Implicit Bias of Gradient Descent on Linear Convolutional Networks (1806.00468v2)

Abstract: We show that gradient descent on full-width linear convolutional networks of depth $L$ converges to a linear predictor related to the $\ell_{2/L}$ bridge penalty in the frequency domain. This is in contrast to linearly fully connected networks, where gradient descent converges to the hard margin linear support vector machine solution, regardless of depth.

• The paper demonstrates that gradient descent on linear convolutional networks converges to solutions that emphasize sparsity in the frequency domain.
• It contrasts this behavior with fully connected networks, where gradient descent aligns with an ℓ2 maximum margin bias across all depths.
• The study leverages Fourier analysis to reveal how network depth influences implicit regularization, offering insights for enhancing CNN generalization.

Implicit Bias of Gradient Descent on Linear Convolutional Networks

The paper "Implicit Bias of Gradient Descent on Linear Convolutional Networks" provides a thorough analysis of how gradient descent informs the learning of linear predictors when applied to linear convolutional networks. At the core, this paper investigates the nuances of implicit regularization introduced by gradient descent optimization on over-parameterized models such as linear networks. An emphasis is placed on contrasting the behaviors exhibited by linear convolutional and fully connected networks.

Key Findings and Theoretical Contributions

The principal result highlights a profound divergence in the implicit bias of gradient descent across different neural architectures: fully connected linear networks versus linear convolutional networks. The authors show that gradient descent on fully connected networks, regardless of depth, converges to solutions influenced by the $\ell_2$ maximum margin implicature, aligning with prior findings on linear logistic regression and the support vector machine paradigm. Specifically, the asymptotic behavior of gradient descent iterates mimics the maximization of margin with respect to the $\ell_2$ norm under linear separability constraints.

In stark contrast, the analysis discovers a different flavor of implicit bias for linear convolutional networks. Here, the bias shifts towards minimizing sparsity in the frequency domain, which is achieved through the $\ell_{2/L}$ bridge penalty related to the Fourier transform of the linear predictors. Intriguingly, this bias becomes more pronounced with network depth, effectively encouraging solutions sparse in their frequency representations. This transformation from a spatial to frequency domain regularization imposes a depth-dependent penalty, yielding solutions with different generalization characteristics compared to their fully connected counterparts.

Methodology

The researchers engage both theoretical examination and framework formulations to support their claims. The mathematical backbone includes formulating the problem in terms of homogeneous polynomial maps from parameters to linear predictors. The alignment with $\ell_2$ bias is demonstrated using established optimization theories, extending these to the field of linear models by exploiting the properties of polynomial homogeneity. Moreover, using Fourier transformation analysis, the paper bridges linear convolutional network dynamics with diagonal networks, elucidating the frequency domain representations.

Implications and Future Directions

Practical repercussions of these findings bear significance particularly for the domain of convolutional neural networks (CNNs) used ubiquitously in visual data processing. The inherent bias towards frequency domain sparsity presents an opportunity to harness gradient descent not only for optimization but implicitly for enhancing generalization through an architecturally tailored regularization.

The paper argues that this unique regulative bias inferred merely from parameterization and optimization, without explicit hand-crafted constraints, accounts for part of the empirical success and generalization prowess observed in deep CNNs. The insights open avenues for refining architectures or designing custom regularizers that might further leverage this phenomenon.

Future work could extend these findings to convolutional architectures used in both constrained convolution scenarios— where filter width substantially undercuts input dimensionality—and more complex network topologies involving nonlinear activations, pooling, and multiple outputs. Exploring the implicit biases under these modifications may potentially unearth more refined understandings or uncover additional implicit biases advantageous for model generalizations across various domains.

In conclusion, the convergence behavior under gradient descent, distinct between convolutional and fully connected linear networks, highlights the implicit inductive biases gravitating convolutional networks towards frequency domain sparsity. This discovery underscores the multidimensional nature of biases intrinsic to deep learning architectures—an insight crucial for future computational advances in machine learning.