Optimized methods for composite optimization: a reduction perspective
本文提出一个框架，将无约束光滑优化的优化方法直接推广到复合优化，并展示了近端梯度下降的步长加速现象、一种比FISTA更快的近端优化梯度方法，以及一种在复合设置下改进梯度范数最小化最新速率的新方法。
Abstract Recent advances in convex optimization have leveraged computer-assisted proofs to develop optimized first-order methods that improve over classical algorithms. However, each optimized method is specially tailored for a particular problem setting, and it is a well-documented challenge to extend optimized methods to other settings due to their highly bespoke design and analysis. We provide a framework that derives optimized methods for composite optimization directly from those for unconstrained smooth optimization. The derived methods naturally extend the original methods, analogous to how proximal gradient descent extends gradient descent. The key to our result is certain algebraic identities that—by leveraging a common structure of optimized methods— provide a unified way of extending convergence analyses from unconstrained to composite settings. As concrete examples, we apply our framework to establish (1) the phenomenon of stepsize acceleration for proximal gradients descent; (2) a convergence rate for the proximal optimized gradient method [2] which is faster than FISTA [3]; (3) a new method that improves the state-of-the-art rate for minimizing gradient norm in the composite setting.