arXiv subjects

Lican Kang

Publications and source records attributed to Lican Kang.

At least 19 recordsLinked to original sources

arXiv2026

Deep Bootstrap

In this work, we propose a novel deep bootstrap framework for nonparametric regression based on conditional diffusion models. Specifically, we construct a conditional diffusion model to learn the distribution of the response variable given the covariates. This model is then used to generate bootstrap samples by pairing the original covariates with newly synthesized responses. We reformulate nonparametric regression as conditional sample mean estimation, which is implemented directly via the learned conditional diffusion model. Unlike traditional bootstrap methods that decouple the estimation of the conditional distribution, sampling, and nonparametric regression, our approach integrates these components into a unified generative framework. With the expressive capacity of diffusion models, our method facilitates both efficient sampling from high-dimensional or multimodal distributions and accurate nonparametric estimation. We establish rigorous theoretical guarantees for the proposed method. In particular, we derive optimal end-to-end convergence rates in the Wasserstein distance between the learned and target conditional distributions. Building on this foundation, we further establish the convergence guarantees of the resulting bootstrap procedure. Numerical studies demonstrate the effectiveness and scalability of our approach for complex regression tasks.

stat.ML↗

arXiv2026

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

Diffusion policies represent multimodal action distributions, but a $Q$-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 action-value function, which defines an exponentially tilted target policy. A Doob $h$-transform expresses this update as a correction to the reference diffusion drift. The target drift is a weighted conditional mean of Brownian-bridge labels. We fit it by weighted regression using endpoints from the current policy and pointwise action-value evaluations. 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 errors from critic estimation, weighted neural drift regression, and diffusion discretization through a reverse-KL actor bound. The average-performance bound depends explicitly on the comparison coefficients over the policy iterations. Synthetic experiments assess approximation to prescribed advantage-tilted targets and sensitivity to the diffusion-step count. Across six offline-to-online continuous-control tasks, the $Q$-guided bridge-regression implementation achieves higher final-checkpoint returns than its BC initialization after $500K$ online interactions.

stat.ML↗

arXiv2026

Neural Actor--Critic Learning with Minimax Critics and Reflected Langevin Sampling

Actor--critic methods rely on critic gradients to improve continuous-action policies, yet finite-sample guarantees must control both value and gradient errors and their propagation through the actor. We develop a neural actor-critic procedure that combines a regularized minimax Bellman estimator over sigmoid networks with a finite reflected Langevin sampler driven by the fitted critic gradient. Under uniform Bellman regularity and covered reset sampling, a positive high-order action-derivative penalty yields value and action-gradient risk bounds with classical nonparametric minimax exponents up to logarithmic factors. We propagate estimation, mixing, discretization, temperature, and policy-iteration errors to obtain return guarantees, with refinements for bounded multimodal potentials and strongly concave regimes. Empirical studies across controlled simulations, logged continuous-control data, and a simulated decision task show favorable performance relative to the considered baselines.

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. We establish the smoothness of $Q^*$ through the Bellman equation using the the Hölder assumptions on the conditional reward and transition densities. With suitable sample sizes, we obtain the oracle value-stage rate $\widetilde{\mathcal O}\bigl(n^{-\frac{2α}{d_x+d_a+2α}}\bigr)$ for the excess Bellman residual risk. We then translate this risk bound into an expected $L^2$ estimation error of order $\widetilde{\mathcal O}\bigl(n^{-\fracα{d_x+d_a+2α}}\bigr)$ under a concentrability condition. For the total offline budget $\widetilde n=n+n_r+n_x$, we obtain a value estimation rate of $\widetilde{\mathcal O}\bigl(\widetilde n^{-\fracα{2d_x+d_a+2α}}\bigr)$ and establish an information-theoretic lower bound of order $\widetilde n^{-\fracα{d_x+d_a+2α}}$. Importantly, our theoretical analysis does not rely on Bellman completeness assumptions commonly used in deep RL theory.

stat.ML↗

arXiv2026

Latent-Space Importance Weighting under Covariate Shift

In this work, we propose a transfer learning algorithm for covariate shift, integrating sample reweighting with representation learning. A key feature of our approach is the use of a lower-dimensional latent space for modeling, in contrast to conventional methods that operate directly in the high-dimensional ambient space. Our method estimates the density ratio of learned latent features for downstream weighted prediction. We establish explicit target-domain excess-risk rates with independent data folds for strictly sufficient learned representations with regular learned-space prediction scores and density ratios. We give dimension comparisons under stated smoothness conditions and a variance comparison for true weights and a fixed predictor. General finite-sample inequalities retain representation-quality errors. Experiments on simulated and real-world datasets demonstrate gains from combining learned representations and latent weighting, and reveal sensitivity to initialization and domain choice.

