Convergence of Gradient Descent for Recurrent Neural Networks: A Nonasymptotic Analysis

We analyze recurrent neural networks trained with gradient descent in the supervised learning setting for dynamical systems, and prove that gradient descent can achieve optimality \emph{without} massive overparameterization. Our in-depth nonasymptotic analysis (i) provides sharp bounds on the network size $m$ and iteration complexity $\tau$ in terms of the sequence length $T$, sample size $n$ and ambient dimension $d$, and (ii) identifies the significant impact of long-term dependencies in the dynamical system on the convergence and network width bounds characterized by a cutoff point that depends on the Lipschitz continuity of the activation function. Remarkably, this analysis reveals that an appropriately-initialized recurrent neural network trained with $n$ samples can achieve optimality with a network size $m$ that scales only logarithmically with $n$. This sharply contrasts with the prior works that require high-order polynomial dependency of $m$ on $n$ to establish strong regularity conditions. Our results are based on an explicit characterization of the class of dynamical systems that can be approximated and learned by recurrent neural networks via norm-constrained transportation mappings, and establishing local smoothness properties of the hidden state with respect to the learnable parameters.

Provably Robust Temporal Difference Learning for Heavy-Tailed Rewards

In a broad class of reinforcement learning applications, stochastic rewards have heavy-tailed distributions, which lead to infinite second-order moments for stochastic (semi)gradients in policy evaluation and direct policy optimization. In such instances, the existing RL methods may fail miserably due to frequent statistical outliers. In this work, we establish that temporal difference (TD) learning with a dynamic gradient clipping mechanism, and correspondingly operated natural actor-critic (NAC), can be provably robustified against heavy-tailed reward distributions. It is shown in the framework of linear function approximation that a favorable tradeoff between bias and variability of the stochastic gradients can be achieved with this dynamic gradient clipping mechanism. In particular, we prove that robust versions of TD learning achieve sample complexities of order $\mathcal{O}(\varepsilon^{-\frac{1}{p}})$ and $\mathcal{O}(\varepsilon^{-1-\frac{1}{p}})$ with and without the full-rank assumption on the feature matrix, respectively, under heavy-tailed rewards with finite moments of order $(1+p)$ for some $p\in(0,1]$, both in expectation and with high probability. We show that a robust variant of NAC based on Robust TD learning achieves $\tilde{\mathcal{O}}(\varepsilon^{-4-\frac{2}{p}})$ sample complexity. We corroborate our theoretical results with numerical experiments.

Policy Mirror Ascent for Efficient and Independent Learning in Mean Field Games

Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous $N$-player games in literature. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Instead, we show that $N$ agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within $\tilde{\mathcal{O}}(\varepsilon^{-2})$ samples from a single sample trajectory without a population generative model, up to a standard $\mathcal{O}(\frac{1}{\sqrt{N}})$ error due to the mean field. Taking a divergent approach from literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. Next, we prove that conditional TD-learning in $N$-agent games can learn value functions within $\tilde{\mathcal{O}}(\varepsilon^{-2})$ time steps. These results allow proving sample complexity guarantees in the oracle-free setting by only relying on a sample path from the $N$ agent simulator. Furthermore, we demonstrate that our methodology allows for independent learning by $N$ agents with finite sample guarantees.

Finite-Time Analysis of Entropy-Regularized Neural Natural Actor-Critic Algorithm

Natural actor-critic (NAC) and its variants, equipped with the representation power of neural networks, have demonstrated impressive empirical success in solving Markov decision problems with large state spaces. In this paper, we present a finite-time analysis of NAC with neural network approximation, and identify the roles of neural networks, regularization and optimization techniques (e.g., gradient clipping and averaging) to achieve provably good performance in terms of sample complexity, iteration complexity and overparametrization bounds for the actor and the critic. In particular, we prove that (i) entropy regularization and averaging ensure stability by providing sufficient exploration to avoid near-deterministic and strictly suboptimal policies and (ii) regularization leads to sharp sample complexity and network width bounds in the regularized MDPs, yielding a favorable bias-variance tradeoff in policy optimization. In the process, we identify the importance of uniform approximation power of the actor neural network to achieve global optimality in policy optimization due to distributional shift.

Learning to Control Partially Observed Systems with Finite Memory

We consider the reinforcement learning problem for partially observed Markov decision processes (POMDPs) with large or even countably infinite state spaces, where the controller has access to only noisy observations of the underlying controlled Markov chain. We consider a natural actor-critic method that employs a finite internal memory for policy parameterization, and a multi-step temporal difference learning algorithm for policy evaluation. We establish, to the best of our knowledge, the first non-asymptotic global convergence of actor-critic methods for partially observed systems under function approximation. In particular, in addition to the function approximation and statistical errors that also arise in MDPs, we explicitly characterize the error due to the use of finite-state controllers. This additional error is stated in terms of the total variation distance between the traditional belief state in POMDPs and the posterior distribution of the hidden state when using a finite-state controller. Further, we show that this error can be made small in the case of sliding-block controllers by using larger block sizes.

