Learning with Simulators: No Regret in a Computationally Bounded World

Understanding the minimal assumptions necessary for generalization is the fundamental question in learning theory. Unfortunately, most results rely heavily on independence (or some proxy thereof) of the data-generating process, while results for strongly dependent data are far more limited. Towards addressing this gap, we introduce the framework of simulatable processes, where the learner has access to a simulator that approximates the distribution generating the data (which may be an arbitrarily complex and dependent process). Surprisingly, given access to such a simulator, we show that we can recover the same learning guarantees as in the classical setting with independent data, namely, error bounds that depend on the VC dimension. Further, we use this framework to study the power of conditional sampling and show strict statistical and computational advantages in this setting. As a highlight of our framework, we exhibit a single algorithm that simultaneously learns any given VC class under all processes samplable in bounded polynomial time, with regret controlled by the time-bounded Kolmogorov complexity of the process. This provides a significant conceptual broadening of the classical PAC model.

The Sample Complexity of Multiclass and Sparse Contextual Bandits

We study contextual bandits in the stochastic i.i.d.\ setting, where a learner observes contexts drawn from an unknown distribution, selects actions from a finite set $A$, and aims to identify an approximately optimal policy from a given class based on bandit feedback. Motivated by bandit multiclass classification with zero-one rewards, we focus on the \emph{$s$-sparse} setting in which, for every context, the reward vector has $L_1$-norm at most $s \ll |A|$. Our main result is the design of algorithms that, with high probability, output an $ε$-optimal policy compared to policy class $Π$ using $\tilde{O} ((s/ε^2 + |A|/ε)\log |Π|/δ)$ samples. We extend this bound to general Natarajan classes and complement it with a matching lower bound (up to logarithmic factors), thereby closing a substantial gap left by prior work (Erez et al., 2024, 2025), which incurred an additional $Θ(|A|^9)$ dependence. We obtain these results via two complementary approaches. First, we analyze contextual bandits through the lens of contextual decision making with structured observations, designing an exploration-by-optimization algorithm whose sample complexity is governed by the \emph{decision-estimation coefficient} (DEC; Foster et al., 2021, 2022). We show that, with $s$-sparse rewards, the induced model class admits a sharp DEC bound that scales with $s$ and directly yields the optimal rate. Since this approach is largely information-theoretic and involves solving complex min-max optimization problems, we also develop a second, more specialized algorithmic method based on a low-variance exploration technique. This approach leads to concrete, tractable algorithms and naturally extends to contextual combinatorial semi-bandits, leading to improved sample complexity guarantees for bandit multiclass list classification.

End-to-End Efficient RL for Linear Bellman Complete MDPs with Deterministic Transitions

We study reinforcement learning (RL) with linear function approximation in Markov Decision Processes (MDPs) satisfying \emph{linear Bellman completeness} -- a fundamental setting where the Bellman backup of any linear value function remains linear. While statistically tractable, prior computationally efficient algorithms are either limited to small action spaces or require strong oracle assumptions over the feature space. We provide a computationally efficient algorithm for linear Bellman complete MDPs with \emph{deterministic transitions}, stochastic initial states, and stochastic rewards. For finite action spaces, our algorithm is end-to-end efficient; for large or infinite action spaces, we require only a standard argmax oracle over actions. Our algorithm learns an $\varepsilon$-optimal policy with sample and computational complexity polynomial in the horizon, feature dimension, and $1/\varepsilon$.

Characterizing Online and Private Learnability under Distributional Constraints via Generalized Smoothness

Understanding minimal assumptions that enable learning and generalization is perhaps the central question of learning theory. Several celebrated results in statistical learning theory, such as the VC theorem and Littlestone's characterization of online learnability, establish conditions on the hypothesis class that allow for learning under independent data and adversarial data, respectively. Building upon recent work bridging these extremes, we study sequential decision making under distributional adversaries that can adaptively choose data-generating distributions from a fixed family $U$ and ask when such problems are learnable with sample complexity that behaves like the favorable independent case. We provide a near complete characterization of families $U$ that admit learnability in terms of a notion known as generalized smoothness i.e. a distribution family admits VC-dimension-dependent regret bounds for every finite-VC hypothesis class if and only if it is generalized smooth. Further, we give universal algorithms that achieve low regret under any generalized smooth adversary without explicit knowledge of $U$. Finally, when $U$ is known, we provide refined bounds in terms of a combinatorial parameter, the fragmentation number, that captures how many disjoint regions can carry nontrivial mass under $U$. These results provide a nearly complete understanding of learnability under distributional adversaries. In addition, building upon the surprising connection between online learning and differential privacy, we show that the generalized smoothness also characterizes private learnability under distributional constraints.

