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

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).