arXiv subjects

Yinan Shen

Publications and source records attributed to Yinan Shen.

8 recordsLinked to original sources

arXiv2026

Validation-Induced Shapley Shifts: How Validation Structure Distorts Data Valuation

Shapley values are widely used to attribute value to training data based on their marginal contribution to performance on a validation set. Existing practice often assumes these values are stable once the training data and model are fixed. In this work, we uncover a systematic vulnerability: even modest changes to the validation set, such as introducing noises, cause directional shifts in Shapley distributions. As noises are added, Shapley values of training samples compress toward zero. We trace this to a noise-induced neighborhood reshuffling effect: perturbations alter the local rank order between validation and training samples, flattening the valuation landscape. Using the KNN-Shapley framework, we show through synthetic and real data that these shifts are consistent and reproducible. Our findings challenge the assumption of Shapley stability and reveal a new axis of fragility in data valuation. We propose normalization and boundary-aware validation strategies to mitigate these distortions and enable more robust, interpretable valuation in machine learning marketplaces.

cs.LG↗

arXiv2026

Sharp Concentration Inequalities: Phase Transition and Mixing of Orlicz Tails with Variance

In this work, we investigate how to develop sharp concentration inequalities for sub-Weibull random variables, including sub-Gaussian and sub-exponential distributions. Although the random variables may not be sub-Guassian, the tail probability around the origin behaves as if they were sub-Gaussian, and the tail probability decays align with the Orlicz $\Psi_\alpha$-tail elsewhere. Specifically, for independent and identically distributed (i.i.d.) $\{X_i\}_{i=1}^n$ with finite Orlicz norm $\|X\|_{\Psi_\alpha}$, our theory unveils that there is an interesting phase transition at $\alpha = 2$ in that $\PP\l(\l|\sum_{i=1}^n X_i \r| \geq t\r)$ with $t > 0$ is upper bounded by $2\exp\l(-C\max\l\{\frac{t^2}{n\|X\|_{\Psi_{\alpha}}^2},\frac{t^{\alpha}}{ n^{\alpha-1} \|X\|_{\Psi_{\alpha}}^{\alpha}}\r\}\r)$ for $\alpha\geq 2$, and by $2\exp\l(-C\min\l\{\frac{t^2}{n\|X\|_{\Psi_{\alpha}}^2},\frac{t^{\alpha}}{ n^{\alpha-1} \|X\|_{\Psi_{\alpha}}^{\alpha}}\r\}\r)$ for $1\leq \alpha\leq 2$ with some positive constant $C$. In many scenarios, it is often necessary to distinguish the standard deviation from the Orlicz norm when the latter can exceed the former greatly. To accommodate this, we build a new theoretical analysis framework, and our sharp, flexible concentration inequalities involve the variance and a mixing of Orlicz $\Psi_\alpha$-tails through the min and max functions. Our theory yields new, improved concentration inequalities even for the cases of sub-Gaussian and sub-exponential distributions with $\alpha = 2$ and $1$, respectively. We further demonstrate our theory on martingales, random vectors, random matrices, and covariance matrix estimation. These sharp concentration inequalities can empower more precise non-asymptotic analyses across different statistical and machine learning applications.

math.ST↗

arXiv2026

SGD with Dependent Data: Optimal Estimation, Regret, and Inference

This work investigates the performance of the final iterate produced by stochastic gradient descent (SGD) under temporally dependent data. We consider two complementary sources of dependence: $(i)$ martingale-type dependence in both the covariate and noise processes, which accommodates non-stationary and non-mixing time series data, and $(ii)$ dependence induced by sequential decision making. Our formulation runs in parallel with classical notions of (local) stationarity and strong mixing, while neither framework fully subsumes the other. Remarkably, SGD is shown to automatically accommodate both independent and dependent information under a broad class of stepsize schedules and exploration rate schemes. Non-asymptotically, we show that SGD simultaneously achieves statistically optimal estimation error and regret, extending and improving existing results. In particular, our tail bounds remain sharp even for potentially infinite horizon $T=+\infty$. Asymptotically, the SGD iterates converge to a Gaussian distribution with only an $O_{\PP}(1/\sqrt{t})$ remainder, demonstrating that the supposed estimation-regret trade-off claimed in prior work can in fact be avoided. We further propose a new ``conic'' approximation of the decision region that allows the covariates to have unbounded support. For online sparse regression, we develop a new SGD-based algorithm that uses only $d$ units of storage and requires $O(d)$ flops per iteration, achieving the long term statistical optimality. Intuitively, each incoming observation contributes to estimation accuracy, while aggregated summary statistics guide support recovery.

