Depth Separations in Neural Networks: Separating the Dimension from the Accuracy (2402.07248v2)

Abstract: We prove an exponential size separation between depth 2 and depth 3 neural networks (with real inputs), when approximating a $\mathcal{O}(1)$-Lipschitz target function to constant accuracy, with respect to a distribution with support in the unit ball, under the mild assumption that the weights of the depth 2 network are exponentially bounded. This resolves an open problem posed in \citet{safran2019depth}, and proves that the curse of dimensionality manifests itself in depth 2 approximation, even in cases where the target function can be represented efficiently using a depth 3 network. Previously, lower bounds that were used to separate depth 2 from depth 3 networks required that at least one of the Lipschitz constant, target accuracy or (some measure of) the size of the domain of approximation scale \emph{polynomially} with the input dimension, whereas in our result these parameters are fixed to be \emph{constants} independent of the input dimension: our parameters are simultaneously optimal. Our lower bound holds for a wide variety of activation functions, and is based on a novel application of a worst- to average-case random self-reducibility argument, allowing us to leverage depth 2 threshold circuits lower bounds in a new domain.

• The paper demonstrates an exponential separation between depth 2 and depth 3 networks by approximating O(1)-Lipschitz functions with exponentially fewer neurons at constant accuracy.
• The authors employ a novel average-to-worst-case random self-reducibility argument along with threshold circuit lower bounds to validate their theoretical claims.
• The findings reveal the practical limitations of shallow networks and emphasize the critical role of deeper architectures in overcoming high-dimensional approximation challenges.

Depth 2 vs. Depth 3 Neural Network Separations: Insights into Approximating Lipschitz Functions

Introduction to Depth Separations

Depth separation in neural networks is a pivotal concept that explores the structural intricacies of neural models, particularly focusing on the comparative capabilities of shallow and deep networks in approximating complex functions. A significant line of inquiry within this domain has been to understand whether increasing the depth of a neural network—by adding more layers—substantially enhances its approximation power, especially for high-dimensional data.

Overview of the Study

The paper conducted by Safran, Reichman, and Valiant introduces a rigorous framework to address a longstanding problem in neural network theory: the separation between depth 2 and depth 3 networks in approximating certain types of functions. Their work proves an exponential separation between these two depths when approximating an O(1)-Lipschitz target function within a constant accuracy, under the constraints of having support in the unit hypercube and assuming exponentially bounded weights.

Key Contributions

• Exponential Separation: The paper underscores a significant finding that depth 3 networks can achieve what depth 2 networks cannot, i.e., approximating the specified class of functions with exponentially fewer neurons. This is particularly notable since the separation persists even when the target accuracy is held constant, highlighting the intrinsic advantage of deeper architectures for certain approximation tasks.
• Methodological Approach: The authors utilize a novel application of an average- to worst-case random self-reducibility argument. This inventive proof technique, divergent from common analytical methods in the field, leverages threshold circuits lower bounds to establish the main result.
• Practical and Theoretical Implications: On a practical level, this separation result emphasizes the potential limitations of employing shallow networks for approximating functions within specified domains. Theoretically, it enriches our understanding of the "curse of dimensionality" in the context of neural network approximations, shedding light on the intrinsic value of depth.

Related Work and Future Directions

The paper positions itself within an ongoing discussion about the depth-width trade-offs in neural networks. Past research, including foundational work by Eldan and Shamir, has hinted at the benefits of depth over width in achieving compact representations of complex functions. However, the current paper establishes a clear exponential advantage under more restricted conditions, contributing to a nuanced understanding of network architecture design choices.

Future research directions opened by this work include exploring the separations between even deeper architectures (beyond depth 3) and under different constraints on weight magnitudes and activation functions. Additionally, understanding the optimization landscape and learning dynamics of networks that exhibit such depth separations could provide valuable insights into training deep neural models more effectively.

Conclusion

Safran et al.'s investigation into depth separations in neural networks offers a compelling expansion of our knowledge regarding the approximation capabilities of shallow versus deep models. By rigorously proving an exponential separation under precise conditions, the paper not only answers a critical theoretical question but also impacts practical considerations in neural network architecture design. Furthermore, the innovative proof strategy adopted here enriches the methodological toolkit available to researchers in the field, paving the way for future explorations of neural network dynamics.