Youheng Zhu

Youheng Zhu, Yiping Lu

In off policy evaluation (OPE) for partially observable Markov decision processes (POMDPs), an agent must infer hidden states from past observations, which exacerbates both the curse of horizon and the curse of memory in existing OPE methods. This paper introduces a novel covering analysis framework that exploits the intrinsic metric structure of the belief space (distributions over latent states) to relax traditional coverage assumptions. By assuming value relevant functions are Lipschitz continuous in the belief space, we derive error bounds that mitigate exponential blow ups in horizon and memory length. Our unified analysis technique applies to a broad class of OPE algorithms, yielding concrete error bounds and coverage requirements expressed in terms of belief space metrics rather than raw history coverage. We illustrate the improved sample efficiency of this framework via case studies: the double sampling Bellman error minimization algorithm, and the memory based future dependent value functions (FDVF). In both cases, our coverage definition based on the belief space metric yields tighter bounds.

Youheng Zhu, Yiping Lu

Inference-time scaling has recently emerged as a powerful paradigm for improving the reasoning capability of large language models. Among various approaches, Sequential Monte Carlo (SMC) has become a particularly important framework, enabling iterative generation, evaluation, rejection, and resampling of intermediate reasoning trajectories. A central component in this process is the reward model, which evaluates partial solutions and guides the allocation of computation during inference. However, in practice, true reward models are never available. All deployed systems rely on approximate reward models, raising a fundamental question: Why and when do approximate reward models suffice for effective inference-time scaling? In this work, we provide a theoretical answer. We identify the Bellman error of the approximate reward model as the key quantity governing the effectiveness of SMC-based inference-time scaling. For a reasoning process of length $T$, we show that if the Bellman error of the approximate reward model is bounded by $O(1/T)$, then combining this reward model with SMC reduces the computational complexity of reasoning from exponential in $T$ to polynomial in $T$. This yields an exponential improvement in inference efficiency despite using only approximate rewards.

Yiping Lu, Youheng Zhu

For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison--Reiman data with a nonsingular $M$-matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular $M$-matrix assumption is structural. In the larger completely-$\mathcal{S}$ class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai--Dieker question has a positive answer in the Harrison--Reiman $M$-matrix class and a negative answer in a natural completely-$\mathcal{S}$ extension.