arXiv subjects

Zhihua Zhang

Publications and source records attributed to Zhihua Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2026

Decoupled Functional Central Limit Theorems for Two-Time-Scale Stochastic Approximation

In two-time-scale stochastic approximation (SA), two iterates are updated at different rates, governed by distinct step sizes, with each update influencing the other. Previous studies have demonstrated that the convergence rates of the error terms for these updates depend solely on their respective step sizes, a property known as decoupled convergence. However, a functional version of this decoupled convergence has not been explored. Our work fills this gap by establishing decoupled functional central limit theorems for two-time-scale SA, offering a more precise characterization of its asymptotic behavior. Our results show that, on each time scale, the limiting dynamics has the same form as in standard SA, and the coupling between the two iterates enters the limit only through the associated coefficients. To achieve these results, we leverage the martingale problem approach and establish tightness as a crucial intermediate step. Furthermore, to address the interdependence between different time scales, we introduce an innovative auxiliary sequence to eliminate the primary influence of the fast-time-scale update on the slow-time-scale update.

math.PR↗

arXiv2026

Layerwise Terminal Discrepancy in Chen's Reverse-Heat Coupling on the Boolean Cube

Recently, Chen \cite{Chen2026} proved that Talagrand's Boolean convolution conjecture holds up to the dimension-free factor $(\log\logη)^{3/2}$, namely for every fixed $τ>0$, \[ μ\{P_τf>η\|f\|_1\} \le C_τ \frac{(\log\logη)^{3/2}}{η\sqrt{\logη}}, \qquad η>e^3. \] We revisit the terminal testing-discrepancy step in Chen's perturbed reverse-heat coupling. Chen estimates this discrepancy globally in terms of the remaining gap to the terminal level. We keep the same coupling and the same reverse-heat formulations, but localize the terminal discrepancy on each remaining-gap layer before summing the layers. This changes the fixed-time anti-concentration cost from order $(\log L)^{3/2}/\sqrt L$ to order $(\log L)/\sqrt L$, where $L=\logη$. Consequently, we obtain a $(\log\logη)^{1/2}$ improvement as \[ μ\{P_τf>η\|f\|_1\} \le C_τ \frac{\log\logη}{η\sqrt{\logη}}, \qquad η>e^3. \]

math.FA↗

arXiv2026

Universality for Products of Random Matrices with i.i.d. Entries and the Fuss--Catalan Number

Let $(w_{ij})_{i,j\ge1}$ be a single infinite array of independent identically distributed real- or complex-valued entries of mean zero, variance $σ^2$, and finite fourth moment. Set $W_n=(w_{ij})_{1\le i,j\le n}$ and $X_n=n^{-1/2}W_n$. For every fixed $k\ge1$, we identify the almost sure limiting operator norm of several fixed products built from this family. Define the $k$-th freeness coefficient by \[ γ_k:=\sqrt{\frac{(k+1)^{k+1}}{k^k}}. \] Then we prove \[ \|X_n^k\|\toσ^kγ_k \qquad \text{almost surely}. \] The same limit holds for products sampled with replacement from any fixed finite pool of independent copies of $X_n$; in particular, it holds for the product of $k$ independent copies. Thus, the freeness coefficient captures the non-commuting characteristic between large random matrices %powers and independent or fixed-pool sampled products under the finite fourth moment assumption. The improvement of the classical Bai--Yin-type power estimate from the scale $σ^k(k{+}1)$ to $σ^k \sqrt{k{+}1}$ is a direct corollary of our result. The main technical challenge is to prove the upper bound using a high-moment expansion of %the upper bound is proved by a high-moment expansion of $\E\Tr((X_n^kX_n^{*k})^m)$. The leading zero-defect trace words are tree-like and are counted by the Fuss--Catalan number \[ F_{k,m}= \frac1{km+1}\binom{(k+1)m}{m}. \] The combinatorial tool helps to devise a defect-sensitive global enumeration: if $L=km$ and \[ r=(L+1-v)+(L-q), \] then the number of admissible word classes with defect $r$ is at most $F_{k,m}(Cm)^{Dr}$. This polynomial-in-$m$ loss, with degree proportional to the defect, is summable in the logarithmic moment range.

math.ST↗

arXiv2026

Finite-Iteration Local Dynamics and Warm Starts for Alternating Power Iteration in Spiked Tensor PCA

