In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning (1412.6614v4)

Abstract: We present experiments demonstrating that some other form of capacity control, different from network size, plays a central role in learning multilayer feed-forward networks. We argue, partially through analogy to matrix factorization, that this is an inductive bias that can help shed light on deep learning.

• The paper demonstrates that implicit regularization, rather than network size, primarily drives generalization in deep networks.
• Experimental results on MNIST and CIFAR-10 show that even highly overparameterized models can achieve robust generalization.
• The study draws a matrix factorization analogy, proposing infinite-sized, bounded-norm networks to explain the implicit bias in optimization.

On the Role of Implicit Regularization in Deep Learning

This paper investigates the concept of implicit regularization as an alternative form of inductive bias in learning multi-layer feed-forward networks. The authors challenge the traditional view that network size strictly governs capacity control and generalization. Through a series of experiments, they observe that merely increasing network size does not necessarily correlate with overfitting as might be expected according to classic statistical learning theory.

Key Insights

The paper begins by addressing the inadequacy of viewing network size as the primary capacity control parameter. It emphasizes that successful generalization in neural networks may largely depend on implicit mechanisms rather than explicit constraints like network architecture or size. Through empirical experiments, particularly on datasets such as MNIST and CIFAR-10, the authors demonstrate that increasing the number of hidden units beyond the number required to fit the training data does not necessarily lead to increased generalization error.

Several counterintuitive phenomena are noted: even when networks are substantially larger than necessary, the generalization error does not worsen. This suggests the presence of an implicit regularization effect that inherently biases the optimization toward simpler solutions, contributing to better generalization performance.

Matrix Factorization Analogy

The authors draw an analogy between neural networks and matrix factorization, suggesting that in the context of linear activations, norm-regularized models might offer a more effective inductive bias than simple dimensionality constraints. For instance, trace-norm regularization in matrix factorization is deemed superior due to its convexity and established generalization capabilities. This comparison implies that the success seen in deep learning might similarly be attributed to some form of implicit norm regularization.

Infinite-Sized Networks and Convex Neural Nets

As a theoretical extension, the paper proposes an infinite-size, bounded-norm network approach, which aligns with the notion of implicit $\ell_2$ regularization equivalency to a convex neural network employing $\ell_1$ regularization on top layers. This perspective hints at the potential model of employing infinite hidden layers with controlled norms, paralleling the construct of convex neural networks.

The theoretical conversion from weight decay in finite networks to $\ell_1$ regularization in convex neural networks is rigorously derived, emphasizing the impact of implicit regularization dynamics over traditional architecture concerns.

Implications and Future Directions

The findings suggest that implicit regularization mechanisms, potentially arising from the optimization processes in deep learning, might be the key factor driving their ability to generalize effectively. This notion opens up new lines of inquiry regarding optimization dynamics and implicit biases in neural networks.

The insights have profound implications on both the theoretical understanding and practical applications of deep learning. For practitioners, this could mean rethinking network design strategies and focusing more on the training processes and optimization pathways. For researchers, exploring the complex interplay between implicit regularization and model generalization offers fertile ground for future work in AI.

Conclusion

The paper's challenges to conventional views on neural network capacity control provoke a reconsideration of how deep learning models achieve generalization. By highlighting implicit regularization's potential role, it encourages a shift in focus from model architecture to the intricacies involved in the training algorithms and regularization dynamics, suggesting a deeper understanding of optimization biases might unlock further advancements in AI.