论文检索

Reinforcement Learning/Function Approximation

Kaiwen Wang, Kevin Zhou, Runzhe Wu, Nathan Kallus, Wen Sun

While distributional reinforcement learning (DistRL) has been empirically effective, the question of when and why it is better than vanilla, non-distributional RL has remained unanswered.This paper explains the benefits of DistRL through the lens of small-loss bounds, which are instance-dependent bounds that scale with optimal achievable cost.Particularly, our bounds converge much faster than those from non-distributional approaches if the optimal cost is small.As warmup, we propose a distributional contextual bandit (DistCB) algorithm, which we show enjoys small-loss regret bounds and empirically outperforms the state-of-the-art on three real-world tasks.In online RL, we propose a DistRL algorithm that constructs confidence sets using maximum likelihood estimation. We prove that our algorithm enjoys novel small-loss PAC bounds in low-rank MDPs.As part of our analysis, we introduce the $\ell_1$ distributional eluder dimension which may be of independent interest. Then, in offline RL, we show that pessimistic DistRL enjoys small-loss PAC bounds that are novel to the offline setting and are more robust to bad single-policy coverage.

Theory/Reinforcement Learning and Planning

Jiayi Huang, Han Zhong, Liwei Wang, Lin Yang

While numerous works have focused on devising efficient algorithms for reinforcement learning (RL) with uniformly bounded rewards, it remains an open question whether sample or time-efficient algorithms for RL with large state-action space exist when the rewards are \emph{heavy-tailed}, i.e., with only finite $(1+\epsilon)$-th moments for some $\epsilon\in(0,1]$. In this work, we address the challenge of such rewards in RL with linear function approximation. We first design an algorithm, \textsc{Heavy-OFUL}, for heavy-tailed linear bandits, achieving an \emph{instance-dependent} $T$-round regret of $\tilde{O}\big(d T^{\frac{1-\epsilon}{2(1+\epsilon)}} \sqrt{\sum_{t=1}^T \nu_t^2} + d T^{\frac{1-\epsilon}{2(1+\epsilon)}}\big)$, the \emph{first} of this kind. Here, $d$ is the feature dimension, and $\nu_t^{1+\epsilon}$ is the $(1+\epsilon)$-th central moment of the reward at the $t$-th round. We further show the above bound is minimax optimal when applied to the worst-case instances in stochastic and deterministic linear bandits. We then extend this algorithm to the RL settings with linear function approximation. Our algorithm, termed as \textsc{Heavy-LSVI-UCB}, achieves the \emph{first} computationally efficient \emph{instance-dependent} $K$-episode regret of $\tilde{O}(d \sqrt{H \mathcal{U}^*} K^\frac{1}{1+\epsilon} + d \sqrt{H \mathcal{V}^* K})$. Here, $H$ is length of the episode, and $\mathcal{U}^*, \mathcal{V}^*$ are instance-dependent quantities scaling with the central moment of reward and value functions, respectively. We also provide a matching minimax lower bound $\Omega(d H K^{\frac{1}{1+\epsilon}} + d \sqrt{H^3 K})$ to demonstrate the optimality of our algorithm in the worst case. Our result is achieved via a novel robust self-normalized concentration inequality that may be of independent interest in handling heavy-tailed noise in general online regression problems.

Theory/Learning Theory

Arthur Jacot

Previous work has shown that DNNs withlarge depth $L$ and $L_{2}$-regularization are biased towards learninglow-dimensional representations of the inputs, which can be interpretedas minimizing a notion of rank $R^{(0)}(f)$ of the learned function$f$, conjectured to be the Bottleneck rank. We compute finite depthcorrections to this result, revealing a measure $R^{(1)}$ of regularitywhich bounds the pseudo-determinant of the Jacobian $\left\|Jf(x)\right\|\_\+$and is subadditive under composition and addition. This formalizesa balance between learning low-dimensional representations and minimizingcomplexity/irregularity in the feature maps, allowing the networkto learn the `right' inner dimension. Finally, we prove the conjecturedbottleneck structure in the learned features as $L\to\infty$: forlarge depths, almost all hidden representations are approximately$R^{(0)}(f)$-dimensional, and almost all weight matrices $W_{\ell}$have $R^{(0)}(f)$ singular values close to 1 while the others are$O(L^{-\frac{1}{2}})$. Interestingly, the use of large learning ratesis required to guarantee an order $O(L)$ NTK which in turns guaranteesinfinite depth convergence of the representations of almost all layers.

