| Popis: |
The first part of this thesis develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy - i.e., the approximation error decays exponentially in the number of nonzero weights in the network - of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function - a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks. The second part of this thesis shows that every d-dimensional probability distribution with bounded support can be generated through deep ReLU networks out of a one-dimensional uniform input distribution. What is more, this is possible without incurring a cost - in terms of approximation error measured in Wasserstein - distance - relative to generating the d-dimensional target distribution from d independent random variables. This is enabled by a vast generalization of the space-filling approach discovered recently in (Bailey & Telgarsky, 2018). The construction we propose elicits the importance of network depth in driving the Wasserstein distance between the target distribution and its neural network approximation to zero. Finally, we find that, for histogram target distributions, the number of bits needed to encode the corresponding generative network equals the fundamental limit for encoding probability distributions as dictated by quantization theory. |
