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Reinforcement Learning

Qiwei Di, Heyang Zhao, Jiafan He, Quanquan Gu

Offline reinforcement learning (RL), where the agent aims to learn the optimal policy based on the data collected by a behavior policy, has attracted increasing attention in recent years. While offline RL with linear function approximation has been extensively studied with optimal results achieved under certain assumptions, many works shift their interest to offline RL with non-linear function approximation.However, limited works on offline RL with non-linear function approximation have instance-dependent regret guarantees. In this paper, we propose an oracle-efficient algorithm, dubbed Pessimistic Nonlinear Least-Square Value Iteration (PNLSVI), for offline RL with non-linear function approximation. Our algorithmic design comprises three innovative components: (1) a variance-based weighted regression scheme that can be applied to a wide range of function classes, (2) a subroutine for variance estimation, and (3) a planning phase that utilizes a pessimistic value iteration approach. Our algorithm enjoys a regret bound that has a tight dependency on the function class complexity and achieves minimax optimal instance-dependent regret when specialized to linear function approximation. Our work extends the previous instance-dependent results within simpler function classes, such as linear and differentiable function to a more general framework. To the best of our knowledge, this is the first statistically optimal algorithm for nonlinear offline RL.

Theory · Reinforcement Learning and Planning

Cassidy Laidlaw, Banghua Zhu, Stuart Russell, Anca Dragan

Reinforcement learning (RL) theory has largely focused on proving minimax sample complexity bounds. These require strategic exploration algorithms that use relatively limited function classes for representing the policy or value function. Our goal is to explain why deep RL algorithms often perform well in practice, despite using random exploration and much more expressive function classes like neural networks. Our work arrives at an explanation by showing that many stochastic MDPs can be solved by performing only a few steps of value iteration on the random policy’s Q function and then acting greedily. When this is true, we find that it is possible to separate the exploration and learning components of RL, making it much easier to analyze. We introduce a new RL algorithm, SQIRL, that iteratively learns a near-optimal policy by exploring randomly to collect rollouts and then performing a limited number of steps of fitted-Q iteration over those roll- outs. We find that any regression algorithm that satisfies basic in-distribution generalization properties can be used in SQIRL to efficiently solve common MDPs. This can explain why deep RL works with complex function approximators like neural networks, since it is empirically established that neural networks generalize well in-distribution. Furthermore, SQIRL explains why random exploration works well in practice, since we show many environments can be solved by effectively estimating the random policy’s Q-function and then applying zero or a few steps of value iteration. We leverage SQIRL to derive instance-dependent sample complexity bounds for RL that are exponential only in an “effective horizon” of lookahead—which is typically much smaller than the full horizon—and on the complexity of the class used for function approximation. Empirically, we also find that SQIRL performance strongly correlates with PPO and DQN performance in a variety of stochastic environments, supporting that our theoretical analysis is predictive of practical performance. Our code and data are available at https://github.com/cassidylaidlaw/effective-horizon.

Probabilistic Methods · Bayesian Models and Methods

Panagiotis Dimitrakopoulos, Giorgos Sfikas, Christophoros Nikou

Bayesian inference is the standard for providing full predictive distributions with well calibrated uncertainty estimates.However, scaling to a modern, overparameterized deep learning setting typically comes at the cost of severe and restrictive approximations, sacrificing model predictive strength.With our approach, we factor model parameters as a function of deterministic and probabilistic components;the model is solved by combining maximum a posteriori estimation of the former,with inference over a low-dimensional, Implicit Neural Representation of the latter.This results in a solution that combines both predictive accuracy and good calibration,as it entails inducing stochasticity over the full set of model weights while being comparatively cheap to compute.Experimentally, our approach compares favorably to the state of the art,including much more expensive methods as well as less expressive posterior approximations over full network parameters.

