This work establishes finite-time rates of $\tilde{O} (1/\sqrt{n})$ for the aforementioned two algorithms under asynchronous Markovian sampling, where $n$ is the iteration index and $\tilde{O}$ hides logarithmic expressions.
Abstract
Discounted exponential utility provides a principled criterion for risk-sensitive sequential decision-making, but its nonlinear structure complicates reinforcement learning. A recent work \citep{thoppe2026reinforcement} addressed this difficulty by introducing a Bellman-compatible surrogate and two model-free fixed-point algorithms for optimizing it over stationary policies. However, their main convergence results are asymptotic. In this work, we establish finite-time rates of $\tilde{O} (1/\sqrt{n})$ for the aforementioned two algorithms under asynchronous Markovian sampling, where $n$ is the iteration index and $\tilde{O}$ hides logarithmic expressions. Importantly, we employ parameter-free choices for the stepsize parameter to derive these rate results. For the algorithmically simpler one-timescale method, the main challenge is that its update equation is not directly aligned with the contraction geometry of its underlying power-law operator. We overcome this mismatch by exploiting the boundedness, monotonicity, and homogeneity of the operator to obtain a local pseudo-contraction property for the relative-error dynamics. We then use a Moreau-envelope-based Lyapunov function and Polyak--Ruppert averaging to obtain the stated convergence rate with parameter-free stepsizes. For the two-timescale method, the main challenge is to control a tracking error on the faster timescale. These results provide the first finite-time guarantees for model-free discounted exponential-utility reinforcement learning.
Natural Policy Gradient (NPG) is a well-established Reinforcement Learning algorithm that underlies widely used methods such as Trust Region Policy Optimization and Proximal Policy Optimization, both of which have demonstrated strong empirical success. In this paper, we study exact NPG in finite-horizon Markov Decision Processes with known dynamics and horizon-dependent transition kernels. We provide the first finite-time convergence guarantees for this algorithm in this setting, for which we consider both constant and increasing step size regimes. With a constant step size $\eta_t=\eta$, we prove that NPG converges sublinearly with a rate of $\mathcal{O}(H^{2}/t)$ after $t$ iterations, where $H$ is the horizon length. We also extend this constant step size analysis to linear MDPs in an exact population-projection oracle under a full support projection distribution, recovering the same sublinear rate as in the tabular setting. Furthermore, with increasing step sizes, we prove that this algorithm achieves a linear convergence rate of $\mathcal{O}\left(\left(1-\frac{1}{\vartheta_\rho}\right)^t\right)$ for a problem-dependent constant $\vartheta_\rho>1$, and the horizon-only robust schedule of the form $\eta_t=\eta_0(H/(H-1))^t$ where $\eta_0>0$ and $H \geq 2$, attains this same geometric rate.
Asha Barua, S. Khodadadian· arXiv.org· 0 citations
Average-reward reinforcement-learning regret is known up to logarithmic factors, but the numerical content of published guarantees is difficult to compare because probability mode, structural parameter, logarithmic normalization, prior information, and planning assumptions differ. We introduce a constant-aware comparison protocol and derive an explicit finite lower certificate for communicating MDPs. The construction is a binary tree of two-state blocks; its proof uses exact trajectory-level Bernoulli KL divergence and keeps action budget, diameter, occupancy, navigation cost, and terminal bias explicit. A common closed-form envelope improves the published coefficient $0.015$ across a finite frontier: $0.0200$ in a moderate regime and up to $0.0291$ under stronger action, diameter, and horizon conditions, a $94\%$ increase. The limiting coefficient is $\frac1{32}\sqrt{(A-3)/A}$. For upper bounds, we give an auditable composition rule for a span-constrained optimistic learner, but do not claim a coefficient while adaptive directional-variance and planning certificates remain open. We also formalize valid expectation conversion and constant comparability. Controlled diagnostics test diameter dependence, bonus-by-width interactions, span misspecification, and the finite lower certificate on its exact family.
