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Vertex-Softmax: Tight Transformer Verification via Exact Softmax Optimization

Summary

A new method called Vertex-Softmax provides tighter bounds for certified verification of transformer attention by exactly optimizing softmax over interval constraints, improving verification rates and lower bounds on multiple datasets while being computationally efficient.

SourcearXiv Machine LearningAuthor: Navid Rezazadeh, Arash Gholami Davoodi
Vertex-Softmax: Tight Transformer Verification via Exact Softmax Optimization
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[2605.10974] Vertex-Softmax: Tight Transformer Verification via Exact Softmax Optimization

[Submitted on 8 May 2026]

Title:Vertex-Softmax: Tight Transformer Verification via Exact Softmax Optimization

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Abstract:Certified verification of transformer attention requires bounding the softmax function over interval constraints on the pre-softmax scores. Existing verifiers relax softmax ndependently of the downstream objective, leaving avoidable slack. We prove that the exact optimum of this score-box problem is attained at a vertex of the constraint box, and establish a threshold structure theorem showing that, after sorting the objective coefficients, the optimum lies among only linearly many candidates, yielding the Vertex-Softmax primitive with log-linear complexity in the sequence length. We further prove a formal optimality result showing that Vertex-Softmax is the tightest sound bound obtainable from score intervals alone, characterizing precisely what additional structure (score correlations, score-value coupling) is needed for further improvement. Integrated into a CROWN Convex Relaxation based Optimization for Worst-case Neurons)-style verifier with a formal soundness guarantee, Vertex-Softmax significantly improves certified rates and substantially tightens lower bounds across MNIST, Fashion-MNIST, and CIFAR-10 attention models, while consistently matching or outperforming alpha-CROWN and branch-and-bound baselines at a fraction of their cost.

Subjects:

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

Cite as: arXiv:2605.10974 [cs.LG]

(or arXiv:2605.10974v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2605.10974

arXiv-issued DOI via DataCite

Submission history

From: Navid Rezazadeh [view email] [v1] Fri, 8 May 2026 20:22:06 UTC (54 KB)

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Key points and analysis

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Key points

  • Proves that the exact optimum of softmax under box constraints is at a vertex, leading to a threshold structure theorem.
  • Introduces Vertex-Softmax primitive with log-linear complexity in sequence length.
  • Achieves the tightest possible sound bounds from score intervals alone.
  • Integrated into a CROWN-style verifier, it outperforms existing methods on MNIST, Fashion-MNIST, and CIFAR-10 at lower cost.

Highlights and analysis are generated automatically and may contain errors. Check the original source.

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