AI 中文总结
研究一阶凸优化算法设计与收敛证明，针对非光滑Lipschitz和光滑问题，在温和条件下表明能保证原始目标差距有界的一阶方法，对原始对偶差距也有更强保证，其隐式最优对偶证书还揭示了动量方法中辅助序列作用。
AI 中文摘要
本文研究一阶凸优化算法的设计与收敛证明。具体而言，分别考虑通过次梯度或梯度预言机访问的非光滑Lipschitz问题和光滑问题。对于固定步长一阶方法的一般类，先前关于性能估计问题（PEPs）的工作表明通常存在结构化、紧密的收敛证明。在温和条件下，我们进一步表明，任何仅假设\(\|x_0 - x_*\|\)有界就能保证原始目标差距\(f(x_N) - f(x_*)\)有界的一阶方法，实际上对显式、可计算的原始对偶差距以相同速率有更强的保证。这些以仿射下界形式出现的隐式最优对偶证书，也为动量方法中辅助序列的作用提供了见解。
英文摘要
This work considers the design of first-order convex optimization algorithms and convergence proofs. In particular, we consider nonsmooth Lipschitz and smooth problems accessed through a subgradient or gradient oracle, respectively. For the general class of fixed-step first-order methods, prior work on Performance Estimation Problems (PEPs) has shown that structured, tight convergence proofs typically exist. Under mild conditions, we further show that any first-order method guaranteeing a bound on the primal objective gap $f(x_N)-f(x_\star)$ assuming only a bound on $\|x_0-x_\star\|$ actually has a stronger guarantee on an explicit, computable primal-dual gap at the same rate. These implicit optimal dual certificates, which take the form of affine lower bounds, also provide insight into the role of auxiliary sequences in momentum methods.
