arXiv subjects

Jerry Zhijian Yang

Publications and source records attributed to Jerry Zhijian Yang.

At least 19 recordsLinked to original sources

arXiv2026

Deep Weighted Bellman Residual Minimization for $Q^*$ Estimation

Off-policy evaluation is a foundational component of offline reinforcement learning, aiming to assess and optimize policy performance using pre-collected datasets. However, such datasets often suffer from pronounced challenges, including distribution shift, $Q$-value overestimation, and low sample utilization efficiency. To address these issues, this paper introduces a weighted Bellman residual minimization framework that incorporates density ratio weighting by effectively integrating expert demonstrations with behavioral data. The proposed weighting scheme departs from the conventional completeness assumption commonly imposed in the theoretical analysis of deep reinforcement learning. We establish a sharp convergence rate for density ratio estimation and derive the convergence rate for the excess risk of resulting deep $Q^*$ estimator. Extensive empirical evaluations demonstrate that, compared to existing methods, our method achieves significant improvements in numerical performance and policy generalization, providing specific guidance for the rational utilization of expert demonstrations.

cs.LG↗

arXiv2026

Schrödinger--Föllmer Actor--Critic: Diffusion Policy Improvement with Finite-Sample Analysis

Diffusion policies represent multimodal action distributions, but an advantage-weighted update does not specify how to sample from the resulting target distribution. We propose Schrödinger--Föllmer Actor--Critic (SFAC), an offline-to-online reinforcement learning (RL) method for Kullback--Leibler (KL)-regularized policy improvement through conditional diffusion. A minimax Bellman critic estimates the advantage function, which defines an exponentially tilted target policy. A Doob $h$-transform expresses this update as a correction to the reference diffusion drift. We derive a posterior-mean representation of the correction and estimate it using paired self-normalized importance sampling (SNIS). Supervised regression on these drift targets updates the neural actor without critic action gradients. In the small-update regime, the KL-regularized update follows the natural policy-gradient direction, and the Doob correction represents the same local change in the space of diffusion drifts. Under suitable conditions, we derive finite-sample bounds that separate the effects of critic estimation, neural drift regression, finite-sample SNIS, diffusion discretization, and inherited actor error on expected average policy suboptimality. Synthetic experiments assess the accuracy of approximation to prescribed advantage-tilted targets and sensitivity to sampling budgets. On six offline-to-online continuous-control tasks, a reference-anchored implementation achieves higher final-window returns than those of its corresponding offline initialization.

stat.ML↗

arXiv2026

Offline Deep Q* Estimation with Diffusion Models

In offline RL, estimating the optimal action-value function $Q^*$ can be formulated as solving the optimal Bellman equation based solely on offline observations. A fundamental challenge is that the reward function and transition kernel are unknown, so the optimal Bellman operator is not directly observable from data. To address this issue, we propose a novel framework that decouples operator estimation from value function learning. In this approach, we first formulate conditional diffusion models to estimate the reward law and transition kernel, which induces a data-driven approximation of the optimal Bellman operator. We then plug these estimators into the Bellman equation and obtain a deep estimator of $Q^*$ by minimizing the empirical Bellman residual over a neural network function class. Theoretically, we first establish sharp nonasymptotic convergence rates for learning the optimal Bellman operator through an end-to-end analysis of conditional diffusion estimation in total variation distance. We then establish the oracle value-stage rate $\widetilde{\mathcal O}\bigl(n^{-\frac{2β}{d_x+d_a+2β}}\bigr)$ for the excess Bellman residual risk. Finally, under a concentrability condition, we translate this residual bound into an $L^2$ convergence rate of $\widetilde{\mathcal O}\bigl(n^{-\fracβ{d_x+d_a+2β}}\bigr)$ for the resulting deep estimator of $Q^*$, where $d_x$ and $d_a$ denote the dimensions of the state and action spaces, respectively, and $β$ denotes the Hölder smoothness index of $Q^*$. Importantly, our theoretical analysis does not rely on completeness assumptions commonly used in deep RL theory. Extensive numerical experiments demonstrate the effectiveness of the proposed method and its strong empirical performance.

stat.ML↗

arXiv2026

Characteristic Learning for Provable One Step Generation

We propose the characteristic generator, an one-step generative model that combines the sampling efficiency of generative adversarial networks (GANs) with the training stability of flow-based models. The proposed model is based on characteristics along which probability-density transport is governed by ordinary differential equations (ODEs). Specifically, we first estimate the underlying velocity field and numerically solve the probability-flow ODE using an exponential integrator, thereby obtaining discrete approximations of the characteristic trajectories. We then train a deep neural network to approximate these trajectories, yielding a one-step transport map from a Gaussian prior to the target distribution. Theoretically, we analyze the errors introduced by velocity-field estimation, numerical discretization, and characteristic approximation, and establish a nonasymptotic convergence rate in the Wasserstein-2 distance under mild assumptions on the data distribution. Moreover, under a low-dimensional linear-subspace assumption, we show that the convergence rate depends on the intrinsic data dimension rather than the potentially much larger ambient dimension, demonstrating the model's ability to alleviate the curse of dimensionality. Experiments on synthetic and real-world datasets show that the characteristic generator produces high-quality, high-resolution samples using only one or a few neural-network evaluations.