Katie Kang, Amrith Setlur, Claire Tomlin, Sergey Levine

Conventional wisdom suggests that neural network predictions tend to be unpredictable and overconfident when faced with out-of-distribution (OOD) inputs. Our work reassesses this assumption for neural networks with high-dimensional inputs. Rather than extrapolating in arbitrary ways, we observe that neural network predictions often tend towards a constant value as input data becomes increasingly OOD. Moreover, we find that this value often closely approximates the optimal constant solution (OCS), i.e., the prediction that minimizes the average loss over the training data without observing the input. We present results showing this phenomenon across 8 datasets with different distributional shifts (including CIFAR10-C and ImageNet-R, S), different loss functions (cross entropy, MSE, and Gaussian NLL), and different architectures (CNNs and transformers). Furthermore, we present an explanation for this behavior, which we first validate empirically and then study theoretically in a simplified setting involving deep homogeneous networks with ReLU activations. Finally, we show how one can leverage our insights in practice to enable risk-sensitive decision-making in the presence of OOD inputs.

Probabilistic Methods · Gaussian Processes

Robert Peach, Matteo Vinao-Carl, Nir Grossman, Michael David, Emma-Jane Mallas, David Sharp, Paresh Malhotra, Pierre Vandergheynst, Adam Gosztolai

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds, appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

Probabilistic Methods · Monte Carlo and Sampling Methods

Dinghuai Zhang, Ricky T. Q. Chen, Chenghao Liu, Aaron Courville, Yoshua Bengio

We tackle the problem of sampling from intractable high-dimensional density functions, a fundamental task that often appears in machine learning and statistics. We extend recent sampling-based approaches that leverage controlled stochastic processes to model approximate samples from these target densities. The main drawback of these approaches is that the training objective requires full trajectories to compute, resulting in sluggish credit assignment issues due to use of entire trajectories and a learning signal present only at the terminal time.In this work, we present Diffusion Generative Flow Samplers (DGFS), a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments, via parameterizing an additional ``flow function''.Our method takes inspiration from the theory developed for generative flow networks (GFlowNets), allowing us to make use of intermediate learning signals.Through various challenging experiments, we demonstrate that DGFS achieves more accurate estimates of the normalization constant than closely-related prior methods.

Optimization · Non-Convex

Zhuqing Liu, Xin Zhang, Jia Liu, Zhengyuan Zhu, Songtao Lu

Learning an accurate value function for a given policy is a critical step in solving reinforcement learning (RL) problems. So far, however, the convergence speed and sample complexity performances of most existing policy evaluation algorithms remain unsatisfactory, particularly with non-linear function approximation. This challenge motivates us to develop a new path-integrated primal-dual stochastic gradient (PILOT) method, that is able to achieve a fast convergence speed for RL policy evaluation with nonlinear function approximation. To further alleviate the periodic full gradient evaluation requirement, we further propose an enhanced method with an adaptive-batch adjustment called PILOT$^+$. The main advantages of our methods include: i) PILOT allows the use of {\em{constant}} step sizes and achieves the $\mathcal{O}(1/K)$ convergence rate to first-order stationary points of non-convex policy evaluation problems; ii) PILOT is a generic {\em{single}}-timescale algorithm that is also applicable for solving a large class of non-convex strongly-concave minimax optimization problems; iii) By adaptively adjusting the batch size via historical stochastic gradient information, PILOT$^+$ is more sample-efficient empirically without loss of theoretical convergence rate. Our extensive numerical experiments verify our theoretical findings and showcase the high efficiency of the proposed PILOT and PILOT$^+$ algorithms compared with the state-of-the-art methods.

Reinforcement Learning · Online

Runzhe Wu, Wen Sun

Reinforcement Learning algorithms that learn from human feedback (RLHF) need to be efficient in terms of *statistical complexity, computational complexity, and query complexity*. In this work, we consider the RLHF setting where the feedback is given in the format of preferences over pairs of trajectories. In the linear MDP model, using randomization in algorithm design, we present an algorithm that is sample efficient (i.e., has near-optimal worst-case regret bounds) and has polynomial running time (i.e., computational complexity is polynomial with respect to relevant parameters). Our algorithm further minimizes the query complexity through a novel randomized active learning procedure. In particular, our algorithm demonstrates a near-optimal tradeoff between the regret bound and the query complexity. To extend the results to more general nonlinear function approximation, we design a model-based randomized algorithm inspired by the idea of Thompson sampling. Our algorithm minimizes Bayesian regret bound and query complexity, again achieving a near-optimal tradeoff between these two quantities. Computation-wise, similar to the prior Thompson sampling algorithms under the regular RL setting, the main computation primitives of our algorithm are Bayesian supervised learning oracles which have been heavily investigated on the empirical side when applying Thompson sampling algorithms to RL benchmark problems.

