arXiv subjects

Oren Mangoubi

Publications and source records attributed to Oren Mangoubi.

22 records · Page 2Linked to original sources

arXiv2018

Convex Optimization with Unbounded Nonconvex Oracles using Simulated Annealing

We consider the problem of minimizing a convex objective function $F$ when one can only evaluate its noisy approximation $\hat{F}$. Unless one assumes some structure on the noise, $\hat{F}$ may be an arbitrary nonconvex function, making the task of minimizing $F$ intractable. To overcome this, prior work has often focused on the case when $F(x)-\hat{F}(x)$ is uniformly-bounded. In this paper we study the more general case when the noise has magnitude $αF(x) + β$ for some $α, β> 0$, and present a polynomial time algorithm that finds an approximate minimizer of $F$ for this noise model. Previously, Markov chains, such as the stochastic gradient Langevin dynamics, have been used to arrive at approximate solutions to these optimization problems. However, for the noise model considered in this paper, no single temperature allows such a Markov chain to both mix quickly and concentrate near the global minimizer. We bypass this by combining "simulated annealing" with the stochastic gradient Langevin dynamics, and gradually decreasing the temperature of the chain in order to approach the global minimizer. As a corollary one can approximately minimize a nonconvex function that is close to a convex function; however, the closeness can deteriorate as one moves away from the optimum.

cs.DS↗

arXiv2017

Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature

We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold $\mathcal{M}$, which we call the $\textit{geodesic walk}$. We prove that the mixing time of this walk on any manifold with positive sectional curvature $C_{x}(u,v)$ bounded both above and below by $0 < \mathfrak{m}_{2} \leq C_{x}(u,v) \leq \mathfrak{M}_2 < \infty$ is $\mathcal{O}^*\left(\frac{\mathfrak{M}_2}{\mathfrak{m}_2}\right)$. In particular, this bound on the mixing time does not depend explicitly on the dimension of the manifold. In the special case that $\mathcal{M}$ is the boundary of a convex body, we give an explicit and computationally tractable algorithm for approximating the exact geodesic walk. As a consequence, we obtain an algorithm for sampling uniformly from the surface of a convex body that has running time bounded solely in terms of the curvature of the body.

math.PR↗

arXiv2017

Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions

We obtain several quantitative bounds on the mixing properties of the Hamiltonian Monte Carlo (HMC) algorithm for a strongly log-concave target distribution $π$ on $\mathbb{R}^{d}$, showing that HMC mixes quickly in this setting. One of our main results is a dimension-free bound on the mixing of an "ideal" HMC chain, which is used to show that the usual leapfrog implementation of HMC can sample from $π$ using only $\mathcal{O}(d^{\frac{1}{4}})$ gradient evaluations. This dependence on dimension is sharp, and our results significantly extend and improve previous quantitative bounds on the mixing of HMC.

math.PR↗

arXiv2015

Integral geometry for Markov chain Monte Carlo: overcoming the curse of search-subspace dimensionality

We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold $\mathcal{M}$. Our method exploits the symmetries of the algorithms' isotropic random search-direction subspaces to analytically average out the variance in the intersection volume caused by the orientation of the search-subspace with respect to the manifold $\mathcal{M}$ it intersects. This variance can grow exponentially with the dimension of the search-subspace, greatly slowing down the algorithm. Eliminating this variance allows us to use search-subspaces of dimensions many times greater than would otherwise be possible, allowing us to sample very rare events that a lower-dimensional search-subspace would be unlikely to intersect. To extend this method to events that are rare for reasons other than their support $\mathcal{M}$ having a lower dimension, we formulate and prove a new theorem in integral geometry that makes use of the curvature form of the Chern-Gauss-Bonnet theorem to reweight sampling probabilities. On the side, we also apply our theorem to obtain new theoretical bounds for the volumes of real algebraic manifolds. Finally, we demonstrate the computational effectiveness and speedup of our method by numerically applying it to the conditional stochastic Airy operator sampling problem in random matrix theory.

math.PR↗