We study simultaneous alternating power iteration for fixed-order asymmetric rank-one spiked tensor models. Our main contribution is a finite-iteration local theory that is independent of any particular initialization. Once the iterates enter a sufficiently small neighborhood of the planted rank-one direction, their error decomposes into a geometrically decaying transient and an intrinsic noise floor caused by fixed orthogonal noise contractions at the planted point. The deterministic finite-sample conditions are stated explicitly, but under a coarse fixed-order multilinear noise event they reduce to a conservative high-signal regime for fixed or slowly expanding local radii. We then separate the warm-start mechanism from any specific spectral construction. A generic one-sweep principle shows that, if a sign-compatible initializer has correlation $γ_N$, first-sweep noise level $a_N$, and $a_N/(γ_N^{d-1}ω_{N,d})\to0$, then one can choose an expanding radius $r_N=o(ω_{N,d})$ for which the first sweep enters the local basin. After entry, the local affine contraction yields convergence to the unique informative local fixed point in that basin. For centered-Gram initialization, we verify the required correlation and same-sample first-sweep noise bound under i.i.d. finite-fourth-moment noise by a signal-preserving noise-only leave-one comparison and an averaged leave-one slice-contraction estimate, which we call a pressed-back estimate. The leave-one comparison keeps the spike fixed and averages over the deleted coordinate, so planted coordinates enter through $\ell_2$-weighted sums rather than worst-case incoherence bounds.

stat.ML↗

arXiv2026

Finite-Time Decoupled Convergence in Nonlinear Two-Time-Scale Stochastic Approximation

In two-time-scale stochastic approximation (SA), two iterates are updated at varying speeds using different step sizes, with each update influencing the other. Previous studies on linear two-time-scale SA have shown that the convergence rates of the mean-square errors for these updates depend solely on their respective step sizes, a phenomenon termed decoupled convergence. However, achieving decoupled convergence in nonlinear SA remains less understood. Our research investigates the potential for finite-time decoupled convergence in nonlinear two-time-scale SA. We demonstrate that, under a nested local linearity assumption, finite-time decoupled convergence rates can be achieved with suitable step size selection. To derive this result, we conduct a convergence analysis of the matrix cross term between the iterates and leverage fourth-order moment convergence rates to control the higher-order error terms induced by local linearity. To further investigate the necessity of local linearity for decoupled convergence, we also construct an example showing that, even when the fast-time-scale update is linear, the nonlinearity of the slow-time-scale update alone can destroy decoupled convergence.

math.OC↗

arXiv2026

Creative Convergence or Imitation? Genre-Specific Homogeneity in LLM-Generated Chinese Literature

Large Language Models (LLMs) have demonstrated remarkable capabilities in narrative generation. However, they often produce structurally homogenized stories, frequently following repetitive arrangements and combinations of plot events along with stereotypical resolutions. In this paper, we propose a novel theoretical framework for analysis by incorporating Proppian narratology and narrative functions. This framework is used to analyze the composition of narrative texts generated by LLMs to uncover their underlying narrative logic. Taking Chinese web literature as our research focus, we extend Propp's narrative theory, defining 34 narrative functions suited to modern web narrative structures. We further construct a human-annotated corpus to support the analysis of narrative structures within LLM-generated text. Experiments reveal that the primary reasons for the singular narrative logic and severe homogenization in generated texts are that current LLMs are unable to correctly comprehend the meanings of narrative functions and instead adhere to rigid narrative generation paradigms.

cs.CL↗

arXiv2026

Universality of General Spiked Tensor Models

We study asymmetric rank-one spiked tensor models in the high-dimensional regime, where the noise entries are independent and identically distributed with zero mean, unit variance, and finite fourth moment. This extends the classical Gaussian framework to a substantially broader class of noise distributions. We analyze the maximum-likelihood estimator associated with the best rank-one approximation of an order-$d$ tensor, for $d\ge 3$. Our approach is formulated along an informative, spectrally separated branch of stationary points of the non-convex maximum-likelihood landscape. In the core order-three asymmetric model, we verify locally in the high-signal regime that such an informative branch exists and remains separated from the bulk. Under this branch-selection framework, we show that the empirical spectral distribution of a suitable block-wise tensor contraction converges almost surely to the same deterministic limit as in the Gaussian case. As a consequence, the asymptotic singular value and the mode-wise alignments between the estimated and planted spike directions admit the same explicit characterizations as under Gaussian noise. These results establish a universality principle for asymmetric spiked tensor models: the high-dimensional spectral behavior and statistical limits of the selected maximum-likelihood stationary point are robust beyond the Gaussian setting. Our proof combines resolvent methods from random matrix theory, cumulant expansions under finite fourth-moment assumptions, and Efron--Stein-type variance bounds. A main technical difficulty is to control the statistical dependence between the estimator and the noise, including the associated cross terms in the non-Gaussian setting.

