Variance Reduction Result for a Projected Adaptive Biasing Force Method

This paper is committed to investigate an extension of the classical adaptive biasing force method, which is used to compute the free energy related to the Boltzmann-Gibbs measure and a reaction coordinate function. The issue of this technique is that the approximated gradient of the free energy, called biasing force, is not a gradient. The commitment to this field is to project the estimated biasing force on a gradient using the Helmholtz decomposition. The variance of the biasing force is reduced using this technique, which makes the algorithm more efficient than the standard ABF method. We prove exponential convergence to equilibrium of the estimated free energy, with a precise rate of convergence in function of Logarithmic Sobolev inequality constants

A Lower-Bound Result on the Power of a Genetic Algorithm

This paper presents a lower-bound result on the computational power of a genetic algorithm in the context of combinatorial optimization. We describe a new genetic algorithm, the merged genetic algorithm, and prove that for the class of monotonic functions, the algorithm finds the optimal solution, and does so with an exponential convergence rate. The analysis pertains to the ideal behavior of the algorithm where the main task reduces to showing convergence of probability distributions over the search space of combinatorial structures to the optimal one. We take exponential convergence to be indicative of efficient solvability for the sample-bounded algorithm, although a sampling theory is needed to better relate the limit behavior to actual behavior. The paper concludes with a discussion of some immediate problems that lie ahead

Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach

Computational optimal transport (OT) has recently emerged as a powerful framework with applications in various fields. In this paper we focus on a relaxation of the original OT problem, the entropic OT problem, which allows to implement efficient and practical algorithmic solutions, even in high dimensional settings. This formulation, also known as the Schr\"odinger Bridge problem, notably connects with Stochastic Optimal Control (SOC) and can be solved with the popular Sinkhorn algorithm. In the case of discrete-state spaces, this algorithm is known to have exponential convergence; however, achieving a similar rate of convergence in a more general setting is still an active area of research. In this work, we analyze the convergence of the Sinkhorn algorithm for probability measures defined on the $d$-dimensional torus $\mathbb{T}_L^d$, that admit densities with respect to the Haar measure of $\mathbb{T}_L^d$. In particular, we prove pointwise exponential convergence of Sinkhorn iterates and their gradient. Our proof relies on the connection between these iterates and the evolution along the Hamilton-Jacobi-Bellman equations of value functions obtained from SOC-problems. Our approach is novel in that it is purely probabilistic and relies on coupling by reflection techniques for controlled diffusions on the torus.Comment: 27 page

Controlled Hopwise Averaging: Bandwidth/Energy-Efficient Asynchronous Distributed Averaging for Wireless Networks

This paper addresses the problem of averaging numbers across a wireless network from an important, but largely neglected, viewpoint: bandwidth/energy efficiency. We show that existing distributed averaging schemes have several drawbacks and are inefficient, producing networked dynamical systems that evolve with wasteful communications. Motivated by this, we develop Controlled Hopwise Averaging (CHA), a distributed asynchronous algorithm that attempts to "make the most" out of each iteration by fully exploiting the broadcast nature of wireless medium and enabling control of when to initiate an iteration. We show that CHA admits a common quadratic Lyapunov function for analysis, derive bounds on its exponential convergence rate, and show that they outperform the convergence rate of Pairwise Averaging for some common graphs. We also introduce a new way to apply Lyapunov stability theory, using the Lyapunov function to perform greedy, decentralized, feedback iteration control. Finally, through extensive simulation on random geometric graphs, we show that CHA is substantially more efficient than several existing schemes, requiring far fewer transmissions to complete an averaging task.Comment: 33 pages, 4 figure

Design of First-Order Optimization Algorithms via Sum-of-Squares Programming

In this paper, we propose a framework based on sum-of-squares programming to design iterative first-order optimization algorithms for smooth and strongly convex problems. Our starting point is to develop a polynomial matrix inequality as a sufficient condition for exponential convergence of the algorithm. The entries of this matrix are polynomial functions of the unknown parameters (exponential decay rate, stepsize, momentum coefficient, etc.). We then formulate a polynomial optimization, in which the objective is to optimize the exponential decay rate over the parameters of the algorithm. Finally, we use sum-of-squares programming as a tractable relaxation of the proposed polynomial optimization problem. We illustrate the utility of the proposed framework by designing a first-order algorithm that shares the same structure as Nesterov's accelerated gradient method

Lookback option pricing using the Fourier transform B-spline method

We derive a new, efficient closed-form formula approximating the price of discrete lookback options, whose underlying asset price is driven by an exponential semimartingale process, which includes (jump) diffusions, Lévy models, affine processes and other models. The derivation of our pricing formula is based on inverting the Fourier transform using B-spline approximation theory. We give an error bound for our formula and establish its fast rate of convergence to the true price. Our method provides lookback option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods. We provide an alternative proof to the Spitzer formula for the characteristic function of the maximum of a discretely observed stochastic process, which yields a numerically efficient algorithm based on convolutions. This is an important result which could have a wide range of applications in which the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete lookback options under Brownian motion, jump diffusion models, and the variance gamma process