Biconvex Landscape In SDP-Related Learning
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Many machine learning problems can be reduced to learning a low-rank positive semidefinite matrix (denoted as Z), which encounters semidefinite program (SDP). Existing SDP solvers are often expensive for large-scale learning. To avoid directly solving SDP, some works convert SDP into a nonconvex program by factorizing Z as XX^. However, this would bring higher-order nonlinearity, resulting in scarcity of structure in subsequent optimization. In this paper, we propose a novel surrogate for SDP-related learning, in which the structure of subproblem is exploited. More specifically, we surrogate unconstrained SDP by a biconvex problem, through factorizing Z as XY^ and using a Courant penalty to penalize the difference of X and Y, in which the resultant subproblems are convex. Furthermore, we provide a theoretical bound for the associated penalty parameter under the assumption that the objective function is Lipschitz-smooth, such that the proposed surrogate will solve the original SDP when the penalty parameter is larger than this bound. Experiments on two SDP-related machine learning applications demonstrate that the proposed algorithm is as accurate as the state-of-the-art, but is faster on large-scale learning.
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