Optimization · Bilevel optimization

Bo Zhou, Ruiwei Jiang, Siqian Shen

Bilevel programs (BPs) find a wide range of applications in fields such as energy, transportation, and machine learning. As compared to BPs with continuous (linear/convex) optimization problems in both levels, the BPs with discrete decision variables have received much less attention, largely due to the ensuing computational intractability and the incapability of gradient-based algorithms for handling discrete optimization formulations. In this paper, we develop deep learning techniques to address this challenge. Specifically, we consider a BP with binary tender, wherein the upper and lower levels are linked via binary variables. We train a neural network to approximate the optimal value of the lower-level problem, as a function of the binary tender. Then, we obtain a single-level reformulation of the BP through a mixed-integer representation of the value function. Furthermore, we conduct a comparative analysis between two types of neural networks: general neural networks and the novel input supermodular neural networks, studying their representational capacities. To solve high-dimensional BPs, we introduce an enhanced sampling method to generate higher-quality samples and implement an iterative process to refine solutions. We demonstrate the performance of these approaches through extensive numerical experiments, whose lower-level problems are linear and mixed-integer programs, respectively.

Deep Learning · Other Representation Learning

Vladimir Kostic, Pietro Novelli, Riccardo Grazzi, Karim Lounici, massimiliano pontil

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

Social Aspects · Accountability, Transparency and Interpretability

Andrew Engel, Zhichao Wang, Natalie Frank, Ioana Dumitriu, Sutanay Choudhury, Anand Sarwate, Tony Chiang

A recent trend in explainable AI research has focused on surrogate modeling, where neural networks are approximated as simpler ML algorithms such as kernel machines. A second trend has been to utilize kernel functions in various explain-by-example or data attribution tasks. In this work, we combine these two trends to analyze approximate empirical neural tangent kernels (eNTK) for data attribution. Approximation is critical for eNTK analysis due to the high computational cost to compute the eNTK. We define new approximate eNTK and perform novel analysis on how well the resulting kernel machine surrogate models correlate with the underlying neural network. We introduce two new random projection variants of approximate eNTK which allow users to tune the time and memory complexity of their calculation. We conclude that kernel machines using approximate neural tangent kernel as the kernel function are effective surrogate models, with the introduced trace NTK the most consistent performer.

Deep Learning · Theory

Yuto Nishimura, Taiji Suzuki

Recent deep learning applications, exemplified by text-to-image tasks, often involve high-dimensional inputs and outputs. While several studies have investigated the function estimation capabilities of deep learning, research on dilated convolutional neural networks (CNNs) has mainly focused on cases where input dimensions are infinite but output dimensions are one-dimensional, similar to many other studies. However, many practical deep learning tasks involve high-dimensional (or even infinite dimensional) inputs and outputs.In this paper, we investigate the optimality of dilated CNNs for estimating a map between infinite-dimensional input and output spaces by analyzing their approximation and estimation abilities. For that purpose, we first show that approximation and estimation errors depend only on the smoothness and decay rate with respect to the infinity norm of the output, and their estimation accuracy actually achieve the {\it minimax optimal} rate of convergence.Second, we demonstrate that the dilated CNNs outperform {\it any} linear estimators including kernel ridge regression and $k$-NN estimators in a minimax error sense, highlighting the usefulness of feature learning realized by deep neural networks.Our theoretical analysis particularly explains the success of deep learning in recent high-dimensional input-output tasks.

