This paper delves into stochastic optimization problems that involve Markovian noise. We present a unified approach for the theoretical analysis of first-order gradient methods for stochastic optimization and variational inequalities. Our approach covers scenarios for both non-convex and strongly convex minimization problems. To achieve an optimal (linear) dependence on the mixing time of the underlying noise sequence, we use the randomized batching scheme, which is based on the multilevel Monte Carlo method. Moreover, our technique allows us to eliminate the limiting assumptions of previous research on Markov noise, such as the need for a bounded domain and uniformly bounded stochastic gradients. Our extension to variational inequalities under Markovian noise is original. Additionally, we provide lower bounds that match the oracle complexity of our method in the case of strongly convex optimization problems.

In this paper, we study the problems of principal Generalized Eigenvector computation and Canonical Correlation Analysis in the stochastic setting. We propose a simple and efficient algorithm, Gen-Oja, for these problems.

In this work, we present the first finite-time analysis of the Q-learning algorithm under time-varying learning policies (i.e., on-policy sampling) with minimal assumptions -- specifically, assuming only the existence of a policy that induces an irreducible Markov chain over the state space. We establish a last-iterate convergence rate for $\mathbb{E}[\|Q_k - Q^*\|_\infty^2]$, implying a sample complexity of order $O(1/ε^2)$ for achieving $\mathbb{E}[\|Q_k - Q^*\|_\infty] \le ε$, matching that of off-policy Q-learning but with a worse dependence on exploration-related parameters. We also derive an explicit rate for $\mathbb{E}[\|Q^{π_k} - Q^*\|_\infty^2]$, where $π_k$ is the learning policy at iteration $k$. These results reveal that on-policy Q-learning exhibits weaker exploration than its off-policy counterpart but enjoys an exploitation advantage, as its policy converges to an optimal one rather than remaining fixed. Numerical simulations corroborate our theory. Technically, the combination of time-varying learning policies (which induce rapidly time-inhomogeneous Markovian noise) and the minimal assumption on exploration presents significant analytical challenges. To address these challenges, we employ a refined approach that leverages the Poisson equation to decompose the Markovian noise corresponding to the lazy transition matrix into a martingale-difference term and residual terms. To control the residual terms under time inhomogeneity, we perform a sensitivity analysis of the Poisson equation solution with respect to both the Q-function estimate and the learning policy. These tools may further facilitate the analysis of general reinforcement learning algorithms with rapidly time-varying learning policies -- such as single-timescale actor--critic methods and learning-in-games algorithms -- and are of independent interest.

This paper delves into stochastic optimization problems that involve Markovian noise. We present a unified approach for the theoretical analysis of first-order gradient methods for stochastic optimization and variational inequalities.