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Curvature information -- particularly, the largest eigenvalue of the loss Hessian, known as the sharpness -- often forms the basis for learning rate tuners. However, recent work has shown that the curvature information undergoes complex dynamics during training, going from a phase of increasing sharpness to eventual stabilization. We analyze the closed-loop feedback effect between learning rate tuning and curvature. We find that classical learning rate tuners may yield greater one-step loss reduction, yet they ultimately underperform in the long term when compared to constant learning rates in the full batch regime. These models break the stabilization of the sharpness, which we explain using a simplified model of the joint dynamics of the learning rate and the curvature. To further investigate these effects, we introduce a new learning rate tuning method, Curvature Dynamics Aware Tuning (CDAT), which prioritizes long term curvature stabilization over instantaneous progress on the objective. In the full batch regime, CDAT shows behavior akin to prefixed warm-up schedules on deep learning objectives, outperforming tuned constant learning rates. In the mini batch regime, we observe that stochasticity introduces confounding effects that explain the previous success of some learning rate tuners at appropriate batch sizes. Our findings highlight the critical role of understanding the joint dynamics of the learning rate and curvature, beyond greedy minimization, to diagnose failures and design effective adaptive learning rate tuners.

We present the findings of the first NeurIPS competition on unlearning, which sought to stimulate the development of novel algorithms and initiate discussions on formal and robust evaluation methodologies. The competition was highly successful: nearly 1,200 teams from across the world participated, and a wealth of novel, imaginative solutions with different characteristics were contributed. In this paper, we analyze top solutions and delve into discussions on benchmarking unlearning, which itself is a research problem. The evaluation methodology we developed for the competition measures forgetting quality according to a formal notion of unlearning, while incorporating model utility for a holistic evaluation. We analyze the effectiveness of different instantiations of this evaluation framework vis-a-vis the associated compute cost, and discuss implications for standardizing evaluation. We find that the ranking of leading methods remains stable under several variations of this framework, pointing to avenues for reducing the cost of evaluation. Overall, our findings indicate progress in unlearning, with top-performing competition entries surpassing existing algorithms under our evaluation framework. We analyze trade-offs made by different algorithms and strengths or weaknesses in terms of generalizability to new datasets, paving the way for advancing both benchmarking and algorithm development in this important area.

Neural network interatomic potentials (NNIPs) are an attractive alternative to ab-initio methods for molecular dynamics (MD) simulations. However, they can produce unstable simulations which sample unphysical states, limiting their usefulness for modeling phenomena occurring over longer timescales. To address these challenges, we present Stability-Aware Boltzmann Estimator (StABlE) Training, a multi-modal training procedure which combines conventional supervised training from quantum-mechanical energies and forces with reference system observables, to produce stable and accurate NNIPs. StABlE Training iteratively runs MD simulations to seek out unstable regions, and corrects the instabilities via supervision with a reference observable. The training procedure is enabled by the Boltzmann Estimator, which allows efficient computation of gradients required to train neural networks to system observables, and can detect both global and local instabilities. We demonstrate our methodology across organic molecules, tetrapeptides, and condensed phase systems, along with using three modern NNIP architectures. In all three cases, StABlE-trained models achieve significant improvements in simulation stability and recovery of structural and dynamic observables. In some cases, StABlE-trained models outperform conventional models trained on datasets 50 times larger. As a general framework applicable across NNIP architectures and systems, StABlE Training is a powerful tool for training stable and accurate NNIPs, particularly in the absence of large reference datasets.

Recent empirical work has revealed an intriguing property of deep learning models by which the sharpness (largest eigenvalue of the Hessian) increases throughout optimization until it stabilizes around a critical value at which the optimizer operates at the edge of stability, given a fixed stepsize (Cohen et al, 2022). We investigate empirically how the sharpness evolves when using stepsize-tuners, the Armijo linesearch and Polyak stepsizes, that adapt the stepsize along the iterations to local quantities such as, implicitly, the sharpness itself. We find that the surprisingly poor performance of a classical Armijo linesearch in the deterministic setting may be well explained by its tendency to ever-increase the sharpness of the objective. On the other hand, we observe that Polyak stepsizes operate generally at the edge of stability or even slightly beyond, outperforming its Armijo and constant stepsizes counterparts in the deterministic setting. We conclude with an analysis that suggests unlocking stepsize tuners requires an understanding of the joint dynamics of the step size and the sharpness., Comment: Presented at the NeurIPS 2023 OPT Wokshop