Social Aspects · Accountability, Transparency and Interpretability

Yongchan Kwon, Eric Wu, Kevin Wu, James Y Zou

Quantifying the impact of training data points is crucial for understanding the outputs of machine learning models and for improving the transparency of the AI pipeline. The influence function is a principled and popular data attribution method, but its computational cost often makes it challenging to use. This issue becomes more pronounced in the setting of large language models and text-to-image models. In this work, we propose DataInf, an efficient influence approximation method that is practical for large-scale generative AI models. Leveraging an easy-to-compute closed-form expression, DataInf outperforms existing influence computation algorithms in terms of computational and memory efficiency. Our theoretical analysis shows that DataInf is particularly well-suited for parameter-efficient fine-tuning techniques such as LoRA. Through systematic empirical evaluations, we show that DataInf accurately approximates influence scores and is orders of magnitude faster than existing methods. In applications to RoBERTa-large, Llama-2-13B-chat, and stable-diffusion-v1.5 models, DataInf effectively identifies the most influential fine-tuning examples better than other approximate influence scores. Moreover, it can help to identify which data points are mislabeled.

Jeongyeol Kwon, Dohyun Kwon, Stephen Wright, Robert Nowak

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter $\sigma > 0$. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be $O(\sigma)$-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as $O(\sigma)$-approximation of the original BO, we propose first-order algorithms that find an $\epsilon$-stationary solution by optimizing the penalty formulation with $\sigma = O(\epsilon)$. When the perturbed lower-level problem uniformly satisfies the {\it small-error} proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an $\epsilon$-stationary point of the penalty function using in total $O(\epsilon^{-7})$ accesses to first-order stochastic gradient oracles. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, {\it i.e.,} with $O(1)$ samples per iteration, and achieves the improved oracle-complexity of $O(\epsilon^{-5})$.

Probabilistic Methods · Monte Carlo and Sampling Methods

Xunpeng Huang, Hanze Dong, Yifan HAO, Yian Ma, Tong Zhang

We propose a Monte Carlo sampler from the reverse diffusion process. Unlike the practice of diffusion models, where the intermediary updates---the score functions---are learned with a neural network, we transform the score matching problem into a mean estimation one.By estimating the means of the regularized posterior distributions, we derive a novel Monte Carlo sampling algorithm called reverse diffusion Monte Carlo (rdMC), which is distinct from the Markov chain Monte Carlo (MCMC) methods. We determine the sample size from the error tolerance and the properties of the posterior distribution to yield an algorithm that can approximately sample the target distribution with any desired accuracy. Additionally, we demonstrate and prove under suitable conditions that sampling with rdMC can be significantly faster than that with MCMC. For multi-modal target distributions such as those in Gaussian mixture models, rdMC greatly improves over the Langevin-style MCMC sampling methods both theoretically and in practice. The proposed rdMC method offers a new perspective and solution beyond classical MCMC algorithms for the challenging complex distributions.

Optimization

Tatjana Chavdarova, Tong Yang, Matteo Pagliardini, Michael Jordan

Yang et al. (2023) recently showed how to use first-order gradient methods to solve general variational inequalities (VIs) under a limiting assumption that analytic solutions of specific subproblems are available. In this paper, we circumvent this assumption via a warm-starting technique where we solve subproblems approximately and initialize variables with the approximate solution found at the previous iteration. We prove the convergence of this method and show that the gap function of the last iterate of the method decreases at a rate of $\mathcal{O}(\frac{1}{\sqrt{K}})$ when the operator is $L$-Lipschitz and monotone. In numerical experiments, we show that this technique can converge much faster than its exact counterpart. Furthermore, for the cases when the inequality constraints are simple, we introduce an alternative variant of ACVI and establish its convergence under the same conditions.Finally, we relax the smoothness assumptions in Yang et al., yielding, to our knowledge, the first convergence result for VIs with general constraints that does not rely on the assumption that the operator is $L$-Lipschitz.

