arXiv subjects

Nicolas Le Roux

Publications and source records attributed to Nicolas Le Roux.

At least 37 records · Page 2Linked to original sources

arXiv2020

An operator view of policy gradient methods

We cast policy gradient methods as the repeated application of two operators: a policy improvement operator $\mathcal{I}$, which maps any policy $π$ to a better one $\mathcal{I}π$, and a projection operator $\mathcal{P}$, which finds the best approximation of $\mathcal{I}π$ in the set of realizable policies. We use this framework to introduce operator-based versions of traditional policy gradient methods such as REINFORCE and PPO, which leads to a better understanding of their original counterparts. We also use the understanding we develop of the role of $\mathcal{I}$ and $\mathcal{P}$ to propose a new global lower bound of the expected return. This new perspective allows us to further bridge the gap between policy-based and value-based methods, showing how REINFORCE and the Bellman optimality operator, for example, can be seen as two sides of the same coin.

cs.LG↗

arXiv2020

To Each Optimizer a Norm, To Each Norm its Generalization

We study the implicit regularization of optimization methods for linear models interpolating the training data in the under-parametrized and over-parametrized regimes. Since it is difficult to determine whether an optimizer converges to solutions that minimize a known norm, we flip the problem and investigate what is the corresponding norm minimized by an interpolating solution. Using this reasoning, we prove that for over-parameterized linear regression, projections onto linear spans can be used to move between different interpolating solutions. For under-parameterized linear classification, we prove that for any linear classifier separating the data, there exists a family of quadratic norms ||.||_P such that the classifier's direction is the same as that of the maximum P-margin solution. For linear classification, we argue that analyzing convergence to the standard maximum l2-margin is arbitrary and show that minimizing the norm induced by the data results in better generalization. Furthermore, for over-parameterized linear classification, projections onto the data-span enable us to use techniques from the under-parameterized setting. On the empirical side, we propose techniques to bias optimizers towards better generalizing solutions, improving their test performance. We validate our theoretical results via synthetic experiments, and use the neural tangent kernel to handle non-linear models.

cs.LG↗

arXiv2020

On the interplay between noise and curvature and its effect on optimization and generalization

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.

cs.LG↗

arXiv2020

The Geometry of Sign Gradient Descent

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.

cs.LG↗

arXiv2019

A Geometric Perspective on Optimal Representations for Reinforcement Learning

We propose a new perspective on representation learning in reinforcement learning based on geometric properties of the space of value functions. We leverage this perspective to provide formal evidence regarding the usefulness of value functions as auxiliary tasks. Our formulation considers adapting the representation to minimize the (linear) approximation of the value function of all stationary policies for a given environment. We show that this optimization reduces to making accurate predictions regarding a special class of value functions which we call adversarial value functions (AVFs). We demonstrate that using value functions as auxiliary tasks corresponds to an expected-error relaxation of our formulation, with AVFs a natural candidate, and identify a close relationship with proto-value functions (Mahadevan, 2005). We highlight characteristics of AVFs and their usefulness as auxiliary tasks in a series of experiments on the four-room domain.

cs.LG↗

arXiv2019

Reducing the variance in online optimization by transporting past gradients

Most stochastic optimization methods use gradients once before discarding them. While variance reduction methods have shown that reusing past gradients can be beneficial when there is a finite number of datapoints, they do not easily extend to the online setting. One issue is the staleness due to using past gradients. We propose to correct this staleness using the idea of implicit gradient transport (IGT) which transforms gradients computed at previous iterates into gradients evaluated at the current iterate without using the Hessian explicitly. In addition to reducing the variance and bias of our updates over time, IGT can be used as a drop-in replacement for the gradient estimate in a number of well-understood methods such as heavy ball or Adam. We show experimentally that it achieves state-of-the-art results on a wide range of architectures and benchmarks. Additionally, the IGT gradient estimator yields the optimal asymptotic convergence rate for online stochastic optimization in the restricted setting where the Hessians of all component functions are equal.

cs.LG↗

arXiv2019

Understanding the impact of entropy on policy optimization

Entropy regularization is commonly used to improve policy optimization in reinforcement learning. It is believed to help with \emph{exploration} by encouraging the selection of more stochastic policies. In this work, we analyze this claim using new visualizations of the optimization landscape based on randomly perturbing the loss function. We first show that even with access to the exact gradient, policy optimization is difficult due to the geometry of the objective function. Then, we qualitatively show that in some environments, a policy with higher entropy can make the optimization landscape smoother, thereby connecting local optima and enabling the use of larger learning rates. This paper presents new tools for understanding the optimization landscape, shows that policy entropy serves as a regularizer, and highlights the challenge of designing general-purpose policy optimization algorithms.

cs.LG↗

arXiv2019

The Value Function Polytope in Reinforcement Learning

We establish geometric and topological properties of the space of value functions in finite state-action Markov decision processes. Our main contribution is the characterization of the nature of its shape: a general polytope (Aigner et al., 2010). To demonstrate this result, we exhibit several properties of the structural relationship between policies and value functions including the line theorem, which shows that the value functions of policies constrained on all but one state describe a line segment. Finally, we use this novel perspective to introduce visualizations to enhance the understanding of the dynamics of reinforcement learning algorithms.

