Deep Neural Network Approximation Theory (1901.02220v4)

Abstract: This paper 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 paper establishes deep neural networks as Kolmogorov-optimal approximants across complex function classes including Besov and modulation spaces.
• The paper finds that finite-width networks achieve exponential error decay with increased connectivity, setting performance limits beyond traditional methods.
• The paper highlights that deeper networks outperform shallow ones in approximating smooth functions, potentially reducing computational and memory demands.

Deep Neural Network Approximation Theory

This paper addresses the intrinsic limits of deep neural networks in learning tasks, considering theoretical constructs like Kolmogorov-optimal approximation. The primary concern is delineating the relationship between the complexity of the target functions and the approximating network complexity in dimensions of connectivity and storage requirements. Such characterization forms a basis for determining how effectively deep neural networks model complex function classes like Besov and modulation spaces, achieving Kolmogorov-optimal approximation conditions.

Core Contributions

1. Kolmogorov-Optimal Approximants: The paper establishes that deep neural networks serve as Kolmogorov-optimal approximants across various function classes. This includes environments like Besov spaces, modulation spaces, polynomials, sinusoidal functions, and even textured oscillations and deterministic fractal functions like the Weierstrass function. These approximations showcase exponential accuracy — the error decreases exponentially concerning the non-zero weights as the network deepens — a critical insight for Fields like machine learning that rely on these structures for functional representation.
2. Error Decay in Network Approximation: The findings demonstrate that neural networks with finite width can achieve exponential approximation accuracy for an array of function types, extending to challenging cases such as oscillatory textures and the Weierstrass function. Remarkably, the approximation error decays exponentially with network connectivity, setting a performance ceiling unparalleled in traditional approximation schemes.
3. Width vs. Depth in Network Performance: The paper contrasts the capability of neural networks in managing both finite width and depth. Networks with finite width provide exponential approximation across function classes typically resistant to such accuracy through traditional means.
4. Deeper Networks for Smoother Functions: A formal case is made for deep networks over shallow architectures in approximating sufficiently smooth functions. Finite-width deep networks achieve exponential approximation rates, whereas finite-depth networks require superior connectivity scaling for similar approximation fidelity.

Implications and Future Directions

The implications of this paper’s findings suggest a horizon where neural networks reduce the computational and memory demands traditionally associated with highly complex function approximation tasks. Applying these theories to machine learning could streamline model training processes, particularly for high-dimensional data sets typifying modern databases. Potential developments could further explore how networks' architectural choices (like connectivity path optimization) and combined learning paradigms could elevate performance in evolving AI systems.

Theoretical frameworks like these offer foundational steps toward understanding and leveraging deep neural network capabilities in data-centric artificial intelligence, proposing future studies into network topological configurations and the effective exchange between depth, connectivity, and aggregate computational efficacy.