cs.LG↗

arXiv2026

Sampling via Stochastic Interpolants by Langevin-based Velocity and Initialization Estimation in Flow ODEs

We propose a novel method for sampling from unnormalized Boltzmann densities based on a probability flow ordinary differential equation (ODE) derived from linear stochastic interpolants. The key innovation of our approach is the use of a sequence of Langevin samplers to enable efficient simulation of the flow. Specifically, these Langevin samplers are employed (i) to generate samples from the interpolant distribution at intermediate times and (ii) to construct, starting from these intermediate times, a robust estimator of the velocity field governing the probability flow ODE. Theoretically, we provide convergence guarantees for both Langevin components, and establish a non-asymptotic convergence rate for the probability flow ODE. Extensive numerical experiments demonstrate the efficiency of the proposed method on challenging multimodal distributions across a range of dimensions, as well as its effectiveness in Bayesian inference tasks.

math.NA↗

arXiv2026

Provable Diffusion Posterior Sampling for Bayesian Inversion

We propose a novel diffusion-based posterior sampling method within a plug-and-play framework. Our approach constructs a probability transport from an easy-to-sample distribution to the target posterior via a diffusion process. To initialize the sampler efficiently, we introduce a warm-start strategy for the particles. The posterior score is then approximated using a Monte Carlo estimator in which samples are generated via Langevin dynamics, avoiding the heuristic approximations prevalent in prior work. The score function driving the Langevin dynamics is learned from data, enabling the model to capture rich structural features of the underlying prior. We also establish non-asymptotic error bounds in Wasserstein-2 distance guaranteeing convergence of the proposed method even for complex, multimodal posterior distributions. We corroborate our theoretical findings with numerical experiments demonstrating the effectiveness of the method across a variety of inverse problems.

stat.ML↗

arXiv2026

Approximation Error Upper and Lower Bounds for Hölder Class with Transformers

We explore the expressive power of Transformers by establishing precise approximation error upper and lower bounds for Hölder class. Specifically, a new approximation upper bound is derived for the standard Transformer architecture equipped with Softmax operators, ReLU activation functions, and residual connections. We prove that a Transformer network composed of at most $\mathcal{O}(\varepsilon^{-{d_{0}}/α})$ blocks can approximate any bounded Hölder function with $d_{0}$-dimensional input and smoothness $α\in(0,1]$ under any accuracy $\varepsilon>0$. In the case of approximation lower bounds, leveraging the VC-dimension upper bound, we are the first to rigorously prove that Transformers demand for at least $Ω(\varepsilon^{-{d_{0}}/({4α})})$ blocks to achieve the $\varepsilon$ approximation accuracy. As a final step, we extend the derived results for standard Transformers to a general regression task and establish the corresponding excess risk rates demonstrating Transformers' empirical effectiveness in real-world settings.

cs.LG↗

arXiv2026

An alternating learning-based collocation method for solving inverse elliptic problems

We propose the Alternating Learning-Based Collocation (ALBC) method for solving inverse elliptic problems. Our approach employs sinusoidal shallow networks as adaptive basis generators. By alternately updating the state variable and the unknown parameter, we decompose the original nonconvex joint optimization problem into a sequence of tractable linear subproblems. This strategy effectively overcomes the fixed-basis limitations of classical collocation methods while avoiding the slow convergence typically encountered in deep learning approaches. Theoretically, we establish stability estimates and prove the convergence of the proposed algorithm. Numerical experiments on five benchmark problems demonstrate the efficacy of ALBC, which consistently outperforms the standard collocation method in accuracy. Furthermore, it achieves performance comparable to or better than that of physics-informed neural networks at a substantially lower computational cost. Finally, the method remains robust under noise levels of up to twenty percent.

math.NA↗

arXiv2026

Robust Fuzzy local k-plane clustering with mixture distance of hinge loss and L1 norm

