标题： 带递归约束的AdaGrad及其在多级和区域分解优化中的应用 显示英文标题

标题： Recursive Bound-Constrained AdaGrad with Applications to Multilevel and Domain Decomposition Minimization

摘要： 两种无需目标函数的噪声容忍算法被提出，它们处理边界约束、不精确梯度并在有二阶信息时加以利用。第一种是一种多级方法，利用问题的分层描述，第二种是一种域分解方法，涵盖标准的加性Schwarz分解。两者都是无约束优化中一阶AdaGrad算法的推广。由于这些算法共享一个共同的理论框架，提供了一个统一的收敛/复杂性理论，涵盖了这两种方法。其主要结果是，以高概率，这两种方法最多需要$O(\epsilon^{-2})$次迭代和噪声梯度评估来计算边界约束问题的一个$\epsilon$-近似的一阶临界点。讨论了在从基于偏微分方程的问题到深度神经网络训练的各种应用中的大量数值实验，展示了它们出色的计算效率。 显示英文摘要

摘要： Two OFFO (Objective-Function Free Optimization) noise tolerant algorithms are presented that handle bound constraints, inexact gradients and use second-order information when available.The first is a multi-level method exploiting a hierarchical description of the problem and the second is a domain-decomposition method covering the standard addditive Schwarz decompositions. Both are generalizations of the first-order AdaGrad algorithm for unconstrained optimization. Because these algorithms share a common theoretical framework, a single convergence/complexity theory is provided which covers them both. Its main result is that, with high probability, both methods need at most $O(\epsilon^{-2})$ iterations and noisy gradient evaluations to compute an $\epsilon$-approximate first-order critical point of the bound-constrained problem. Extensive numerical experiments are discussed on applications ranging from PDE-based problems to deep neural network training, illustrating their remarkable computational efficiency.