Directional convergence and alignment in deep learning (2006.06657v2)

Abstract: In this paper, we show that although the minimizers of cross-entropy and related classification losses are off at infinity, network weights learned by gradient flow converge in direction, with an immediate corollary that network predictions, training errors, and the margin distribution also converge. This proof holds for deep homogeneous networks -- a broad class of networks allowing for ReLU, max-pooling, linear, and convolutional layers -- and we additionally provide empirical support not just close to the theory (e.g., the AlexNet), but also on non-homogeneous networks (e.g., the DenseNet). If the network further has locally Lipschitz gradients, we show that these gradients also converge in direction, and asymptotically align with the gradient flow path, with consequences on margin maximization, convergence of saliency maps, and a few other settings. Our analysis complements and is distinct from the well-known neural tangent and mean-field theories, and in particular makes no requirements on network width and initialization, instead merely requiring perfect classification accuracy. The proof proceeds by developing a theory of unbounded nonsmooth Kurdyka-{\L}ojasiewicz inequalities for functions definable in an o-minimal structure, and is also applicable outside deep learning.

• The paper demonstrates that despite network weights diverging to infinity, their directions converge to stable limits, ensuring robust prediction margins.
• It employs the Kurdyka-Łojasiewicz inequality for functions in o-minimal structures to prove finite path lengths of normalized weight vectors without relying on width constraints.
• The work shows that locally Lipschitz network gradients align with parameter directions, enhancing saliency map reliability and achieving maximum margin conditions.

Overview of "Directional Convergence and Alignment in Deep Learning"

The paper "Directional Convergence and Alignment in Deep Learning" by Ziwei Ji and Matus Telgarsky provides a rigorous examination of the convergence behavior of network weights in deep learning. Specifically, the paper addresses key theoretical aspects related to the directional convergence of network parameters and the alignment of gradients in deep learning models. These notions are explored within the context of deep homogeneous networks, which encompass a variety of architectures such as ReLU, max-pooling, linear, and convolutional layers but exclude non-homogeneous elements like skip connections and biases.

Directional Convergence

A central claim of the paper is that despite the divergence of network weights to infinity during training, the direction of these weights converges. This directional convergence implies that normalized weight vectors reach a stable limit, which has significant implications for the stability of the prediction margins and thereby the generalization capabilities and adversarial robustness of the model. The authors confirm this behavior not only through theoretical proofs but also via empirical investigations on models like AlexNet and DenseNet.

The primary theoretical tool used to establish directional convergence is the Kurdyka-\L{}ojasiewicz (K-L) inequality adapted for functions definable in an o-minimal structure. This technique assures finite length for the path swept by the normalized weight vectors, which guarantees convergence. Unlike previous theories—such as neural tangent and mean-field—that require assumptions on network width and initialization, this work only necessitates perfect classification accuracy, broadening its applicability.

Gradient Alignment

In addition to demonstrating directional convergence, the paper further claims that if the network gradients are locally Lipschitz, these gradients converge and align with the direction of the parameters. This alignment suggests several practical and theoretical outcomes: the reliability of saliency maps in interpretability studies, robust margin maximization, and convergence in the deep linear and 2-homogeneous network scenarios presented in the analysis.

Saliency map convergence is crucial for interpretability as it ensures that visualization does not arbitrarily change after extended training. The paper outlines how gradient alignment contributes to achieving maximum margin conditions under specific network configurations, notably without imposing width constraints.

Theoretical Implications and Future Directions

The analysis and results presented in this paper diverge from existing frameworks, offering new perspectives and tools relevant for understanding deep learning dynamics, particularly in late-stage training when small risk has been achieved. This work suggests several pathways for further research, such as extending these results beyond binary classification to multi-class settings, exploring non-homogeneous networks with skip connections (like ResNet), and developing convergence guarantees for stochastic gradient descent methods.

Conclusion

In summary, Ji and Telgarsky make significant contributions to the theoretical understanding of deep learning models by proving directional convergence and alignment of parameters and gradients. Their findings pave the way for addressing stability and robustness concerns in deep network models, which are critical for practical applications. By advancing the mathematical framework surrounding these models, the paper opens up new doors for future investigations into the fundamental mechanics of deep learning training processes.