提出新方法提升矩阵补全精度，逼近理论最优解。
GL-LowPopArt: A Nearly Instance-Wise Minimax-Optimal Estimator for Generalized Low-Rank Trace Regression
- 两阶段设计：先核范数正则化，再用矩阵Catoni估计
- 误差界优于现有方法，接近理论最优
- 适用于矩阵补全与双人对弈强化学习场景
我们提出GL-LowPopArt，一种用于广义低秩迹回归的新型Catoni型估计器。基于LowPopArt（Jang等，2024），采用两阶段策略：先进行核范数正则化，再实施矩阵Catoni估计。建立了当前最优的估计误差界，超越了已有结果（Fan等，2019；Kang等，2022），并揭示了一个新的实验设计目标GL(π)。关键技术挑战在于控制非线性逆链接函数带来的偏差，通过两阶段方法得以解决。我们证明了局部极小极大下界，表明GL-LowPopArt在近似实例最优意义上表现优异，仅受真实海森矩阵条件数影响。该方法立即实现了广义线性矩阵补全的改进弗罗贝尼乌斯误差保证。此外，我们引入了一种新问题设定——双线性对弈老虎机，即带有通用偏好模型的上下文对弈老虎机。通过使用探索-然后确定策略结合GL-LowPopArt，我们展示了相比朴素向量化方法（Wu等，2024）更优的Borda后悔界。
原文摘要 · Abstract (English)
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, $\mathrm{GL}(π)$. 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 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 naïve vectorization (Wu et al., 2024).
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