Starting Small - Learning with Adaptive Sample Sizes

Hadi Daneshmand
Aurélien Lucchi
Thomas Hofmann 0001

For many machine learning problems, data is abundant and it may be prohibitive to make multiple passes through the full training set. In this context, we investigate strategies for dynamically increasing the effective sample size, when using iterative methods such as stochastic gradient descent. Our interest is motivated by the rise of variance-reduced methods, which achieve linear convergence rates that scale favorably for smaller sample sizes. Exploiting this feature, we show - theoretically and empirically - how to obtain significant speed-ups with a novel algorithm that reaches statistical accuracy on an n-sample in 2n, instead of n log n steps.

ICML Conference 2013 Conference Paper

An Optimal Policy for Target Localization with Application to Electron Microscopy

Raphael Sznitman
Aurélien Lucchi
Peter I. Frazier
Bruno Jedynak
Pascal Fua

This paper considers the task of finding a target location by making a limited number of sequential observations. Each observation results from evaluating an imperfect classifier of a chosen cost and accuracy on an interval of chosen length and position. Within a Bayesian framework, we study the problem of minimizing an objective that combines the entropy of the posterior distribution with the cost of the questions asked. In this problem, we show that the one-step lookahead policy is Bayes-optimal for any arbitrary time horizon. Moreover, this one-step lookahead policy is easy to compute and implement. We then use this policy in the context of localizing mitochondria in electron microscope images, and experimentally show that significant speed ups in acquisition can be gained, while maintaining near equal image quality at target locations, when compared to current policies.