This paper proposes a computationally tractable algorithm for learning infinite-horizon average-reward linear mixture Markov decision processes (MDPs) under the Bellman optimality condition. Our algorithm for linear mixture MDPs achieves a nearly minimax optimal regret upper bound of $\widetilde{\mathcal{O}}(d\sqrt{\mathrm{sp}(v^*)T})$ over $T$ time steps where $\mathrm{sp}(v^*)$ is the span of the optimal bias function $v^*$ and $d$ is the dimension of the feature mapping. Our algorithm applies the recently developed technique of running value iteration on a discounted-reward MDP approximation with clipping by the span. We prove that the value iteration procedure, even with the clipping operation, converges. Moreover, we show that the associated variance term due to random transitions can be bounded even under clipping. Combined with the weighted ridge regression-based parameter estimation scheme, this leads to the nearly minimax optimal regret guarantee.

We study the problem of infinite-horizon average-reward reinforcement learning with linear Markov decision processes (MDPs). The associated Bellman operator of the problem not being a contraction makes the algorithm design challenging. Previous approaches either suffer from computational inefficiency or require strong assumptions on dynamics, such as ergodicity, for achieving a regret bound of $\widetilde{\mathcal{O}}(\sqrt{T})$. In this paper, we propose the first algorithm that achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret with computational complexity polynomial in the problem parameters, without making strong assumptions on dynamics. Our approach approximates the average-reward setting by a discounted MDP with a carefully chosen discounting factor, and then applies an optimistic value iteration. We propose an algorithmic structure that plans for a nonstationary policy through optimistic value iteration and follows that policy until a specified information metric in the collected data doubles. Additionally, we introduce a value function clipping procedure for limiting the span of the value function for sample efficiency.

We study model-based reinforcement learning with non-linear function approximation where the transition function of the underlying Markov decision process (MDP) is given by a multinomial logit (MNL) model. We develop a provably efficient discounted value iteration-based algorithm that works for both infinite-horizon average-reward and discounted-reward settings. For average-reward communicating MDPs, the algorithm guarantees a regret upper bound of $\tilde{\mathcal{O}}(dD\sqrt{T})$ where $d$ is the dimension of feature mapping, $D$ is the diameter of the underlying MDP, and $T$ is the horizon. For discounted-reward MDPs, our algorithm achieves $\tilde{\mathcal{O}}(d(1-\gamma)^{-2}\sqrt{T})$ regret where $\gamma$ is the discount factor. Then we complement these upper bounds by providing several regret lower bounds. We prove a lower bound of $\Omega(d\sqrt{DT})$ for learning communicating MDPs of diameter $D$ and a lower bound of $\Omega(d(1-\gamma)^{-3/2}\sqrt{T})$ for learning discounted-reward MDPs with discount factor $\gamma$. Lastly, we show a regret lower bound of $\Omega(dH^{3/2}\sqrt{K})$ for learning $H$-horizon episodic MDPs with MNL function approximation where $K$ is the number of episodes, which improves upon the best-known lower bound for the finite-horizon setting.

Online safe reinforcement learning (RL) plays a key role in dynamic environments, with applications in autonomous driving, robotics, and cybersecurity. The objective is to learn optimal policies that maximize rewards while satisfying safety constraints modeled by constrained Markov decision processes (CMDPs). Existing methods achieve sublinear regret under stochastic constraints but often fail in adversarial settings, where constraints are unknown, time-varying, and potentially adversarially designed. In this paper, we propose the Optimistic Mirror Descent Primal-Dual (OMDPD) algorithm, the first to address online CMDPs with anytime adversarial constraints. OMDPD achieves optimal regret $\tilde{\mathcal{O}}(\sqrt{K})$ and strong constraint violation $\tilde{\mathcal{O}}(\sqrt{K})$ without relying on Slater’s condition or the existence of a strictly known safe policy. We further show that access to accurate estimates of rewards and transitions can further improve these bounds. Our results offer practical guarantees for safe decision-making in adversarial environments.

This paper considers stochastic-constrained stochastic optimization where the stochastic constraint is to satisfy that the expectation of a random function is below a certain threshold. In particular, we study the setting where data samples are drawn from a Markov chain and thus are not independent and identically distributed. We generalize the drift-plus-penalty framework, a primal-dual stochastic gradient method developed for the i.i.d. case, to the Markov chain sampling setting. We propose three variants of drift-plus-penalty; two are for the case when the mixing time of the underlying Markov chain is known while the other is for the case of unknown mixing time. In fact, our algorithms apply to a more general setting of constrained online convex optimization where the sequence of constraint functions follows a Markov chain. The algorithms are adaptive in that the first two work without knowledge of the time horizon while the third uses AdaGrad-style algorithm parameters, which is of independent interest. We demonstrate the effectiveness of our proposed methods through numerical experiments on classification with fairness constraints.

This paper proposes a learning and scheduling algorithm to minimize the expected cumulative holding cost incurred by jobs, where statistical parameters defining their individual holding costs are unknown a priori. In each time slot, the server can process a job while receiving the realized random holding costs of the jobs remaining in the system. Our algorithm is a learning-based variant of the $c\mu$ rule for scheduling: it starts with a preemption period of fixed length which serves as a learning phase, and after accumulating enough data about individual jobs, it switches to nonpreemptive scheduling mode. The algorithm is designed to handle instances with large or small gaps in jobs' parameters and achieves near-optimal performance guarantees. The performance of our algorithm is captured by its regret, where the benchmark is the minimum possible cost attained when the statistical parameters of jobs are fully known. We prove upper bounds on the regret of our algorithm, and we derive a regret lower bound that is almost matching the proposed upper bounds. Our numerical results demonstrate the effectiveness of our algorithm and show that our theoretical regret analysis is nearly tight.

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that every 4-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it in the binary case. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.