arXiv subjects

Zhihua Zhang

Publications and source records attributed to Zhihua Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Mean Velocity Matching: Rethinking Generative Dynamics in Diffusion Models

This work studies prediction parameterization for stochastic generative dynamics in diffusion models. Existing velocity-based generative models provide the simplicity of learning a single transport field, but their standard formulation is deterministic, whereas stochastic extensions generally require additional score information or an intermediate velocity-to-score reconstruction. To retain single-field prediction while directly supporting stochastic reverse dynamics, this paper introduces Mean Velocity Matching (MVM). MVM constructs a Gaussian perturbation process for which the conditional expectation of a restoration-oriented velocity, $(x_0-x_t)/t$, directly forms the reverse-SDE drift. Consequently, a single learned field is sufficient to parameterize the stochastic reverse process without separately estimating or reconstructing the score. Because direct regression of this velocity becomes unbounded near $t=0$, MVM further introduces a $\sqrt{t}$-scaled parameterization that preserves the reverse dynamics while yielding a bounded training target. The same learned field also induces a deterministic probability-flow ODE, enabling stochastic and deterministic sampling to be studied within a unified formulation. Experiments with Transformer-based generative models achieve an FID of $\MVMImageNetThirtyTwoFID$ at \MVMImageNetThirtyTwoNFE\ NFE on ImageNet $32\times32$ and $\MVMImageNetTwoFiftySixFID$ at \MVMImageNetTwoFiftySixNFE\ NFE on ImageNet $256\times256$. Controlled SDE--ODE comparisons further show that the ODE performs better under very low NFE, whereas the stochastic reverse process achieves lower FID when sufficient function evaluations are available. These results demonstrate that MVM provides a direct single-field parameterization of stochastic reverse dynamics while maintaining competitive generation quality.

cs.LG↗

arXiv2026

A Finite-Sample Analysis of Quantile Temporal-Difference Learning

Quantile temporal-difference learning (QTD) is an effective method for learning return distributions through quantile approximation, yet its finite-time behavior remains poorly understood. Its update is nonlinear and nonsmooth, and the stability needed for a sharp convergence rate holds only near the target. We establish a global high-probability last-iterate guarantee for synchronous tabular QTD under general positive, nonincreasing step-size sequences and arbitrary initialization in the natural parameter range. For polynomially decaying step sizes with exponent $a\in(0,1)$, the last iterate converges to the target at rate $T^{-a/2}$ in the infinity norm, up to logarithmic and lower-order terms. A suitably tuned harmonic schedule recovers the $T^{-1/2}$ statistical rate up to logarithmic factors. For the $m$-quantile representation, its $\infty$-Wasserstein error scales as $\sqrt{m/T}$ up to logarithmic factors, matching the leading polynomial dependence on the quantile resolution and sample size of the corresponding model-based estimator. The proof uses a two-stage global-to-local argument. From arbitrary initialization, Bellman contraction and CDF monotonicity first bring the iterate close to the target, after which, a novel variance--drift matching argument sharpens the control of accumulated noise and local contraction reduces the remaining errors, yielding the sharp rate. Simulations verify the predicted polynomial decay and assess the finite-time entrance bound.

stat.ML↗

arXiv2026

Poisson-Corrector Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling

We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $π(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:\mathbb{R}^d\to\mathbb{R}$ is convex and globally $G$-Lipschitz. For the Moreau-smoothed target $π_λ$ and the MYULA invariant law $\widehatπ_{λ,h}$, we prove \[ \sqrt m\,W_2(π_λ,\widehatπ_{λ,h}) =O(h)+\widetilde O(h^{3/4}) \] under $0<h(L_f+λ^{-1})\le c$, with only logarithmic dependence on $λ^{-1}$ in the error coefficients. Combining this estimate with the Moreau approximation bias yields $\widetilde O(\varepsilon^{-4/3})$ iterations to achieve $\sqrt m\,W_2(μ_N,π)\le\varepsilon$, for fixed model parameters and initialization. The proof combines a discrete Poisson corrector with active-trace estimates and a shared-noise bound for the exact--Euler two-point curvature.

cs.LG↗

arXiv2026

Rényi Tracking Bounds for Langevin Dynamics with Moving Targets

We study Langevin diffusion and Langevin Monte Carlo (LMC) when the target distribution changes over time. Under a log-Sobolev inequality (LSI), we derive non-asymptotic Rényi-divergence guarantees for tracking the current target. The framework covers continuous-time Langevin diffusion and its discretizations. We then apply the results to nonsmooth sampling based on successive Moreau envelopes. For this scheme, we give explicit choices of the smoothing parameters and step sizes, together with corresponding complexity bounds. To our knowledge, these are the first non-asymptotic Rényi-divergence tracking bounds for Langevin dynamics with discrete target updates.