Many problems in machine learning involve bilevel optimization (BLO), including hyperparameter optimization, meta-learning, and dataset distillation. Bilevel problems consist of two nested sub-problems, called the outer and inner problems, respectively. In practice, often at least one of these sub-problems is overparameterized. In this case, there are many ways to choose among optima that achieve equivalent objective values. Inspired by recent studies of the implicit bias induced by optimization algorithms in single-level optimization, we investigate the implicit bias of gradient-based algorithms for bilevel optimization. We delineate two standard BLO methods -- cold-start and warm-start -- and show that the converged solution or long-run behavior depends to a large degree on these and other algorithmic choices, such as the hypergradient approximation. We also show that the inner solutions obtained by warm-start BLO can encode a surprising amount of information about the outer objective, even when the outer parameters are low-dimensional. We believe that implicit bias deserves as central a role in the study of bilevel optimization as it has attained in the study of single-level neural net optimization., Comment: ICML 2022

Many machine learning problems encode their data as a matrix with a possibly very large number of rows and columns. In several applications like neuroscience, image compression or deep reinforcement learning, the principal subspace of such a matrix provides a useful, low-dimensional representation of individual data. Here, we are interested in determining the $d$-dimensional principal subspace of a given matrix from sample entries, i.e. from small random submatrices. Although a number of sample-based methods exist for this problem (e.g. Oja's rule \citep{oja1982simplified}), these assume access to full columns of the matrix or particular matrix structure such as symmetry and cannot be combined as-is with neural networks \citep{baldi1989neural}. In this paper, we derive an algorithm that learns a principal subspace from sample entries, can be applied when the approximate subspace is represented by a neural network, and hence can be scaled to datasets with an effectively infinite number of rows and columns. Our method consists in defining a loss function whose minimizer is the desired principal subspace, and constructing a gradient estimate of this loss whose bias can be controlled. We complement our theoretical analysis with a series of experiments on synthetic matrices, the MNIST dataset \citep{lecun2010mnist} and the reinforcement learning domain PuddleWorld \citep{sutton1995generalization} demonstrating the usefulness of our approach., Comment: 8 pages in main content, 2 pages of bibliography and 5 pages in Appendix

The extragradient method has gained popularity due to its robust convergence properties for differentiable games. Unlike single-objective optimization, game dynamics involve complex interactions reflected by the eigenvalues of the game vector field's Jacobian scattered across the complex plane. This complexity can cause the simple gradient method to diverge, even for bilinear games, while the extragradient method achieves convergence. Building on the recently proven accelerated convergence of the momentum extragradient method for bilinear games \citep{azizian2020accelerating}, we use a polynomial-based analysis to identify three distinct scenarios where this method exhibits further accelerated convergence. These scenarios encompass situations where the eigenvalues reside on the (positive) real line, lie on the real line alongside complex conjugates, or exist solely as complex conjugates. Furthermore, we derive the hyperparameters for each scenario that achieve the fastest convergence rate.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

The recently developed average-case analysis of optimization methods allows a more fine-grained and representative convergence analysis than usual worst-case results. In exchange, this analysis requires a more precise hypothesis over the data generating process, namely assuming knowledge of the expected spectral distribution (ESD) of the random matrix associated with the problem. This work shows that the concentration of eigenvalues near the edges of the ESD determines a problem's asymptotic average complexity. This a priori information on this concentration is a more grounded assumption than complete knowledge of the ESD. This approximate concentration is effectively a middle ground between the coarseness of the worst-case scenario convergence and the restrictive previous average-case analysis. We also introduce the Generalized Chebyshev method, asymptotically optimal under a hypothesis on this concentration and globally optimal when the ESD follows a Beta distribution. We compare its performance to classical optimization algorithms, such as gradient descent or Nesterov's scheme, and we show that, in the average-case context, Nesterov's method is universally nearly optimal asymptotically., Comment: To be published in ICML 2022

Tuning the step size of stochastic gradient descent is tedious and error prone. This has motivated the development of methods that automatically adapt the step size using readily available information. In this paper, we consider the family of SPS (Stochastic gradient with a Polyak Stepsize) adaptive methods. These are methods that make use of gradient and loss value at the sampled points to adaptively adjust the step size. We first show that SPS and its recent variants can all be seen as extensions of the Passive-Aggressive methods applied to nonlinear problems. We use this insight to develop new variants of the SPS method that are better suited to nonlinear models. Our new variants are based on introducing a slack variable into the interpolation equations. This single slack variable tracks the loss function across iterations and is used in setting a stable step size. We provide extensive numerical results supporting our new methods and a convergence theory., Comment: 48 pages, 7 figures

