Adaptive Smoothing Algorithms for Nonsmooth Composite Convex Minimization
This work addresses optimization challenges in machine learning and engineering by offering an incremental improvement in adaptive parameter tuning for nonsmooth problems.
The authors tackled the problem of nonsmooth composite convex minimization by proposing an adaptive smoothing algorithm that achieves an O(1/ε) worst-case iteration complexity while maintaining the same per-iteration complexity as Nesterov's method, with numerical examples provided for validation.
We propose an adaptive smoothing algorithm based on Nesterov's smoothing technique in \cite{Nesterov2005c} for solving "fully" nonsmooth composite convex optimization problems. Our method combines both Nesterov's accelerated proximal gradient scheme and a new homotopy strategy for smoothness parameter. By an appropriate choice of smoothing functions, we develop a new algorithm that has the $\mathcal{O}\left(\frac{1}{\varepsilon}\right)$-worst-case iteration-complexity while preserves the same complexity-per-iteration as in Nesterov's method and allows one to automatically update the smoothness parameter at each iteration. Then, we customize our algorithm to solve four special cases that cover various applications. We also specify our algorithm to solve constrained convex optimization problems and show its convergence guarantee on a primal sequence of iterates. We demonstrate our algorithm through three numerical examples and compare it with other related algorithms.