stat.ML↗

arXiv2026

Matrix Moment and Concentration Inequalities for Martingales and Ergodic Markov Chains with Applications in Statistical Learning

In this paper, we study moment and concentration inequalities for the spectral norm of sums of dependent random matrices. We establish novel Rosenthal-Burkholder inequalities for the spectral norm of discrete-time matrix local martingales, Burkholder-Davis-Gundy inequality for the spectral norm of continuous matrix local martingales, as well as matrix Rosenthal, Hoeffding, and Bernstein inequalities under the spectral norm for ergodic Markov chains. Compared with previous work on matrix concentration inequalities for Markov chains, which assume a non-zero absolute $L^2$-spectral gap or the stronger $ψ$-mixing condition, our results assume geometric ergodicity, a condition commonly used in statistical applications. Furthermore, our results have leading terms that match the Markov chain central limit theorem, rather than relying on suboptimal variance proxies. We also give dimension-free versions of the inequalities, which are independent of the ambient dimension $d$ and relies on the effective rank instead. This enables the generalization of our results to linear operators in infinite-dimensional Hilbert spaces. Our results have extensive applications in statistics and machine learning; in particular, we obtain improved bounds in covariance estimation and principal component analysis on Markovian data.

math.PR↗

arXiv2026

Conditionally Resampled Sliding-Window Count Kernels: Spectral-Gap Bounds and Poincaré Inequalities

We study the conditionally resampled sliding-window count kernel associated with the empirical counts of length-$n$ windows from a stationary finite-state reversible Markov chain. Although the resulting count process is generally not Markov, its stationary one-step conditional law defines a genuine Markov kernel. For every fixed strictly positive reversible kernel $P$ on a finite state space, we present a Poincaré inequality for the induced count kernel $\tP_n$ of length $n$. In other words, we derive the lower bound of the spectral gap $\Gap(\tP_n)$ of $\tP_n$ as \[ \Gap(\tP_n)\ge \frac{c(P)}{n}, \] where $c(P)>0$ depends only on $P$. The proof combines a martingale oscillation inequality for the stationary path law with a direct comparison of coordinate oscillations to the Dirichlet form of the count kernel. A linear statistic of the count vector gives the matching $O(1/n)$ upper bound, so for every fixed strictly positive reversible $P$ one has $\Gap(\tP_n)=Θ_P(1/n)$. The resulting count-space Poincaré inequality yields a local-to-global variance bound for finite-window count statistics and, together with a general matrix-concentration principle, operator-norm concentration for matrix-valued empirical averages.

math.ST↗

arXiv2026

Last-Iterate Analyses of FTRL with the 1/2-Tsallis Entropy in Stochastic Bandits

The convergence analysis of online learning algorithms is central to machine learning theory, where the last-iterate convergence is particularly important, as it captures the learner's actual decisions and describes the evolution of the learning process over time. However, in multi-armed bandits, most existing algorithmic analyses mainly focus on the order of regret, while the last-iterate (simple regret) convergence rate remains less explored---especially for the widely studied Follow-the-Regularized-Leader (FTRL) algorithms. Recently, FTRL with the $1/2$-Tsallis entropy regularizer $Ψ(p) = -4\sum_{i=1}^d \sqrt{p_i}$ (the $1/2$-Tsallis-INF algorithm, by arXiv:1807.07623) was shown to achieve the desirable Best-of-Both-Worlds (BOBW) guarantees and perform well in both adversarial and stochastic settings. Nevertheless, its last-iterate convergence rate has not yet been fully studied. This paper studies the $1/2$-Tsallis-INF algorithm in stochastic bandits and shows that its sampling simple regret decays at rate $\mathcal{O}(t^{-1})$, without requiring the optimal arm to be unique. Under a unique optimal arm, we further show that the expected Bregman divergence induced by $Ψ$ between the point mass on the optimal arm and the sampling distribution at iteration $t$ decays at rate $\mathcal{O}(t^{-1/2})$. Matching lower bounds under the same uniqueness condition show that both exponents of $t$ are tight.

cs.LG↗

arXiv2026

Does 1/2-Tsallis-INF Also Work Well for Best-Arm Identification?

