Improved Regret Bound for Safe Reinforcement Learning via Tighter Cost Pessimism and Reward Optimism

This paper studies the safe reinforcement learning problem formulated as an episodic finite-horizon tabular constrained Markov decision process with an unknown transition kernel and stochastic reward and cost functions. We propose a model-based algorithm based on novel cost and reward function estimators that provide tighter cost pessimism and reward optimism. While guaranteeing no constraint violation in every episode, our algorithm achieves a regret upper bound of $\widetilde{\mathcal{O}}((\bar C - \bar C_b)^{-1}H^{2.5} S\sqrt{AK})$ where $\bar C$ is the cost budget for an episode, $\bar C_b$ is the expected cost under a safe baseline policy over an episode, $H$ is the horizon, and $S$, $A$ and $K$ are the number of states, actions, and episodes, respectively. This improves upon the best-known regret upper bound, and when $\bar C- \bar C_b=\Omega(H)$, it nearly matches the regret lower bound of $\Omega(H^{1.5}\sqrt{SAK})$. We deduce our cost and reward function estimators via a Bellman-type law of total variance to obtain tight bounds on the expected sum of the variances of value function estimates. This leads to a tighter dependence on the horizon in the function estimators. We also present numerical results to demonstrate the computational effectiveness of our proposed framework.

Provably Efficient Infinite-Horizon Average-Reward Reinforcement Learning with Linear Function Approximation

This paper proposes a computationally tractable algorithm for learning infinite-horizon average-reward linear Markov decision processes (MDPs) and linear mixture MDPs under the Bellman optimality condition. While guaranteeing computational efficiency, our algorithm for linear MDPs achieves the best-known regret upper bound of $\widetilde{\mathcal{O}}(d^{3/2}\mathrm{sp}(v^*)\sqrt{T})$ 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. For linear mixture MDPs, our algorithm attains a regret bound of $\widetilde{\mathcal{O}}(d\cdot\mathrm{sp}(v^*)\sqrt{T})$. The algorithm applies novel techniques to control the covering number of the value function class and the span of optimistic estimators of the value function, which is of independent interest.

Reinforcement Learning for Infinite-Horizon Average-Reward MDPs with Multinomial Logistic Function Approximation

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 logistic (MNL) model. In this paper, we develop two algorithms for the infinite-horizon average reward setting. Our first algorithm \texttt{UCRL2-MNL} applies to the class of communicating MDPs and achieves an $\tilde{\mathcal{O}}(dD\sqrt{T})$ regret, where $d$ is the dimension of feature mapping, $D$ is the diameter of the underlying MDP, and $T$ is the horizon. The second algorithm \texttt{OVIFH-MNL} is computationally more efficient and applies to the more general class of weakly communicating MDPs, for which we show a regret guarantee of $\tilde{\mathcal{O}}(d^{2/5} \mathrm{sp}(v^*)T^{4/5})$ where $\mathrm{sp}(v^*)$ is the span of the associated optimal bias function. We also prove a lower bound of $\Omega(d\sqrt{DT})$ for learning communicating MDPs with MNL transitions of diameter at most $D$. Furthermore, 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.

Parameter-Free Algorithms for Performative Regret Minimization under Decision-Dependent Distributions

This paper studies performative risk minimization, a formulation of stochastic optimization under decision-dependent distributions. We consider the general case where the performative risk can be non-convex, for which we develop efficient parameter-free optimistic optimization-based methods. Our algorithms significantly improve upon the existing Lipschitz bandit-based method in many aspects. In particular, our framework does not require knowledge about the sensitivity parameter of the distribution map and the Lipshitz constant of the loss function. This makes our framework practically favorable, together with the efficient optimistic optimization-based tree-search mechanism. We provide experimental results that demonstrate the numerical superiority of our algorithms over the existing method and other black-box optimistic optimization methods.

