  In this work we consider a variant of adversarial online learning where in
each round one picks $B$ out of $N$ arms and incurs cost equal to the
$\textit{minimum}$ of the costs of each arm chosen. We propose an algorithm
called Follow the Perturbed Multiple Leaders (FPML) for this problem, which we
show (by adapting the techniques of Kalai and Vempala [2005]) achieves expected
regret $\mathcal{O}(T^{\frac{1}{B+1}}\ln(N)^{\frac{B}{B+1}})$ over time horizon
$T$ relative to the $\textit{single}$ best arm in hindsight. This introduces a
trade-off between the budget $B$ and the single-best-arm regret, and we proceed
to investigate several applications of this trade-off. First, we observe that
algorithms which use standard regret minimizers as subroutines can sometimes be
adapted by replacing these subroutines with FPML, and we use this to generalize
existing algorithms for Online Submodular Function Maximization [Streeter and
Golovin, 2008] in both the full feedback and semi-bandit feedback settings.
Next, we empirically evaluate our new algorithms on an online black-box
hyperparameter optimization problem. Finally, we show how FPML can lead to new
algorithms for Linear Programming which require stronger oracles at the benefit
of fewer oracle calls.