Regret minimization (RM) and best-arm identification (BAI) are two fundamental objectives in multi-armed bandits. Among regret-minimizing algorithms, $1/2$-Tsallis-INF is a canonical best-of-both-worlds FTRL algorithm: it achieves logarithmic pseudo-regret in stochastic bandits while retaining minimax-optimal regret in adversarial bandits, without knowing the environment in advance. This raises a natural question: can the same algorithm, without additional exploration, also identify the best arm reliably? We study this question in stochastic bandits by analyzing the failure probability $\operatorname{Err}_t$, defined as the probability that the empirical best arm determined by the cumulative importance-weighted loss estimates of 1/2-Tsallis-INF differs from the true optimal arm. The main difficulty is that, at the logarithmic-regret scale, suboptimal arms are sampled with probability heuristically of order $1/t$. Consequently, importance weighting causes the cumulative estimator to fluctuate on the same linear scale as its mean separation. To overcome this obstacle, guided by a diffusion toy model, we construct a Lyapunov function for the gap process between the estimated cumulative loss of the optimal arm and that of the best competing arm. This leads to polynomial upper bounds on $\operatorname{Err}_t$: for learning rate $η_t=α/\sqrt t$, $\operatorname{Err}_t$ decays at rate $t^{-2+α^2μ_{i_*}/4+ρ}$ for any $ρ>0$, where $μ_{i_*}$ denotes the mean loss of the true optimal arm. We also establish a lower bound $Ω(t^{-2-\varepsilon})$ for any $\varepsilon>0$, showing that the exponent $2$ is essentially tight.

cs.LG↗

arXiv2026

Online Inference for Quantile Temporal Difference Learning in Distributional Reinforcement Learning

In this paper, we study how to perform statistical inference for quantile temporal difference learning (QTD) in distributional reinforcement learning. Assuming access to a generative model, we first establish functional central limit theorems for both synchronous and asynchronous QTD, which show that the averaged iterates of QTD converge weakly to a rescaled Brownian motion. We next provide online inference methods. Based on random scaling, the inference procedure constructs an asymptotically pivotal statistic for inference by using the information along the whole QTD path. Meanwhile, the proposed statistic can be computed online without storing the entire trajectory of QTD iterates. This substantially reduces the memory requirement and enables efficient statistical inference in distributional reinforcement learning.

stat.ML↗

arXiv2026

Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling

We study the Moreau--Yosida unadjusted Langevin algorithm (MYULA) for the nonsmooth composite target \[ π(dx)\propto \exp\{-f(x)-g(x)\}\,dx, \qquad x\in\mathbb R^d, \] where $f$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g$ is convex and $G$-Lipschitz. Let $g_λ$ be the Moreau envelope of $g$, $π_λ$ the corresponding smoothed target, and $a_λ=\operatorname{tr}H_λ$, where $H_λ$ is the a.e./weak Hessian of $g_λ$. We show that the leading MYULA discretization error is controlled by the reference active trace $B_{\mathrm{ref}}$, the average of $a_λ$ along the heat substep of one MYULA update started from $π_λ$, rather than by the global curvature bound $d/λ$. If $M_λ$ is an a.e. upper bound for $a_λ$, then, up to logarithmic factors, \[ N \lesssim \frac{1}{m} \left[ L_f + \frac{ τ_f+G^2+B_{\mathrm{ref}} }{ \varepsilon_{\mathrm{alg}}^2 } + \frac{M_λ}{\varepsilon_{\mathrm{alg}}} \right], \qquad τ_f:= \sup_x\operatorname{tr}\nabla^2 f(x), \] iterations suffice to ensure $\sqrt m\,W_2(μ_N,π_λ)\leq\varepsilon_{\mathrm{alg}}$, where $μ_N$ is the law of the $N$-th iterate and $W_2$ is the quadratic Wasserstein distance. We also prove the Moreau-bias bound \[ \sqrt m\,W_2(π_λ,π) \leq \frac{G^2λ}{4}. \] Thus, choosing $λ\asymp\varepsilon/G^2$ gives an end-to-end guarantee for $π$. The universal estimate $B_{\mathrm{ref}}\leq d/λ$ yields $\widetilde O(\varepsilon^{-3})$ accuracy dependence. For the structured piecewise-linear, lasso-type, group, and total-variation penalties considered here, curvature--tube estimates make $B_{\mathrm{ref}}$ independent of $λ$, yielding $\widetilde O(\varepsilon^{-2})$ for the same classical MYULA kernel.

cs.LG↗

arXiv2026

Online Statistical Inference for Nonlinear Stochastic Approximation with Markovian Data

