Learning Boolean Circuits with Neural Networks (1910.11923v2)

Abstract: While on some natural distributions, neural-networks are trained efficiently using gradient-based algorithms, it is known that learning them is computationally hard in the worst-case. To separate hard from easy to learn distributions, we observe the property of local correlation: correlation between local patterns of the input and the target label. We focus on learning deep neural-networks using a gradient-based algorithm, when the target function is a tree-structured Boolean circuit. We show that in this case, the existence of correlation between the gates of the circuit and the target label determines whether the optimization succeeds or fails. Using this result, we show that neural-networks can learn the (log n)-parity problem for most product distributions. These results hint that local correlation may play an important role in separating easy/hard to learn distributions. We also obtain a novel depth separation result, in which we show that a shallow network cannot express some functions, while there exists an efficient gradient-based algorithm that can learn the very same functions using a deep network. The negative expressivity result for shallow networks is obtained by a reduction from results in communication complexity, that may be of independent interest.

• The paper introduces the local correlation property that enables gradient-based algorithms to learn tree-structured Boolean circuits efficiently.
• The paper demonstrates that deep networks can efficiently learn complex functions that shallow architectures cannot represent.
• The paper shows that, under most product distributions, gradient-based methods can solve the log n-parity problem, highlighting practical algorithm design.

Learning Boolean Circuits with Neural Networks

Overview

The paper by Eran Malach and Shai Shalev-Shwartz fundamentally addresses the challenge of computationally efficient learning in neural networks, particularly focusing on learning Boolean circuits. Though neural networks, particularly deep ones, show empirical success in various applications, theoretically, their training can be computationally hard in the worst case. This paper aims to identify data distribution properties that allow for efficient learning, bridging the gap between theoretical challenges and practical successes.

Key Contributions

1. Local Correlation Property: The paper introduces the concept of local correlation, which involves the correlation between local patterns of the input data and the target labels. This property is posited to distinguish between data distributions that are easy or hard to learn.
2. Tree-Structured Boolean Circuits: The research focuses on learning tree-structured Boolean circuits with neural networks. A significant result is the observation that the success of a gradient-based algorithm in learning these circuits is contingent on the existence of local correlations between the circuit gates and the target labels.
3. Learning $(\log n)$-Parity Problems: For most product distributions, the gradient-based neural network algorithm can learn the $(\log n)$-parity problem. This underscores the practical importance of local correlations in facilitating efficient learning processes.
4. Depth Separation: The paper establishes a novel depth-separation result. Specifically, it demonstrates that certain functions, which cannot be efficiently expressed by shallow networks, can be effectively learned by deep networks using a gradient-based algorithm. This result leverages reductions from communication complexity literature and highlights instances where deep networks provide distinct computational advantages.

Theoretical Implications

The theoretically significant aspect of the paper is its depth separation result. It stands as the first of its kind to show that for specific distributions, computationally efficient learning with gradient-based algorithms is possible with deep networks but not with shallow ones. This result advocates for the necessity of depth in neural network architectures, particularly when dealing with complex target functions.

Practical Implications

1. Algorithm Design: The findings suggest that leveraging local consistency within data can significantly enhance the performance of gradient-based algorithms. This can inform the design of new learning algorithms and neural network architectures tailored for specific classes of problems.
2. Data Distribution Analysis: Understanding that local correlations can facilitate efficient learning offers a new perspective for analyzing and preprocessing data. By ensuring that training data inherently possesses such correlations, one could potentially achieve more effective learning outcomes.

Future Directions

1. Generalization to Other Circuit Structures: Future research can explore whether similar results hold for more complex circuit structures beyond tree-structured Boolean circuits. Generalizing the theoretical framework could broaden the applicability of these findings.
2. Application to Broader Neural Architectures: Extending the analysis of local correlations to other neural network architectures, such as convolutional neural networks (CNNs) used in image processing, could yield insights into their success mechanisms.
3. Empirical Evaluation on Natural Data: While the present work is primarily theoretical, future studies could empirically validate these findings on real-world datasets to observe the practical significance of local correlation properties.

In conclusion, the research by Malach and Shalev-Shwartz offers a robust theoretical framework that elucidates the role of depth in neural network architecture and the importance of local correlations in effective learning. This not only advances the theoretical understanding of neural network learning but also has potential implications for practical applications and future research in machine learning and artificial intelligence.