Stochastic-Constrained Stochastic Optimization with Markovian Data

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 two variants of drift-plus-penalty; one is 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. Both algorithms are adaptive in that the first works without knowledge of the time horizon while the second 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.

Online Resource Allocation in Episodic Markov Decision Processes

This paper studies a long-term resource allocation problem over multiple periods where each period requires a multi-stage decision-making process. We formulate the problem as an online resource allocation problem in an episodic finite-horizon Markov decision process with unknown non-stationary transitions and stochastic non-stationary reward and resource consumption functions for each episode. We provide an equivalent online linear programming reformulation based on occupancy measures, for which we develop an online mirror descent algorithm. Our online dual mirror descent algorithm for resource allocation deals with uncertainties and errors in estimating the true feasible set, which is of independent interest. We prove that under stochastic reward and resource consumption functions, the expected regret of the online mirror descent algorithm is bounded by $O(\rho^{-1}{H^{3/2}}S\sqrt{AT})$ where $\rho\in(0,1)$ is the budget parameter, $H$ is the length of the horizon, $S$ and $A$ are the numbers of states and actions, and $T$ is the number of episodes.

Projection-Free Online Convex Optimization with Stochastic Constraints

This paper develops projection-free algorithms for online convex optimization with stochastic constraints. We design an online primal-dual projection-free framework that can take any projection-free algorithms developed for online convex optimization with no long-term constraint. With this general template, we deduce sublinear regret and constraint violation bounds for various settings. Moreover, for the case where the loss and constraint functions are smooth, we develop a primal-dual conditional gradient method that achieves $O(\sqrt{T})$ regret and $O(T^{3/4})$ constraint violations. Furthermore, for the setting where the loss and constraint functions are stochastic and strong duality holds for the associated offline stochastic optimization problem, we prove that the constraint violation can be reduced to have the same asymptotic growth as the regret.

Online Convex Optimization with Stochastic Constraints: Zero Constraint Violation and Bandit Feedback

This paper studies online convex optimization with stochastic constraints. We propose a variant of the drift-plus-penalty algorithm that guarantees $O(\sqrt{T})$ expected regret and zero constraint violation, after a fixed number of iterations, which improves the vanilla drift-plus-penalty method with $O(\sqrt{T})$ constraint violation. Our algorithm is oblivious to the length of the time horizon $T$, in contrast to the vanilla drift-plus-penalty method. This is based on our novel drift lemma that provides time-varying bounds on the virtual queue drift and, as a result, leads to time-varying bounds on the expected virtual queue length. Moreover, we extend our framework to stochastic-constrained online convex optimization under two-point bandit feedback. We show that by adapting our algorithmic framework to the bandit feedback setting, we may still achieve $O(\sqrt{T})$ expected regret and zero constraint violation, improving upon the previous work for the case of identical constraint functions. Numerical results demonstrate our theoretical results.

Learning to Schedule

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.

Test Score Algorithms for Budgeted Stochastic Utility Maximization

Motivated by recent developments in designing algorithms based on individual item scores for solving utility maximization problems, we study the framework of using test scores, defined as a statistic of observed individual item performance data, for solving the budgeted stochastic utility maximization problem. We extend an existing scoring mechanism, namely the replication test scores, to incorporate heterogeneous item costs as well as item values. We show that a natural greedy algorithm that selects items solely based on their replication test scores outputs solutions within a constant factor of the optimum for a broad class of utility functions. Our algorithms and approximation guarantees assume that test scores are noisy estimates of certain expected values with respect to marginal distributions of individual item values, thus making our algorithms practical and extending previous work that assumes noiseless estimates. Moreover, we show how our algorithm can be adapted to the setting where items arrive in a streaming fashion while maintaining the same approximation guarantee. We present numerical results, using synthetic data and data sets from the Academia.StackExchange Q&A forum, which show that our test score algorithm can achieve competitiveness, and in some cases better performance than a benchmark algorithm that requires access to a value oracle to evaluate function values.