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arXiv2026

Conditional Flow Models for Offline Maximum Entropy Reinforcement Learning

Offline maximum entropy reinforcement learning seeks to estimate the optimal soft action-value function $Q^*$ from a fixed dataset without further interaction with the environment. Although $Q^*$ is characterized by the soft optimal Bellman equation, the corresponding Bellman operator cannot be evaluated directly because both the reward distribution and the transition kernel are unknown. Consequently, estimating $Q^*$ requires the joint treatment of environment modeling and value function learning. We propose a framework that addresses these tasks separately. Conditional diffusion models based on probability flow ODEs estimate the conditional reward distribution and transition kernel. The resulting distributional estimators define a data driven approximation of the soft optimal Bellman operator, whose empirical Bellman residual is minimized over a class of deep neural networks to estimate $Q^*$. We establish nonasymptotic guarantees for the proposed method. For operator learning, we derive total variation error bounds for the conditional flow estimators that account explicitly for score estimation, numerical discretization, and terminal smoothing. For soft value learning, we first use the soft Bellman equation to show that $Q^*$ inherits Hölder regularity from the conditional reward and transition densities. For any Hölder index $β$, the value learning stage achieves an excess Bellman residual risk bound of order $\widetilde{\mathcal O}\big(n^{-\frac{2β}{d+2β}}\big),$ where $d$ is the total dimension of the state and action spaces. In contrast to many existing theoretical analyses of deep reinforcement learning, our results do not require Bellman completeness. Numerical experiments support the theoretical findings and demonstrate the effectiveness of the proposed method.

stat.ML↗

arXiv2026

Annealed Lévy Diffusion via Gaussian Surrogate Reverse Dynamics

Gaussian diffusion models offer a practical framework for generative modeling. Denoising Lévy Probabilistic Models (DLPM) introduce heavy-tailed perturbations with a fixed tail index throughout denoising. We introduce Annealed Lévy Diffusion (ALD). ALD varies the stability index $α(t)$ to combine heavy-tailed exploration with Gaussian refinement. Our marginal-first formulation represents stable noise as a scalar Gaussian scale mixture. This formulation enables standard Gaussian noise-prediction training with one additional scale input. We study two sampling processes. ALD-renoise estimates the clean sample and injects fresh marginal noise. ALD-posterior preserves dependence between successive states through an accumulated conditional variance and a Gaussian bridge update. Both samplers reuse the same trained predictor with different scalar inputs. We evaluate ALD on four 2D benchmarks and an imbalanced nine-mode mixture. We also examine training efficiency and the effects of prediction perturbations on clipped Holtsmark data. Additional experiments evaluate MNIST and CelebA64 generation and CIFAR-10LT augmentation. ALD preserves local geometry on the evaluated light-tailed and manifold-structured targets.

cs.LG↗

arXiv2026

Joint Contrastive-Adversarial Representation Learning for Unsupervised Domain Adaptation

Effective unsupervised domain adaptation (UDA) requires a representation that captures transferable structure from both the labeled source domain and the unlabeled target domain. We introduce Joint Contrastive-Adversarial Representation Learning (JCAR), a two-stage framework that learns a common representation from the union of unlabeled source and target samples by combining contrastive learning with adversarial domain alignment, and then freezes the representation to fit a source-supervised predictor. We establish a non-asymptotic excess-risk bound for this procedure that couples representation and domain-discriminator estimation while separating downstream prediction, with explicit neural-network approximation errors. Under the stated assumptions and common Hölder smoothness $β\geq1$, the bound identifies a sufficiently large unlabeled-sample regime in which the downstream rate is $\widetilde{\mathcal{O}}(m^{-β/(2β+d)})$, where $m$ counts labeled source samples and logarithmic factors are suppressed. The labeled-sample exponent is governed by the latent dimension $d$ rather than the ambient dimension $p$, which enters the pretraining cost. Experiments on Office-31, Office-Home, and DomainNet demonstrate clear transfer gains across diverse domain shifts, with particularly strong performance when representations are learned from scratch.

stat.ML↗

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

Conditional Diffusion for Nonparametric Instrumental Variable Quantile Regression

