论文标题
通过在线学习学习最小估计器
Learning Minimax Estimators via Online Learning
论文作者
论文摘要
我们考虑设计最小估计器来估计概率分布的参数的问题。与经典方法（例如MLE和最小距离估计器）不同，我们考虑了一种构建此类估计器的算法方法。我们认为设计最小估计器的问题是找到零和游戏的混合策略NASH平衡。通过利用非convex损失的在线学习的最新结果，我们提供了一种一般算法，用于查找一般非convex non-Concave non-concave no-Concave Zero Zero Zero Games的混合策略NASH平衡。我们的算法需要访问两个子例程：（a）输出与给定先前概率分布相对应的贝叶斯估计器的一个，以及（b）计算任何给定估计器的最坏风险的一个。给定对这两个子例程的访问，我们表明我们的算法输出既有最小值估计器，又是最不利的先验。为了证明这种方法的力量，我们使用它来构建可证明的最小估计器，以解决经典问题，例如有限的高斯序列模型中的估计和线性回归。
We consider the problem of designing minimax estimators for estimating the parameters of a probability distribution. Unlike classical approaches such as the MLE and minimum distance estimators, we consider an algorithmic approach for constructing such estimators. We view the problem of designing minimax estimators as finding a mixed strategy Nash equilibrium of a zero-sum game. By leveraging recent results in online learning with non-convex losses, we provide a general algorithm for finding a mixed-strategy Nash equilibrium of general non-convex non-concave zero-sum games. Our algorithm requires access to two subroutines: (a) one which outputs a Bayes estimator corresponding to a given prior probability distribution, and (b) one which computes the worst-case risk of any given estimator. Given access to these two subroutines, we show that our algorithm outputs both a minimax estimator and a least favorable prior. To demonstrate the power of this approach, we use it to construct provably minimax estimators for classical problems such as estimation in the finite Gaussian sequence model, and linear regression.
