On Structured Filtering-Clustering: Global Error Bound and Optimal First-Order Algorithms

In recent years, the filtering-clustering problems have been a central topic in statistics and machine learning, especially the $\ell_1$-trend filtering and $\ell_2$-convex clustering problems. In practice, such structured problems are typically solved by first-order algorithms despite the extremely ill-conditioned structures of difference operator matrices. Inspired by the desire to analyze the convergence rates of these algorithms, we show that for a large class of filtering-clustering problems, a \textit{global error bound} condition is satisfied for the dual filtering-clustering problems when a certain regularization is chosen. Based on this result, we show that many first-order algorithms attain the \textit{optimal rate of convergence} in different settings. In particular, we establish a generalized dual gradient ascent (GDGA) algorithmic framework with several subroutines. In deterministic setting when the subroutine is accelerated gradient descent (AGD), the resulting algorithm attains the linear convergence. This linear convergence also holds for the finite-sum setting in which the subroutine is the Katyusha algorithm. We also demonstrate that the GDGA with stochastic gradient descent (SGD) subroutine attains the optimal rate of convergence up to the logarithmic factor, shedding the light to the possibility of solving the filtering-clustering problems efficiently in online setting. Experiments conducted on $\ell_1$-trend filtering problems illustrate the favorable performance of our algorithms over other competing algorithms.