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Reinforcement Learning · Everything Else

Ziyi Chen, Yan Wen, Zhengmian Hu, Heng Huang

Reinforcement Learning (RL) problem with general utility is a powerful decision making framework that covers standard RL with cumulative cost, exploration problems, and demonstration learning. Existing works on RL with general utility do not consider the robustness under environmental perturbation, which is important to adapt RL system in the real-world environment that differs from the training environment. To train a robust policy, we propose a robust RL framework with general utility, which subsumes many existing RL frameworks including RL, robust RL, RL with general utility, constrained RL, robust constrained RL, pure exploration, robust entropy regularized RL, etc. Then we focus on popular convex utility functions, with which our proposed learning framework is a challenging nonconvex-nonconcave minimax optimization problem, and design a two-phase stochastic policy gradient type algorithm and obtain its sample complexity result for gradient convergence. Furthermore, for convex utility on a widely used polyhedral ambiguity set, we design an algorithm and obtain its convergence rate to a global optimal solution.

Optimization · Bilevel optimization

Franck Iutzeler, Edouard Pauwels, Samuel Vaiter

We consider stochastic optimization problems where the objective depends on some parameter, as commonly found in hyperparameter optimization for instance. We investigate the behavior of the derivatives of the iterates of Stochastic Gradient Descent (SGD) with respect to that parameter and show that they are driven by an inexact SGD recursion on a different objective function, perturbed by the convergence of the original SGD. This enables us to establish that the derivatives of SGD converge to the derivative of the solution mapping in terms of mean squared error whenever the objective is strongly convex. Specifically, we demonstrate that with constant step-sizes, these derivatives stabilize within a noise ball centered at the solution derivative, and that with vanishing step-sizes they exhibit $O(\log(k)^2 / k)$ convergence rates. Additionally, we prove exponential convergence in the interpolation regime. Our theoretical findings are illustrated by numerical experiments on synthetic tasks.

Ilias Diakonikolas, Nikos Zarifis

We study the problem of PAC learning $\gamma$-margin halfspaces in the presence of Massart noise. Without computational considerations, the sample complexity of this learning problem is known to be $\widetilde{\Theta}(1/(\gamma^2 \epsilon))$. Prior computationally efficient algorithms for the problem incur sample complexity $\tilde{O}(1/(\gamma^4 \epsilon^3))$ and achieve 0-1 error of $\eta+\epsilon$, where $\eta<1/2$ is the upper bound on the noise rate.Recent work gave evidence of an information-computation tradeoff, suggesting that a quadratic dependence on $1/\epsilon$ is required for computationally efficient algorithms. Our main result is a computationally efficient learner with sample complexity $\widetilde{\Theta}(1/(\gamma^2 \epsilon^2))$, nearly matching this lower bound. In addition, our algorithm is simple and practical, relying on online SGD on a carefully selected sequence of convex losses.

Theory · Statistics

Mathieu Even, Luca Ganassali, Jakob Maier, Laurent Massoulié

The Procrustes-Wasserstein problem consists in matching two high-dimensional point clouds in an unsupervised setting, and has many applications in natural language processing and computer vision. We consider a planted model with two datasets $X,Y$ that consist of $n$ datapoints in $\mathbb{R}^d$, where $Y$ is a noisy version of $X$, up to an orthogonal transformation and a relabeling of the data points. This setting is related to the graph alignment problem in geometric models.In this work, we focus on the euclidean transport cost between the point clouds as a measure of performance for the alignment. We first establish information-theoretic results, in the high ($d \gg \log n$) and low ($d \ll \log n$) dimensional regimes. We then study computational aspects and propose the ‘Ping-Pong algorithm', alternatively estimating the orthogonal transformation and the relabeling, initialized via a Franke-Wolfe convex relaxation. We give sufficient conditions for the method to retrieve the planted signal after one single step. We provide experimental results to compare the proposed approach with the state-of-the-art method of Grave et al. (2019).

