Zixuan Xie

Xinyu Liu, Zixuan Xie, Shangtong Zhang

Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-η})$ with $η\in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2η})$. For a harmonic learning rate $O(n^{-1})$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{-1})$, which we argue is a strong result because it is close to the optimal rate $O(n^{-1}\log\log n)$ given by the law of the iterated logarithm (for a special case of i.i.d. noise). Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.

Xiaochi Qian, Zixuan Xie, Xinyu Liu et al.

This paper establishes the first almost sure convergence rate and the first maximal concentration bound with exponential tails for general contractive stochastic approximation algorithms with Markovian noise. As a corollary, we also obtain convergence rates in $L^p$. Key to our successes is a novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing (instead of constant) length. As applications, we provide the first almost sure convergence rate for $Q$-learning with Markovian samples without count-based learning rates. We also provide the first concentration bound for off-policy temporal difference learning with Markovian samples.

Xinyu Liu, Zixuan Xie, Shangtong Zhang

$Q$-learning is one of the most fundamental reinforcement learning algorithms. It is widely believed that $Q$-learning with linear function approximation (i.e., linear $Q$-learning) suffers from possible divergence until the recent work Meyn (2024) which establishes the ultimate almost sure boundedness of the iterates of linear $Q$-learning. Building on this success, this paper further establishes the first $L^2$ convergence rate of linear $Q$-learning iterates (to a bounded set). Similar to Meyn (2024), we do not make any modification to the original linear $Q$-learning algorithm, do not make any Bellman completeness assumption, and do not make any near-optimality assumption on the behavior policy. All we need is an $ε$-softmax behavior policy with an adaptive temperature. The key to our analysis is the general result of stochastic approximations under Markovian noise with fast-changing transition functions. As a side product, we also use this general result to establish the $L^2$ convergence rate of tabular $Q$-learning with an $ε$-softmax behavior policy, for which we rely on a novel pseudo-contraction property of the weighted Bellman optimality operator.

Changan Liu, Zixuan Xie, Ahad N. Zehmakan et al.

Random walk centrality is a fundamental metric in graph mining for quantifying node importance and influence, defined as the weighted average of hitting times to a node from all other nodes. Despite its ability to capture rich graph structural information and its wide range of applications, computing this measure for large networks remains impractical due to the computational demands of existing methods. In this paper, we present a novel formulation of random walk centrality, underpinning two scalable algorithms: one leveraging approximate Cholesky factorization and sparse inverse estimation, while the other sampling rooted spanning trees. Both algorithms operate in near-linear time and provide strong approximation guarantees. Extensive experiments on large real-world networks, including one with over 10 million nodes, demonstrate the efficiency and approximation quality of the proposed algorithms.