  Multi-agent systems are being increasingly deployed in challenging
environments for performing complex tasks such as multi-target tracking,
search-and-rescue, and intrusion detection. Notwithstanding the computational
limitations of individual robots, such systems rely on collaboration to sense
and react to the environment. This paper formulates the generic target tracking
problem as a time-varying optimization problem and puts forth an inexact online
gradient descent method for solving it sequentially. The performance of the
proposed algorithm is studied by characterizing its dynamic regret, a notion
common to the online learning literature. Building upon the existing results,
we provide improved regret rates that not only allow non-strongly convex costs
but also explicating the role of the cumulative gradient error. Two distinct
classes of problems are considered: one in which the objective function adheres
to a quadratic growth condition, and another where the objective function is
convex but the variable belongs to a compact domain. For both cases, results
are developed while allowing the error to be either adversarial or arising from
a white noise process. Further, the generality of the proposed framework is
demonstrated by developing online variants of existing stochastic gradient
algorithms and interpreting them as special cases of the proposed inexact
gradient method. The efficacy of the proposed inexact gradient framework is
established on a multi-agent multi-target tracking problem, while its
flexibility is exemplified by generating online movie recommendations for
Movielens $10$M dataset.