This work proposes deep nonparametric Instrumental variable quantile regression (IVQR), a two-stage estimator that combines conditional diffusion modeling with a kernel-smoothed conditional moment formulation. In the first stage, we estimate the joint conditional distribution of the outcome and endogenous covariates given the instrument using a variance-preserving conditional diffusion model. In the second stage, we approximate the conditional moment operator through Monte Carlo sampling and a kernel-smoothed surrogate for the indicator function, and then estimate the structural quantile function by empirical risk minimization over deep neural networks. We establish an excess-risk bound for the proposed estimator and derive end-to-end total variation guarantees for the conditional diffusion model under unbounded support, explicitly accounting for score estimation, early stopping, and discretization errors. Our theory is developed under a polynomial-tail envelope on the data distribution and degenerates continuously to the exponential setting: as the tail index grows, the obtained excess-risk rate converges to the minimax-optimal rate of nonparametric regression, thus our heavy-tailed theory covers the classical light-tailed nonparametric guarantees as a limiting case. Simulation studies and a real-data application demonstrate that the proposed method outperforms existing nonparametric IVQR approaches, with gains that become increasingly pronounced as the dimensionality of the covariates and instruments increases.

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arXiv2025

Opening the black box of deep learning: Validating the statistical association between explainable artificial intelligence (XAI) and clinical domain knowledge in fundus image-based glaucoma diagnosis

While deep learning has exhibited remarkable predictive capabilities in various medical image tasks, its inherent black-box nature has hindered its widespread implementation in real-world healthcare settings. Our objective is to unveil the decision-making processes of deep learning models in the context of glaucoma classification by employing several Class Activation Map (CAM) techniques to generate model focus regions and comparing them with clinical domain knowledge of the anatomical area (optic cup, optic disk, and blood vessels). Four deep neural networks, including VGG-11, ResNet-18, DeiT-Tiny, and Swin Transformer-Tiny, were developed using binary diagnostic labels of glaucoma and five CAM methods (Grad-CAM, XGrad-CAM, Score-CAM, Eigen-CAM, and Layer-CAM) were employed to highlight the model focus area. We applied the paired-sample t-test to compare the percentage of anatomies in the model focus area to the proportion of anatomies in the entire image. After that, Pearson's and Spearman's correlation tests were implemented to examine the relationship between model predictive ability and the percentage of anatomical structures in the model focus area. On five public glaucoma datasets, all deep learning models consistently displayed statistically significantly higher percentages of anatomical structures in the focus area than the proportions of anatomical structures in the entire image. Also, we validated the positive relationship between the percentage of anatomical structures in the focus area and model predictive performance. Our study provides evidence of the convergence of decision logic between deep neural networks and human clinicians through rigorous statistical tests. We anticipate that it can help alleviate clinicians' concerns regarding the trustworthiness of deep learning in healthcare. For reproducibility, the code and dataset have been released at GitHub.

cs.CV↗

arXiv2024

Latent Schr{ö}dinger Bridge Diffusion Model for Generative Learning

This paper aims to conduct a comprehensive theoretical analysis of current diffusion models. We introduce a novel generative learning methodology utilizing the Schr{ö}dinger bridge diffusion model in latent space as the framework for theoretical exploration in this domain. Our approach commences with the pre-training of an encoder-decoder architecture using data originating from a distribution that may diverge from the target distribution, thus facilitating the accommodation of a large sample size through the utilization of pre-existing large-scale models. Subsequently, we develop a diffusion model within the latent space utilizing the Schr{ö}dinger bridge framework. Our theoretical analysis encompasses the establishment of end-to-end error analysis for learning distributions via the latent Schr{ö}dinger bridge diffusion model. Specifically, we control the second-order Wasserstein distance between the generated distribution and the target distribution. Furthermore, our obtained convergence rates effectively mitigate the curse of dimensionality, offering robust theoretical support for prevailing diffusion models.

stat.ML↗

arXiv2024

Model Free Prediction with Uncertainty Assessment

Deep nonparametric regression, characterized by the utilization of deep neural networks to learn target functions, has emerged as a focus of research attention in recent years. Despite considerable progress in understanding convergence rates, the absence of asymptotic properties hinders rigorous statistical inference. To address this gap, we propose a novel framework that transforms the deep estimation paradigm into a platform conducive to conditional mean estimation, leveraging the conditional diffusion model. Theoretically, we develop an end-to-end convergence rate for the conditional diffusion model and establish the asymptotic normality of the generated samples. Consequently, we are equipped to construct confidence regions, facilitating robust statistical inference. Furthermore, through numerical experiments, we empirically validate the efficacy of our proposed methodology.

