Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks (1610.09887v3)

Abstract: We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower networks are much larger. This includes indicators of balls and ellipses; non-linear functions which are radial with respect to the $L_1$ norm; and smooth non-linear functions. We also show that these gaps can be observed experimentally: Increasing the depth indeed allows better learning than increasing width, when training neural networks to learn an indicator of a unit ball.

• The paper proves that deeper architectures can approximate natural functions, such as indicators of balls and ellipsoids, using polynomial width while shallow networks require exponential width.
• It validates these theoretical insights with experiments demonstrating that 3-layer networks efficiently learn unit ball indicator functions compared to 2-layer counterparts.
• It also shows that depth significantly improves approximation for radial L1 and smooth C^2 functions, offering new guidelines for neural network design.

Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks

The paper "Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks" by Itay Safran and Ohad Shamir investigates the advantages of neural network depth over width in approximating certain natural functions. The core premise is grounded in theoretical findings and experimental validation, demonstrating the pivotal role of depth in expressiveness and approximation efficiency for specific problem settings. The results distill into rigorous mathematical findings alongside practical implications, enriching the understanding of neural network architecture design principles.

Theoretical Insights

The paper embarks on a series of constructive proofs highlighting depth-based separation results. It examines the expressive power of feed-forward neural networks by analyzing functions that can be approximated better using deeper networks, despite potentially increased network size when expanding width alone.

• Indicators of Balls and Ellipsoids: The research demonstrates that indicators of Euclidean unit balls and general ellipsoids necessitate depth for high-fidelity approximation. Specifically, it proves that such functions cannot be approximated to a high degree of accuracy by 2-layer networks unless the network width grows exponentially with the dimension $d$. Conversely, 3-layer networks can approximate these functions efficiently with polynomially scaled width and depth with respect to the approximation error.
• Experiments Validating Depth Efficiency: Complementing theoretical results, the experimental paper supports the assertion that 3-layer networks outperform wider 2-layer networks in learning the indicator function of a unit ball, corroborating the theoretical implications regarding approximation error barriers encountered by shallow networks.
• Radial $L_1$ Norm Functions: Extending beyond $L_2$-norm functions, the paper tackles $L_1$ radial functions, showcasing that a piecewise-linear target function cannot be efficiently approximated using shallow networks without significant width scaling, establishing substantial size lower bounds.
• Complex Smooth Functions: For complex $C^2$ functions, an intricate depth-width tradeoff is elaborated. The paper conveys that depth can exponentially enhance accuracy relative to wide but shallow architectures. It highlights that functions realizable with bounded arithmetic complexity can be precisely approximated by networks with moderate depth, asserting depth's superiority in managing approximation errors.

Implications and Future Work

The theoretical findings in this paper have nuanced implications for understanding neural network design. Depth emerges as a critical factor in efficiently capturing nonlinear patterns and structural intricacies inherent in data. This work paves the way for designing architectures tailored to specific approximation tasks, especially where depth can leverage compositional expressivity for better results.

The paper sets the stage for further exploration into the depth-width dynamics across a broader range of functions and network architectures. Future developments may refine bounds on network configurations, reinforcing depth’s necessity in practice, particularly for high-dimensional data or tasks requiring precision.

In conclusion, Itay Safran and Ohad Shamir's exploration explores fundamental questions on neural network architecture, laying a robust theoretical foundation and guiding empirical exploration. Through comprehensive analysis, it underscores the non-trivial depth requirements for approximating targeted functions, ushering advancements in neural network research and application.