GL-LowPopArt: A Nearly Instance-Wise Minimax-Optimal Estimator for Generalized Low-Rank Trace Regression

Junghyun Lee, Kyoungseok Jang, Kwang-Sung Jun, Milan Vojnovic, Se-Young Yun

Abstract: We present **GL-LowPopArt**, a novel Catoni-style estimator for generalized 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(π)**. The key technical challenge is controlling bias from the nonlinear inverse link function, which we address with our two-stage approach. We prove a *local minimax lower bound*, showing that **GL-LowPopArt** enjoys instance-wise optimality up to the condition number of the ground-truth Hessian. Our method immediately achieves an improved Frobenius error guarantee for generalized linear matrix completion. We also introduce a new problem setting called **bilinear dueling bandits**, a contextualized version of dueling bandits with a general preference model. Using an explore-then-commit approach with **GL-LowPopArt**, we show an improved Borda regret bound over naïve vectorization (Wu et al., 2024).

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Code Dataset Url: https://github.com/nick-jhlee/GL-LowPopArt

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Submission Number: 1272