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arXiv2023

Lipschitz Transport Maps via the Follmer Flow

Inspired by the construction of the F{ö}llmer process, we construct a unit-time flow on the Euclidean space, termed the F{ö}llmer flow, whose flow map at time 1 pushes forward a standard Gaussian measure onto a general target measure. We study the well-posedness of the F{ö}llmer flow and establish the Lipschitz property of the flow map at time 1. We apply the Lipschitz mapping to several rich classes of probability measures on deriving dimension-free functional inequalities and concentration inequalities for the empirical measure.

math.PR↗

arXiv2023

On Newton Screening

Screening and working set techniques are important approaches to reducing the size of an optimization problem. They have been widely used in accelerating first-order methods for solving large-scale sparse learning problems. In this paper, we develop a new screening method called Newton screening (NS) which is a generalized Newton method with a built-in screening mechanism. We derive an equivalent KKT system for the Lasso and utilize a generalized Newton method to solve the KKT equations. Based on this KKT system, a built-in working set with a relatively small size is first determined using the sum of primal and dual variables generated from the previous iteration, then the primal variable is updated by solving a least-squares problem on the working set and the dual variable updated based on a closed-form expression. Moreover, we consider a sequential version of Newton screening (SNS) with a warm-start strategy. We show that NS possesses an optimal convergence property in the sense that it achieves one-step local convergence. Under certain regularity conditions on the feature matrix, we show that SNS hits a solution with the same signs as the underlying true target and achieves a sharp estimation error bound with high probability. Simulation studies and real data analysis support our theoretical results and demonstrate that SNS is faster and more accurate than several state-of-the-art methods in our comparative studies.

stat.ML↗

arXiv2023

An empirical study of the effect of background data size on the stability of SHapley Additive exPlanations (SHAP) for deep learning models

Nowadays, the interpretation of why a machine learning (ML) model makes certain inferences is as crucial as the accuracy of such inferences. Some ML models like the decision tree possess inherent interpretability that can be directly comprehended by humans. Others like artificial neural networks (ANN), however, rely on external methods to uncover the deduction mechanism. SHapley Additive exPlanations (SHAP) is one of such external methods, which requires a background dataset when interpreting ANNs. Generally, a background dataset consists of instances randomly sampled from the training dataset. However, the sampling size and its effect on SHAP remain to be unexplored. In our empirical study on the MIMIC-III dataset, we show that the two core explanations - SHAP values and variable rankings fluctuate when using different background datasets acquired from random sampling, indicating that users cannot unquestioningly trust the one-shot interpretation from SHAP. Luckily, such fluctuation decreases with the increase of the background dataset size. Also, we notice an U-shape in the stability assessment of SHAP variable rankings, demonstrating that SHAP is more reliable in ranking the most and least important variables compared to moderately important ones. Overall, our results suggest that users should take into account how background data affects SHAP results, with improved SHAP stability as the background sample size increases.

cs.LG↗

arXiv2022

Global Optimization via Schr{ö}dinger-F{ö}llmer Diffusion

We study the problem of finding global minimizers of $V(x):\mathbb{R}^d\rightarrow\mathbb{R}$ approximately via sampling from a probability distribution $μ_σ$ with density $p_σ(x)=\dfrac{\exp(-V(x)/σ)}{\int_{\mathbb R^d} \exp(-V(y)/σ) dy }$ with respect to the Lebesgue measure for $σ\in (0,1]$ small enough. We analyze a sampler based on the Euler-Maruyama discretization of the Schr{ö}dinger-F{ö}llmer diffusion processes with stochastic approximation under appropriate assumptions on the step size $s$ and the potential $V$. We prove that the output of the proposed sampler is an approximate global minimizer of $V(x)$ with high probability at cost of sampling $\mathcal{O}(d^{3})$ standard normal random variables. Numerical studies illustrate the effectiveness of the proposed method and its superiority to the Langevin method.

math.OC↗

arXiv2021

A Data-Driven Line Search Rule for Support Recovery in High-dimensional Data Analysis

In this work, we consider the algorithm to the (nonlinear) regression problems with $\ell_0$ penalty. The existing algorithms for $\ell_0$ based optimization problem are often carried out with a fixed step size, and the selection of an appropriate step size depends on the restricted strong convexity and smoothness for the loss function, hence it is difficult to compute in practical calculation. In sprite of the ideas of support detection and root finding \cite{HJK2020}, we proposes a novel and efficient data-driven line search rule to adaptively determine the appropriate step size. We prove the $\ell_2$ error bound to the proposed algorithm without much restrictions for the cost functional. A large number of numerical comparisons with state-of-the-art algorithms in linear and logistic regression problems show the stability, effectiveness and superiority of the proposed algorithms.

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