[Paper Review] Regularized Risk Minimization by Nesterov's Accelerated Gradient Methods: Algorithmic Extensions and Empirical Studies

Xinhua Zhang, Ankan Saha+1 more|arXiv (Cornell University)|Nov 1, 2010

Sparse and Compressive Sensing Techniques44 references5 citations

TL;DR

This paper extends Nesterov's accelerated gradient method (AGM) to handle strongly convex and composite regularized risk minimization problems using Bregman divergence-based prox-functions, enabling adaptive Lipschitz constant estimation and duality gap bounding. The authors demonstrate that smoothing the empirical risk improves AGM's performance on max-margin models, enabling it to outperform state-of-the-art solvers through efficient gradient computation and convergence guarantees.

ABSTRACT

Nesterov's accelerated gradient methods (AGM) have been successfully applied in many machine learning areas. However, their empirical performance on training max-margin models has been inferior to existing specialized solvers. In this paper, we first extend AGM to strongly convex and composite objective functions with Bregman style prox-functions. Our unifying framework covers both the $\infty$-memory and 1-memory styles of AGM, tunes the Lipschiz constant adaptively, and bounds the duality gap. Then we demonstrate various ways to apply this framework of methods to a wide range of machine learning problems. Emphasis will be given on their rate of convergence and how to efficiently compute the gradient and optimize the models. The experimental results show that with our extensions AGM outperforms state-of-the-art solvers on max-margin models.

Motivation & Objective

• Address the poor empirical performance of standard Nesterov's accelerated gradient methods (AGM) on max-margin models despite strong theoretical convergence rates.
• Develop a unifying framework that extends AGM to strongly convex and composite objectives using Bregman-style prox-functions.
• Enable adaptive estimation of the Lipschitz constant and tight duality gap bounds in the optimization process.
• Demonstrate that smoothing the empirical risk improves both convergence speed and practical performance of AGM on max-margin problems.
• Show that the proposed framework generalizes to a wide range of machine learning problems with efficient gradient computation and linear convergence under strong convexity.

Proposed method

• Introduce a smoothing technique for the non-smooth empirical risk $ R_{ ext{emp}}(oldsymbol{w}) $, transforming it into a smooth approximation $ ilde{R}_{ ext{emp}}(oldsymbol{w}) $ to enable efficient gradient-based optimization.
• Formulate the regularized risk minimization problem as a composite objective $ J(oldsymbol{w}) = rac{1}{2} orm{oldsymbol{w}}_2^2 + ilde{R}_{ ext{emp}}(oldsymbol{w}) $, where the smoothness enables $ O(1/ heta) $ convergence with $ heta $-accuracy.
• Integrate Bregman divergence-based proximal functions into AGM to handle non-smooth regularizers like $ L_1 $-norms and elastic net, enabling composite optimization.
• Derive a Hessian-based analysis of the smoothed dual function $ g_ u^ullet(Aoldsymbol{w}) $, showing that its Lipschitz constant depends on the maximum eigenvalue of a weighted sum of $ oldsymbol{a}_i oldsymbol{a}_i^ op $, which is minimized via optimal $ b_i $-selection.
• Use power iteration to efficiently compute the dominant eigenvector of $ oldsymbol{A}^ op oldsymbol{A} $, which informs the optimal choice of smoothing parameters $ b_i $ to minimize the Lipschitz constant.
• Apply a homotopy scheme (annealing the smoothing parameter) to progressively refine the solution, leveraging the fact that most dual variables $ oldsymbol{ u}_i $ are zero at optimum, improving approximation quality.

Experimental results

Research questions

• RQ1 Can Nesterov’s accelerated gradient method be effectively extended to composite and strongly convex regularized risk minimization problems using Bregman divergence-based prox-functions?
• RQ2 Does smoothing the empirical risk in max-margin models lead to better practical performance of AGM compared to direct dual or primal-dual formulations?
• RQ3 Can adaptive estimation of the Lipschitz constant and duality gap bounds be achieved within the AGM framework for non-smooth, composite objectives?
• RQ4 How does the choice of smoothing parameters affect the convergence rate and Lipschitz constant of the smoothed objective?
• RQ5 Can the proposed framework outperform specialized solvers in training max-margin models like SVMs while maintaining theoretical convergence guarantees?

Key findings

• Smoothing the empirical risk $ R_{ ext{emp}}(oldsymbol{w}) $ results in a smooth, unconstrained optimization problem that enables $ O(1/ heta) $ convergence with $ heta $-accuracy, significantly improving practical performance of AGM on max-margin models.
• The proposed framework achieves linear convergence when the regularizer $ rac{1}{2} orm{oldsymbol{w}}_2^2 $ is strongly convex, which is critical for fast convergence in practice.
• The Lipschitz constant of the smoothed dual function is minimized by choosing $ b_i^2 = |oldsymbol{a}_i^ op oldsymbol{v}^*| $, where $ oldsymbol{v}^* $ is the dominant eigenvector of $ oldsymbol{A}^ op oldsymbol{A} $, leading to improved convergence speed.
• The homotopy scheme (annealing the smoothing parameter) allows the algorithm to converge quickly to a high-quality solution, as most dual variables are zero at the optimum, making the approximation tight.
• Empirical results show that the improved AGM framework outperforms state-of-the-art specialized solvers on max-margin models, despite AGM’s prior underperformance in this domain.
• The framework generalizes to $ L_1 $-norm, $ L_{1, ext{infty}} $, and elastic net regularizers by leveraging composite optimization techniques, maintaining $ O(1/ heta) $ convergence even when the regularizer is non-differentiable.