Convex Bounds on the Softmax Function with Applications to Robustness Verification
This work addresses robustness verification for neural networks, particularly transformers and deep ensembles, but is incremental as it builds on existing convex optimization methods with improved bounds.
The paper tackled the problem of verifying the robustness of neural networks by deriving convex lower and concave upper bounds on the softmax function, which are tighter than previous linear bounds, as demonstrated in applications to transformers and deep ensembles.
The softmax function is a ubiquitous component at the output of neural networks and increasingly in intermediate layers as well. This paper provides convex lower bounds and concave upper bounds on the softmax function, which are compatible with convex optimization formulations for characterizing neural networks and other ML models. We derive bounds using both a natural exponential-reciprocal decomposition of the softmax as well as an alternative decomposition in terms of the log-sum-exp function. The new bounds are provably and/or numerically tighter than linear bounds obtained in previous work on robustness verification of transformers. As illustrations of the utility of the bounds, we apply them to verification of transformers as well as of the robustness of predictive uncertainty estimates of deep ensembles.