The architecture and the parameters of neural networks are often optimized independently, which requires costly retraining of the parameters whenever the architecture is modified. In this work we instead focus on growing the architecture without requiring costly retraining. We present a method that adds new neurons during training without impacting what is already learned, while improving the training dynamics. We achieve the latter by maximizing the gradients of the new weights and find the optimal initialization efficiently by means of the singular value decomposition (SVD). We call this technique Gradient Maximizing Growth (GradMax) and demonstrate its effectiveness in variety of vision tasks and architectures., Comment: ICLR 2022

Automatic differentiation (autodiff) has revolutionized machine learning. It allows to express complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers, and in bi-level problems such as hyper-parameter optimization and meta-learning. However, so far, implicit differentiation remained difficult to use for practitioners, as it often required case-by-case tedious mathematical derivations and implementations. In this paper, we propose automatic implicit differentiation, an efficient and modular approach for implicit differentiation of optimization problems. In our approach, the user defines directly in Python a function $F$ capturing the optimality conditions of the problem to be differentiated. Once this is done, we leverage autodiff of $F$ and the implicit function theorem to automatically differentiate the optimization problem. Our approach thus combines the benefits of implicit differentiation and autodiff. It is efficient as it can be added on top of any state-of-the-art solver and modular as the optimality condition specification is decoupled from the implicit differentiation mechanism. We show that seemingly simple principles allow to recover many existing implicit differentiation methods and create new ones easily. We demonstrate the ease of formulating and solving bi-level optimization problems using our framework. We also showcase an application to the sensitivity analysis of molecular dynamics., Comment: V3: added more related work and Jacobian precision figure

Variational Inference makes a trade-off between the capacity of the variational family and the tractability of finding an approximate posterior distribution. Instead, Boosting Variational Inference allows practitioners to obtain increasingly good posterior approximations by spending more compute. The main obstacle to widespread adoption of Boosting Variational Inference is the amount of resources necessary to improve over a strong Variational Inference baseline. In our work, we trace this limitation back to the global curvature of the KL-divergence. We characterize how the global curvature impacts time and memory consumption, address the problem with the notion of local curvature, and provide a novel approximate backtracking algorithm for estimating local curvature. We give new theoretical convergence rates for our algorithms and provide experimental validation on synthetic and real-world datasets.

We demonstrate that the choice of optimizer, neural network architecture, and regularizer significantly affect the adversarial robustness of linear neural networks, providing guarantees without the need for adversarial training. To this end, we revisit a known result linking maximally robust classifiers and minimum norm solutions, and combine it with recent results on the implicit bias of optimizers. First, we show that, under certain conditions, it is possible to achieve both perfect standard accuracy and a certain degree of robustness, simply by training an overparametrized model using the implicit bias of the optimization. In that regime, there is a direct relationship between the type of the optimizer and the attack to which the model is robust. To the best of our knowledge, this work is the first to study the impact of optimization methods such as sign gradient descent and proximal methods on adversarial robustness. Second, we characterize the robustness of linear convolutional models, showing that they resist attacks subject to a constraint on the Fourier-$\ell_\infty$ norm. To illustrate these findings we design a novel Fourier-$\ell_\infty$ attack that finds adversarial examples with controllable frequencies. We evaluate Fourier-$\ell_\infty$ robustness of adversarially-trained deep CIFAR-10 models from the standard RobustBench benchmark and visualize adversarial perturbations., Comment: New CIFAR-10 experiments and Fourier attack variations

We propose a new framework, inspired by random matrix theory, for analyzing the dynamics of stochastic gradient descent (SGD) when both number of samples and dimensions are large. This framework applies to any fixed stepsize and the finite sum setting. Using this new framework, we show that the dynamics of SGD on a least squares problem with random data become deterministic in the large sample and dimensional limit. Furthermore, the limiting dynamics are governed by a Volterra integral equation. This model predicts that SGD undergoes a phase transition at an explicitly given critical stepsize that ultimately affects its convergence rate, which we also verify experimentally. Finally, when input data is isotropic, we provide explicit expressions for the dynamics and average-case convergence rates (i.e., the complexity of an algorithm averaged over all possible inputs). These rates show significant improvement over the worst-case complexities.