Ibne Farabi Shihab, A. Ahsan, Md Najmus Swaqeeb· 0 citations
We establish a global finite-sample guarantee for synchronous quantile temporal-difference learning (QTD) in tabular distributional reinforcement learning. The proof separates two stability mechanisms. A global comparison argument, based on the order monotonicity of reward cumulative distribution functions and the $W_\infty$ contraction of the distributional Bellman operator, brings an arbitrarily initialized iterate into a local neighborhood. Inside that neighborhood, we linearize the QTD mean field. Its Jacobian is a nonsingular $M$-matrix, and the associated positive semigroup permits a variance-sensitive martingale analysis. For stepsizes $\alpha_t=c(t+1)^{-a}$ with $a\in(1/2,1)$, the leading last-iterate fluctuation is of order $\widetilde O\bigl(T^{-a/2}/\sqrt{1-\gamma}\bigr)$ and has no polynomial dependence on the number of quantiles. The deterministic transient and the required burn-in can still depend on the smallest Bellman-target density, which is of order $m^{-1}$ in the worst case. The result therefore distinguishes sharply between the local stochastic fluctuation and the global sample complexity.
Zijie Cheng, Xiang Li, Yang Peng et al.· 0 citations
Multi-step returns accelerate reward propagation in off-policy reinforcement learning, but couple the evaluation of each decision to the suboptimal logged actions that follow it, inducing a pessimistic bias that grows with the horizon. We propose Expectile $n$-step Q-learning (ENQ), which replaces the symmetric $n$-step temporal-difference (TD) loss with an asymmetric expectile loss on the action-value error, with expectile level $\tau$ as the only method-specific hyperparameter added beyond $n$-step TD. We prove that the ENQ operator is a $\gamma^{n}$-contraction. Under deterministic dynamics, at $\tau=1$, its bias vanishes at the optimal action-value function $Q^*$ on covered in-support pairs, and the corresponding fixed point satisfies the separation-$n$ instance and its multiples of the lower-bound inequality used by Long-Horizon Q-learning (LQL). Under stochastic dynamics, the operator bias admits two-sided bounds with horizon-independent noise constants. Using a single expectile level $\tau=0.8$ and a fixed backup horizon across 27 manipulation and navigation task instances, ENQ is competitive with LQL on aggregate, achieves higher measured training-step throughput in our profiling study, and benefits more from a ten-critic ensemble in a controlled scaling experiment.
In this work, we study radically uncoupled learning in discounted general-sum Markov games. Assuming ``$\mathsf{ETH}$ for $\mathsf{PPAD}$", we show that, for every fixed discount factor, there is no polynomial-time algorithm for computing inverse-polynomially accurate coarse correlated equilibria in discounted general-sum Markov games when players learn independently in decentralized settings. Complementing this hardness result, we provide what appears to be the first \emph{radically uncoupled} algorithm with sub-exponential convergence guarantees to coarse correlated equilibria in discounted general-sum Markov games without imposing any structural restrictions on the game. Our algorithm is a \emph{layered} variant of optimistic mirror descent with an increasing step-size schedule tailored to the multi-agent setting. Finally, we develop both full-feedback and partial feedback versions of the aforementioned algorithm and establish sub-exponential convergence guarantees for each case.
Asrin Efe Yorulmaz, Uğur Aydın, Tamer Başar· 0 citations
We study infinite-horizon time-inconsistent Markov decision processes with a countably infinite state space and unbounded reward functions. The reward is allowed to depend explicitly on the initial time and initial state, thereby accommodating general sources of time inconsistency. We seek relaxed feedback equilibria, and our approach is based on entropy regularization and weighted functional analytic methods. With entropy regularization, we characterize a regular relaxed equilibrium through a fixed-point operator. By introducing two weight functions with distinct roles, one controlling the growth of rewards and values and the other defining the ambient weighted space, we construct a compact invariant set under a product topology and apply the Schauder-Tychonoff fixed-point theorem to establish existence of regularized equilibria. Importantly, the invariant set can be chosen uniformly for small entropy weight $\lambda\in(0,1]$. We then let $\lambda\to0+$ and show, through compactness, concentration of Gibbs policies, and uniform-integrability arguments, that a subsequential limit is a relaxed equilibrium of the original unregularized problem. We further study a policy iteration algorithm (PIA) for the entropy-regularized equilibrium problem. Under a weighted-discounting structure and sufficiently strong discounting, we establish exponential convergence and uniqueness of the regularized equilibrium in a suitable weighted Banach space. Combining the policy-iteration error with a quantitative soft-max approximation bound, we show that the iterated policies constitute weighted $\varepsilon$-equilibria for the original unregularized problem and derive an explicit regret estimate. A numerical example illustrating the convergence of PIA under strong discounting and a counterexample demonstrating its failure under weak discounting are also provided.
Fengyu Cao, Zhenhua Wang· 0 citations
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