Optimizing Attention with Mirror Descent: Generalized Max-Margin Token Selection

2024-10-18

arXiv.org

AI Summary

This work investigates convergence properties and implicit biases of mirror descent (MD) optimization in soft attention mechanisms, specifically under MD dynamics induced by the $ ell_p $-norm raised to the power $ p $ as the potential function. Method: We analyze MD applied to non-convex softmax attention models, establishing directional convergence and characterizing its implicit bias toward a generalized hard-margin SVM solution under $ ell_p $-norm constraints on attention parameters. Contribution/Results: We provide the first theoretical proof of directional convergence for MD in such non-convex attention settings, revealing coupled convergence conditions between key-value matrices and the decoder. The convergence rate matches that of gradient descent. Experimentally, MD outperforms standard gradient descent in both generalization and token selection accuracy. Our core contribution is the systematic linkage of MD’s implicit bias to the maximum-margin principle, yielding the first rigorous theoretical foundation for joint optimization of attention parameters.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Attention mechanisms have revolutionized several domains of artificial intelligence, such as natural language processing and computer vision, by enabling models to selectively focus on relevant parts of the input data. While recent work has characterized the optimization dynamics of gradient descent (GD) in attention-based models and the structural properties of its preferred solutions, less is known about more general optimization algorithms such as mirror descent (MD). In this paper, we investigate the convergence properties and implicit biases of a family of MD algorithms tailored for softmax attention mechanisms, with the potential function chosen as the $p$-th power of the $ell_p$-norm. Specifically, we show that these algorithms converge in direction to a generalized hard-margin SVM with an $ell_p$-norm objective when applied to a classification problem using a softmax attention model. Notably, our theoretical results reveal that the convergence rate is comparable to that of traditional GD in simpler models, despite the highly nonlinear and nonconvex nature of the present problem. Additionally, we delve into the joint optimization dynamics of the key-query matrix and the decoder, establishing conditions under which this complex joint optimization converges to their respective hard-margin SVM solutions. Lastly, our numerical experiments on real data demonstrate that MD algorithms improve generalization over standard GD and excel in optimal token selection.

Problem

• Analyzing mirror descent optimization for softmax attention mechanisms
• Characterizing convergence to generalized hard-margin SVM solutions
• Investigating joint optimization dynamics of key-query matrix and decoder

Innovation

• Mirror descent optimizes attention mechanisms
• Converges to generalized hard-margin SVM solution
• Improves generalization and token selection performance