Theory · Game Theory

Yiling Chen, Tao Lin, Ariel Procaccia, Aaditya Ramdas, Itai Shapira

We introduce and study the problem of detecting whether an agent is updating their prior beliefs given new evidence in an optimal way that is Bayesian, or whether they are biased towards their own prior. In our model, biased agents form posterior beliefs that are a convex combination of their prior and the Bayesian posterior, where the more biased an agent is, the closer their posterior is to the prior. Since we often cannot observe the agent's beliefs directly, we take an approach inspired by *information design*. Specifically, we measure an agent's bias by designing a *signaling scheme* and observing the actions they take in response to different signals, assuming that they are maximizing their own expected utility; our goal is to detect bias with a minimum number of signals. Our main results include a characterization of scenarios where a single signal suffices and a computationally efficient algorithm to compute optimal signaling schemes.

Learning Theory

Jun Chen, Hong Chen, Bin Gu

Stochastic compositional optimization (SCO) problem constitutes a class of optimization problems characterized by the objective function with a compositional form, including the tasks with known derivatives, such as AUC maximization, and the derivative-free tasks exemplified by black-box vertical federated learning (VFL). From the learning theory perspective, the learning guarantees of SCO algorithms with known derivatives have been studied in the literature. However, the potential impacts of the derivative-free setting on the learning guarantees of SCO remains unclear and merits further investigation. This paper aims to reveal the impacts by developing a theoretical analysis for two derivative-free algorithms, black-box SCGD and SCSC. Specifically, we first provide the sharper generalization upper bounds of convex SCGD and SCSC based on a new stability analysis framework more effective than prior work under some milder conditions, which is further developed to the non-convex case using the almost co-coercivity property of smooth function. Then, we derive the learning guarantees of three black-box variants of non-convex SCGD and SCSC with additional optimization analysis. Comparing these results, we theoretically uncover the impacts that a better gradient estimation brings a tighter learning guarantee and a larger proportion of unknown gradients may lead to a stronger dependence on the gradient estimation quality. Finally, our analysis is applied to two SCO algorithms, FOO-based vertical VFL and VFL-CZOFO, to build the first learning guarantees for VFL that align with the findings of SCGD and SCSC.

Generative Models · New Approaches

Nazar Buzun, Maksim Bobrin, Dmitry V. Dylov

We present a new approach for Neural Optimal Transport (NOT) training procedure, capable of accurately and efficiently estimating optimal transportation plan via specific regularization on dual Kantorovich potentials. The main bottleneck of existing NOT solvers is associated with the procedure of finding a near-exact approximation of the conjugate operator (i.e., the c-transform), which is done either by optimizing over non-convex max-min objectives or by the computationally intensive fine-tuning of the initial approximated prediction. We resolve both issues by proposing a new theoretically justified loss in the form of expectile regularization which enforces binding conditions on the learning process of the dual potentials. Such a regularization provides the upper bound estimation over the distribution of possible conjugate potentials and makes the learning stable, completely eliminating the need for additional extensive fine-tuning. Proposed method, called Expectile-Regularized Neural Optimal Transport (ENOT), outperforms previous state-of-the-art approaches in the established Wasserstein-2 benchmark tasks by a large margin (up to a 3-fold improvement in quality and up to a 10-fold improvement in runtime). Moreover, we showcase performance of ENOT for various cost functions in different tasks, such as image generation, demonstrating generalizability and robustness of the proposed algorithm.

Optimization · Non-Convex

Lingkai Kong, Molei Tao

Explicit, momentum-based dynamics that optimize functions defined on Lie groups can be constructed via variational optimization and momentum trivialization. Structure preserving time discretizations can then turn this dynamics into optimization algorithms. This article investigates two types of discretization, Lie Heavy-Ball, which is a known splitting scheme, and Lie NAG-SC, which is newly proposed. Their convergence rates are explicitly quantified under $L$-smoothness and \emph{local} strong convexity assumptions. Lie NAG-SC provides acceleration over the momentumless case, i.e. Riemannian gradient descent, but Lie Heavy-Ball does not. When compared to existing accelerated optimizers for general manifolds, both Lie Heavy-Ball and Lie NAG-SC are computationally cheaper and easier to implement, thanks to their utilization of group structure. Only gradient oracle and exponential map are required, but not logarithm map or parallel transport which are computational costly.

