Local Linear Convergence of Forward–Backward under Partial Smoothness
Liang, Jingwei; Fadili, Jalal; Peyré, Gabriel (2014), Local Linear Convergence of Forward–Backward under Partial Smoothness, in Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence, K.Q. Weinberger, Advances in Neural Information Processing Systems 27 (NIPS 2014), Neural Information Processing Systems Foundation, Inc.
TypeCommunication / Conférence
External document linkhttps://arxiv.org/abs/1407.5611v6
Conference titleAdvances in Neural Information Processing Systems 27 (NIPS 2014)
Book titleAdvances in Neural Information Processing Systems 27 (NIPS 2014)
Book authorZ. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence, K.Q. Weinberger
MetadataShow full item record
Abstract (EN)In this paper, we consider the Forward–Backward proximal splitting algorithm to minimize the sum of two proper closed convex functions, one of which having a Lipschitz–continuous gradient and the other being partly smooth relatively to an active manifoldM.We propose a unified framework in which we show that the Forward–Backward (i) correctlyidentifies the active manifoldMin a finite number of iterations, and then (ii) enters a locallinear convergence regime that we characterize precisely.This explains the typical behaviourthat has been observed numerically for many problems encompassed in our framework, including the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularizationto name a few. These results may have numerous applications including in signal/imageprocessing processing, sparse recovery and machine learning.
Subjects / KeywordsActivity identification; Partial smoothnes; Local linear convergence; Forward– Backward splitting
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