GL-LowPopArt: A Nearly Instance-Wise Minimax Estimator for (Adaptive) Generalized Linear Low-Rank Trace Regression

Abstract

We present GL-LowPopArt, a novel Catoni-style estimator for generalized linear low-rank trace regression. Building on LowPopArt (Jang et al., 2024), it employs a two-stage approach – nuclear norm regularization followed by matrix Catoni estimation. We establish state-of-the-art estimation error bounds, surpassing existing guarantees (Fan et al., 2019; Kang et al., 2022), and reveal a novel experimental design objective, GL(\pi). The key technical challenge is controlling bias from the nonlinear inverse link function, which we address by our two-stage approach. We prove a local minimax lower bound, showing that our GL-LowPopArt, enjoys instance-wise optimality up to the condition number of the ground-truth Hessian. Applications include generalized linear matrix completion, where GL-LowPopArt, achieves a state-of-the-art Frobenius error guarantee, and bilinear dueling bandits, a novel setting inspired by general preference learning (Zhang et al., 2024). Our analysis of a GL-LowPopArt,-based explore-then-commit algorithm reveals a new, potentially interesting problem-dependent quantity, along with improved Borda regret bound than vectorization (Wu et al., 2024).

Junghyun Lee

PhD Student

PhD student at GSAI, KAIST, jointly advised by Profs. Se-Young Yun and Chulhee Yun. Research focuses on interactive machine learning, particularly at the intersection of RLHF and preference learning, and statistical analyses of large networks, with an emphasis on community detection. Broadly interested in mathematical and theoretical AI and related problems in mathematics.