Oblivious Sketching-based Central Path Method for Solving Linear Programming Problems

Zhao Song, Zheng Yu

Keywords: optimization, sketching, linear programming, central path method, running time complexity

Abstract: In this work, we propose a sketching-based central path method for solving linear programmings, whose running time matches the state of art results [Cohen, Lee, Song STOC 19; Lee, Song, Zhang COLT 19]. Our method opens up the iterations of the central path method and deploys an "iterate and sketch" approach towards the problem by introducing a new coordinate-wise embedding technique, which may be of independent interest. Compare to previous methods, the work [Cohen, Lee, Song STOC 19] enjoys feasibility while being non-oblivious, and [Lee, Song, Zhang COLT 19] is oblivious but infeasible, and relies on $\mathit{dense}$ sketching matrices such as subsampled randomized Hadamard/Fourier transform matrices. Our method enjoys the benefits of being both oblivious and feasible, and can use $\mathit{sparse}$ sketching matrix [Nelson, Nguyen FOCS 13] to speed up the online matrix-vector multiplication. Our framework for solving LP naturally generalizes to a broader class of convex optimization problems including empirical risk minimization.

One-sentence Summary: We propose a sketching-based central path method for solving linear programs, which has the same running time as the state of art algorithms and enjoys the advantages of being "oblivious" and "feasible".

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