Title: An Online Convex Optimization Approach to Dynamic Network Resource Allocation,
Abstract: Existing approaches to online convex optimization (OCO) make sequential
one-slot-ahead decisions, which lead to (possibly adversarial) losses that
drive subsequent decision iterates. Their performance is evaluated by the
so-called regret that measures the difference of losses between the online
solution and the best yet fixed overall solution in hindsight. The present
paper deals with online convex optimization involving adversarial loss
functions and adversarial constraints, where the constraints are revealed after
making decisions, and can be tolerable to instantaneous violations but must be
satisfied in the long term. Performance of an online algorithm in this setting
is assessed by: i) the difference of its losses relative to the best dynamic
solution with one-slot-ahead information of the loss function and the
constraint (that is here termed dynamic regret); and, ii) the accumulated
amount of constraint violations (that is here termed dynamic fit). In this
context, a modified online saddle-point (MOSP) scheme is developed, and proved
to simultaneously yield sub-linear dynamic regret and fit, provided that the
accumulated variations of per-slot minimizers and constraints are sub-linearly
growing with time. MOSP is also applied to the dynamic network resource
allocation task, and it is compared with the well-known stochastic dual
gradient method. Under various scenarios, numerical experiments demonstrate the
performance gain of MOSP relative to the state-of-the-art.