Stochastic Block Mirror Descent Methods for Nonsmooth and Stochastic Optimization

In this paper, we present a new stochastic algorithm, namely the stochastic block mirror descent (SBMD) method for solving large-scale nonsmooth and stochastic optimization problems. The basic idea of this algorithm is to incorporate the block-coordinate decomposition and an incremental block averaging scheme into the classic (stochastic) mirror-descent method, in order to significantly reduce the cost per iteration of the latter algorithm. We establish the rate of convergence of the SBMD method along with its associated large-deviation results for solving general nonsmooth and stochastic optimization problems. We also introduce different variants of this method and establish their rate of convergence for solving strongly convex, smooth, and composite optimization problems, as well as certain nonconvex optimization problems. To the best of our knowledge, all these developments related to the SBMD methods are new in the stochastic optimization literature. Moreover, some of our results also seem to be new for block coordinate descent methods for deterministic optimization

Is demagnetization an efficient optimization method?

Demagnetization, commonly employed to study ferromagnets, has been proposed as the basis for an optimization tool, a method to find the ground state of a disordered system. Here we present a detailed comparison between the ground state and the demagnetized state in the random field Ising model, combing exact results in $d=1$ and numerical solutions in $d=3$. We show that there are important differences between the two states that persist in the thermodynamic limit and thus conclude that AC demagnetization is not an efficient optimization method.Comment: 2 pages, 1 figur

Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets

The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections - the computational bottleneck in many applications - replacing it by a linear optimization step. Despite this advantage, the known convergence rates of the FW method fall behind standard first order methods for most settings of interest. It is an active line of research to derive faster linear optimization-based algorithms for various settings of convex optimization. In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of $\frac{1}{t^2}$. This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order $\frac{1}{t}$, and known to be tight. We show that various balls induced by $\ell_p$ norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. We further show how several previous fast-rate results for the FW method follow easily from our analysis

Path optimization method for the sign problem

We propose a path optimization method (POM) to evade the sign problem in the Monte-Carlo calculations for complex actions. Among many approaches to the sign problem, the Lefschetz-thimble path-integral method and the complex Langevin method are promising and extensively discussed. In these methods, real field variables are complexified and the integration manifold is determined by the flow equations or stochastically sampled. When we have singular points of the action or multiple critical points near the original integral surface, however, we have a risk to encounter the residual and global sign problems or the singular drift term problem. One of the ways to avoid the singular points is to optimize the integration path which is designed not to hit the singular points of the Boltzmann weight. By specifying the one-dimensional integration-path as $z=t+if(t) (f\in \mathbb{R})$ and by optimizing $f(t)$ to enhance the average phase factor, we demonstrate that we can avoid the sign problem in a one-variable toy model for which the complex Langevin method is found to fail. In this proceedings, we propose POM and discuss how we can avoid the sign problem in a toy model. We also discuss the possibility to utilize the neural network to optimize the path.Comment: Talk given at the 35th International Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spain. 8 pages, 4 figures (references are updated in v2

Leitmann's direct method for fractional optimization problems

Based on a method introduced by Leitmann [Internat. J. Non-Linear Mech. {\bf 2} (1967), 55--59], we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control.Comment: Submitted June 16, 2009 and accepted March 15, 2010 for publication in Applied Mathematics and Computation

The q-gradient method for global optimization

The q-gradient is an extension of the classical gradient vector based on the concept of Jackson's derivative. Here we introduce a preliminary version of the q-gradient method for unconstrained global optimization. The main idea behind our approach is the use of the negative of the q-gradient of the objective function as the search direction. In this sense, the method here proposed is a generalization of the well-known steepest descent method. The use of Jackson's derivative has shown to be an effective mechanism for escaping from local minima. The q-gradient method is complemented with strategies to generate the parameter q and to compute the step length in a way that the search process gradually shifts from global in the beginning to almost local search in the end. For testing this new approach, we considered six commonly used test functions and compared our results with three Genetic Algorithms (GAs) considered effective in optimizing multidimensional unimodal and multimodal functions. For the multimodal test functions, the q-gradient method outperformed the GAs, reaching the minimum with a better accuracy and with less function evaluations.Comment: 12 pages, 1 figur