K-plane clustering (KPC), hyperplane clustering, and mixture regression all essentially fall within the same class of problems. This problem can be conceptualized as clustering in relatively high-dimensional K subspaces or K linear manifolds. Traditional KPC or fuzzy KPC models demonstrate a pronounced susceptibility to outliers, as they presuppose that the projection distance between data points and the plane normal vector adheres to the L2 distance. Meanwhile, the assumption of infinitely extending clusters adversely affects clustering performance. To solve these problems, this paper proposed a new robust fuzzy local k-plane clustering (RFLkPC) method that combines the mixture distance of hinge loss and L1 norm. The RFLkPC model assumes that each plane cluster is bounded to a finite area, which can flexibly and robustly handle plane clustering tasks with outliers or not. The corresponding model and optimization algorithms of RFLkPC were provided. Compared to other related models on this topic, a large number of experiments verify the efficiency of RFLkPC on simulated data and real data. The source code for the proposed RFLkPC method is publicly available at https://github.com/xuelin-xie/RFLkPC.

cs.LG↗

arXiv2026

Nonlinear Assimilation via Score-based Sequential Langevin Sampling

This paper introduces score-based sequential Langevin sampling (SSLS), a novel approach to nonlinear data assimilation within a recursive Bayesian filtering framework. The proposed method decomposes the assimilation process into alternating prediction and update steps, using dynamic models for state prediction and incorporating observational data via score-based Langevin Monte Carlo during the updates. To overcome inherent challenges in highly non-log-concave posterior sampling, we integrate an annealing strategy into the update mechanism. Theoretically, we establish convergence guarantees for SSLS in total variation (TV) distance, yielding concrete insights into the algorithm's error behavior with respect to key hyperparameters. Crucially, our derived error bounds demonstrate the asymptotic stability of SSLS, guaranteeing that local posterior sampling errors do not accumulate indefinitely over time. Extensive numerical experiments across challenging scenarios, including high-dimensional systems, strong nonlinearity, and sparse observations, highlight the robust performance of the proposed method. Furthermore, SSLS effectively quantifies the uncertainty associated with state estimates, rendering it particularly valuable for reliable error calibration.

math.NA↗

arXiv2026

Near-optimal Prediction Error Estimation for Quantum Machine Learning Models

Understanding the theoretical capabilities and limitations of quantum machine learning (QML) models to solve machine learning tasks is crucial to advancing both quantum software and hardware developments. Similarly to the classical setting, the performance of QML models can be significantly affected by the limited access to the underlying data set. Previous studies have focused on proving generalization error bounds for any QML models trained on a limited finite training set. We focus on the optimal QML models obtained by training them on a finite training set and establish a tight prediction error bound in terms of the number of trainable gates and the size of training sets. To achieve this, we derive covering number upper bounds and packing number lower bounds for the data re-uploading QML models and linear QML models, respectively, which may be of independent interest. We support our theoretical findings by numerically simulating the QML strategies for function approximation and quantum phase recognition.

quant-ph↗

arXiv2026

Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matching

Inference-time alignment for diffusion models aims to adapt a pre-trained reference diffusion model toward a target distribution without retraining the reference score network, thereby preserving the generative capacity of the reference model while enforcing desired properties at the inference time. A central mechanism for achieving such alignment is guidance, which modifies the sampling dynamics through an additional drift term. In this work, we introduce variationally stable Doob's matching, a novel framework for provable guidance estimation grounded in Doob's $h$-transform. Our approach formulates guidance as the gradient of logarithm of an underlying Doob's $h$-function and employs gradient-regularized regression to simultaneously estimate both the $h$-function and its gradient, resulting in a consistent estimator of the guidance. Theoretically, we establish non-asymptotic convergence rates for the estimated guidance. Moreover, we analyze the resulting controllable diffusion processes and prove non-asymptotic convergence guarantees for the generated distributions in the 2-Wasserstein distance. Finally, we show that variationally stable guidance estimators are adaptive to unknown low dimensionality, effectively mitigating the curse of dimensionality under low-dimensional subspace assumptions.

stat.ML↗

arXiv2025

Toward fast, accurate and robust AI prediction of ground states in rotating BEC

We propose an unsupervised deep learning approach for computing the ground state (GS) of rotating Bose-Einstein condensation. To minimize the energy under a mass constraint, our approach introduces two key and novel ingredients: a normalized loss function that exactly enforces the mass constraint, and a training strategy named virtual rotation acceleration that is essential for avoiding local minima and guiding the learning process to the correct quantized vortex phase. Extensive numerical experiments demonstrate the proposed approach as an effective and accurate method to predict GS across physical conditions--from slow to fast rotation and from isotropic to anisotropic confinement. Through further distillation, we establish a unified operator network capable of efficiently generalizing physical parameters across different phases. It enables rapid GS predictions while correctly capturing phase transitions and is applied for inverse problems.

cond-mat.quant-gas↗

arXiv2025

Adv-SSL: Adversarial Self-Supervised Representation Learning with Theoretical Guarantees

