Abstract:We introduce a nearly parameter-free algorithm for minimizing piecewise smooth (PWS) convex functions under the quadratic-growth (QG) condition, where the locations and structure of the smooth regions are entirely unknown. Our algorithm, APEX (Accelerated Prox-Level method for Exploring Piecewise Smoothness), is an accelerated bundle-level method designed to adaptively exploit the underlying PWS structure. For this setting, APEX achieves the best-known oracle-complexity result among existing first-order methods, improving the dependence on the condition number relative to prior bundle-level guarantees. Furthermore, APEX generates a verifiable and accurate termination certificate, enabling a robust, nearly parameter-free implementation. To the best of our knowledge, APEX is the first algorithm to simultaneously achieve the best-known first-order oracle complexity for PWS optimization and provide certificate guarantees.
From: Zhenwei Lin [view email]
[v1]
Wed, 21 Jan 2026 05:51:32 UTC (173 KB)
[v2]
Sun, 22 Feb 2026 21:34:06 UTC (174 KB)
[v3]
Tue, 2 Jun 2026 02:28:42 UTC (182 KB)
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