math.ST↗

arXiv2026

CAST-LUT: Tokenizer-Guided HSV Look-Up Tables for Purple Flare Removal

Purple flare, a diffuse chromatic aberration artifact commonly found around highlight areas, severely degrades the tone transition and color of the image. Existing traditional methods are based on hand-crafted features, which lack flexibility and rely entirely on fixed priors, while the scarcity of paired training data critically hampers deep learning. To address this issue, we propose a novel network built upon decoupled HSV Look-Up Tables (LUTs). The method aims to simplify color correction by adjusting the Hue (H), Saturation (S), and Value (V) components independently. This approach resolves the inherent color coupling problems in traditional methods. Our model adopts a two-stage architecture: First, a Chroma-Aware Spectral Tokenizer (CAST) converts the input image from RGB space to HSV space and independently encodes the Hue (H) and Value (V) channels into a set of semantic tokens describing the Purple flare status; second, the HSV-LUT module takes these tokens as input and dynamically generates independent correction curves (1D-LUTs) for the three channels H, S, and V. To effectively train and validate our model, we built the first large-scale purple flare dataset with diverse scenes. We also proposed new metrics and a loss function specifically designed for this task. Extensive experiments demonstrate that our model not only significantly outperforms existing methods in visual effects but also achieves state-of-the-art performance on all quantitative metrics.

cs.CV↗

arXiv2025

Accelerated Distributional Temporal Difference Learning with Linear Function Approximation

In this paper, we study the finite-sample statistical rates of distributional temporal difference (TD) learning with linear function approximation. The purpose of distributional TD learning is to estimate the return distribution of a discounted Markov decision process for a given policy. Previous works on statistical analysis of distributional TD learning focus mainly on the tabular case. We first consider the linear function approximation setting and conduct a fine-grained analysis of the linear-categorical Bellman equation. Building on this analysis, we further incorporate variance reduction techniques in our new algorithms to establish tight sample complexity bounds independent of the support size $K$ when $K$ is large. Our theoretical results imply that, when employing distributional TD learning with linear function approximation, learning the full distribution of the return function from streaming data is no more difficult than learning its expectation. This work provide new insights into the statistical efficiency of distributional reinforcement learning algorithms.

stat.ML↗

arXiv2025

Convergence and Inference of Stream SGD, with Applications to Queueing Systems and Inventory Control

Stream stochastic gradient descent (SGD) is a simple and efficient method for solving online optimization problems in operations research (OR), where data is generated by parameter-dependent Markov chains. Unlike traditional approaches which require increasing batch sizes during iterations, stream SGD uses a single sample per iteration, significantly improving sample efficiency. This paper establishes a systematic framework for analyzing stream SGD, leveraging the Poisson equation solution to address gradient bias and statistical dependence. We prove optimal O(1/T) convergence rates and the state-of-the-art O(log T) regret, while also introducing an online inference method for uncertainty quantification and supporting it by a novel functional central limit theorem. We propose a novel Wasserstein-type divergence to describe the framework's conditions, which makes the assumptions in question directly verified via coupling techniques tailored to underlying OR models. We consider applications in queueing systems and inventory management, demonstrating the practicality and broad relevance, as well as providing new insights into the effectiveness of stream SGD in OR fields.

math.OC↗

arXiv2025

Follow-the-Perturbed-Leader Approaches Best-of-Both-Worlds for the m-Set Semi-Bandit Problems

We consider a common case of the combinatorial semi-bandit problem, the $m$-set semi-bandit, where the learner exactly selects $m$ arms from the total $d$ arms. In the adversarial setting, the best regret bound, known to be $\mathcal{O}(\sqrt{nmd})$ for time horizon $n$, is achieved by the well-known Follow-the-Regularized-Leader (FTRL) policy. However, this requires to explicitly compute the arm-selection probabilities via optimizing problems at each time step and sample according to them. This problem can be avoided by the Follow-the-Perturbed-Leader (FTPL) policy, which simply pulls the $m$ arms that rank among the $m$ smallest (estimated) loss with random perturbation. In this paper, we show that FTPL with a Fréchet perturbation also enjoys the near optimal regret bound $\mathcal{O}(\sqrt{nm}(\sqrt{d\log(d)}+m^{5/6}))$ in the adversarial setting and approaches best-of-both-world regret bounds, i.e., achieves a logarithmic regret for the stochastic setting. Moreover, our lower bounds show that the extra factors are unavoidable with our approach; any improvement would require a fundamentally different and more challenging method.