Many stochastic approximation (SA) algorithms evolve along a single trajectory, making uncertainty quantification challenging under nonlinear dynamics and Markov dependence. We develop an online inference framework for nonlinear SA with decreasing step sizes. Under local stability and verifiable conditions, we establish a functional central limit theorem for the partial-sum path. The proof uses a Poisson-equation decomposition to handle Markov dependence and a uniform bound to control endpoint-dependent remainders induced by decreasing step sizes. The resulting path limit yields self-normalized confidence intervals without estimating the asymptotic variance. Our primary construction uses a five-dimensional polynomial-series normalizer, has an asymptotic Student $t_5$ pivot, and requires only constant memory. We apply the framework to Q-learning, including asynchronous tabular, projected linear, and entropy-regularized updates, as well as SGD for generalized linear models with Markov data and inference for the identified product in low-rank adaptation (LoRA). Across four settings, the polynomial-series method achieves near-nominal coverage, with shorter confidence intervals and lower computational cost than online bootstrap.

math.ST↗

arXiv2026

Strong Convergence for a General Class of Random Matrix Models

Let $X_{1,n},\ldots,X_{d,n}$ be $n\times n$ random matrices built from independent i.i.d. entry arrays, with centered entries, normalized by $n^{-1/2}$. We prove that, if every entry law has finite fourth moment, then this tuple converges almost surely strongly in $*$-distribution to a free circular family with the matching variances. Equivalently, normalized traces and operator norms converge for every fixed noncommutative $*$-polynomial, including polynomials with fixed matrix coefficients. No assumption is imposed on the pseudo-variances of the complex entries. The bounded-entry argument applies the spectrum and moment universality estimates of Brailovskaya and van Handel to all self-adjoint linear pencils. The matching Gaussian pencils are reduced to independent Wigner matrices and identified by Anderson's strong convergence theorem. A fixed-level centered truncation, followed by the Bai--Yin norm bound, transfers the result to finite fourth moments.

math.PR↗

arXiv2026

PinEqualizer: Full Funnel Content Exploration and Debiasing System at Pinterest

In this paper, we propose a new solution for addressing the content cold-start problem in industry-scale search and recommender systems. Compared to prior approaches, we have made the following new contributions: 1) our solution spans the entire multi-stage funnel and generalizes well for both search and recommendation surfaces, 2) our solution reduces bias favoring existing content, allowing more accurate model prediction across content types and reducing short-term tradeoffs associated with high volumes of explicit content exploration, 3) our solution is evaluated with a scalable measurement framework that enables fast short-term experimentation while validating long-term impact. We have iteratively built and successfully deployed this new system at Pinterest in the past two years and observed significant improvements in fresh content exploration, overall user engagement, and content ecosystem health.

cs.IR↗

arXiv2026

Statistical Efficiency and Inference of Quantile Distributional Reinforcement Learning

In this paper, we study quantile-based distributional reinforcement learning from the perspective of statistical efficiency. We focus on distributional policy evaluation, whose goal is to characterize the return distribution, namely the distribution of discounted cumulative rewards under a given policy. To obtain a finite-dimensional representation of the return distribution, we consider the quantile fixed point $η_m$ induced by the quantile-projected distributional Bellman equation. Assuming access to a generative model, we construct an estimator $η_m^{(n)}$ based on an empirical Markov decision process. For a fixed number of quantiles $m$, we establish a non-asymptotic error bound for $η_m^{(n)}$ and $η_m$ under the supremum $W_\infty$ metric, showing that the estimation error scales as $\widetilde{O}(\sqrt{m/n})$ with respect to $m$ and $n$. This implies that the quantile-based distributional policy evaluation problem can be solved with sample efficiency, achieving the optimal parametric $\sqrt{n}$ convergence rate. We derive the asymptotic distribution of the quantile parameters $\sqrt{n}(θ_m^{(n)}-θ_m)$ and characterize the semiparametric efficiency bound, which is attained by our estimator. Beyond the fixed-dimensional setting, we investigate the asymptotic regime in which the number of quantiles diverges. We characterize the limit covariance structure and show that it matches the semiparametric efficiency bound of the nonparametric model for distributional policy evaluation, showing that quantile-based estimators remain asymptotically efficient in the infinite-dimensional limit. Finally, we establish a Berry--Esseen theorem for smooth functionals $\sqrt{n}(η_m^{(n)}(s)-η_m(s))f$, thereby providing a foundation for statistically valid inference on functionals of the quantile-projected return distribution.

stat.ML↗

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↗