Flavors of Margin: Implicit Bias of Steepest Descent in Homogeneous Neural Networks

Abstract: We study the implicit bias of the general family of steepest descent algorithms with infinitesimal learning rate in deep homogeneous neural networks. We show that: (a) an algorithm-dependent geometric margin starts increasing once the networks reach perfect training accuracy, and (b) any limit point of the training trajectory corresponds to a KKT point of the corresponding margin-maximization problem. We experimentally zoom into the trajectories of neural networks optimized with various steepest descent algorithms, highlighting connections to the implicit bias of popular adaptive methods (Adam and Shampoo).

• The paper shows steepest descent methods, including gradient, coordinate, and sign descent, increase an algorithm-dependent margin when achieving perfect training accuracy in homogeneous neural networks.
• It introduces a "soft" geometric margin concept crucial for tracking progress and demonstrating margin monotonicity beyond perfect training accuracy.
• The research proves asymptotic limit points of steepest descent flows align with generalized KKT points of a margin-maximization problem, validated by experiments.

An Analysis of Implicit Bias in Steepest Descent for Homogeneous Neural Networks

The paper "Flavors of Margin: Implicit Bias of Steepest Descent in Homogeneous Neural Networks" explores the implicit bias exhibited by a broader class of optimization algorithms, namely steepest descent methods, when applied to deep, homogeneous neural networks. This family of algorithms encapsulates gradient descent, coordinate descent, and sign descent, among others. The research presents a detailed theoretical framework, extending our understanding of optimization-induced biases beyond what is typically associated with standard gradient descent.

Key Contributions and Insights

1. Implicit Bias Characterization: The paper postulates that when neural networks achieve perfect training accuracy, steepest descent methods initiate an increase in an algorithm-dependent geometric margin. This is similar to, but more general than, the behavior observed in gradient descent. The authors rigorously prove that these algorithms reduce a generalized Bregman divergence, thus moving towards generalized stationary points.
2. Algorithm-Dependent Margin: In neural networks, the notion of simplicity varies depending on the algorithm's geometry. The paper introduces a "soft" geometric margin to track algorithm progress, a critical component in demonstrating margin monotonicity beyond the point of perfect training accuracy.
3. Convergence to Generalized KKT Points: It is demonstrated that any asymptotic limit point of steepest descent flows is aligned with a generalized KKT point of a margin-maximization problem. This theoretical result generalizes previous findings for gradient descent to a wider array of descent methods by leveraging a notion of directional convergence defined in terms of a generalized Bregman distance.
4. Experimentation and Empirical Validation: The experimental component reinforces theoretical claims by training neural networks under various steepest descent algorithms. The outcomes underscore the progressive enhancement in corresponding margins and highlight the nuanced differences in implicit bias imposed by each descent method. Particularly, they shed light on the connection between Adam and sign descent methods.

Implications and Future Directions

The implications of establishing such a generalized framework for implicit bias are twofold. Practically, understanding the biases inherent in different optimization strategies allows practitioners to select algorithms that naturally align with desired model properties, such as robustness or efficiency in specific applications (e.g., language processing with Adam). Theoretically, these insights expand the bias-variance tradeoff discourse by attributing expected model behavior directly to algorithmic choices rather than architectural constraints alone.

Future inquiries might explore extending this analysis to architectures beyond homogeneous networks or integrating insights derived from this implicit bias understanding into the design of new, potentially more effective, optimization algorithms. Moreover, further exploration of how these biases interact with the dynamically adaptive properties of algorithms like Adam could yield richer directional insights.

In conclusion, this paper substantiates the inherent biases of steepest descent algorithms in maximizing geometric margins within the context of deep learning, elucidating a pivotal aspect of neural network training dynamics that holds substantial promise for both theoretical advancement and practical enhancement in machine learning disciplines.