Minimal Communication-Cost Statistical Learning

A client device which has access to
$n$
training data samples needs to obtain a statistical hypothesis or model W and then to send it to a remote server. The client and the server devices share some common randomness sequence as well as a prior on the hypothesis space. In this problem a suitable hypothesis or model
$W$
should meet two distinct design criteria simultaneously: (i) small (population) risk during the inference phase and (ii) small ‘complexity’ for it to be conveyed to the server with minimum communication cost. In this paper, we propose a joint training and source coding scheme with provable in-expectation guarantees, where the expectation is over the encoder's output message. Specifically, we show that by imposing a constraint on a suitable Kullback-Leibler divergence between the conditional distribution induced by a compressed learning model
$\hat{W}$
given
$W$
and the prior, one guarantees simultaneously small average empirical risk (aka training loss), small average generalization error and small average communication cost. We also consider a one-shot scenario in which the guarantees on the empirical risk and generalization error are obtained for every encoder's output message.