Vertex-Softmax: Tight Transformer Verification via Exact Softmax Optimization
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916001661059072 |
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| author | Rezazadeh, Navid Davoodi, Arash Gholami |
| author_facet | Rezazadeh, Navid Davoodi, Arash Gholami |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10974 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex-Softmax: Tight Transformer Verification via Exact Softmax Optimization Rezazadeh, Navid Davoodi, Arash Gholami Machine Learning Artificial Intelligence 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. |
| title | Vertex-Softmax: Tight Transformer Verification via Exact Softmax Optimization |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2605.10974 |