  We study the problem of solving Packing Integer Programs (PIPs) in the online
setting, where columns in $[0,1]^d$ of the constraint matrix are revealed
sequentially, and the goal is to pick a subset of the columns that sum to at
most $B$ in each coordinate while maximizing the objective. Excellent results
are known in the secretary setting, where the columns are adversarially chosen,
but presented in a uniformly random order. However, these existing algorithms
are susceptible to adversarial attacks: they try to "learn" characteristics of
a good solution, but tend to over-fit to the model, and hence a small number of
adversarial corruptions can cause the algorithm to fail.
  In this paper, we give the first robust algorithms for Packing Integer
Programs, specifically in the recently proposed Byzantine Secretary framework.
Our techniques are based on a two-level use of online learning, to robustly
learn an approximation to the optimal value, and then to use this robust
estimate to pick a good solution. These techniques are general and we use them
to design robust algorithms for PIPs in the prophet model as well, specifically
in the Prophet-with-Augmentations framework. We also improve known results in
the Byzantine Secretary framework: we make the non-constructive results
algorithmic and improve the existing bounds for single-item and matroid
constraints.