Theory/Learning Theory

Maria-Florina Balcan, Anh Nguyen, Dravyansh Sharma

The task of tuning regularization coefficients in regularized regression models with provable guarantees across problem instances still poses a significant challenge in the literature. This paper investigates the sample complexity of tuning regularization parameters in linear and logistic regressions under $\ell_1$ and $\ell_2$-constraints in the data-driven setting. For the linear regression problem, by more carefully exploiting the structure of the dual function class, we provide a new upper bound for the pseudo-dimension of the validation loss function class, which significantly improves the best-known results on the problem. Remarkably, we also instantiate the first matching lower bound, proving our results are tight. For tuning the regularization parameters of logistic regression, we introduce a new approach to studying the learning guarantee via an approximation of the validation loss function class. We examine the pseudo-dimension of the approximation class and construct a uniform error bound between the validation loss function class and its approximation, which allows us to instantiate the first learning guarantee for the problem of tuning logistic regression regularization coefficients.

Theory/Reinforcement Learning and Planning

Thanh Nguyen-Tang, Raman Arora

We seek to understand what facilitates sample-efficient learning from historical datasets for sequential decision-making, a problem that is popularly known as offline reinforcement learning (RL). Further, we are interested in algorithms that enjoy sample efficiency while leveraging (value) function approximation. In this paper, we address these fundamental questions by (i) proposing a notion of data diversity that subsumes the previous notions of coverage measures in offline RL and (ii) using this notion to \emph{unify} three distinct classes of offline RL algorithms based on version spaces (VS), regularized optimization (RO), and posterior sampling (PS). We establish that VS-based, RO-based, and PS-based algorithms, under standard assumptions, achieve \emph{comparable} sample efficiency, which recovers the state-of-the-art sub-optimality bounds for finite and linear model classes with the standard assumptions. This result is surprising, given that the prior work suggested an unfavorable sample complexity of the RO-based algorithm compared to the VS-based algorithm, whereas posterior sampling is rarely considered in offline RL due to its explorative nature. Notably, our proposed model-free PS-based algorithm for offline RL is \emph{novel}, with sub-optimality bounds that are \emph{frequentist} (i.e., worst-case) in nature.

Reinforcement Learning

Ruida Zhou, Tao Liu, Min Cheng, Dileep Kalathil, P. R. Kumar, Chao Tian

We study robust reinforcement learning (RL) with the goal of determining a well-performing policy that is robust against model mismatch between the training simulator and the testing environment. Previous policy-based robust RL algorithms mainly focus on the tabular setting under uncertainty sets that facilitate robust policy evaluation, but are no longer tractable when the number of states scales up. To this end, we propose two novel uncertainty set formulations, one based on double sampling and the other on an integral probability metric. Both make large-scale robust RL tractable even when one only has access to a simulator. We propose a robust natural actor-critic (RNAC) approach that incorporates the new uncertainty sets and employs function approximation. We provide finite-time convergence guarantees for the proposed RNAC algorithm to the optimal robust policy within the function approximation error. Finally, we demonstrate the robust performance of the policy learned by our proposed RNAC approach in multiple MuJoCo environments and a real-world TurtleBot navigation task.

Theory/Learning Theory

Anand Brahmbhatt, Rishi Saket, Aravindan Raghuveer

Learning from label proportions (LLP) is a generalization of supervised learning in which the training data is available as sets or bags of feature-vectors (instances) along with the average instance-label of each bag. The goal is to train a good instance classifier. While most previous works on LLP have focused on training models on such training data, computational learnability of LLP was onlyrecently explored by Saket (2021, 2022) who showed worst case intractability of properly learning linear threshold functions (LTFs) from label proportions. However, their work did not rule out efficient algorithms for this problem for natural distributions.In this work we show that it is indeed possible to efficiently learn LTFs using LTFs when given access to random bags of some label proportion in which feature-vectors are, conditioned on their labels, independently sampled from a Gaussian distribution $N(µ, Σ)$. Our work shows that a certain matrix – formed using covariances of the differences of feature-vectors sampled from the bags with and without replacement – necessarily has its principal component, after a transformation, in the direction of the normal vector of the LTF. Our algorithm estimates the means and covariance matrices using subgaussian concentration bounds which we show can be applied to efficiently sample bags for approximating the normal direction. Using this in conjunction with novel generalization error bounds in the bag setting, we show that a low error hypothesis LTF can be identified. For some special cases of the $N(0, I)$ distribution we provide a simpler mean estimation based algorithm. We include an experimental evaluation of our learning algorithms along with a comparison with those of Saket (2021, 2022) and random LTFs, demonstrating the effectiveness of our techniques.

