A Novelty in Blahut-Arimoto Type Algorithms: Optimal Control Over Noisy Communication Channels

Abstract

A probability-theoretic problem under information constraints for the concept of optimal control over a noisy-memoryless channel is considered. For our Observer-Controller block, i.e., the lossy joint-source-channel-coding (JSCC) scheme, after providing the relative mathematical expressions, we propose a Blahut-Arimoto -type algorithm $-$ which is, to the best of our knowledge, for the first time. The algorithm efficiently finds the probability-mass-functions (PMFs) required for $ \mathop {{\rm min}} _{\mathscr {P}(i), i \in \lbrace \mathscr {Y}, \hat{\mathscr {S}}, \mathscr {X},{\mathscr {S}},\hat{\mathscr {X}}\rbrace } {\rm \; } \phi _1 \mathscr {I}(\mathscr {Y};\hat{\mathscr {S}}|\mathscr {X,S})-\phi _2 \mathscr {I}(\mathscr {Y};\hat{\mathscr {X}}|\mathscr {X,S})$ . This problem is an $NP-$ hard and non-convex multi-objective optimisation (MOO) one, were the objective functions are respectively the distortion function $dim (Null (\mathscr {I}(\hat{\mathscr {S}};{\mathscr {S}})) \rightarrow \infty$ and the memoryless-channel capacity $dim (Null (\mathscr {I}(\mathscr {X};\hat{\mathscr {X}})) \rightarrow 0$ . Our novel algorithm applies an Alternating optimisation method. Subsequently, a robust version of the algorithm is discussed with regard to the perturbed PMFs $-$ parameter uncertainties in general. The aforementioned robustness is actualised by exploiting the simultaneous-perturbation-stochastic-approximation (SPSA). The principles of detectability-and-stabilisability as well as synchronisability are explored, in addition to providing the simulations - by which the efficiency of our work is shown. We also calculate the total complexity of our proposed algorithms respectively as $\mathscr {O} (\mathscr {T}\mathscr {K}\mathscr {M}_0(\mathscr {K} \log \mathscr {K}))$ and $\mathscr {O} (\mathscr {T}\mathscr {K}\mathscr {M}_0(\mathscr {K} \log \mathscr {K}+0.33 \mathscr {K}))$ . Our methodology is generic which can be applied to other fields of studies which are optimisation-driven.