We consider online convex optimization (OCO) over a heterogeneous network with communication delay, where multiple workers together with a master execute a sequence of decisions to minimize the accumulation of time-varying global costs. The local dat
a may not be independent or identically distributed, and the global cost functions may not be locally separable. Due to communication delay, neither the master nor the workers have in-time information about the current global cost function. We propose a new algorithm, termed Hierarchical OCO (HiOCO), which takes full advantage of the network heterogeneity in information timeliness and computation capacity to enable multi-step gradient descent at both the workers and the master. We analyze the impacts of the unique hierarchical architecture, multi-slot delay, and gradient estimation error to derive upper bounds on the dynamic regret of HiOCO, which measures the gap of costs between HiOCO and an offline globally optimal performance benchmark.
We show that for any von Neumann measurement, we can construct a logically reversible measurement such that Shannon entropies and quantum discords induced by the two measurements have compact connections. In particular, we prove that quantum discord
for the logically reversible measurement is never less than that for the von Neumann measurement.
