arXiv:2410.14581v4 Announce Type: replace-cross 
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.

دراسة حديثة نُشرت على arXiv تستكشف ديناميات تحسين خوارزميات الانحدار المرآتي (MD) المخصصة لآليات الانتباه softmax، موضحة خصائص تقاربها نحو SVM ذو هامش صلب عام مع هدف معيار $	ext{l}_p$ في مهام التصنيف. تسلط هذه الأبحاث الضوء على إمكانية MD في تحسين آليات الانتباه، التي تعتبر حيوية في تطبيقات الذكاء الاصطناعي مثل معالجة اللغة الطبيعية والرؤية الحاسوبية.

Un estudio reciente publicado en arXiv explora las dinámicas de optimización de los algoritmos de descenso espejo (MD) adaptados para mecanismos de atención softmax, demostrando sus propiedades de convergencia hacia un SVM de margen duro generalizado con un objetivo de norma $	ext{l}_p$ en tareas de clasificación. Esta investigación resalta el potencial del MD para mejorar los mecanismos de atención, que son fundamentales en aplicaciones de IA como el procesamiento del lenguaje natural y la visión por computadora.

Une étude récente publiée sur arXiv explore les dynamiques d'optimisation des algorithmes de descente miroir (MD) adaptés aux mécanismes d'attention softmax, démontrant leurs propriétés de convergence vers un SVM à marge dure généralisée avec un objectif de norme $	ext{l}_p$ dans les tâches de classification. Cette recherche met en lumière le potentiel du MD pour améliorer les mécanismes d'attention, qui sont essentiels dans les applications d'IA telles que le traitement du langage naturel et la vision par ordinateur.

A recent study published on arXiv explores the optimization dynamics of mirror descent (MD) algorithms tailored for softmax attention mechanisms, demonstrating their convergence properties towards a generalized hard-margin SVM with an $	ext{l}_p$-norm objective in classification tasks. This research highlights the potential of MD in enhancing attention mechanisms, which are pivotal in AI applications such as natural language processing and computer vision.

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

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