论文标题
最佳在线广义线性回归，并具有随机噪声及其在异质匪内的应用
Optimal Online Generalized Linear Regression with Stochastic Noise and Its Application to Heteroscedastic Bandits
论文作者
论文摘要
我们研究了在随机环境中在线广义线性回归的问题，在随机环境中，该标签是由具有无界添加噪声的广义线性模型生成的。我们对经典的遵循规范领导者（FTRL）算法进行了尖锐的分析，以应对标签噪声。更具体地说，对于$σ$ -sub-gaussian标签噪声，我们的分析提供了$ O（σ^2 d \ log t） + O（\ log t）$的遗憾上限，其中$ d $是输入矢量的维度，$ t $是回合总数。我们还证明了一个$ω（σ^2d \ log（t/d））$下限的随机在线线性回归，这表明我们的上限几乎是最佳的。此外，我们将分析扩展到更精致的伯恩斯坦噪声条件。作为一种应用，我们研究了具有异性噪声的广义线性匪徒，并提出了基于FTRL的算法，以实现第一个方差感知的遗憾结合。
We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for $σ$-sub-Gaussian label noise, our analysis provides a regret upper bound of $O(σ^2 d \log T) + o(\log T)$, where $d$ is the dimension of the input vector, $T$ is the total number of rounds. We also prove a $Ω(σ^2d\log(T/d))$ lower bound for stochastic online linear regression, which indicates that our upper bound is nearly optimal. In addition, we extend our analysis to a more refined Bernstein noise condition. As an application, we study generalized linear bandits with heteroscedastic noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.