Optimization · Everything Else

Shangmin Guo, YI REN, Stefano Albrecht, Kenny Smith

Although much research has been done on proposing new models or loss functions to improve the generalisation of artificial neural networks (ANNs), less attention has been directed to the impact of the training data on generalisation. In this work, we start from approximating the interaction between samples, i.e. how learning one sample would modify the model's prediction on other samples. Through analysing the terms involved in weight updates in supervised learning, we find that labels influence the interaction between samples. Therefore, we propose the labelled pseudo Neural Tangent Kernel (lpNTK) which takes label information into consideration when measuring the interactions between samples. We first prove that lpNTK asymptotically converges to the empirical neural tangent kernel in terms of the Frobenius norm under certain assumptions. Secondly, we illustrate how lpNTK helps to understand learning phenomena identified in previous work, specifically the learning difficulty of samples and forgetting events during learning. Moreover, we also show that using lpNTK to identify and remove poisoning training samples does not hurt the generalisation performance of ANNs.

Theory · Game Theory

Nikolas Patris, Ioannis Panageas

Learning Nash equilibria (NE) in games has garnered significant attention, particularly in the context of training Generative Adversarial Networks (GANs) and multi-agent Reinforcement Learning. The current state-of-the-art in efficiently learning games focuses on landscapes that meet the (weak) Minty property or games characterized by a unique function, often referred to as potential games. A significant challenge in this domain is that computing Nash equilibria is a computationally intractable task [Daskalakis et al. 2009]. In this paper we focus on bimatrix games (A,B) called rank-1. These are games in which the sum of the payoff matrices A+B is a rank 1 matrix; note that standard zero-sum games are rank 0. We show that optimistic gradient descent/ascent converges to an \epsilon-approximate NE after 1/\epsilon^2 log(1/\epsilon) iterates in rank-1 games. We achieve this by leveraging structural results about the NE landscape of rank-1 games Adsul et al. 2021. Notably, our approach bypasses the fact that these games do not satisfy the MVI property.

Probabilistic Methods · Monte Carlo and Sampling Methods

Kejun Tang, Jiayu Zhai, Xiaoliang Wan, Chao Yang

Solving partial differential equations (PDEs) is a central task in scientific computing. Recently, neural network approximation of PDEs has received increasing attention due to its flexible meshless discretization and its potential for high-dimensional problems. One fundamental numerical difficulty is that random samples in the training set introduce statistical errors into the discretization of the loss functional which may become the dominant error in the final approximation, and therefore overshadow the modeling capability of the neural network. In this work, we propose a new minmax formulation to optimize simultaneously the approximate solution, given by a neural network model, and the random samples in the training set, provided by a deep generative model. The key idea is to use a deep generative model to adjust the random samples in the training set such that the residual induced by the neural network model can maintain a smooth profile in the training process. Such an idea is achieved by implicitly embedding the Wasserstein distance between the residual-induced distribution and the uniform distribution into the loss, which is then minimized together with the residual. A nearly uniform residual profile means that its variance is small for any normalized weight function such that the Monte Carlo approximation error of the loss functional is reduced significantly for a certain sample size. The adversarial adaptive sampling (AAS) approach proposed in this work is the first attempt to formulate two essential components, minimizing the residual and seeking the optimal training set, into one minmax objective functional for the neural network approximation of PDEs.

Deep Learning · Self-Supervised Learning

Runtian Zhai, Bingbin Liu, Andrej Risteski, J Kolter, Pradeep K Ravikumar

Data augmentation is critical to the empirical success of modern self-supervised representation learning, such as contrastive learning and masked language modeling.However, a theoretical understanding of the exact role of the augmentation remains limited.Recent work has built the connection between self-supervised learning and the approximation of the top eigenspace of a graph Laplacian operator, suggesting that learning a linear probe atop such representation can be connected to RKHS regression.Building on this insight, this work delves into a statistical analysis of augmentation-based pretraining.Starting from the isometry property, a geometric characterization of the target function given by the augmentation, we disentangle the effects of the model and the augmentation,and prove two generalization bounds that are free of model complexity.Our first bound works for an arbitrary encoder, and it is the sum of an estimation error bound incurred by fitting a linear probe, and an approximation error bound by RKHS approximation.Our second bound specifically addresses the casewhere the encoder extracts the top-d eigenspace of a finite-sample-based approximation of the underlying RKHS.A key ingredient in our analysis is the *augmentation complexity*,which we use to quantitatively compare different augmentations and analyze their impact on downstream performance.