Deep Learning/Theory

Yahong Yang, Haizhao Yang, Yang Xiang

This paper addresses the problem of nearly optimal Vapnik--Chervonenkis dimension (VC-dimension) and pseudo-dimension estimations of the derivative functions of deep neural networks (DNNs). Two important applications of these estimations include: 1) Establishing a nearly tight approximation result of DNNs in the Sobolev space; 2) Characterizing the generalization error of machine learning methods with loss functions involving function derivatives. This theoretical investigation fills the gap of learning error estimations for a wide range of physics-informed machine learning models and applications including generative models, solving partial differential equations, operator learning, network compression, distillation, regularization, etc.

Probabilistic Methods/Bayesian Models and Methods

Balhae Kim, Hyungi Lee, Juho Lee

A Bayesian pseudocoreset is a compact synthetic dataset summarizing essential information of a large-scale dataset and thus can be used as a proxy dataset for scalable Bayesian inference. Typically, a Bayesian pseudocoreset is constructed by minimizing a divergence measure between the posterior conditioning on the pseudocoreset and the posterior conditioning on the full dataset. However, evaluating the divergence can be challenging, particularly for the models like deep neural networks having high-dimensional parameters. In this paper, we propose a novel Bayesian pseudocoreset construction method that operates on a function space. Unlike previous methods, which construct and match the coreset and full data posteriors in the space of model parameters (weights), our method constructs variational approximations to the coreset posterior on a function space and matches it to the full data posterior in the function space. By working directly on the function space, our method could bypass several challenges that may arise when working on a weight space, including limited scalability and multi-modality issue. Through various experiments, we demonstrate that the Bayesian pseudocoresets constructed from our method enjoys enhanced uncertainty quantification and better robustness across various model architectures.

Reinforcement Learning

Yuankun Jiang, Nuowen Kan, Chenglin Li, Wenrui Dai, Junni Zou, Hongkai Xiong

Meta-reinforcement learning (Meta-RL), though enabling a fast adaptation to learn new skills by exploiting the common structure shared among different tasks, suffers performance degradation in the sparse-reward setting. Current hindsight-based sample transfer approaches can alleviate this issue by transferring relabeled trajectories from other tasks to a new task so as to provide informative experience for the target reward function, but are unfortunately constrained with the unrealistic assumption that tasks differ only in reward functions. In this paper, we propose a doubly robust augmented transfer (DRaT) approach, aiming at addressing the more general sparse reward meta-RL scenario with both dynamics mismatches and varying reward functions across tasks. Specifically, we design a doubly robust augmented estimator for efficient value-function evaluation, which tackles dynamics mismatches with the optimal importance weight of transition distributions achieved by minimizing the theoretically derived upper bound of mean squared error (MSE) between the estimated values of transferred samples and their true values in the target task. Due to its intractability, we then propose an interval-based approximation to this optimal importance weight, which is guaranteed to cover the optimum with a constrained and sample-independent upper bound on the MSE approximation error. Based on our theoretical findings, we finally develop a DRaT algorithm for transferring informative samples across tasks during the training of meta-RL. We implement DRaT on an off-policy meta-RL baseline, and empirically show that it significantly outperforms other hindsight-based approaches on various sparse-reward MuJoCo locomotion tasks with varying dynamics and reward functions.

Theory/Reinforcement Learning and Planning

Qinghua Liu, Gellert Weisz, András György, Chi Jin, Csaba Szepesvari

While policy optimization algorithms have played an important role in recent empirical success of Reinforcement Learning (RL), the existing theoretical understanding of policy optimization remains rather limited---they are either restricted to tabular MDPs or suffer from highly suboptimal sample complexity, especial in online RL where exploration is necessary. This paper proposes a simple efficient policy optimization framework---Optimistic NPG for online RL. Optimistic NPG can be viewed as simply combining of the classic natural policy gradient (NPG) algorithm [Kakade, 2001] with optimistic policy evaluation subroutines to encourage exploration. For $d$-dimensional linear MDPs, Optimistic NPG is computationally efficient, and learns an $\epsilon$-optimal policy within $\tilde{\mathcal{O}}(d^2/\epsilon^3)$ samples, which is the first computationally efficient algorithm whose sample complexity has the optimal dimension dependence $\tilde{\Theta}(d^2)$. It also improves over state-of-the-art results of policy optimization algorithms [Zanette et al., 2021] by a factor of $d$. For general function approximation that subsumes linear MDPs, Optimistic NPG, to our best knowledge, is also the first policy optimization algorithm that achieves the polynomial sample complexity for learning near-optimal policies.