Advances in generative modeling and adversarial learning have given rise to renewed interest in smooth games. However, the absence of symmetry in the matrix of second derivatives poses challenges that are not present in the classical minimization framework. While a rich theory of average-case analysis has been developed for minimization problems, little is known in the context of smooth games. In this work we take a first step towards closing this gap by developing average-case optimal first-order methods for a subset of smooth games. We make the following three main contributions. First, we show that for zero-sum bilinear games the average-case optimal method is the optimal method for the minimization of the Hamiltonian. Second, we provide an explicit expression for the optimal method corresponding to normal matrices, potentially non-symmetric. Finally, we specialize it to matrices with eigenvalues located in a disk and show a provable speed-up compared to worst-case optimal algorithms. We illustrate our findings through benchmarks with a varying degree of mismatch with our assumptions., Comment: 24 pages, 1 figure

We propose a novel Stochastic Frank-Wolfe (a.k.a. conditional gradient) algorithm for constrained smooth finite-sum minimization with a generalized linear prediction/structure. This class of problems includes empirical risk minimization with sparse, low-rank, or other structured constraints. The proposed method is simple to implement, does not require step-size tuning, and has a constant per-iteration cost that is independent of the dataset size. Furthermore, as a byproduct of the method we obtain a stochastic estimator of the Frank-Wolfe gap that can be used as a stopping criterion. Depending on the setting, the proposed method matches or improves on the best computational guarantees for Stochastic Frank-Wolfe algorithms. Benchmarks on several datasets highlight different regimes in which the proposed method exhibits a faster empirical convergence than related methods. Finally, we provide an implementation of all considered methods in an open-source package., Comment: Proceedings of the 37th International Conference on Machine Learning, 2020

Sign-based optimization methods have become popular in machine learning due to their favorable communication cost in distributed optimization and their surprisingly good performance in neural network training. Furthermore, they are closely connected to so-called adaptive gradient methods like Adam. Recent works on signSGD have used a non-standard "separable smoothness" assumption, whereas some older works study sign gradient descent as steepest descent with respect to the $\ell_\infty$-norm. In this work, we unify these existing results by showing a close connection between separable smoothness and $\ell_\infty$-smoothness and argue that the latter is the weaker and more natural assumption. We then proceed to study the smoothness constant with respect to the $\ell_\infty$-norm and thereby isolate geometric properties of the objective function which affect the performance of sign-based methods. In short, we find sign-based methods to be preferable over gradient descent if (i) the Hessian is to some degree concentrated on its diagonal, and (ii) its maximal eigenvalue is much larger than the average eigenvalue. Both properties are common in deep networks.

We develop a framework for the average-case analysis of random quadratic problems and derive algorithms that are optimal under this analysis. This yields a new class of methods that achieve acceleration given a model of the Hessian's eigenvalue distribution. We develop explicit algorithms for the uniform, Marchenko-Pastur, and exponential distributions. These methods are momentum-based algorithms, whose hyper-parameters can be estimated without knowledge of the Hessian's smallest singular value, in contrast with classical accelerated methods like Nesterov acceleration and Polyak momentum. Through empirical benchmarks on quadratic and logistic regression problems, we identify regimes in which the the proposed methods improve over classical (worst-case) accelerated methods.

Determining the extent to which different cognitive modalities (understood here as the set of cognitive processes underlying the elaboration of a stimulus by the brain) rely on overlapping neural representations is a fundamental issue in cognitive neuroscience. In the last decade, the identification of shared activity patterns has been mostly framed as a supervised learning problem. For instance, a classifier is trained to discriminate categories (e.g. faces vs. houses) in modality I (e.g. perception) and tested on the same categories in modality II (e.g. imagery). This type of analysis is often referred to as cross-modal decoding. In this paper we take a different approach and instead formulate the problem of assessing shared patterns across modalities within the framework of statistical hypothesis testing. We propose both an appropriate test statistic and a scheme based on permutation testing to compute the significance of this test while making only minimal distributional assumption. We denote this test cross-modal permutation test (CMPT). We also provide empirical evidence on synthetic datasets that our approach has greater statistical power than the cross-modal decoding method while maintaining low Type I errors (rejecting a true null hypothesis). We compare both approaches on an fMRI dataset with three different cognitive modalities (perception, imagery, visual search). Finally, we show how CMPT can be combined with Searchlight analysis to explore spatial distribution of shared activity patterns., Comment: 5 figures, tables after References (as required by SciRep template)