cs.LG↗

arXiv2025

The Empirical Spectral Distribution of i.i.d. Random Matrices with Random Perturbations

A large i.i.d. random matrix with deterministic low-rank perturbation has been extensively studied, particularly in the aspects of the ESD (Empirical Spectral Distribution) and the outliers of eigenvalues. In this work, we investigate the analogous scenario where the perturbation is random and extend the previous results from the deterministic perturbation to the random case. Specifically, we consider an i.i.d. matrix with random perturbation, $\mathbf{M}$. Our results show that: (i) the eigenvalue outliers of $\mathbf{M}$ converge to the eigenvalues of its perturbation; (ii) the ESD of $\mathbf{M}$ converges to the circular law; (iii) the eigenvector alignment holds for specific perturbations. As an application of the above random matrices, we present the first optimal query complexity lower bound for approximating the top eigenvector of asymmetric matrices. In the inverse polynomial accuracy regime, the complexity matches the upper bounds that can be obtained via the power method. As far as we know, it is the first lower bound for approximating the eigenvector of an asymmetric matrix.

cs.IT↗

arXiv2025

Entropy-based Training Methods for Scalable Neural Implicit Sampler

Efficiently sampling from un-normalized target distributions is a fundamental problem in scientific computing and machine learning. Traditional approaches such as Markov Chain Monte Carlo (MCMC) guarantee asymptotically unbiased samples from such distributions but suffer from computational inefficiency, particularly when dealing with high-dimensional targets, as they require numerous iterations to generate a batch of samples. In this paper, we introduce an efficient and scalable neural implicit sampler that overcomes these limitations. The implicit sampler can generate large batches of samples with low computational costs by leveraging a neural transformation that directly maps easily sampled latent vectors to target samples without the need for iterative procedures. To train the neural implicit samplers, we introduce two novel methods: the KL training method and the Fisher training method. The former method minimizes the Kullback-Leibler divergence, while the latter minimizes the Fisher divergence between the sampler and the target distributions. By employing the two training methods, we effectively optimize the neural implicit samplers to learn and generate from the desired target distribution. To demonstrate the effectiveness, efficiency, and scalability of our proposed samplers, we evaluate them on three sampling benchmarks with different scales.

stat.ML↗

arXiv2025

A Finite Sample Analysis of Distributional TD Learning with Linear Function Approximation

In this paper, we study the finite-sample statistical rates of distributional temporal difference (TD) learning with linear function approximation. The aim of distributional TD learning is to estimate the return distribution of a discounted Markov decision process for a given policy π. Previous works on statistical analysis of distributional TD learning mainly focus on the tabular case. In contrast, we first consider the linear function approximation setting and derive sharp finite-sample rates. Our theoretical results demonstrate that the sample complexity of linear distributional TD learning matches that of classic linear TD learning. This implies that, with linear function approximation, learning the full distribution of the return from streaming data is no more difficult than learning its expectation (value function). To derive tight sample complexity bounds, we conduct a fine-grained analysis of the linear-categorical Bellman equation and employ the exponential stability arguments for products of random matrices. Our results provide new insights into the statistical efficiency of distributional reinforcement learning algorithms.

stat.ML↗

arXiv2025

AgentPolyp: Accurate Polyp Segmentation via Image Enhancement Agent

Since human and environmental factors interfere, captured polyp images usually suffer from issues such as dim lighting, blur, and overexposure, which pose challenges for downstream polyp segmentation tasks. To address the challenges of noise-induced degradation in polyp images, we present AgentPolyp, a novel framework integrating CLIP-based semantic guidance and dynamic image enhancement with a lightweight neural network for segmentation. The agent first evaluates image quality using CLIP-driven semantic analysis (e.g., identifying ``low-contrast polyps with vascular textures") and adapts reinforcement learning strategies to dynamically apply multi-modal enhancement operations (e.g., denoising, contrast adjustment). A quality assessment feedback loop optimizes pixel-level enhancement and segmentation focus in a collaborative manner, ensuring robust preprocessing before neural network segmentation. This modular architecture supports plug-and-play extensions for various enhancement algorithms and segmentation networks, meeting deployment requirements for endoscopic devices.