Deep Learning/Robustness

Yuriy Biktairov, Jyotirmoy Deshmukh

Finding tight linear bounds for activation functions in neural networksis an essential part of several state of the art neural network robustness certification tools. An activation function is an arbitrary, nonlinear,scalar function $f: \mathbb{R}^d \rightarrow \mathbb{R}$. In the existing work on robustness certification, such bounds have been computed using human ingenuity for a handful of the most popular activation functions. While a number of heuristics have been proposed for bounding arbitrary functions,no analysis of the tightness optimality for general scalar functions has been offered yet, to the best of our knowledge. We fill this gap by formulating a concise optimality criterion for tightness of the approximation which allows us tobuild optimal bounds for any function convex in the region of interest $R$. Fora more general class of functions Lipshitz-continuous in $R$ we propose a sampling-based approach (SOL) which, given an instance of the bounding problem, efficiently computes the tightest linear bounds within a given $\varepsilon > 0$ threshold. We leverage an adaptive sampling technique to iteratively build a setof sample points suitable for representing the target activation function. While the theoretical worst case time complexity of our approach is$O(\varepsilon^{-2d})$,it typically only takes $O(\log^{\beta} \frac{1}{\varepsilon})$ time for some $\beta \ge 1$ and isthus sufficiently fast in practice. We provide empirical evidence of SOL's practicalityby incorporating it into a robustness certifier and observing that itproduces similar or higher certification rates while taking as low as quarter of the time compared to the other methods.

Optimization/Non-Convex

Morteza Boroun, Erfan Yazdandoost Hamedani, Afrooz Jalilzadeh

In this paper, we investigate a class of constrained saddle point (SP) problems where the objective function is nonconvex-concave and smooth. This class of problems has wide applicability in machine learning, including robust multi-class classification and dictionary learning. Several projection-based primal-dual methods have been developed to tackle this problem; however, the availability of methods with projection-free oracles remains limited. To address this gap, we propose efficient single-loop projection-free methods reliant on first-order information. In particular, using regularization and nested approximation techniques, we propose a primal-dual conditional gradient method that solely employs linear minimization oracles to handle constraints. Assuming that the constraint set in the maximization is strongly convex, our method achieves an $\epsilon$-stationary solution within $\mathcal{O}(\epsilon^{-6})$ iterations. When the projection onto the constraint set of maximization is easy to compute, we propose a one-sided projection-free method that achieves an $\epsilon$-stationary solution within $\mathcal{O}(\epsilon^{-4})$ iterations. Moreover, we present improved iteration complexities of our methods under a strong concavity assumption. To the best of our knowledge, our proposed algorithms are among the first projection-free methods with convergence guarantees for solving nonconvex-concave SP problems.

Reinforcement Learning

Nikki Lijing Kuang, Ming Yin, Mengdi Wang, Yu-Xiang Wang, Yian Ma

Recent studies in reinforcement learning (RL) have made significant progress by leveraging function approximation to alleviate the sample complexity hurdle for better performance. Despite the success, existing provably efficient algorithms typically rely on the accessibility of immediate feedback upon taking actions. The failure to account for the impact of delay in observations can significantly degrade the performance of real-world systems due to the regret blow-up. In this work, we tackle the challenge of delayed feedback in RL with linear function approximation by employing posterior sampling, which has been shown to empirically outperform the popular UCB algorithms in a wide range of regimes. We first introduce \textit{Delayed-PSVI}, an optimistic value-based algorithm that effectively explores the value function space via noise perturbation with posterior sampling. We provide the first analysis for posterior sampling algorithms with delayed feedback in RL and show our algorithm achieves $\widetilde{O}(\sqrt{d^3H^3 T} + d^2H^2 \mathbb{E}[\tau])$ worst-case regret in the presence of unknown stochastic delays. Here $\mathbb{E}[\tau]$ is the expected delay. To further improve its computational efficiency and to expand its applicability in high-dimensional RL problems, we incorporate a gradient-based approximate sampling scheme via Langevin dynamics for \textit{Delayed-LPSVI}, which maintains the same order-optimal regret guarantee with $\widetilde{O}(dHK)$ computational cost. Empirical evaluations are performed to demonstrate the statistical and computational efficacy of our algorithms.

Theory/Everything Else

Marina Knittel, Max Springer, John Dickerson, MohammadTaghi Hajiaghayi

Research in fair machine learning, and particularly clustering, has been crucial in recent years given the many ethical controversies that modern intelligent systems have posed. Ahmadian et al. [2020] established the study of fairness in hierarchical clustering, a stronger, more structured variant of its well-known flat counterpart, though their proposed algorithm that optimizes for Dasgupta's [2016] famous cost function was highly theoretical. Knittel et al. [2023] then proposed the first practical fair approximation for cost, however they were unable to break the polynomial-approximate barrier they posed as a hurdle of interest. We break this barrier, proposing the first truly polylogarithmic-approximate low-cost fair hierarchical clustering, thus greatly bridging the gap between the best fair and vanilla hierarchical clustering approximations.

