The softmax policy gradient (PG) method, which performs gradient ascent under softmax policy parameterization, is arguably one of the de facto implementations of policy optimization in modern reinforcement learning. For
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Abstract discounted infinitehorizon tabular Markov decision processes (MDPs), remarkable progress has recently been achieved towards establishing global convergence of softmax PG methods in finding a nearoptimal policy. However, prior results fall short of delineating clear dependencies of convergence rates on salient parameters such as the cardinality of the state space$$\gamma $$ $\gamma $ and the effective horizon$${\mathcal {S}}$$ $S$ , both of which could be excessively large. In this paper, we deliver a pessimistic message regarding the iteration complexity of softmax PG methods, despite assuming access to exact gradient computation. Specifically, we demonstrate that the softmax PG method with stepsize$$\frac{1}{1\gamma }$$ $\frac{1}{1\gamma}$ can take$$\eta $$ $\eta $ to converge, even in the presence of a benign policy initialization and an initial state distribution amenable to exploration (so that the distribution mismatch coefficient is not exceedingly large). This is accomplished by characterizing the algorithmic dynamics over a carefullyconstructed MDP containing only three actions. Our exponential lower bound hints at the necessity of carefully adjusting update rules or enforcing proper regularization inmore »$$\begin{aligned} \frac{1}{\eta } {\mathcal {S}}^{2^{\Omega \big (\frac{1}{1\gamma }\big )}} ~\text {iterations} \end{aligned}$$ $\begin{array}{c}\frac{1}{\eta}{\leftS\right}^{{2}^{\Omega (\frac{1}{1\gamma})}}\phantom{\rule{0ex}{0ex}}\text{iterations}\end{array}$ 
Free, publiclyaccessible full text available June 30, 2024

Abstract Achieving sample efficiency in online episodic reinforcement learning (RL) requires optimally balancing exploration and exploitation. When it comes to a finitehorizon episodic Markov decision process with $S$ states, $A$ actions and horizon length $H$, substantial progress has been achieved toward characterizing the minimaxoptimal regret, which scales on the order of $\sqrt{H^2SAT}$ (modulo log factors) with $T$ the total number of samples. While several competing solution paradigms have been proposed to minimize regret, they are either memoryinefficient, or fall short of optimality unless the sample size exceeds an enormous threshold (e.g. $S^6A^4 \,\mathrm{poly}(H)$ for existing modelfree methods). To overcome such a large sample size barrier to efficient RL, we design a novel modelfree algorithm, with space complexity $O(SAH)$, that achieves nearoptimal regret as soon as the sample size exceeds the order of $SA\,\mathrm{poly}(H)$. In terms of this sample size requirement (also referred to the initial burnin cost), our method improves—by at least a factor of $S^5A^3$—upon any prior memoryefficient algorithm that is asymptotically regretoptimal. Leveraging the recently introduced variance reduction strategy (also called referenceadvantage decomposition), the proposed algorithm employs an earlysettled reference update rule, with the aid of two Qlearning sequences with upper and lower confidence bounds. The design principlemore »Free, publiclyaccessible full text available February 21, 2024

We study a noisy tensor completion problem of broad practical interest, namely, the reconstruction of a lowrank tensor from highly incomplete and randomly corrupted observations of its entries. Whereas a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for largescale applications or come with suboptimal statistical guarantees. Focusing on “incoherent” and wellconditioned tensors of a constant canonical polyadic rank, we propose a twostage nonconvex algorithm—(vanilla) gradient descent following a rough initialization—that achieves the best of both worlds. Specifically, the proposed nonconvex algorithm faithfully completes the tensor and retrieves all individual tensor factors within nearly linear time, while at the same time enjoying nearoptimal statistical guarantees (i.e., minimal sample complexity and optimal estimation accuracy). The estimation errors are evenly spread out across all entries, thus achieving optimal [Formula: see text] statistical accuracy. We also discuss how to extend our approach to accommodate asymmetric tensors. The insight conveyed through our analysis of nonconvex optimization might have implications for other tensor estimation problems.

Natural policy gradient (NPG) methods are among the most widely used policy optimization algorithms in contemporary reinforcement learning. This class of methods is often applied in conjunction with entropy regularization—an algorithmic scheme that encourages exploration—and is closely related to soft policy iteration and trust region policy optimization. Despite the empirical success, the theoretical underpinnings for NPG methods remain limited even for the tabular setting. This paper develops nonasymptotic convergence guarantees for entropyregularized NPG methods under softmax parameterization, focusing on discounted Markov decision processes (MDPs). Assuming access to exact policy evaluation, we demonstrate that the algorithm converges linearly—even quadratically, once it enters a local region around the optimal policy—when computing optimal value functions of the regularized MDP. Moreover, the algorithm is provably stable visàvis inexactness of policy evaluation. Our convergence results accommodate a wide range of learning rates and shed light upon the role of entropy regularization in enabling fast convergence.