arXiv subjects

Luwei Bai

Publications and source records attributed to Luwei Bai.

5 recordsLinked to original sources

arXiv2026

Optimal Two-Step Stepsize Schedule for Stochastic Gradient Methods

Structured nonconstant large stepsizes can improve the convergence of gradient descent in the deterministic setting. However, in stochastic optimization, aggressive stepsizes can amplify oracle noise and hinder the convergence of stochastic gradient methods. We characterize the globally optimal two-step stepsize schedule for stochastic gradient methods applied to strongly convex and smooth functions, assuming access only to unbiased stochastic gradient estimates with finite support and bounded variance. The optimal schedule depends on the ratio of the initial optimality gap to the noise level and exhibits several distinct regimes. As the influence of stochastic noise diminishes, the optimal two-step stepsizes become larger, reflecting a balance between the benefits of faster iterate convergence and the perturbations induced by stochastic noise.

math.OC↗

arXiv2025

Generalization of Silver Stepsize Schedule to Stochastic Optimization

This work introduces a two-step stepsize schedule for stochastic gradient methods minimizing smooth strongly convex functions. We consider the setting where only stochastic gradient approximations, which are unbiased, of bounded variance, and supported on a finite set, are accessible. When the variance bound is relatively smaller than a ratio of the initial optimality gap, the proposed stepsize schedule achieves better convergence performance compared to the well-regarded constant stepsize {\alpha} = 2/(M+m), where m and M denote the strong convexity and gradient-Lipschitz parameters, respectively. Our stepsize schedule can be viewed as a generalization of the well-known two-step silver stepsize schedule in [J. M. Altschuler and P. A. Parrilo, Journal of the ACM, 72(2):1-38, 2025] from deterministic setting to stochastic optimization.

math.OC↗

arXiv2025

Exploring Effective Strategies for Building a User-Configured GPT for Coding Classroom Dialogues

This study investigated effective strategies for developing a custom GPT to code classroom dialogue. While classroom dialogue is widely recognised as a crucial element of education, its analysis remains challenging due to the need for a nuanced understanding of dialogic functions and the labour-intensive nature of manual transcript coding. Recent advancements in large language models (LLMs) offer promising avenues for automating this process. However, existing studies predominantly focus on training large-scale models or evaluating pre-trained models with fixed codebooks, the outcomes of which are often not applicable, or the methods are not replicable for dialogue researchers working with small datasets or employing customised coding schemes. Using MyGPT - a GPT-4-based customised GPT system configured for dialogue analysis - as a case, this study evaluates its baseline performance in coding classroom dialogue with a human codebook and examines how performance varies with different example inputs under a controlled variable design. Through a design-based research approach, this study explores a set of practical strategies - based upon MyGPT's unique features - for configuring an effective tool with limited data. The findings suggest that, despite a few limitations, a custom GPT developed using these specific strategies can serve as a useful coding assistant by generating coding suggestions.

cs.AI↗

arXiv2024

Anderson Acceleration in Nonsmooth Problems: Local Convergence via Active Manifold Identification

Anderson acceleration is an effective technique for enhancing the efficiency of fixed-point iterations; however, analyzing its convergence in nonsmooth settings presents significant challenges. In this paper, we investigate a class of nonsmooth optimization algorithms characterized by the active manifold identification property. This class includes a diverse array of methods such as the proximal point method, proximal gradient method, proximal linear method, proximal coordinate descent method, Douglas-Rachford splitting (or the alternating direction method of multipliers), and the iteratively reweighted $\ell_1$ method, among others. Under the assumption that the optimization problem possesses an active manifold at a stationary point, we establish a local R-linear convergence rate for the Anderson-accelerated algorithm. Our extensive numerical experiments further highlight the robust performance of the proposed Anderson-accelerated methods.

math.OC↗

arXiv2024

Avoiding strict saddle points of nonconvex regularized problems

In this paper, we consider a class of non-convex and non-smooth sparse optimization problems, which encompass most existing nonconvex sparsity-inducing terms. We show the second-order optimality conditions only depend on the nonzeros of the stationary points. We propose two damped iterative reweighted algorithms including the iteratively reweighted $\ell_1$ algorithm (DIRL$_1$) and the iteratively reweighted $\ell_2$ (DIRL$_2$) algorithm, to solve these problems. For DIRL$_1$, we show the reweighted $\ell_1$ subproblem has support identification property so that DIRL$_1$ locally reverts to a gradient descent algorithm around a stationary point. For DIRL$_2$, we show the solution map of the reweighted $\ell_2$ subproblem is differentiable and Lipschitz continuous everywhere. Therefore, the map of DIRL$_1$ and DIRL$_2$ and their inverse are Lipschitz continuous, and the strict saddle points are their unstable fixed points. By applying the stable manifold theorem, these algorithms are shown to converge only to local minimizers with randomly initialization when the strictly saddle point property is assumed.

math.OC↗