Frank E. Curtis

Regional Complexity Analysis of Algorithms for Nonconvex Smooth Optimization

  • Frank E. Curtis
  • Daniel P. Robinson
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an objective function, an upper bound on the number of iterations (or function or derivative evaluations) required until a pth-order stationarity condition is approximately satisfied. This arguably … Read more

    A Stochastic Trust Region Algorithm Based on Careful Step Normalization

  • Frank E. Curtis
  • Katya Scheinberg
  • Rui Shi
    • Categories Nonlinear Optimization, Stochastic Programming, Unconstrained Optimization Tags , , , , , ,

    An algorithm is proposed for solving stochastic and finite sum minimization problems. Based on a trust region methodology, the algorithm employs normalized steps, at least as long as the norms of the stochastic gradient estimates are within a specified interval. The complete algorithm—which dynamically chooses whether or not to employ normalized steps—is proved to have … Read more

    An Accelerated Communication-Efficient Primal-Dual Optimization Framework for Structured Machine Learning

  • Frank E. Curtis
  • Martin Jaggi
  • Chenxin Ma
  • Nathan Srebro
  • Martin Takac
    • Categories Nonlinear Optimization, Parallel Algorithms

    Distributed optimization algorithms are essential for training machine learning models on very large-scale datasets. However, they often suffer from communication bottlenecks. Confronting this issue, a communication-efficient primal-dual coordinate ascent framework (CoCoA) and its improved variant CoCoA+ have been proposed, achieving a convergence rate of $\mathcal{O}(1/t)$ for solving empirical risk minimization problems with Lipschitz continuous losses. … Read more

    A Self-Correcting Variable-Metric Algorithm Framework for Nonsmooth Optimization

  • Frank E. Curtis
  • Daniel P. Robinson
  • Baoyu Zhou
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    An algorithm framework is proposed for minimizing nonsmooth functions. The framework is variable-metric in that, in each iteration, a step is computed using a symmetric positive definite matrix whose value is updated as in a quasi-Newton scheme. However, unlike previously proposed variable-metric algorithms for minimizing nonsmooth functions, the framework exploits self-correcting properties made possible through … Read more

    An Inexact Regularized Newton Framework with a Worst-Case Iteration Complexity of $\mathcal{O}(\epsilon^{-3/2})$ for Nonconvex Optimization

  • Frank E. Curtis
  • Daniel P. Robinson
  • Mohammadreza Samadi
    • Categories Nonlinear Optimization, Unconstrained Optimization Tags , , , , ,

    An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes $\mathcal{O}(\epsilon^{-3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive real number $\epsilon$ and can take $\mathcal{O}(\epsilon^{-3})$ iterations to drive the leftmost eigenvalue of the Hessian of the objective above $-\epsilon$. The proposed … Read more

    Exploiting Negative Curvature in Deterministic and Stochastic Optimization

  • Frank E. Curtis
  • Daniel P. Robinson
    • Categories Stochastic Programming, Unconstrained Optimization Tags , , ,

    This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although some prior work has established convergence results for algorithms that integrate both descent and negative curvature directions, there has not yet been numerical evidence showing that such methods offer significant performance improvements. In … Read more

    Complexity Analysis of a Trust Funnel Algorithm for Equality Constrained Optimization

  • Frank E. Curtis
  • Daniel P. Robinson
  • Mohammadreza Samadi
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    A method is proposed for solving equality constrained nonlinear optimization problems involving twice continuously differentiable functions. The method employs a trust funnel approach consisting of two phases: a first phase to locate an $\epsilon$-feasible point and a second phase to seek optimality while maintaining at least $\epsilon$-feasibility. A two-phase approach of this kind based on … Read more

    R-Linear Convergence of Limited Memory Steepest Descent

  • Frank E. Curtis
  • Wei Guo
    • Categories Nonlinear Optimization, Unconstrained Optimization

    The limited memory steepest descent method (LMSD) proposed by Fletcher is an extension of the Barzilai-Borwein “two-point step size” strategy for steepest descent methods for solving unconstrained optimization problems. It is known that the Barzilai-Borwein strategy yields a method with an R-linear rate of convergence when it is employed to minimize a strongly convex quadratic. … Read more

    A Sequential Algorithm for Solving Nonlinear Optimization Problems with Chance Constraints

  • Frank E. Curtis
  • Victor M. Zavala
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , , ,

    An algorithm is presented for solving nonlinear optimization problems with chance constraints, i.e., those in which a constraint involving an uncertain parameter must be satisfied with at least a minimum probability. In particular, the algorithm is designed to solve cardinality-constrained nonlinear optimization problems that arise in sample average approximations of chance-constrained problems, as well as … Read more

    Optimization Methods for Large-Scale Machine Learning

  • Léon Bottou
  • Frank E. Curtis
  • Jorge Nocedal
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Programming Tags , , , , ,

    This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of … Read more