Optimization

Xixi Jia, Fangchen FENG, Deyu Meng, Defeng Sun

This paper focuses on the optimization of overparameterized, non-convex low-rank matrix sensing (LRMS)—an essential component in contemporary statistics and machine learning. Recent years have witnessed significant breakthroughs in first-order methods, such as gradient descent, for tackling this non-convex optimization problem. However, the presence of numerous saddle points often prolongs the time required for gradient descent to overcome these obstacles. Moreover, overparameterization can markedly decelerate gradient descent methods, transitioning its convergence rate from linear to sub-linear. In this paper, we introduce an approximated Gauss-Newton (AGN) method for tackling the non-convex LRMS problem. Notably, AGN incurs a computational cost comparable to gradient descent per iteration but converges much faster without being slowed down by saddle points. We prove that, despite the non-convexity of the objective function, AGN achieves Q-linear convergence from random initialization to the global optimal solution. The global Q-linear convergence of AGN represents a substantial enhancement over the convergence of the existing methods for the overparameterized non-convex LRMS. The code for this paper is available at \url{https://github.com/hsijiaxidian/AGN}.

Optimization

Ruichen Jiang, Ali Kavis, Qiujiang Jin, Sujay Sanghavi, Aryan Mokhtari

We propose adaptive, line-search-free second-order methods with optimal rate of convergence for solving convex-concave min-max problems. By means of an adaptive step size, our algorithms feature a simple update rule that requires solving only one linear system per iteration, eliminating the need for line-search or backtracking mechanisms. Specifically, we base our algorithms on the optimistic method and appropriately combine it with second-order information. Moreover, distinct from common adaptive schemes, we define the step size recursively as a function of the gradient norm and the prediction error in the optimistic update. We first analyze a variant where the step size requires knowledge of the Lipschitz constant of the Hessian. Under the additional assumption of Lipschitz continuous gradients, we further design a parameter-free version by tracking the Hessian Lipschitz constant locally and ensuring the iterates remain bounded. We also evaluate the practical performance of our algorithm by comparing it to existing second-order algorithms for minimax optimization.

Theory · Game Theory

Zhiyuan Fan, Christian Kroer, Gabriele Farina

First-order methods (FOMs) are arguably the most scalable algorithms for equilibrium computation in large extensive-form games. To operationalize these methods, a distance-generating function, acting as a regularizer for the strategy space, must be chosen. The ratio between the strong convexity modulus and the diameter of the regularizer is a key parameter in the analysis of FOMs.A natural question is then: what is the optimal distance-generating function for extensive-form decision spaces? In this paper, we make a number of contributions, ultimately establishing that the weight-one dilated entropy (DilEnt) distance-generating function is optimal up to logarithmic factors. The DilEnt regularizer is notable due to its iterate-equivalence with Kernelized OMWU (KOMWU)---the algorithm with state-of-the-art dependence on the game tree size in extensive-form games---when used in conjunction with the online mirror descent (OMD) algorithm. However, the standard analysis for OMD is unable to establish such a result; the only current analysis is by appealing to the iterate equivalence to KOMWU. We close this gap by introducing a pair of primal-dual treeplex norms, which we contend form the natural analytic viewpoint for studying the strong convexity of DilEnt. Using these norm pairs, we recover the diameter-to-strong-convexity ratio that predicts the same performance as KOMWU. Along with a new regret lower bound for online learning in sequence-form strategy spaces, we show that this ratio is nearly optimal.Finally, we showcase our analytic techniques by refining the analysis of Clairvoyant OMD when paired with DilEnt, establishing an $\mathcal{O}(n \log |\mathcal{V}| \log T/T)$ approximation rate to coarse correlated equilibrium in $n$-player games, where $|\mathcal{V}|$ is the number of reduced normal-form strategies of the players, establishing the new state of the art.

Optimization

Ipsita Ghosh, Abiy Tasissa, Christian Kümmerle

The problem of finding suitable point embedding or geometric configurations given only Euclidean distance information of point pairs arises both as a core task and as a sub-problem in a variety of machine learning applications. In this paper, we aim to solve this problem given a minimal number of distance samples. To this end, we leverage continuous and non-convex rank minimization formulations of the problem and establish a local convergence guarantee for a variant of iteratively reweighted least squares (IRLS), which applies if a minimal random set of observed distances is provided. As a technical tool, we establish a restricted isometry property (RIP) restricted to a tangent space of the manifold of symmetric rank-$r$ matrices given random Euclidean distance measurements, which might be of independent interest for the analysis of other non-convex approaches. Furthermore, we assess data efficiency, scalability and generalizability of different reconstruction algorithms through numerical experiments with simulated data as well as real-world data, demonstrating the proposed algorithm's ability to identify the underlying geometry from fewer distance samples compared to the state-of-the-art. The Matlab code can be found at \href{https://github.com/ipsita-ghosh-1/EDG-IRLS}{github\_SEGRED}

