Phase Transitions in Rate Distortion Theory and Deep Learning

TL;DR

This paper investigates phase transitions in rate distortion theory and their implications for deep learning, demonstrating that for certain signal classes, encoding errors beyond a critical rate are almost surely unattainable, with applications to neural network approximation.

Contribution

It establishes a phase transition phenomenon in rate distortion theory for specific signal classes and connects this to the sharpness of neural network approximation results.

Findings

Phase transition occurs where encoding errors become almost surely impossible beyond a critical rate.

Sharpness results for neural network approximation are shown to be generically optimal.

Provides probabilistic bounds on encoding success for neural networks with constrained weights.

Abstract

Rate distortion theory is concerned with optimally encoding a given signal class $\mathcal{S}$ using a budget of $R$ bits, as $R\to\infty$. We say that $\mathcal{S}$ can be compressed at rate $s$ if we can achieve an error of $\mathcal{O}(R^{-s})$ for encoding $\mathcal{S}$; the supremal compression rate is denoted $s^\ast(\mathcal{S})$. Given a fixed coding scheme, there usually are elements of $\mathcal{S}$ that are compressed at a higher rate than $s^\ast(\mathcal{S})$ by the given coding scheme; we study the size of this set of signals. We show that for certain "nice" signal classes $\mathcal{S}$, a phase transition occurs: We construct a probability measure $\mathbb{P}$ on $\mathcal{S}$ such that for every coding scheme $\mathcal{C}$ and any $s >s^\ast(\mathcal{S})$, the set of signals encoded with error $\mathcal{O}(R^{-s})$ by $\mathcal{C}$ forms a $\mathbb{P}$-null-set. In particular our results apply to balls in Besov and Sobolev spaces that embed compactly into $L^2(\Omega)$ for a bounded Lipschitz domain $\Omega$. 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 $f\in\mathcal{S}$ can be encoded to within accuracy $\varepsilon$ using $R$ bits. This result is applied to the problem of approximately representing $f\in\mathcal{S}$ to within accuracy $\varepsilon$ 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 $s >s^\ast(\mathcal{S})$ there are constants $c,C$ such that, no matter how we choose the "learning" procedure, the probability of success is bounded from above by $\min\big\{1,2^{C\cdot W\lceil\log_2(1+W)\rceil^2 -c\cdot\varepsilon^{-1/s}}\big\}$.