Goal-Oriented Quantization: Applications to Convex Cost Functions with Polyhedral Decision Space

In this paper, the situation in which a receiver has to execute a task from a quantized version of the information source of interest is considered. The task is modeled by the minimization problem of a general cost function f(x;g) for which the decision x has to be taken from quantized parameters g. Especially, we focus on the particular scenario where the decision space is a convex polyhedron with cost function being convex. Furthermore, we propose a new goal-oriented quantization algorithm by combining the procedure of iteratively expanding and reinstating decision set together with Jensen’s inequality. Proposed method could also be extended to some non-convex scenarios, namely, weakly convex cost function whose eigenvalues of Hessian matrix w.r.t decision x are lower-bounded. Numerical results show that proposed algorithm can considerably reduce the optimality loss (OL) compared to conventional approaches or the required number of quantization bits to achieve a certain relative optimality loss.