{"ID":6023464,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-10T08:47:37.263450031Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.06013","arxiv_id":"2607.06013","title":"Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data","abstract":"Adaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, we prove that the normalized iterates converge to the minimizer of a convex Burg-type barrier over a margin slice. The proof rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, controls the dual gap by a KL recursion, and yields an explicit S_t^{-1/2} normalized-iterate envelope. The static barrier geometry is fully characterized, including KKT conditions and both endpoint limits. Experiments validate the exact dual identities to floating-point error, illustrate the predicted path and rate diagram, and show an empirical fixed-epsilon crossover scaling in cumulative time. We further report robustness and boundary diagnostics for logistic tails, fixed-epsilon crossover, and adaptive-method variants, delineating the scope of the proved smoothed-sign theory.","short_abstract":"Adaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability...","url_abs":"https://arxiv.org/abs/2607.06013","url_pdf":"https://arxiv.org/pdf/2607.06013v1","authors":"[\"Xiangwu Wang\",\"Chengwei Cao\",\"Yicheng Song\",\"Ran Bi\",\"Peilin Yu\"]","published":"2026-07-07T08:55:35Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.OC\"]","methods":"[]","has_code":false}