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An Improved Deterministic Rescaling for Linear Programming Algorithms

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10328)

Abstract

The perceptron algorithm for linear programming, arising from machine learning, has been around since the 1950s. While not a polynomial-time algorithm, it is useful in practice due to its simplicity and robustness. In 2004, Dunagan and Vempala showed that a randomized rescaling turns the perceptron method into a polynomial time algorithm, and later Peña and Soheili gave a deterministic rescaling. In this paper, we give a deterministic rescaling for the perceptron algorithm that improves upon the previous rescaling methods by making it possible to rescale much earlier. This results in a faster running time for the rescaled perceptron algorithm. We will also demonstrate that the same rescaling methods yield a polynomial time algorithm based on the multiplicative weights update method. This draws a connection to an area that has received a lot of recent attention in theoretical computer science.

Keywords

  • Polynomial Time
  • Unit Ball
  • Convex Body
  • Gradient Descent
  • Symmetric Positive Definite Matrix

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

T. Rothvoss—Supported by an Alfred P. Sloan Research Fellowship. Both authors supported by NSF grant 1420180 with title “Limitations of convex relaxations in combinatorial optimization”.

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Notes

  1. 1.

    The \(\tilde{O}\)-notation suppresses any \({\text {polylog}}(m,n)\) terms.

  2. 2.

    It suffices here to consider the trivial example with \(\lambda _1=\ldots = \lambda _n = \frac{1}{n}\) and \(A_i = e_i\) being the standard basis. Then \(\Vert \sum _{i \in J} \lambda _i A_i\Vert _2 \le \frac{1}{\sqrt{n}}\) for any subset J. The optimality of our rescaling can also be seen since the cone in the last iteration is \(\tilde{O}(n)\)-well rounded, which is optimal up to \(\tilde{O}\)-terms.

  3. 3.

    Recall that a function \(F : {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) is Lipschitz with Lipschitz constant 1 if \(|F(x)-F(y)| \le \Vert x-y\Vert _2\) for all \(x,y \in {\mathbb {R}}^n\). A famous concentration inequality by Sudakov, Tsirelson, Borell states that \(\Pr [|F(g)-\mu | \ge t] \le e^{-t^2/\pi ^2}\), where g is a random Gaussian and \(\mu \) is the mean of F under g.

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Correspondence to Rebecca Hoberg .

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Hoberg, R., Rothvoss, T. (2017). An Improved Deterministic Rescaling for Linear Programming Algorithms. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_22

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  • DOI: https://doi.org/10.1007/978-3-319-59250-3_22

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