  We consider online optimization in the 1-lookahead setting, where the
objective does not decompose additively over the rounds of the online game. The
resulting formulation enables us to deal with non-stationary and/or long-term
constraints , which arise, for example, in online display advertising problems.
We propose an on-line primal-dual algorithm for which we obtain dynamic
cumulative regret guarantees. They depend on the convexity and the smoothness
of the non-additive penalty, as well as terms capturing the smoothness with
which the residuals of the non-stationary and long-term constraints vary over
the rounds. We conduct experiments on synthetic data to illustrate the benefits
of the non-additive penalty and show vanishing regret convergence on live
traffic data collected by a display advertising platform in production.
