arXiv subjects

Wendao Wu

Publications and source records attributed to Wendao Wu.

8 recordsLinked to original sources

arXiv2026

Matching Upper and Lower Bounds for Higher-Order Nonconvex Finite-Sum Optimization

We establish tight randomized higher-order oracle complexity for finding first-order stationary points of nonconvex finite sums. Let $n$ be the number of components, $Δ>0$ the initial objective-gap bound, $L_p>0$ an individual $p$-th derivative Lipschitz bound, and $ε>0$ the target gradient norm. For every fixed integer $p\ge 2$, the minimax number of exact component queries returning the value and all derivatives through order $p$, with success probability at least $2/3$, is \[ Θ_p\!\left( n+ΔL_p^{1/p}n^{1-1/(2p)} ε^{-(p+1)/p} \right), \] where the constants depend only on $p$ and the worst case ranges over all finite dimensions. The lower bound holds for unrestricted randomized adaptive algorithms and closes the $\sqrt{n}$ gap between the previously known general-order upper and lower bounds in their dependence on $n$. We extend dense weak hiding to complete higher-order replies while keeping each component's regularity independent of the chain length. The matching upper bound retains the known finite-sum exponent, requires only mean-squared $p$-th derivative increments, and removes the fixed-confidence logarithmic loss by verifying entire recursive-estimation epochs with exact function values. The characterization includes the additive $n$ term for every positive parameter regime; it counts oracle calls with unrestricted internal computation.

math.OC↗

arXiv2026

Second-Order Stationarity with Common Random Losses: Matching Tolerance Bounds

We establish tight polynomial tolerance bounds for stochastic second-order stationarity when each fresh oracle response is a derivative of one common random scalar loss. For a population objective $F$ with Lipschitz gradient and Hessian, the target is $\|\nabla F(x)\|\le ε$ and $λ_{\min}(\nabla^2 F(x))\ge -γ$, with independent tolerances $ε,γ>0$. Under bounded gradient variance and almost-surely bounded Hessian error, the minimax number of fresh gradient or Hessian-vector-product calls is $\widetildeΘ\!\left(ε^{-3}+γ^{-5}\right)$. The characterization fixes positive gap, smoothness, and noise parameters, suppresses logarithmic factors, and allows dimension to grow within an explicit polynomial envelope. The upper bound removes the mixed term $ε^{-2}γ^{-2}$ from the earlier fresh-HVP guarantee. Direct random-line Hessian estimates and a dyadic gradient tracker separate gradient drift from randomly signed curvature motion. The lower bound realizes the endpoint costs through globally defined smooth random losses: a smooth partition localizes scalar noise without a chain-length penalty, while exact cancellation limits the information in the entire response. Consequently, the same tolerance exponents hold even for joint value, gradient, and full-Hessian responses with bounded value variance. For a population Hessian with Hölder exponent $ν\in(0,1]$, fresh gradient/HVP complexity becomes $\widetildeΘ\!\left(ε^{-3}+γ^{-(3+2/ν)}\right)$ under the corresponding dimension envelope.

math.OC↗

arXiv2026

Near-Optimal Higher-Order Oracle Complexity for Convex--Concave Minimax Optimization

For smooth convex--concave minimax optimization, the higher-order lower bound of Chen et al. (2026) applies to a restricted tensor-algorithm class with prescribed regularized Taylor-model updates. We establish the same bound for arbitrary adaptive deterministic and randomized algorithms, matching, up to logarithmic factors, the upper bound of Zhang et al. (2026). Fix an integer $p\ge 2$ and let $L_p>0$ bound the Lipschitz constant of the objective's $p$-th derivative on a compact convex product domain of diameter at most $D_Z>0$. Each feasible query returns the objective value and all derivatives through order $p$. For accuracy $ε>0$, set $Q_{\mathrm{tan}}=L_pD_Z^p/ε$ for tangent residual and $Q_{\mathrm{gap}}=L_pD_Z^{p+1}/ε$ for saddle gap. Let $T_E^{\mathrm{det}}(ε)$ and $T_E^{\mathrm{rand}}(ε)$ denote the high-dimensional minimax query complexities for criterion $E\in\{\mathrm{tan},\mathrm{gap}\}$, with randomized success probability at least $2/3$ on every instance. Our lower bounds and the existing upper bound give $c_pQ_E^{2/(3p-1)}\le T_E^{\mathrm{rand}}(ε)\le T_E^{\mathrm{det}}(ε)\le C_pQ_E^{2/(3p-1)}[1+\log(3+Q_E)]^{6(p-1)}$ for sufficiently large $Q_E$, where $c_p,C_p>0$ depend only on $p$. Thus the same accuracy exponent holds beyond tensor update rules, even for randomized queries and arbitrary feasible outputs. The proof constructs a scalar convex--concave chain with exactly flat gates that hide complete derivative information. Direct product-domain error witnesses and adaptive transcript arguments establish the lower bounds for both criteria.

math.OC↗

arXiv2026

Near-Optimal Deterministic Exact-Value Complexity for Smooth Convex Optimization

We study the deterministic oracle complexity of smooth convex optimization when the algorithm receives only exact function values. The objective is a globally $β$-smooth convex function, all queries and the final output are restricted to the Euclidean ball of radius $R$, and the unique minimizer lies in the ball of radius $R/2$. We establish an upper bound of $O(d\sqrt{βR^2/ε})$ using coordinate finite differences together with an error-robust accelerated projected method. Our main contribution is a matching lower bound, up to the high-accuracy saturation of the construction: any deterministic adaptive value-oracle algorithm requires $Ω\!\left(d\min\{\sqrt{βR^2/ε},(d/\log(ed))^{1/3}\}\right)$ queries. Consequently, the minimax oracle complexity is $Θ(d\sqrt{βR^2/ε})$ throughout the moderate-accuracy regime $βR^2(\log(ed)/d)^{2/3}\leqε\leq cβR^2$ for a universal constant $c>0$. The lower bound must account for the fact that a single exact real value can encode arbitrarily much information. To overcome this difficulty, we construct a single fixed smooth convex hard instance using a Moreau-smoothed biased max chain, an exact prefix-shielding mechanism, and batched delayed rotations. These techniques preserve consistency with the full adaptive transcript and establish the optimality of the square-root complexity branch for deterministic bounded-query algorithms.

