用后悔值最小化方法构造新迭代法，可更快求解非自映射的不动点。
A regret minimization approach to fixed-point iterations
- 将后悔值最小算法转化为不动点迭代，理论收敛性由后悔界保证
- 基于AdaGrad的迭代比经典Krasnoselskii-Mann方法收敛更快
- 适用于需要自适应步长的非自映射不动点求解场景
我们提出一种转换机制，将后悔值最小化算法转化为不动点迭代，其收敛性由后悔界直接保证。该方法可视为经典Krasnoselskii--Mann迭代的广义扩展，后者可通过转换在线梯度下降算法得到。该框架可构造新的简单迭代法用于求解非自映射的不动点。我们还重点研究了从AdaGrad家族后悔值最小化器出发的转换，从而获得具有新型自适应性质的不动点迭代。在多种问题上的数值实验表明，基于AdaGrad的不动点迭代相比Krasnoselskii--Mann迭代具有更快的收敛速度。
原文摘要 · Abstract (English)
We propose a conversion scheme that turns regret minimizing algorithms into fixed point iterations, with convergence guarantees following from regret bounds. The resulting iterations can be seen as a grand extension of the classical Krasnoselskii--Mann iterations, as the latter are recovered by converting the Online Gradient Descent algorithm. This approach yields new simple iterations for finding fixed points of non-self operators. We also focus on converting algorithms from the AdaGrad family of regret minimizers, and thus obtain fixed point iterations with adaptive guarantees of a new kind. Numerical experiments on various problems demonstrate faster convergence of AdaGrad-based fixed point iterations over Krasnoselskii--Mann iterations.
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