We investigate the difficulties of training sparse neural networks and make new observations about optimization dynamics and the energy landscape within the sparse regime. Recent work of \citep{Gale2019, Liu2018} has shown that sparse ResNet-50 architectures trained on ImageNet-2012 dataset converge to solutions that are significantly worse than those found by pruning. We show that, despite the failure of optimizers, there is a linear path with a monotonically decreasing objective from the initialization to the "good" solution. Additionally, our attempts to find a decreasing objective path from "bad" solutions to the "good" ones in the sparse subspace fail. However, if we allow the path to traverse the dense subspace, then we consistently find a path between two solutions. These findings suggest traversing extra dimensions may be needed to escape stationary points found in the sparse subspace., Comment: sparse networks, pruning, energy landscape, sparsity

The speed at which one can minimize an expected loss using stochastic methods depends on two properties: the curvature of the loss and the variance of the gradients. While most previous works focus on one or the other of these properties, we explore how their interaction affects optimization speed. Further, as the ultimate goal is good generalization performance, we clarify how both curvature and noise are relevant to properly estimate the generalization gap. Realizing that the limitations of some existing works stems from a confusion between these matrices, we also clarify the distinction between the Fisher matrix, the Hessian, and the covariance matrix of the gradients., Comment: Accepted to AISTATS 2020

We propose and analyze an adaptive step-size variant of the Davis-Yin three operator splitting. This method can solve optimization problems composed by a sum of a smooth term for which we have access to its gradient and an arbitrary number of potentially non-smooth terms for which we have access to their proximal operator. The proposed method sets the step-size based on local information of the objective --hence allowing for larger step-sizes--, only requires two extra function evaluations per iteration and does not depend on any step-size hyperparameter besides an initial estimate. We provide an iteration complexity analysis that matches the best known results for the non-adaptive variant: sublinear convergence for general convex functions and linear convergence under strong convexity of the smooth term and smoothness of one of the proximal terms. Finally, an empirical comparison with related methods on 6 different problems illustrates the computational advantage of the proposed method.

As datasets continue to increase in size and multi-core computer architectures are developed, asynchronous parallel optimization algorithms become more and more essential to the field of Machine Learning. Unfortunately, conducting the theoretical analysis asynchronous methods is difficult, notably due to the introduction of delay and inconsistency in inherently sequential algorithms. Handling these issues often requires resorting to simplifying but unrealistic assumptions. Through a novel perspective, we revisit and clarify a subtle but important technical issue present in a large fraction of the recent convergence rate proofs for asynchronous parallel optimization algorithms, and propose a simplification of the recently introduced "perturbed iterate" framework that resolves it. We demonstrate the usefulness of our new framework by analyzing three distinct asynchronous parallel incremental optimization algorithms: Hogwild (asynchronous SGD), KROMAGNON (asynchronous SVRG) and ASAGA, a novel asynchronous parallel version of the incremental gradient algorithm SAGA that enjoys fast linear convergence rates. We are able to both remove problematic assumptions and obtain better theoretical results. Notably, we prove that ASAGA and KROMAGNON can obtain a theoretical linear speedup on multi-core systems even without sparsity assumptions. We present results of an implementation on a 40-core architecture illustrating the practical speedups as well as the hardware overhead. Finally, we investigate the overlap constant, an ill-understood but central quantity for the theoretical analysis of asynchronous parallel algorithms. We find that it encompasses much more complexity than suggested in previous work, and often is order-of-magnitude bigger than traditionally thought., Comment: 67 pages, published in JMLR, can be found online at http://jmlr.org/papers/v19/17-650.html. arXiv admin note: substantial text overlap with arXiv:1606.04809

Most models in machine learning contain at least one hyperparameter to control for model complexity. Choosing an appropriate set of hyperparameters is both crucial in terms of model accuracy and computationally challenging. In this work we propose an algorithm for the optimization of continuous hyperparameters using inexact gradient information. An advantage of this method is that hyperparameters can be updated before model parameters have fully converged. We also give sufficient conditions for the global convergence of this method, based on regularity conditions of the involved functions and summability of errors. Finally, we validate the empirical performance of this method on the estimation of regularization constants of L2-regularized logistic regression and kernel Ridge regression. Empirical benchmarks indicate that our approach is highly competitive with respect to state of the art methods., Comment: Fixes error in proof of Theorem 2