math.OC↗

arXiv2026

Near-Optimal Exact-Value Zeroth-Order Complexity for Smooth Strongly Convex Optimization

We study deterministic adaptive optimization of globally $β$-smooth, $μ$-strongly convex functions using exact scalar function values. Queries and outputs lie in $B_2^d(R)$, and the minimizer lies in $B_2^d(R/2)$. Set $κ=β/μ$, $Q=βR^2/ε$, and $D_d=(d/\log(ed))^{1/3}$. For sufficiently large $d$ and $0<ε\le c_εβR^2$, the minimax value complexity $N_ε$ satisfies \[ \begin{aligned} N_ε&\ge c d\min\{\sqrt Q,\sqrtκ,D_d\},\\ N_ε&\le C d\min\left\{ \sqrt Q,\sqrtκ[1+\log_+(Q/κ)] \right\}, \end{aligned} \] where $c,C,c_ε>0$ are universal constants and $\log_+(t)=\max\{0,\log t\}$. The lower bound uses an exactly shielded smooth chain and batched delayed rotations; the upper bound combines finite differences, acceleration, and restart. When $\min\{Q,κ\}\le D_d^2$, these bounds match up to constants in the accuracy-dominated regime $Q\leκ$ and at constant relative accuracy $ε=Θ(μR^2)$. For arbitrarily higher accuracy in the range $κ\le D_d^2$, the bounds differ by at most $1+\log(μR^2/ε)$; the optimal accuracy dependence remains unresolved in general.

math.OC↗

arXiv2026

Matching Higher-Order Oracle Complexity for Smooth Monotone Variational Inequalities

We establish near-optimal higher-order oracle bounds for smooth monotone variational inequalities. For fixed $p\ge2$, let $F$ be monotone on a known compact convex set $X$ of diameter at most $D$, with $\operatorname{Lip}(D^{p-1}F)\le L_p$. Each feasible query returns the complete jet $(F,DF,\ldots,D^{p-1}F)$, and the goal is to find $x$ with tangent residual $\operatorname{dist}(0,F(x)+N_X(x))\le\varepsilon$. Writing $Q=L_pD^p/\varepsilon$, we improve the $\widetilde O_p(Q^{1/p})$ upper bound of Chen et al. to $\widetilde O_p(Q^{2/(3p-1)})$ via a dimension-independent deterministic algorithm that returns an explicit tangent-residual certificate. We prove a matching $Ω_p(Q^{2/(3p-1)})$ lower bound for arbitrary adaptive deterministic algorithms and randomized algorithms with per-instance success probability at least $2/3$, without span or tensor-update restrictions. Hence the high-dimensional worst-case oracle complexity is $\widetildeΘ_p((L_pD^p/\varepsilon)^{2/(3p-1)})$. The same method applies to smooth convex--concave minimax problems, improving the fixed-geometry accuracy exponent from $4/(3p+1)$ to $2/(3p-1)$ and matching the known lower-bound exponent.

math.OC↗

arXiv2026

What Makes a Strong Model? A Unified Spectral Analysis of Knowledge Transfer over High-dimensional Linear Regression

Teacher-Student Knowledge Transfer (KT) is ubiquitous in modern machine learning, ranging from classical model compression via Knowledge Distillation (KD) to the emergent phenomenon of Weak-to-Strong (W2S) generalization. While existing studies offer isolated insights, a unified theoretical framework explaining the efficacy of KT across these disparate regimes remains lacking. In this work, we establish a unified spectral analysis of SGD dynamics in high-dimensional linear regression, elucidating the efficiency of KT across seemingly disparate regimes. We characterize KT efficiency through two distinct mechanisms: \emph{Spectral Horizon Expansion} in KD, which enables the capture of statistically inaccessible high-frequency signals, and \emph{Spectral Denoising} in W2S, where the student acts as a filter for optimization noise. Our framework unifies these phenomena, revealing that the efficacy of transfer is governed by the interplay between implicit regularization and heterogeneous spectral learning speeds over the spectrum.

cs.LG↗

arXiv2025

Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias

Recent advances in machine learning have inspired a surge of research into reconstructing specific quantities of interest from measurements that comply with certain physical laws. These efforts focus on inverse problems that are governed by partial differential equations (PDEs). In this work, we develop an asymptotic Sobolev norm learning curve for kernel ridge(less) regression when addressing (elliptical) linear inverse problems. Our results show that the PDE operators in the inverse problem can stabilize the variance and even behave benign overfitting for fixed-dimensional problems, exhibiting different behaviors from regression problems. Besides, our investigation also demonstrates the impact of various inductive biases introduced by minimizing different Sobolev norms as a form of implicit regularization. For the regularized least squares estimator, we find that all considered inductive biases can achieve the optimal convergence rate, provided the regularization parameter is appropriately chosen. The convergence rate is actually independent to the choice of (smooth enough) inductive bias for both ridge and ridgeless regression. Surprisingly, our smoothness requirement recovered the condition found in Bayesian setting and extend the conclusion to the minimum norm interpolation estimators.

stat.ML↗