We consider the problem of heteroskedastic generalized linear bandits (GLBs) with adversarial corruptions, which subsumes various stochastic contextual bandit settings, including heteroskedastic linear bandits and logistic/Poisson bandits. We propose HCW-GLB-OMD, which consists of two components: an online mirror descent (OMD)-based estimator and Hessian-based confidence weights to achieve corruption robustness. This is computationally efficient in that it only requires ${O}(1)$ space and time complexity per iteration. Under the self-concordance assumption on the link function, we show a regret bound of $\tilde{O}\left( d \sqrt{\sum_t g(τ_t) \dotμ_{t,\star}} + d^2 g_{\max} κ+ d κC \right)$, where $\dotμ_{t,\star}$ is the slope of $μ$ around the optimal arm at time $t$, $g(τ_t)$'s are potentially exogenously time-varying dispersions (e.g., $g(τ_t) = σ_t^2$ for heteroskedastic linear bandits, $g(τ_t) = 1$ for Bernoulli and Poisson), $g_{\max} = \max_{t \in [T]} g(τ_t)$ is the maximum dispersion, and $C \geq 0$ is the total corruption budget of the adversary. We complement this with a lower bound of $\tildeΩ(d \sqrt{\sum_t g(τ_t) \dotμ_{t,\star}} + d C)$, unifying previous problem-specific lower bounds. Thus, our algorithm achieves, up to a $κ$-factor in the corruption term, instance-wise minimax optimality simultaneously across various instances of heteroskedastic GLBs with adversarial corruptions.

Policy inference plays an essential role in the contextual bandit problem. In this paper, we use empirical likelihood to develop a Bayesian inference method for the joint analysis of multiple contextual bandit policies in finite sample regimes. The proposed inference method is robust to small sample sizes and is able to provide accurate uncertainty measurements for policy value evaluation. In addition, it allows for flexible inferences on policy comparison with full uncertainty quantification. We demonstrate the effectiveness of the proposed inference method using Monte Carlo simulations and its application to an adolescent body mass index data set.

This work addresses the problem of offline safe imitation learning (IL), where the goal is to learn safe and reward-maximizing policies from demonstrations that do not have per-timestep safety cost or reward information. In many real-world domains, online learning in the environment can be risky, and specifying accurate safety costs can be difficult. However, it is often feasible to collect trajectories that reflect undesirable or unsafe behavior, implicitly conveying what the agent should avoid. We refer to these as non-preferred trajectories. We propose a novel offline safe IL algorithm, OSIL, that infers safety from non-preferred demonstrations. We formulate safe policy learning as a Constrained Markov Decision Process (CMDP). Instead of relying on explicit safety cost and reward annotations, OSIL reformulates the CMDP problem by deriving a lower bound on reward maximizing objective and learning a cost model that estimates the likelihood of non-preferred behavior. Our approach allows agents to learn safe and reward-maximizing behavior entirely from offline demonstrations. We empirically demonstrate that our approach can learn safer policies that satisfy cost constraints without degrading the reward performance, thus outperforming several baselines.

We develop a theoretical analysis of LLM-guided formal theorem proving in interactive proof assistants (e.g., Lean) by modeling tactic proposal as a stochastic policy in a finite-horizon deterministic MDP. To capture modern representation learning, we treat the state and action spaces as general compact metric spaces and assume Lipschitz policies. To explain the gap between worst-case hardness and empirical success, we introduce problem distributions generated by a reference policy $q$, including a latent-variable model in which proofs exhibit reusable cut/lemma/sketch structure represented by a proof DAG. Under a top-$k$ search protocol and Tsybakov-type margin conditions, we derive lower bounds on finite-horizon success probability that decompose into search and learning terms, with learning controlled by sequential Rademacher/covering complexity. Our main separation result shows that when cut elimination expands a DAG of depth $D$ into a cut-free tree of size $Ω(Λ^D)$ while the cut-aware hierarchical process has size $O(λ^D)$ with $λ\llΛ$, a flat (cut-free) learner provably requires exponentially more data than a cut-aware hierarchical learner. This provides a principled justification for subgoal decomposition in recent agentic theorem provers.

Understanding model's sensitivity to its training data is crucial but can also be challenging and costly, especially during training. To simplify such issues, we present the Memory-Perturbation Equation (MPE) which relates model's sensitivity to perturbation in its training data.