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

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, $\mathrm{GL}(\pi)$. 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).