Model-based reinforcement learning (MBRL) agents typically learn world models by minimizing predictive loss. However, powerful RL optimizers inevitably exploit minor model inaccuracies, leading to simulator exploitation and a reality gap where policies succeed in simulation but fail in the real world. We propose that the objective for learning simulators should be strategic robustness rather than predictive accuracy, and formulate this as a zero-sum minimax game between a model player and an adversarial policy player. We provide a comprehensive theoretical analysis: (1) an online learning guarantee showing the game is learnable with sublinear regret bounds; (2) a tractable critic-based simplification bounding the global policy-value gap by the local critic's loss; and (3) an Error-MDP duality, proving that finding the worst-case policy is formally dual to a standard RL problem where the reward is the one-step critic error. This duality yields a provably convergent active data selection algorithm. Experiments on continuous control tasks demonstrate that our approach reduces prediction error in strategically important regions by $1.5$-$2.2\times$ and enables policies trained purely in simulation to match near-optimal real-world performance.

We revisit the classical problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss. Framing the task as a stochastic convex optimization problem over the space of $ M $-component mixture distributions, we propose a family of estimators derived from the stochastic mirror descent (SMD) algorithm. This optimization-based approach provides a principled and flexible framework that generalizes traditional estimators and proposes a variety of novel estimators through the choice of Bregman divergences. A key advantage of our method is that it scales efficiently with the number of candidate components $ f_i $; that is, one can employ a large set of basis distributions in the mixture model without incurring significant computational overhead. This enables richer approximations and improved estimation accuracy. Moreover, in the case of categorical distribution (discrete outcomes) our estimators do not require a strict lower bound, in other words our framework does not require the precise knowledge of the support of the distribution. We demonstrate that, under mild conditions, the proposed $ φ$-SMD estimators achieve near-optimal convergence rates in both Kullback-Leibler (KL) divergence and $ \ell_2 $-norm and offer practical benefits when computation is expensive. Our numerical analysis highlights improved performance guaranties over classical estimators, particularly in terms of sample efficiency and scalability.

We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in Rényi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

Reinforcement Learning from Human Feedback (RLHF) has become a cornerstone technique for post-training large language models. While most existing approaches rely on the reverse KL-regularization, recent empirical studies have begun exploring alternative divergences (e.g., forward KL, chi-squared) as regularizers in RLHF. However, a unified theoretical understanding of general $f$-divergence regularization remains under-explored. To fill this gap, this work develops a comprehensive theoretical framework for online RLHF with a general $f$-divergence regularized objective. Rather than treating each possible divergence function individually, we adopt a holistic perspective across the entire function class and propose two algorithms based on distinct sampling principles. The first extends the classical optimism principle with a carefully designed exploration bonus, while the second introduces a new method that exploits the sensitivity of the optimal policy to reward perturbations under $f$-divergence regularization. Theoretical analysis shows that $O(\log T)$ regret and $O(1/T)$ sub-optimality gap are achievable, establishing provable efficiency of both algorithms and, to the best of our knowledge, the first performance bounds for online RLHF under general $f$-divergence regularization.

Bayesian predictive inference propagates parameter uncertainty to quantities of interest through the posterior-predictive distribution. In practice, this is typically performed using a two-stage procedure: first approximating the posterior distribution of model parameters, and then propagating posterior samples through the predictive model via Monte Carlo simulation. This sequential workflow can be computationally demanding, particularly for high-fidelity models such as those governed by partial differential equations. We propose a variational Bayesian framework that directly targets the posterior-predictive distribution and jointly learns variational approximations of both the posterior and the corresponding predictive distribution. The formulation introduces a variational upper bound on the Kullback--Leibler divergence together with moment-based regularization terms. The variational distributions are trained in an amortized manner, shifting computational effort to an offline stage and enabling efficient online inference. Numerical experiments ranging from analytical benchmarks to a finite-element solid mechanics problem demonstrate that the proposed method achieves more accurate predictive distributions than conventional two-stage variational inference, while substantially reducing the cost of online predictive inference.

The Distributional Alignment Game framework provides a powerful variational perspective on Answer-Level Fine-Tuning (ALFT). However, standard algorithms for these games rely on estimating logarithmic rewards from small batches, introducing a systematic bias due to Jensen's inequality that can destabilize training. In this paper, we systematically resolve this structural estimation bias. First, we generalize the alignment game to arbitrary Bregman divergences, showing that for a family of geometries inducing polynomial rewards, we can construct provably exact and unbiased estimators using U-statistics. Second, for the canonical KL divergence game where an exact solution is impossible, we derive a globally robust minimax polynomial estimator that is provably optimal, achieving the fundamental statistical error limit of $Θ(1/K^2)$, which we establish via the Ditzian-Totik theorem. Finally, we synthesize these two approaches to propose a novel Variance-Optimal Augmented Polynomial Optimization Program (AQP) Estimator, proving that by systematically reducing variance, our method achieves not only optimal bias but also provably accelerated game convergence, leading to more efficient and stable training with zero online computational overhead.