Fast Regularized Policy Mirror Descent with One-Step TD Updates

Value iteration-type methods have been extensively studied for computing a nearly optimal value function in reinforcement learning (RL). Under a generative sampling model, these methods can achieve sharper sample complexity than policy optimization approaches, particularly in their dependence on the discount factor. In practice, they are often employed for offline training. In this paper, we consider discounted Markov decision processes with state space S, action space A, discount factor $γ\in(0,1)$ and costs in $[0,1]$. We introduce a novel value optimization method, termed value mirror descent (VMD), which integrates mirror descent from convex optimization into the classical value iteration framework. In the deterministic setting with known transition kernels, we show that VMD converges linearly. For the stochastic setting with a generative model, we develop a stochastic variant, SVMD, which incorporates variance reduction commonly used in stochastic value iteration-type methods. For RL problems with general convex regularizers, SVMD attains a near-optimal sample complexity of $\tilde{O}(|S||A|(1-γ)^{-3}ε^{-2})$. Moreover, we establish that the Bregman divergence between the generated and optimal policies remains bounded throughout the iterations, even under the presence of model misspecification. This property is absent in existing stochastic value iteration-type methods but is important for enabling effective online (continual) learning following offline training. Under a strongly convex regularizer, SVMD achieves sample complexity of $\tilde{O}(|S||A|(1-γ)^{-5}ε^{-1})$, improving performance in the high-accuracy regime. Furthermore, we prove convergence of the generated policy to the optimal policy. Overall, the proposed method, its analysis, and the resulting guarantees, constitute new contributions to the RL and optimization literature.

Gradient temporal-difference methods provide stable off-policy prediction with linear function approximation, but their practical performance is strongly affected by the geometry induced by the auxiliary-variable metric. Existing Mirror-Prox TD methods typically use the feature covariance metric, whereas hybrid TD methods suggest that behavior-policy transition information can provide a more informative update geometry. This paper proposes a behavior-induced Mirror-Prox temporal-difference method, called STHTD-MP, which replaces the covariance metric in the primal-dual saddle-point formulation with the symmetric part of the behavior-policy Bellman matrix. The method keeps a single learning rate for the primal and auxiliary variables and applies a Mirror-Prox prediction-correction step to the resulting hybrid saddle-point operator. We provide a formal convergence analysis for fixed-policy linear prediction under standard stochastic approximation assumptions: the behavior-induced metric is positive definite, the joint mean system is Hurwitz, boundedness follows from a Lyapunov argument, and the stochastic recursion converges by the ODE method. We further derive projected-oracle ergodic gap bounds and an exact mean-operator comparison with GTD2-MP based on the spectral radius of the deterministic Mirror-Prox error matrix. The analysis shows that STHTD-MP can have a smaller mean contraction factor than GTD2-MP when the behavior-induced metric improves the saddle-point geometry. Exact numerical mean-operator analysis on two-state, Random Walk, and Boyan Chain benchmarks supports this condition, while Baird's counterexample is identified as a singular boundary case where the strict assumptions fail.

Mirror Descent (MD) extends Gradient Descent (GD) beyond Euclidean geometry and has recently reappeared as a lens for KL-regularized policy optimization in reinforcement learning and LLM post-training. This raises a basic robustness question, crucial to reproducibility and reliability: how sensitive are MD dynamics to their inputs? We focus on initialization, often itself a pretrained or previously aligned model. Quadratic-regularized MD, including GD and Mahalanobis geometries, is well-known to be stable for convex smooth objectives. We show a sharp contrast: once the regularizer is non-quadratic, MD can be exponentially more sensitive to initialization than GD, even with a well-conditioned regularizer in Euclidean norm. We give a three-dimensional construction with a convex, smooth objective and a strongly convex, smooth, well-conditioned regularizer where an initial $\varepsilon$ perturbation is quickly amplified to $\min\{\text{polylog}^{-1}(1/\varepsilon), \varepsilon e^{Ω(ηT)}\}$ after $T$ iterations of MD with step size $η$. For canonical KL-regularized MD on the simplex, we show that even linear objectives can amplify an initial $\varepsilon$ perturbation exponentially fast in high-dimensional or near-boundary regimes. Finally, we show that adding a Bregman regularization term toward an anchor point can stabilize the dynamics while largely preserving the optimization guarantees, and that the choice of anchor is crucial: anchoring at the initialization only partially mitigates the instability, whereas anchoring at a fixed point yields a more stable mechanism.