One of the challenges in analyzing learning algorithms is the circular entanglement between the objective value and the stochastic noise. This is also known as the "chicken and egg" phenomenon and traditionally, there is no principled way to tackle this issue. People solve the problem by utilizing the special structure of the dynamic, and hence the analysis would be difficult to generalize.
In this work, we present a streamlined three-step recipe to tackle the "chicken and egg" problem and give a general framework for analyzing stochastic dynamics in learning algorithms. Our framework composes standard techniques from probability theory, such as stopping time and martingale concentration. We demonstrate the power and flexibility of our framework by giving a unifying analysis for three very different learning problems with the last iterate and the strong uniform high probability convergence guarantee. The problems are stochastic gradient descent for strongly convex functions, streaming principal component analysis, and linear bandit with stochastic gradient descent updates. We either improve or match the state-of-the-art bounds on all three dynamics.
Transversality Conditions for Stochastic Higher-Order Optimality: Continuous and Discrete Time Problems
在研究有限长度专利的效果时，如Futagami and Iwaisako（2007）的开创性工作那样，自然会出现高阶最优化问题。在本文中，我们建立了欧拉方程和横截性条件分析这样的高阶最优化问题。我们发展我们的随机一般简化形式模型的结果，并考虑连续和离散时间的情况下。我们使用我们的结果来建立简化的家庭最大化问题的欧拉方程和横截性条件在Futagami和Iwaisako（2007）。
Multilevel Stochastic Gradient Descent for Risk-Averse PDE-Constrained Optimization
Heidelberg University（海德堡大学）；Johann Radon Institute for Computational and Applied Mathematics（约翰·拉多纳计算与应用数学研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Politecnico di Torino（都灵理工大学）
Mini-Batch Stochastic Conjugate Gradient Algorithms with Minimal Variance
Beijing University of Posts and Telecommunications（北京邮电大学）；Zhejiang Gongshang University Hangzhou College of Commerce（浙江工商大学杭州商学院）；Chinese Academy of Sciences（中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）