Learning transferable data representations from abundant unlabeled data remains a central challenge in machine learning. Although numerous self-supervised learning methods have been proposed to address this challenge, a significant class of these approaches aligns the covariance or correlation matrix with the identity matrix. Despite impressive performance across various downstream tasks, these methods often suffer from biased sample risk, leading to substantial optimization shifts in mini-batch settings and complicating theoretical analysis. In this paper, we introduce a novel \underline{\bf Adv}ersarial \underline{\bf S}elf-\underline{\bf S}upervised Representation \underline{\bf L}earning (Adv-SSL) for unbiased transfer learning with no additional cost compared to its biased counterparts. Our approach not only outperforms the existing methods across multiple benchmark datasets but is also supported by comprehensive end-to-end theoretical guarantees. Our analysis reveals that the minimax optimization in Adv-SSL encourages representations to form well-separated clusters in the embedding space, provided there is sufficient upstream unlabeled data. As a result, our method achieves strong classification performance even with limited downstream labels, shedding new light on few-shot learning.

stat.ML↗

arXiv2025

Deep conditional distribution learning via conditional Föllmer flow

We introduce an ordinary differential equation (ODE) based deep generative method for learning conditional distributions, named Conditional Föllmer Flow. Starting from a standard Gaussian distribution, the proposed flow could approximate the target conditional distribution very well when the time is close to 1. For effective implementation, we discretize the flow with Euler's method where we estimate the velocity field nonparametrically using a deep neural network. Furthermore, we also establish the convergence result for the Wasserstein-2 distance between the distribution of the learned samples and the target conditional distribution, providing the first comprehensive end-to-end error analysis for conditional distribution learning via ODE flow. Our numerical experiments showcase its effectiveness across a range of scenarios, from standard nonparametric conditional density estimation problems to more intricate challenges involving image data, illustrating its superiority over various existing conditional density estimation methods.

stat.ML↗

arXiv2025

Online Quantum State Tomography via Stochastic Gradient Descent

We initiate the study of online quantum state tomography (QST), where the matrix representation of an unknown quantum state is reconstructed by sequentially performing a batch of measurements and updating the state estimate using only the measurement statistics from the current round. Motivated by recent advances in non-convex optimization algorithms for solving low-rank QST, we propose non-convex mini-batch stochastic gradient descent (SGD) algorithms to tackle online QST, which leverage the low-rank structure of the unknown quantum state and are well-suited for practical applications. Our main technical contribution is a rigorous convergence analysis of these algorithms. With proper initialization, we demonstrate that the SGD algorithms for online low-rank QST achieve linear convergence both in expectation and with high probability. Our algorithms achieve nearly optimal sample complexity while remaining highly memory-efficient. In particular, their time complexities are better than the state-of-the-art non-convex QST algorithms, in terms of the rank and the logarithm of the dimension of the unknown quantum state.

quant-ph↗

arXiv2025

CRPE: Expanding The Reasoning Capability of Large Language Model for Code Generation

We introduce CRPE (Code Reasoning Process Enhancer), an innovative three-stage framework for data synthesis and model training that advances the development of sophisticated code reasoning capabilities in large language models (LLMs). Building upon existing system-1 models, CRPE addresses the fundamental challenge of enhancing LLMs' analytical and logical processing in code generation tasks. Our framework presents a methodologically rigorous yet implementable approach to cultivating advanced code reasoning abilities in language models. Through the implementation of CRPE, we successfully develop an enhanced COT-Coder that demonstrates marked improvements in code generation tasks. Evaluation results on LiveCodeBench (20240701-20240901) demonstrate that our COT-Coder-7B-StepDPO, derived from Qwen2.5-Coder-7B-Base, with a pass@1 accuracy of 21.88, exceeds all models with similar or even larger sizes. Furthermore, our COT-Coder-32B-StepDPO, based on Qwen2.5-Coder-32B-Base, exhibits superior performance with a pass@1 accuracy of 35.08, outperforming GPT4O on the benchmark. Overall, CRPE represents a comprehensive, open-source method that encompasses the complete pipeline from instruction data acquisition through expert code reasoning data synthesis, culminating in an autonomous reasoning enhancement mechanism.

cs.SE↗

arXiv2025

Semi-Supervised Deep Sobolev Regression: Estimation and Variable Selection by ReQU Neural Network

We propose SDORE, a Semi-supervised Deep Sobolev Regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep ReQU neural networks to minimize the empirical risk with gradient norm regularization, allowing the approximation of the regularization term by unlabeled data. Our study includes a thorough analysis of the convergence rates of SDORE in $L^{2}$-norm, achieving the minimax optimality. Further, we establish a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift. These theoretical findings offer valuable insights for selecting regularization parameters and determining the size of the neural network, while showcasing the provable advantage of leveraging unlabeled data in semi-supervised learning. To the best of our knowledge, SDORE is the first provable neural network-based approach that simultaneously estimates the regression function and its gradient, with diverse applications such as nonparametric variable selection. The effectiveness of SDORE is validated through an extensive range of numerical simulations.

stat.ML↗