Phase Transitions in Rate Distortion Theory and Deep Learning

Kurzfassung/Abstract

Rate distortion theory is concerned with optimally encoding a given signal class S using a budget of R bits, as R→∞. We say that S can be compressed at rate s if we can achieve an error of O(R−s) for encoding S; the supremal compression rate is denoted s∗(S). Given a fixed coding scheme, there usually are elements of S that are compressed at a higher rate than s∗(S) by the given coding scheme; we study the size of this set of signals. We show that for certain "nice" signal classes S, a phase transition occurs: We construct a probability measure P on S such that for every coding scheme C and any s>s∗(S), the set of signals encoded with error O(R−s) by C forms a P-null-set. In particular our results apply to balls in Besov and Sobolev spaces that embed compactly into L2(Ω) for a bounded Lipschitz domain Ω. As an application, we show that several existing sharpness results concerning function approximation using deep neural networks are generically sharp.
We also provide quantitative and non-asymptotic bounds on the probability that a random f∈S can be encoded to within accuracy ε using R bits. This result is applied to the problem of approximately representing f∈S to within accuracy ε by a (quantized) neural network that is constrained to have at most W nonzero weights and is generated by an arbitrary "learning" procedure. We show that for any s>s∗(S) there are constants c,C such that, no matter how we choose the "learning" procedure, the probability of success is bounded from above by min{1,2C⋅W⌈log2(1+W)⌉2−c⋅ε−1/s}.