Optimizing costly functions with simple constraints: a limited-memory projected quasi-Newton algorithm

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Abstract

An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus low-rank quadratic approximation to the function. The quadratic approximation is constructed using a limited-memory quasi-Newton update. The method is suitable for large-scale problems where evaluation of the function is substantially more expensive than projection onto the constraint set. Numerical experiments on one-norm regularized test problems indicate that the proposed method is competitive with state-of-the-art methods such as bound-constrained L-BFGS and orthant-wise descent. We further show that the method generalizes to a wide class of problems, and substantially improves on state-of-the-art methods for problems such as learning the structure of Gaussian graphical models and Markov random fields.

BiBTeX

@InProceedings{SchmidtBergFriedlanderMurphy:2009,
  author =       {M. Schmidt, E. van den Berg, M. P. Friedlander,
                  K. Murphy},
  title =        {Optimizing Costly Functions with Simple Constraints:
                  A Limited-Memory Projected Quasi-Newton Algorithm},
  booktitle =    {Proceedings of The Twelfth International Conference
                  on Artificial Intelligence and Statistics (AISTATS)
                  2009},
  pages =        {456-463},
  year =         2009,
  editor =       {D. van Dyk and M. Welling},
  volume =       5,
  address =      {Clearwater Beach, Florida},
  month =        {April},
}