Online Decision Making with History-Average Dependent Costs (Extended)
Vijeth Hebbar Cedric Langbort
In many online sequential decision-making scenarios, a learner's choices
affect not just their current costs but also the future ones. In this work, we
look at one particular case of such a situation where the costs depend on the
time average of past decisions over a history horizon. We first recast this
problem with history dependent costs as a problem of decision making under
stage-wise constraints. To tackle this, we then propose the novel
Follow-The-Adaptively-Regularized-Leader (FTARL) algorithm. Our innovative
algorithm incorporates adaptive regularizers that depend explicitly on past
decisions, allowing us to enforce stage-wise constraints while simultaneously
enabling us to establish tight regret bounds. We also discuss the implications
of the length of history horizon on design of no-regret algorithms for our
problem and present impossibility results when it is the full learning horizon.