Reinforcement Learning · Deep RL

Aneesh Muppidi, Zhiyu Zhang, Heng Yang

A key challenge in lifelong reinforcement learning (RL) is the loss of plasticity, where previous learning progress hinders an agent's adaptation to new tasks. While regularization and resetting can help, they require precise hyperparameter selection at the outset and environment-dependent adjustments. Building on the principled theory of online convex optimization, we present a parameter-free optimizer for lifelong RL, called TRAC, which requires no tuning or prior knowledge about the distribution shifts. Extensive experiments on Procgen, Atari, and Gym Control environments show that TRAC works surprisingly well—mitigating loss of plasticity and rapidly adapting to challenging distribution shifts—despite the underlying optimization problem being nonconvex and nonstationary.

Optimization

Aaron Defazio, Xingyu Yang, Ahmed Khaled, Konstantin Mishchenko, Harsh Mehta, Ashok Cutkosky

Existing learning rate schedules that do not require specification of the optimization stopping step $T$ are greatly out-performed by learning rate schedules that depend on $T$. We propose an approach that avoids the need for this stopping time by eschewing the use of schedules entirely, while exhibiting state-of-the-art performance compared to schedules across a wide family of problems ranging from convex problems to large-scale deep learning problems. Our Schedule-Free approach introduces no additional hyper-parameters over standard optimizers with momentum. Our method is a direct consequence of a new theory we develop that unifies scheduling and iterate averaging. An open source implementation of our method is available at https://github.com/facebookresearch/schedule_free. Schedule-Free AdamW is the core algorithm behind our winning entry to the MLCommons 2024 AlgoPerf Algorithmic Efficiency Challenge Self-Tuning track.

Optimization · Convex

Kanan Gupta, Jonathan W. Siegel, Stephan Wojtowytsch

We present a generalization of Nesterov's accelerated gradient descent algorithm. Our algorithm (AGNES) provably achieves acceleration for smooth convex and strongly convex minimization tasks with noisy gradient estimates if the noise intensity is proportional to the magnitude of the gradient at every point. Nesterov's method converges at an accelerated rate if the constant of proportionality is below 1, while AGNES accommodates any signal-to-noise ratio. The noise model is motivated by applications in overparametrized machine learning. AGNES requires only two parameters in convex and three in strongly convex minimization tasks, improving on existing methods. We further provide clear geometric interpretations and heuristics for the choice of parameters.

Optimization · Convex

Yura Malitsky, Konstantin Mishchenko

In this paper, we explore two fundamental first-order algorithms in convex optimization, namely, gradient descent (GD) and proximal gradient method (ProxGD). Our focus is on making these algorithms entirely adaptive by leveraging local curvature information of smooth functions. We propose adaptive versions of GD and ProxGD that are based on observed gradient differences and, thus, have no added computational costs. Moreover, we prove convergence of our methods assuming only local Lipschitzness of the gradient. In addition, the proposed versions allow for even larger stepsizes than those initially suggested in [MM20].

Deep Learning · Algorithms

Kaizheng Wang, Fabio Cuzzolin, Shireen Kudukkil Manchingal -, Keivan Shariatmadar, David Moens, Hans Hallez

This paper introduces an innovative approach to classification called Credal Deep Ensembles (CreDEs), namely, ensembles of novel Credal-Set Neural Networks (CreNets). CreNets are trained to predict a lower and an upper probability bound for each class, which, in turn, determine a convex set of probabilities (credal set) on the class set. The training employs a loss inspired by distributionally robust optimization which simulates the potential divergence of the test distribution from the training distribution, in such a way that the width of the predicted probability interval reflects the epistemic uncertainty about the future data distribution. Ensembles can be constructed by training multiple CreNets, each associated with a different random seed, and averaging the outputted intervals. Extensive experiments are conducted on various out-of-distributions (OOD) detection benchmarks (CIFAR10/100 vs SVHN/Tiny-ImageNet, CIFAR10 vs CIFAR10-C, ImageNet vs ImageNet-O) and using different network architectures (ResNet50, VGG16, and ViT Base). Compared to Deep Ensemble baselines, CreDEs demonstrate higher test accuracy, lower expected calibration error, and significantly improved epistemic uncertainty estimation.