Deep Learning/Algorithms

Bhaskar Mukhoty, Velibor Bojkovic, William de Vazelhes, Xiaohan Zhao, Giulia De Masi, Huan Xiong, Bin Gu

Spiking neural networks are becoming increasingly popular for their low energy requirement in real-world tasks with accuracy comparable to traditional ANNs. SNN training algorithms face the loss of gradient information and non-differentiability due to the Heaviside function in minimizing the model loss over model parameters. To circumvent this problem, the surrogate method employs a differentiable approximation of the Heaviside function in the backward pass, while the forward pass continues to use the Heaviside as the spiking function. We propose to use the zeroth-order technique at the local or neuron level in training SNNs, motivated by its regularizing and potential energy-efficient effects and establish a theoretical connection between it and the existing surrogate methods. We perform experimental validation of the technique on standard static datasets (CIFAR-10, CIFAR-100, ImageNet-100) and neuromorphic datasets (DVS-CIFAR-10, DVS-Gesture, N-Caltech-101, NCARS) and obtain results that offer improvement over the state-of-the-art results. The proposed method also lends itself to efficient implementations of the back-propagation method, which could provide 3-4 times overall speedup in training time. The code is available at \url{https://github.com/BhaskarMukhoty/LocalZO}.

Deep Learning/Sequential Models, Time series

Zihao Zhou, Rose Yu

Learning continuous-time point processes is essential to many discrete event forecasting tasks. However, integration poses a major challenge, particularly for spatiotemporal point processes (STPPs), as it involves calculating the likelihood through triple integrals over space and time. Existing methods for integrating STPP either assume a parametric form of the intensity function, which lacks flexibility; or approximating the intensity with Monte Carlo sampling, which introduces numerical errors. Recent work by Omi et al. proposes a dual network approach for efficient integration of flexible intensity function. However, their method only focuses on the 1D temporal point process. In this paper, we introduce a novel paradigm: `Auto-STPP` (Automatic Integration for Spatiotemporal Neural Point Processes) that extends the dual network approach to 3D STPP. While previous work provides a foundation, its direct extension overly restricts the intensity function and leads to computational challenges. In response, we introduce a decomposable parametrization for the integral network using ProdNet. This approach, leveraging the product of simplified univariate graphs, effectively sidesteps the computational complexities inherent in multivariate computational graphs. We prove the consistency of `Auto-STPP` and validate it on synthetic data and benchmark real-world datasets. `Auto-STPP` shows a significant advantage in recovering complex intensity functions from irregular spatiotemporal events, particularly when the intensity is sharply localized. Our code is open-source at https://github.com/Rose-STL-Lab/AutoSTPP.

Probabilistic Methods/Monte Carlo and Sampling Methods

Omar Chehab, Aapo Hyvarinen, Andrej Risteski

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions which interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.

Miscellaneous Aspects of Machine Learning/Kernel methods

Peter Macgregor, He Sun

Constructing a similarity graph from a set $X$ of data points in $ \mathbb{R}^d$ is the first step of many modern clustering algorithms. However, typical constructions of a similarity graph have high time complexity, and a quadratic space dependency with respect to $|X|$. We address this limitation and present a new algorithmic framework that constructs a sparse approximation of the fully connected similarity graph while preserving its cluster structure. Our presented algorithm is based on the kernel density estimation problem, and is applicable for arbitrary kernel functions. We compare our designed algorithm with the well-known implementations from the scikit-learn library and the FAISS library, and find that our method significantly outperforms the implementation from both libraries on a variety of datasets.

Deep Learning/Theory

Taiji Suzuki, Denny Wu, Atsushi Nitanda

The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the Langevin dynamics that incorporates a distribution-dependent drift, and it naturally arises from the optimization of two-layer neural networks via (noisy) gradient descent. Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures. However, all prior analyses assumed the infinite-particle or continuous-time limit, and cannot handle stochastic gradient updates. We provide a general framework to prove a uniform-in-time propagation of chaos for MFLD that takes into account the errors due to finite-particle approximation, time-discretization, and stochastic gradient. To demonstrate the wide applicability of our framework, we establish quantitative convergence rate guarantees to the regularized global optimal solution for $(i)$ a wide range of learning problems such as mean-field neural network and MMD minimization, and $(ii)$ different gradient estimators including SGD and SVRG. Despite the generality of our results, we achieve an improved convergence rate in both the SGD and SVRG settings when specialized to the standard Langevin dynamics.