math.ST↗

arXiv2025

Online Quantile Regression

This paper addresses the challenge of integrating sequentially arriving data within the quantile regression framework, where the number of features is allowed to grow with the number of observations, the horizon is unknown, and memory is limited. We employ stochastic sub-gradient descent to minimize the empirical check loss and study its statistical properties and regret performance. In our analysis, we unveil the delicate interplay between updating iterates based on individual observations versus batches of observations, revealing distinct regularity properties in each scenario. Our method ensures long-term optimal estimation irrespective of the chosen update strategy. Importantly, our contributions go beyond prior works by achieving exponential-type concentration inequalities and attaining optimal regret and error rates that exhibit only \textsf{ short-term} sensitivity to initial errors. A key insight from our study is the delicate statistical analyses and the revelation that appropriate stepsize schemes significantly mitigate the impact of initial errors on subsequent errors and regrets. This underscores the robustness of stochastic sub-gradient descent in handling initial uncertainties, emphasizing its efficacy in scenarios where the sequential arrival of data introduces uncertainties regarding both the horizon and the total number of observations. Additionally, when the initial error rate is well-controlled, there is a trade-off between short-term error rate and long-term optimality. Due to the lack of delicate statistical analysis for squared loss, we also briefly discuss its properties and proper schemes. Extensive simulations support our theoretical findings.

math.ST↗

arXiv2024

Improved performance guarantees for Tukey's median

Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with John Tukey, is among the most popular. Tukey's depth has found applications in robust statistics, graph theory, and the study of elections and social choice. We present improved performance guarantees for empirical Tukey's median, a deepest point associated with a given sample, when the data-generating distribution is elliptically symmetric and possibly anisotropic. Some of our results remain valid in the wider class of affine equivariant estimators. As a corollary of our bounds, we show that the typical diameter of the set of all empirical Tukey's medians scales like $o(n^{-1/2})$ where $n$ is the sample size. Moreover, when the data follow the bivariate normal distribution, we prove that with high probability, the diameter is of order $O(n^{-3/4}\log^{1/2}(n))$. On the technical side, we show how affine equivariance can be leveraged to improve concentration bounds; moreover, we develop sharp strong approximation results for empirical processes indexed by halfspaces that could be of independent interest.

math.ST↗

arXiv2023

Quantile and pseudo-Huber Tensor Decomposition

This paper studies the computational and statistical aspects of quantile and pseudo-Huber tensor decomposition. The integrated investigation of computational and statistical issues of robust tensor decomposition poses challenges due to the non-smooth loss functions. We propose a projected sub-gradient descent algorithm for tensor decomposition, equipped with either the pseudo-Huber loss or the quantile loss. In the presence of both heavy-tailed noise and Huber's contamination error, we demonstrate that our algorithm exhibits a so-called phenomenon of two-phase convergence with a carefully chosen step size schedule. The algorithm converges linearly and delivers an estimator that is statistically optimal with respect to both the heavy-tailed noise and arbitrary corruptions. Interestingly, our results achieve the first minimax optimal rates under Huber's contamination model for noisy tensor decomposition. Compared with existing literature, quantile tensor decomposition removes the requirement of specifying a sparsity level in advance, making it more flexible for practical use. We also demonstrate the effectiveness of our algorithms in the presence of missing values. Our methods are subsequently applied to the food balance dataset and the international trade flow dataset, both of which yield intriguing findings.

math.ST↗

arXiv2023

Computationally Efficient and Statistically Optimal Robust Low-rank Matrix and Tensor Estimation

Low-rank matrix estimation under heavy-tailed noise is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a novel Riemannian sub-gradient (RsGrad) algorithm which is not only computationally efficient with linear convergence but also is statistically optimal, be the noise Gaussian or heavy-tailed. Convergence theory is established for a general framework and specific applications to absolute loss, Huber loss, and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of dual-phase convergence. In phase one, RsGrad behaves as in a typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator which is already observed in the existing literature. Interestingly, during phase two, RsGrad converges linearly as if minimizing a smooth and strongly convex objective function and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Lastly, RsGrad is applicable for low-rank tensor estimation under heavy-tailed noise where a statistically optimal rate is attainable with the same phenomenon of dual-phase convergence, and a novel shrinkage-based second-order moment method is guaranteed to deliver a warm initialization. Numerical simulations confirm our theoretical discovery and showcase the superiority of RsGrad over prior methods.

math.ST↗

arXiv2023

Computationally Efficient and Statistically Optimal Robust High-Dimensional Linear Regression

High-dimensional linear regression under heavy-tailed noise or outlier corruption is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since the robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a projected sub-gradient descent algorithm for both the sparse linear regression and low-rank linear regression problems. The algorithm is not only computationally efficient with linear convergence but also statistically optimal, be the noise Gaussian or heavy-tailed with a finite 1 + epsilon moment. The convergence theory is established for a general framework and its specific applications to absolute loss, Huber loss and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of two-phase convergence. In phase one, the algorithm behaves as in typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator, which is already observed in the existing literature. Interestingly, during phase two, the algorithm converges linearly as if minimizing a smooth and strongly convex objective function, and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Numerical simulations confirm our theoretical discovery and showcase the superiority of our algorithm over prior methods.

math.ST↗