Lower Bounds on Cross-Entropy Loss in the Presence of Test-time Adversaries

Arjun Nitin Bhagoji, Daniel Cullina, Vikash Sehwag, Prateek Mittal

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations


Understanding the fundamental limits of robust supervised learning has emerged as a problem of immense interest, from both practical and theoretical standpoints. In particular, it is critical to determine classifier-agnostic bounds on the training loss to establish when learning is possible. In this paper, we determine optimal lower bounds on the cross-entropy loss in the presence of test-time adversaries, along with the corresponding optimal classification outputs. Our formulation of the bound as a solution to an optimization problem is general enough to encompass any loss function depending on soft classifier outputs. We also propose and provide a proof of correctness for a bespoke algorithm to compute this lower bound efficiently, allowing us to determine lower bounds for multiple practical datasets of interest. We use our lower bounds as a diagnostic tool to determine the effectiveness of current robust training methods and find a gap from optimality at larger budgets. Finally, we investigate the possibility of using of optimal classification outputs as soft labels to empirically improve robust training.

Original languageEnglish (US)
Title of host publicationProceedings of the 38th International Conference on Machine Learning, ICML 2021
PublisherML Research Press
Number of pages11
ISBN (Electronic)9781713845065
StatePublished - 2021
Event38th International Conference on Machine Learning, ICML 2021 - Virtual, Online
Duration: Jul 18 2021Jul 24 2021

Publication series

NameProceedings of Machine Learning Research
ISSN (Electronic)2640-3498


Conference38th International Conference on Machine Learning, ICML 2021
CityVirtual, Online

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability


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