The effectiveness of layer-by-layer training using the information bottleneck principle

The recently proposed information bottleneck (IB) theory of deep nets suggests that during training, each layer attempts to maximize its mutual information (MI) with the target labels (so as to allow good prediction accuracy), while minimizing its MI with the input (leading to effective compression and thus good generalization). To date, evidence of this phenomenon has been indirect and aroused controversy due to theoretical and practical complications. In particular, it has been pointed out that the MI with the input is theoretically infinite in many cases of interest, and that the MI with the target is fundamentally difficult to estimate in high dimensions. As a consequence, the validity of this theory has been questioned. In this paper, we overcome these obstacles by two means. First, as previously suggested, we replace the MI with the input by a noise-regularized version, which ensures it is finite. As we show, this modified penalty in fact acts as a form of weight decay regularization. Second, to obtain accurate (noise regularized) MI estimates between an intermediate representation and the input, we incorporate the strong prior-knowledge we have about their relation, into the recently proposed MI estimator of Belghazi et al. (2018). With this scheme, we are able to stably train each layer independently to explicitly optimize the IB functional. Surprisingly, this leads to enhanced prediction accuracy, thus directly validating the IB theory of deep nets for the first time.