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Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data

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Xiangwu Wang, Chengwei Cao, Yicheng Song, Ran Bi, Peilin Yu

Machine Learning · Submitted 7 Jul 2026 · 35 references

Pith's verdict: Clean middle-regime implicit-bias result for annealed smoothed-sign: Burg barrier via exact dual rewrite; residual control under time-varying stability is the step to pressure.

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arXiv ID2607.06013 (version 1)
Pith IDOBDA63IX, Pith's permanent record of this paper and its review
CitationChengwei Cao, Peilin Yu, Ran Bi, Xiangwu Wang, Yicheng Song. "Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data." arXiv:2607.06013 (2026). https://pith.science/paper/OBDA63IX
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The paper

The paper claims

Stability annealing steers smoothed sign descent onto a Burg barrier path for max-margin separators

The reading

Adaptive gradient methods can settle on different max-margin separators than plain gradient descent, but a fixed positive stability constant eventually warps the geometry again. This paper isolates the rate-controlled middle regime for full-batch linear classification on separable data: memoryless stability-annealed smoothed-sign descent with weighted exponential loss. The authors prove that the normalized iterates converge to the minimizer of a convex Burg-type barrier over a margin slice. They rewrite the dynamics exactly as entropic mirror ascent on a concave dual objective, control the dual gap by a KL recursion, and obtain an explicit envelope of order S_t^{-1/2} on the normalized iterates, where S_t is cumulative stability mass. The static barrier geometry is fully characterized, including KKT conditions and both endpoint limits. Experiments confirm the dual identities to floating-point error, map the predicted path and rate diagram, and show an empirical fixed-epsilon crossover scaling in cumulative time, while diagnostics mark the boundaries for logistic tails, fixed epsilon, and adaptive second-moment memory.

What carries the argument

The exact dual rewrite of annealed smoothed-sign updates as entropic mirror ascent on a concave dual, paired with a KL recursion that controls the dual gap and produces the S_t^{-1/2} normalized-iterate envelope; the static object is the Burg-type barrier minimized over the margin slice.

Extended reading notes

Core claim

For memoryless stability-annealed smoothed-sign descent with weighted exponential loss on full-batch linearly separable data, the normalized iterates converge to the unique minimizer of a convex Burg-type barrier over a margin slice. The dynamics admit an exact rewrite as entropic mirror ascent on a concave dual objective, a KL recursion controls the dual gap, and the normalized-iterate error is bounded by an explicit envelope of order S_t^{-1/2} under the rate-controlled annealing schedule.

Load-bearing premise

The theory needs memoryless smoothed-sign updates under a stability schedule whose cumulative mass diverges while instantaneous stability goes to zero, together with weighted exponential loss on full-batch linearly separable data.

Original 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.

Figures
Figure 1. Two-dimensional barrier geometry. Contours show
Figure 2. Theorem validation dashboard. All panels use theorem-compatible weighted exponential loss,
Figure 3. Mechanism scorecard. Panel (a) compares final finite-window error ratios against the proposed exponential
Figure 4. Fixed-ϵ crossover scaling for the memoryless proxy, with τϵ(c) = min{t : ∥ − ∇L(wt)∥∞ ≤ cϵ}. Panel (a) shows residual collapse for the empirical law Sτϵ = γ −1 ∞ log(1/ϵ) + O(1). Panel (b) reports ρ = b1γ∞ by learning-rate power and family, and panel (c) compares normalized threshold-induced intercept shifts with − log c. This remains empirical and conditional; the two-sided tail bound needed for a proof is open. would require two-sided gradient-tail control up to the thresh￾old window. The appendix records these missing obliga￾tions explicitly. Thus the crossover evidence clarifies…
Figure 5. Adaptive-method boundary diagnostics. The main paper reports a compact summary; these appendix panels show
Figure 6. Expanded fixed-ϵ crossover diagnostics. These panels stress-test the empirical slope relation by threshold subgroup, ϵ-range removal, and secondary iteration-time scaling. They are empirical sensitivity checks; the two-sided tail bound remains open. E3: optimizer boundary. E3 compares memoryless smoothed-sign dynamics, RMSProp, full Adam, and Adam with and without bias correction. The prespecified grid is β1 ∈ {0, 0.9}, β2 ∈ {0.9, 0.99, 0.999}, and κ/γ∞ ∈ {0.20, 0.50, 0.65}, using the same aggregate dimensions and seeds as E1/E2 where applicable. E4: fixed-ϵ crossover. E4 uses d =…
Figure 7. Extended Theorem 1 validation diagnostics. These panels expand the main theorem-validation dashboard by showing
Figure 8. Primary-window finite-horizon margin diagnostics. Complementing the main dashboard and the extended validation
Figure 9. Finite-window convergence-factor scorecard. The two panels compare final/first factors for

Figures reproduced from arXiv:2607.06013 by the authors.

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