Online Learning

Wenhao Yang, Yibo Wang, Peng Zhao, Lijun Zhang

To address the uncertainty in function types, recent progress in online convex optimization (OCO) has spurred the development of universal algorithms that simultaneously attain minimax rates for multiple types of convex functions. However, for a $T$-round online problem, state-of-the-art methods typically conduct $O(\log T)$ projections onto the domain in each round, a process potentially time-consuming with complicated feasible sets. In this paper, inspired by the black-box reduction of Cutkosky and Orabona [2018], we employ a surrogate loss defined over simpler domains to develop universal OCO algorithms that only require $1$ projection. Embracing the framework of prediction with expert advice, we maintain a set of experts for each type of functions and aggregate their predictions via a meta-algorithm. The crux of our approach lies in a uniquely designed expert-loss for strongly convex functions, stemming from an innovative decomposition of the regret into the meta-regret and the expert-regret. Our analysis sheds new light on the surrogate loss, facilitating a rigorous examination of the discrepancy between the regret of the original loss and that of the surrogate loss, and carefully controlling meta-regret under the strong convexity condition. With only $1$ projection per round, we establish optimal regret bounds for general convex, exponentially concave, and strongly convex functions simultaneously. Furthermore, we enhance the expert-loss to exploit the smoothness property, and demonstrate that our algorithm can attain small-loss regret for multiple types of convex and smooth functions.

Optimization · Convex

Yu-Hu Yan, Peng Zhao, Zhi-Hua Zhou

We investigate the problem of universal online learning with gradient-variation regret. Universal online learning aims to achieve regret guarantees without prior knowledge of the curvature of the online functions. Moreover, we study the problem-dependent gradient-variation regret as it plays a crucial role in bridging stochastic and adversarial optimization as well as game theory. In this work, we design a universal approach with the *optimal* gradient-variation regret simultaneously for strongly convex, exp-concave, and convex functions, thus addressing an open problem highlighted by [Yan et al. [2023]](https://openreview.net/forum?id=AA1xrgAP5z). Our approach is *simple* since it is algorithmically efficient-to-implement with a two-layer online ensemble structure and only $1$ gradient query per round, and theoretically easy-to-analyze with a novel and alternative analysis to the gradient-variation regret. Concretely, previous works on gradient variations require controlling the algorithmic stability, which is challenging and leads to sub-optimal regret and less efficient algorithm design. Our analysis overcomes this issue by using a Bregman divergence negative term from linearization and a useful smoothness property.

Optimization · Non-Convex

Oscar Leong, Eliza O&#x27;Reilly, Yong Sheng Soh

Variational regularization is a classical technique to solve statistical inference tasks and inverse problems, with modern data-driven approaches parameterizing regularizers via deep neural networks showcasing impressive empirical performance. Recent works along these lines learn task-dependent regularizers. This is done by integrating information about the measurements and ground-truth data in an unsupervised, critic-based loss function, where the regularizer attributes low values to likely data and high values to unlikely data. However, there is little theory about the structure of regularizers learned via this process and how it relates to the two data distributions. To make progress on this challenge, we initiate a study of optimizing critic-based loss functions to learn regularizers over a particular family of regularizers: gauges (or Minkowski functionals) of star-shaped bodies. This family contains regularizers that are commonly employed in practice and shares properties with regularizers parameterized by deep neural networks. We specifically investigate critic-based losses derived from variational representations of statistical distances between probability measures. By leveraging tools from star geometry and dual Brunn-Minkowski theory, we illustrate how these losses can be interpreted as dual mixed volumes that depend on the data distribution. This allows us to derive exact expressions for the optimal regularizer in certain cases. Finally, we identify which neural network architectures give rise to such star body gauges and when do such regularizers have favorable properties for optimization. More broadly, this work highlights how the tools of star geometry can aid in understanding the geometry of unsupervised regularizer learning.