Dynamic Regret of Distributed Online Saddle Point Problem

This article is concerned with the distributed online saddle point problem for multiagent networks over time-varying graphs. The objective is to design distributed online algorithms for optimizing the accumulation of the time-varying loss functions, and further to analyze the efficiency of the algorithms by utilizing the so-called dynamic regret as a performance measure. To this end, two kinds of distributed online optimization algorithms are developed by considering full-information feedback and bandit feedback, respectively. Under some certain assumptions and resorting to appropriate diminishing step sizes, it is proved that the two developed algorithms yield upper bounds of the dynamic regret in order
$\mathcal {O}(\max \lbrace T^{\lambda _{1}}, T^{\lambda _{2}} (V_{T}+1) \rbrace)$
and order
$ \mathcal {O}(\max \lbrace k T^{\lambda _{3}}, k T^{\lambda _{4}} (V_{T}+1) \rbrace)$
, respectively, where
$T$
is the horizon (total iteration number) of the algorithm,
$V_{T}$
is the path-variation level,
$k$
is the maximum dimension of the state variables, and
$\lambda _{i}$
,
$i=1,2,3,4$
, are positive tuning parameters less than 1. Clearly, both of the two developed algorithms can guarantee that the considered dynamic regret is sublinear with respect to
$T$
, provided that the path-variation level
$V_{T}$
is sublinear with respect to
$T$
. This theoretically shows the efficiency of the developed algorithms. Moreover, the efficiency of the developed algorithms is also illustrated numerically by taking an example of a specific bilinear matrix game problem.