Duksang Lee

Kihyun Yu, Duksang Lee, William Overman et al.

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 $\bar C- \bar C_b=Ω(H)$, it nearly matches the regret lower bound of $Ω(H^{1.5}\sqrt{SAK})$. 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.

Duksang Lee, William Overman, Dabeen Lee

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 allocation problem in an episodic finite-horizon constrained Markov decision process with an unknown non-stationary transition function and stochastic non-stationary reward and resource consumption functions. We propose the observe-then-decide regime and improve the existing decide-then-observe regime, while the two settings differ in how the observations and feedback about the reward and resource consumption functions are given to the decision-maker. We develop an online dual mirror descent algorithm that achieves near-optimal regret bounds for both settings. For the observe-then-decide regime, we prove that the expected regret against the dynamic clairvoyant optimal policy is bounded by $\tilde O(ρ^{-1}{H^{3/2}}S\sqrt{AT})$ where $ρ\in(0,1)$ 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. For the decide-then-observe regime, we show that the regret against the static optimal policy that has access to the mean reward and mean resource consumption functions is bounded by $\tilde O(ρ^{-1}{H^{3/2}}S\sqrt{AT})$ with high probability. We test the numerical efficiency of our method for a variant of the resource-constrained inventory management problem.

Duksang Lee, Nam Ho-Nguyen, Dabeen Lee

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.