arXiv subjects

Yanyi Li

Publications and source records attributed to Yanyi Li.

7 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

PRAC: Principal-Random Subspace for LLM Activation Compression and Memory-Efficient Training

Activations have become the primary memory bottleneck in large-batch LLM training. However, existing compression methods fail to exploit the spectral structure of activations, resulting in slow convergence or limited compression. To address this, we bridge the relationship between the algorithm's fast convergence and the requirements for subspace projection, and show that an effective compression should yield an unbiased estimate of the original activation with low variance. We propose Principal-Random Subspace for LLM Activation Compression (PRAC), which novelly decomposes activations into two components: a principal subspace captured via SVD to retain dominant information, and a random subspace sampled from the orthogonal complement to approximate the tail. By introducing a precise scaling factor, we prove that PRAC yields an unbiased gradient estimator with minimum variance under certain conditions. Extensive experiments on pre-training and fine-tuning tasks demonstrate that PRAC achieves up to 36% total memory reduction with negligible performance degradation and minimal computational cost.

cs.LG↗

arXiv2023

MS-UNet-v2: Adaptive Denoising Method and Training Strategy for Medical Image Segmentation with Small Training Data

Models based on U-like structures have improved the performance of medical image segmentation. However, the single-layer decoder structure of U-Net is too "thin" to exploit enough information, resulting in large semantic differences between the encoder and decoder parts. Things get worse if the number of training sets of data is not sufficiently large, which is common in medical image processing tasks where annotated data are more difficult to obtain than other tasks. Based on this observation, we propose a novel U-Net model named MS-UNet for the medical image segmentation task in this study. Instead of the single-layer U-Net decoder structure used in Swin-UNet and TransUnet, we specifically design a multi-scale nested decoder based on the Swin Transformer for U-Net. The proposed multi-scale nested decoder structure allows the feature mapping between the decoder and encoder to be semantically closer, thus enabling the network to learn more detailed features. In addition, we propose a novel edge loss and a plug-and-play fine-tuning Denoising module, which not only effectively improves the segmentation performance of MS-UNet, but could also be applied to other models individually. Experimental results show that MS-UNet could effectively improve the network performance with more efficient feature learning capability and exhibit more advanced performance, especially in the extreme case with a small amount of training data, and the proposed Edge loss and Denoising module could significantly enhance the segmentation performance of MS-UNet.

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