Multiple kernel learning, conic duality, and the SMO algorithm

TLDR

Classical SVMs use a single kernel, but combining multiple kernels is desirable; however, optimizing conic combinations leads to a QCQP that current convex solvers can handle only for few kernels and data points, and SMO methods cannot be applied due to non‑differentiability. The authors reformulate the QCQP as a second‑order cone program and apply Moreau‑Yosida regularization, enabling the use of SMO techniques for efficient optimization. They propose a novel dual formulation of the QCQP as a second‑order cone programming problem and show how to exploit Moreau‑Yosida regularization to yield a formulation to which SMO techniques can be applied. Experiments demonstrate that the SMO‑based algorithm outperforms general‑purpose interior‑point solvers in efficiency. Lanckriet et al.

Abstract

While classical kernel-based classifiers are based on a single kernel, in practice it is often desirable to base classifiers on combinations of multiple kernels. Lanckriet et al. (2004) considered conic combinations of kernel matrices for the support vector machine (SVM), and showed that the optimization of the coefficients of such a combination reduces to a convex optimization problem known as a quadratically-constrained quadratic program (QCQP). Unfortunately, current convex optimization toolboxes can solve this problem only for a small number of kernels and a small number of data points; moreover, the sequential minimal optimization (SMO) techniques that are essential in large-scale implementations of the SVM cannot be applied because the cost function is non-differentiable. We propose a novel dual formulation of the QCQP as a second-order cone programming problem, and show how to exploit the technique of Moreau-Yosida regularization to yield a formulation to which SMO techniques can be applied. We present experimental results that show that our SMO-based algorithm is significantly more efficient than the general-purpose interior point methods available in current optimization toolboxes.