Minimax PAC Bounds for Learning in Exogenous Contextual MDPs

We study PAC learning in tabular discounted Markov decision processes with exogenous i.i.d. contexts, with discount factor $γ$, finite state space $\mathcal X$, action space $\mathcal A$, and context space $\mathcal Z$. At each time step, a context is drawn independently from an unknown distribution $μ$ and revealed before the agent acts. This context may affect both rewards and transitions, while remaining uncontrolled by the agent. Depending on the regime, the learner has access either to a sampling oracle for $μ$, to a sampling oracle for the transition kernel conditioned on state-context-action tuples, or to both. Oracles can be accessed before and during policy execution. The sample complexity is measured by a couple $(n,m)$, where $n$ is the number of calls to the sampling oracles before execution and $m$ is the number of calls to the sampling oracles during execution. When rewards and transitions are known and only the context distribution $μ$ is sampled, we give a variance-reduced algorithm that solves policy evaluation (PE), best-value estimation (BVE), and best-policy extraction (BPE) with $\left(\widetilde O\left(1/((1-γ)^3\varepsilon^2)\right), 0 \right) $ sample complexity. The rate is independent of $|\mathcal Z|$ and minimax optimal up to logarithmic factors. As a corollary, we also obtain tight rates in the case of one-step perfect look-ahead, improving upon the existing guarantees. In the fully unknown regime, where both $μ$ and P must be learned, we show that PE remains $|\mathcal Z|$-free, with matching upper and lower bounds $\bigl(\widetilde O(|\mathcal X|/((1-γ)^3\varepsilon^2)),\, \widetilde O(1/((1-γ)^2\varepsilon^2))\bigr)$.

Learning to Allocate Resources with Censored Feedback

We study the online resource allocation problem in which at each round, a budget $B$ must be allocated across $K$ arms under censored feedback. An arm yields a reward if and only if two conditions are satisfied: (i) the arm is activated according to an arm-specific Bernoulli random variable with unknown parameter, and (ii) the allocated budget exceeds a random threshold drawn from a parametric distribution with unknown parameter. Over $T$ rounds, the learner must jointly estimate the unknown parameters and allocate the budget so as to maximize cumulative reward facing the exploration--exploitation trade-off. We prove an information-theoretic regret lower bound $Ω(T^{1/3})$, demonstrating the intrinsic difficulty of the problem. We then propose RA-UCB, an optimistic algorithm that leverages non-trivial parameter estimation and confidence bounds. When the budget $B$ is known at the beginning of each round, RA-UCB achieves a regret of order $\widetilde{\mathcal{O}}(\sqrt{T})$, and even $\mathcal{O}(\mathrm{poly}\text{-}\log T)$ under stronger assumptions. As for unknown, round dependent budget, we introduce MG-UCB, which allows within-round switching and infinitesimal allocations, and matches the regret guarantees of RA-UCB. We then validate our theoretical results through experiments on real-world datasets.

Distribution-Aware Mean Estimation under User-level Local Differential Privacy

We consider the problem of mean estimation under user-level local differential privacy, where $n$ users are contributing through their local pool of data samples. Previous work assume that the number of data samples is the same across users. In contrast, we consider a more general and realistic scenario where each user $u \in [n]$ owns $m_u$ data samples drawn from some generative distribution $\mu$; $m_u$ being unknown to the statistician but drawn from a known distribution $M$ over $\mathbb{N}^\star$. Based on a distribution-aware mean estimation algorithm, we establish an $M$-dependent upper bounds on the worst-case risk over $\mu$ for the task of mean estimation. We then derive a lower bound. The two bounds are asymptotically matching up to logarithmic factors and reduce to known bounds when $m_u = m$ for any user $u$.