A Lyapunov-Based Methodology for Constrained Optimization with Bandit Feedback

In a wide variety of applications including online advertising, contractual hiring, and wireless scheduling, the controller is constrained by a stringent budget constraint on the available resources, which are consumed in a random amount by each action, and a stochastic feasibility constraint that may impose important operational limitations on decision-making. In this work, we consider a general model to address such problems, where each action returns a random reward, cost, and penalty from an unknown joint distribution, and the decision-maker aims to maximize the total reward under a budget constraint $B$ on the total cost and a stochastic constraint on the time-average penalty. We propose a novel low-complexity algorithm based on Lyapunov optimization methodology, named ${\tt LyOn}$, and prove that it achieves $O(\sqrt{B\log B})$ regret and $O(\log B/B)$ constraint-violation. The low computational cost and sharp performance bounds of ${\tt LyOn}$ suggest that Lyapunov-based algorithm design methodology can be effective in solving constrained bandit optimization problems.

Linear Convergence of Entropy-Regularized Natural Policy Gradient with Linear Function Approximation

Natural policy gradient (NPG) methods with function approximation achieve impressive empirical success in reinforcement learning problems with large state-action spaces. However, theoretical understanding of their convergence behaviors remains limited in the function approximation setting. In this paper, we perform a finite-time analysis of NPG with linear function approximation and softmax parameterization, and prove for the first time that widely used entropy regularization method, which encourages exploration, leads to linear convergence rate. We adopt a Lyapunov drift analysis to prove the convergence results and explain the effectiveness of entropy regularization in improving the convergence rates.

Sample Complexity and Overparameterization Bounds for Projection-Free Neural TD Learning

We study the dynamics of temporal-difference learning with neural network-based value function approximation over a general state space, namely, \emph{Neural TD learning}. Existing analysis of neural TD learning relies on either infinite width-analysis or constraining the network parameters in a (random) compact set; as a result, an extra projection step is required at each iteration. This paper establishes a new convergence analysis of neural TD learning \emph{without any projection}. We show that the projection-free TD learning equipped with a two-layer ReLU network of any width exceeding $poly(\overline{\nu},1/\epsilon)$ converges to the true value function with error $\epsilon$ given $poly(\overline{\nu},1/\epsilon)$ iterations or samples, where $\overline{\nu}$ is an upper bound on the RKHS norm of the value function induced by the neural tangent kernel. Our sample complexity and overparameterization bounds are based on a drift analysis of the network parameters as a stopped random process in the lazy training regime.

Continuous-Time Multi-Armed Bandits with Controlled Restarts

Time-constrained decision processes have been ubiquitous in many fundamental applications in physics, biology and computer science. Recently, restart strategies have gained significant attention for boosting the efficiency of time-constrained processes by expediting the completion times. In this work, we investigate the bandit problem with controlled restarts for time-constrained decision processes, and develop provably good learning algorithms. In particular, we consider a bandit setting where each decision takes a random completion time, and yields a random and correlated reward at the end, with unknown values at the time of decision. The goal of the decision-maker is to maximize the expected total reward subject to a time constraint $\tau$. As an additional control, we allow the decision-maker to interrupt an ongoing task and forgo its reward for a potentially more rewarding alternative. For this problem, we develop efficient online learning algorithms with $O(\log(\tau))$ and $O(\sqrt{\tau\log(\tau)})$ regret in a finite and continuous action space of restart strategies, respectively. We demonstrate an applicability of our algorithm by using it to boost the performance of SAT solvers.

Group-Fair Online Allocation in Continuous Time

The theory of discrete-time online learning has been successfully applied in many problems that involve sequential decision-making under uncertainty. However, in many applications including contractual hiring in online freelancing platforms and server allocation in cloud computing systems, the outcome of each action is observed only after a random and action-dependent time. Furthermore, as a consequence of certain ethical and economic concerns, the controller may impose deadlines on the completion of each task, and require fairness across different groups in the allocation of total time budget $B$. In order to address these applications, we consider continuous-time online learning problem with fairness considerations, and present a novel framework based on continuous-time utility maximization. We show that this formulation recovers reward-maximizing, max-min fair and proportionally fair allocation rules across different groups as special cases. We characterize the optimal offline policy, which allocates the total time between different actions in an optimally fair way (as defined by the utility function), and impose deadlines to maximize time-efficiency. In the absence of any statistical knowledge, we propose a novel online learning algorithm based on dual ascent optimization for time averages, and prove that it achieves $\tilde{O}(B^{-1/2})$ regret bound.