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

TL;DR

GL-LowPopArt is a new estimator for generalized low-rank trace regression that achieves near-optimal error bounds and introduces a novel experimental design objective, improving upon previous methods.

Contribution

It introduces `GL-LowPopArt`, a two-stage estimator with state-of-the-art guarantees and a new experimental design objective, addressing bias control and instance-wise optimality.

Findings

Surpasses existing estimation error bounds in low-rank trace regression.

Achieves an improved Frobenius error guarantee for matrix completion.

Introduces bilinear dueling bandits with better regret bounds.

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 with 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. 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 na\"ive vectorization (Wu et al., 2024).