cs.LG↗

arXiv2019

Anytime Tail Averaging

Tail averaging consists in averaging the last examples in a stream. Common techniques either have a memory requirement which grows with the number of samples to average, are not available at every timestep or do not accomodate growing windows. We propose two techniques with a low constant memory cost that perform tail averaging with access to the average at every time step. We also show how one can improve the accuracy of that average at the cost of increased memory consumption.

cs.LG↗

arXiv2019

Distributional reinforcement learning with linear function approximation

Despite many algorithmic advances, our theoretical understanding of practical distributional reinforcement learning methods remains limited. One exception is Rowland et al. (2018)'s analysis of the C51 algorithm in terms of the Cramér distance, but their results only apply to the tabular setting and ignore C51's use of a softmax to produce normalized distributions. In this paper we adapt the Cramér distance to deal with arbitrary vectors. From it we derive a new distributional algorithm which is fully Cramér-based and can be combined to linear function approximation, with formal guarantees in the context of policy evaluation. In allowing the model's prediction to be any real vector, we lose the probabilistic interpretation behind the method, but otherwise maintain the appealing properties of distributional approaches. To the best of our knowledge, ours is the first proof of convergence of a distributional algorithm combined with function approximation. Perhaps surprisingly, our results provide evidence that Cramér-based distributional methods may perform worse than directly approximating the value function.

cs.LG↗

arXiv2019

Negative eigenvalues of the Hessian in deep neural networks

The loss function of deep networks is known to be non-convex but the precise nature of this nonconvexity is still an active area of research. In this work, we study the loss landscape of deep networks through the eigendecompositions of their Hessian matrix. In particular, we examine how important the negative eigenvalues are and the benefits one can observe in handling them appropriately.

cs.LG↗

arXiv2019

A comparative study of counterfactual estimators

We provide a comparative study of several widely used off-policy estimators (Empirical Average, Basic Importance Sampling and Normalized Importance Sampling), detailing the different regimes where they are individually suboptimal. We then exhibit properties optimal estimators should possess. In the case where examples have been gathered using multiple policies, we show that fused estimators dominate basic ones but can still be improved.

stat.ML↗

arXiv2018

Tracking the gradients using the Hessian: A new look at variance reducing stochastic methods

Our goal is to improve variance reducing stochastic methods through better control variates. We first propose a modification of SVRG which uses the Hessian to track gradients over time, rather than to recondition, increasing the correlation of the control variates and leading to faster theoretical convergence close to the optimum. We then propose accurate and computationally efficient approximations to the Hessian, both using a diagonal and a low-rank matrix. Finally, we demonstrate the effectiveness of our method on a wide range of problems.

math.OC↗

arXiv2017

Distributed SAGA: Maintaining linear convergence rate with limited communication

In recent years, variance-reducing stochastic methods have shown great practical performance, exhibiting linear convergence rate when other stochastic methods offered a sub-linear rate. However, as datasets grow ever bigger and clusters become widespread, the need for fast distribution methods is pressing. We propose here a distribution scheme for SAGA which maintains a linear convergence rate, even when communication between nodes is limited.

math.OC↗

arXiv2016

Tighter bounds lead to improved classifiers

The standard approach to supervised classification involves the minimization of a log-loss as an upper bound to the classification error. While this is a tight bound early on in the optimization, it overemphasizes the influence of incorrectly classified examples far from the decision boundary. Updating the upper bound during the optimization leads to improved classification rates while transforming the learning into a sequence of minimization problems. In addition, in the context where the classifier is part of a larger system, this modification makes it possible to link the performance of the classifier to that of the whole system, allowing the seamless introduction of external constraints.

cs.LG↗

arXiv2016

Efficient iterative policy optimization

We tackle the issue of finding a good policy when the number of policy updates is limited. This is done by approximating the expected policy reward as a sequence of concave lower bounds which can be efficiently maximized, drastically reducing the number of policy updates required to achieve good performance. We also extend existing methods to negative rewards, enabling the use of control variates.

cs.AI↗

arXiv2016

Minimizing Finite Sums with the Stochastic Average Gradient

We propose the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG) methods, the SAG method's iteration cost is independent of the number of terms in the sum. However, by incorporating a memory of previous gradient values the SAG method achieves a faster convergence rate than black-box SG methods. The convergence rate is improved from O(1/k^{1/2}) to O(1/k) in general, and when the sum is strongly-convex the convergence rate is improved from the sub-linear O(1/k) to a linear convergence rate of the form O(p^k) for p \textless{} 1. Further, in many cases the convergence rate of the new method is also faster than black-box deterministic gradient methods, in terms of the number of gradient evaluations. Numerical experiments indicate that the new algorithm often dramatically outperforms existing SG and deterministic gradient methods, and that the performance may be further improved through the use of non-uniform sampling strategies.

math.OC↗

arXiv2013

Fast Convergence of Stochastic Gradient Descent under a Strong Growth Condition

We consider optimizing a function smooth convex function $f$ that is the average of a set of differentiable functions $f_i$, under the assumption considered by Solodov [1998] and Tseng [1998] that the norm of each gradient $f_i'$ is bounded by a linear function of the norm of the average gradient $f'$. We show that under these assumptions the basic stochastic gradient method with a sufficiently-small constant step-size has an $O(1/k)$ convergence rate, and has a linear convergence rate if $g$ is strongly-convex.

math.OC↗