High-accuracy log-concave sampling with stochastic queries

We show that high-accuracy guarantees for log-concave sampling -- that is, iteration and query complexities which scale as $\mathrm{poly}\log(1/δ)$, where $δ$ is the desired target accuracy -- are achievable using stochastic gradients with subexponential tails. Notably, this exhibits a separation with the problem of convex optimization, where stochasticity (even additive Gaussian noise) in the gradient oracle incurs $\mathrm{poly}(1/δ)$ queries. We also give an information-theoretic argument that light-tailed stochastic gradients are necessary for high accuracy: for example, in the bounded variance case, we show that the minimax-optimal query complexity scales as $Θ(1/δ)$. Our framework also provides similar high accuracy guarantees under stochastic zeroth order (value) queries.

High-accuracy sampling for diffusion models and log-concave distributions

We present algorithms for diffusion model sampling which obtain $δ$-error in $\mathrm{polylog}(1/δ)$ steps, given access to $\widetilde O(δ)$-accurate score estimates in $L^2$. This is an exponential improvement over all previous results. Specifically, under minimal data assumptions, the complexity is $\widetilde O(d\,\mathrm{polylog}(1/δ))$ where $d$ is the dimension of the data; under a non-uniform $L$-Lipschitz condition, the complexity is $\widetilde O(\sqrt{dL}\,\mathrm{polylog}(1/δ))$; and if the data distribution has intrinsic dimension $d_\star$, then the complexity reduces to $\widetilde O(d_\star\,\mathrm{polylog}(1/δ))$. Our approach also yields the first $\mathrm{polylog}(1/δ)$ complexity sampler for general log-concave distributions using only gradient evaluations.

Outcome-Based Online Reinforcement Learning: Algorithms and Fundamental Limits

Reinforcement learning with outcome-based feedback faces a fundamental challenge: when rewards are only observed at trajectory endpoints, how do we assign credit to the right actions? This paper provides the first comprehensive analysis of this problem in online RL with general function approximation. We develop a provably sample-efficient algorithm achieving $\widetilde{O}({C_{\rm cov} H^3}/{\epsilon^2})$ sample complexity, where $C_{\rm cov}$ is the coverability coefficient of the underlying MDP. By leveraging general function approximation, our approach works effectively in large or infinite state spaces where tabular methods fail, requiring only that value functions and reward functions can be represented by appropriate function classes. Our results also characterize when outcome-based feedback is statistically separated from per-step rewards, revealing an unavoidable exponential separation for certain MDPs. For deterministic MDPs, we show how to eliminate the completeness assumption, dramatically simplifying the algorithm. We further extend our approach to preference-based feedback settings, proving that equivalent statistical efficiency can be achieved even under more limited information. Together, these results constitute a theoretical foundation for understanding the statistical properties of outcome-based reinforcement learning.

Trajectory Bellman Residual Minimization: A Simple Value-Based Method for LLM Reasoning

Policy-based methods currently dominate reinforcement learning (RL) pipelines for large language model (LLM) reasoning, leaving value-based approaches largely unexplored. We revisit the classical paradigm of Bellman Residual Minimization and introduce Trajectory Bellman Residual Minimization (TBRM), an algorithm that naturally adapts this idea to LLMs, yielding a simple yet effective off-policy algorithm that optimizes a single trajectory-level Bellman objective using the model's own logits as $Q$-values. TBRM removes the need for critics, importance-sampling ratios, or clipping, and operates with only one rollout per prompt. We prove convergence to the near-optimal KL-regularized policy from arbitrary off-policy data via an improved change-of-trajectory-measure analysis. Experiments on standard mathematical-reasoning benchmarks show that TBRM consistently outperforms policy-based baselines, like PPO and GRPO, with comparable or lower computational and memory overhead. Our results indicate that value-based RL might be a principled and efficient alternative for enhancing reasoning capabilities in LLMs.

Near-Optimal Private Learning in Linear Contextual Bandits

We analyze the problem of private learning in generalized linear contextual bandits. Our approach is based on a novel method of re-weighted regression, yielding an efficient algorithm with regret of order $\sqrt{T}+\frac{1}{\alpha}$ and $\sqrt{T}/\alpha$ in the joint and local model of $\alpha$-privacy, respectively. Further, we provide near-optimal private procedures that achieve dimension-independent rates in private linear models and linear contextual bandits. In particular, our results imply that joint privacy is almost "for free" in all the settings we consider, partially addressing the open problem posed by Azize and Basu (2024).

Decision Making in Changing Environments: Robustness, Query-Based Learning, and Differential Privacy

We study the problem of interactive decision making in which the underlying environment changes over time subject to given constraints. We propose a framework, which we call \textit{hybrid Decision Making with Structured Observations} (hybrid DMSO), that provides an interpolation between the stochastic and adversarial settings of decision making. Within this framework, we can analyze local differentially private (LDP) decision making, query-based learning (in particular, SQ learning), and robust and smooth decision making under the same umbrella, deriving upper and lower bounds based on variants of the Decision-Estimation Coefficient (DEC). We further establish strong connections between the DEC's behavior, the SQ dimension, local minimax complexity, learnability, and joint differential privacy. To showcase the framework's power, we provide new results for contextual bandits under the LDP constraint.