eess.IV↗

arXiv2025

PolypFlow: Reinforcing Polyp Segmentation with Flow-Driven Dynamics

Accurate polyp segmentation remains challenging due to irregular lesion morphologies, ambiguous boundaries, and heterogeneous imaging conditions. While U-Net variants excel at local feature fusion, they often lack explicit mechanisms to model the dynamic evolution of segmentation confidence under uncertainty. Inspired by the interpretable nature of flow-based models, we present \textbf{PolypFLow}, a flow-matching enhanced architecture that injects physics-inspired optimization dynamics into segmentation refinement. Unlike conventional cascaded networks, our framework solves an ordinary differential equation (ODE) to progressively align coarse initial predictions with ground truth masks through learned velocity fields. This trajectory-based refinement offers two key advantages: 1) Interpretable Optimization: Intermediate flow steps visualize how the model corrects under-segmented regions and sharpens boundaries at each ODE-solver iteration, demystifying the ``black-box" refinement process; 2) Boundary-Aware Robustness: The flow dynamics explicitly model gradient directions along polyp edges, enhancing resilience to low-contrast regions and motion artifacts. Numerous experimental results show that PolypFLow achieves a state-of-the-art while maintaining consistent performance in different lighting scenarios.

eess.IV↗

arXiv2025

A Regularized Online Newton Method for Stochastic Convex Bandits with Linear Vanishing Noise

We study a stochastic convex bandit problem where the subgaussian noise parameter is assumed to decrease linearly as the learner selects actions closer and closer to the minimizer of the convex loss function. Accordingly, we propose a Regularized Online Newton Method (RONM) for solving the problem, based on the Online Newton Method (ONM) of arXiv:2406.06506. Our RONM reaches a polylogarithmic regret in the time horizon $n$ when the loss function grows quadratically in the constraint set, which recovers the results of arXiv:2402.12042 in linear bandits. Our analyses rely on the growth rate of the precision matrix $Σ_t^{-1}$ in ONM and we find that linear growth solves the question exactly. These analyses also help us obtain better convergence rates when the loss function grows faster. We also study and analyze two new bandit models: stochastic convex bandits with noise scaled to a subgaussian parameter function and convex bandits with stochastic multiplicative noise.

math.OC↗

arXiv2025

Statistical Efficiency of Distributional Temporal Difference Learning and Freedman's Inequality in Hilbert Spaces

Distributional reinforcement learning (DRL) has achieved empirical success in various domains. One core task in DRL is distributional policy evaluation, which involves estimating the return distribution $η^π$ for a given policy $π$. Distributional temporal difference learning has been accordingly proposed, which extends the classic temporal difference learning (TD) in RL. In this paper, we focus on the non-asymptotic statistical rates of distributional TD. To facilitate theoretical analysis, we propose non-parametric distributional TD (NTD). For a $γ$-discounted infinite-horizon tabular Markov decision process, we show that for NTD with a generative model, we need $\tilde{O}(\varepsilon^{-2}μ_{\min}^{-1}(1-γ)^{-3})$ interactions with the environment to achieve an $\varepsilon$-optimal estimator with high probability, when the estimation error is measured by the $1$-Wasserstein. This sample complexity bound is minimax optimal up to logarithmic factors. In addition, we revisit categorical distributional TD (CTD), showing that the same non-asymptotic convergence bounds hold for CTD in the case of the $1$-Wasserstein distance. We also extend our analysis to the more general setting where the data generating process is Markovian. In the Markovian setting, we propose variance-reduced variants of NTD and CTD, and show that both can achieve a $\tilde{O}(\varepsilon^{-2} μ_{π,\min}^{-1}(1-γ)^{-3}+t_{mix}μ_{π,\min}^{-1}(1-γ)^{-1})$ sample complexity bounds in the case of the $1$-Wasserstein distance, which matches the state-of-the-art statistical results for classic policy evaluation. To achieve the sharp statistical rates, we establish a novel Freedman's inequality in Hilbert spaces. This new Freedman's inequality would be of independent interest for statistical analysis of various infinite-dimensional online learning problems.

stat.ML↗