  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.
