Vanishing Step Size Research Articles

Distributed Nash Equilibrium Seeking for Aggregative Games With Quantization Constraints

The problem of seeking Nash equilibrium (NE) based on aggregative games under quantization constraints is full of challenges. Although the NE seeking algorithm in continuous-time systems has been studied, this problem in discrete-time systems still needs to be solved urgently. To address this problem, three distributed algorithms are first proposed under three quantization cases, adaptive, random, and time-varying quantizations, based on doubly stochastic communication topology networks. Then, the actions of players would eventually converge to NE under the conditions of vanishing step size and strong monotonicity are proved. Moreover, the convergence rate of the three quantization cases are analyzed, respectively. Finally, numerical experiments are implemented on plug-in hybrid electric vehicles (PHEVs) to validate the effectiveness of the proposed distributed algorithms. Comparing the convergence rates of the three proposed algorithms, the convergence effect of the adaptive quantization is better than that of the other two quantization cases.

Examples of Pathological Dynamics of the Subgradient Method for Lipschitz Path-Differentiable Functions

We show that the vanishing step size subgradient method—widely adopted for machine learning applications—can display rather messy behavior even in the presence of favorable assumptions. We establish that convergence of bounded subgradient sequences may fail even with a Whitney stratifiable objective function satisfying the Kurdyka-Łojasiewicz inequality. Moreover, when the objective function is path-differentiable, we show that various properties all may fail to occur: criticality of the limit points, convergence of the sequence, convergence in values, codimension one of the accumulation set, equality of the accumulation and essential accumulation sets, connectedness of the essential accumulation set, spontaneous slowdown, oscillation compensation, and oscillation perpendicularity to the accumulation set.

Distributed Adaptive Optimization With Weight-Balancing

This article addresses the continuous-time distributed optimization of a strictly convex summation-separable cost function with possibly nonconvex local functions over strongly connected digraphs. Distributed optimization methods in the literature require convexity of local functions, or balanced weights, or vanishing step sizes, or algebraic information (eigenvalues or eigenvectors) of the Laplacian matrix. The solution proposed here covers both weight-balanced and unbalanced digraphs in a unified way, without any of the aforementioned requirements.

On the Convergence of Discrete-Time Linear Systems: A Linear Time-Varying Mann Iteration Converges IFF Its Operator Is Strictly Pseudocontractive

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete-time linear systems. We mainly focus on the so-called Krasnoselskij–Mann iteration, ${x}$ ( $k + 1$ ) = ( $1-\alpha _{k}$ ) ${x}$ ( ${k}$ ) + $\alpha _{k} A~{x}$ ( ${k}$ ), which is relevant for distributed computation in optimization and game theory, when $A$ is not available in a centralized way. We show that strict pseudocontractiveness of the linear operator $x \mapsto Ax$ is not only sufficient (as known) but also necessary for the convergence to a vector in the kernel of $I-A$ . We also characterize some relevant operator-theoretic properties of linear operators via eigenvalue location and linear matrix inequalities. We apply the convergence conditions to multi-agent linear systems with vanishing step sizes, in particular, to linear consensus dynamics and equilibrium seeking in monotone linear-quadratic games.

A piecewise-constant congestion taxing policy for repeated routing games

In this paper, we consider repeated routing games with piecewise-constant congestion taxing in which a central planner sets and announces the congestion taxes for fixed windows of time in advance. Specifically, congestion taxes are calculated using marginal congestion pricing based on the flow of the vehicles on each road prior to the beginning of the taxing window (and, hence, there is a time-varying delay in setting the congestion taxes). We motivate the piecewise-constant taxing policy by that users or drivers may dislike fast-changing prices and that they also prefer prior knowledge of the prices. We prove for this model that the multiplicative update rule and the discretized replicator dynamics converge to a socially optimal flow when using vanishing step sizes. Considering that the algorithm cannot adapt itself to a changing environment when using vanishing step sizes, we propose adopting constant step sizes in this case. Then, however, we can only prove the convergence of the dynamics to a neighborhood of the socially optimal flow (with the size of the neighbourhood being of the order of the selected step size). The results are illustrated on a nonlinear version of Pigou’s famous routing game.

Difference Methods for Quasilinear 2D-diffusion Problems in Toroidal Configurations

Implicit one-step difference methods for quasilinear parabolic initial boundary value problems in two space variables and in polar coordinates are considered. Problems of this kind are suitable for the description of diffusion processes in plasma physics. Therefore, because of its special physical interest the dependence of the coefficients both on the solution and on its space derivatives (the fluxes) is included. The first part of this paper deals with proving convergence of the discrete approximations for vanishing step sizes. For this purpose, bounds for the inverse difference operators have to be derived previously. The nonlinear systems arising from the discretizations have exactly one solution for all step sizes being sufficiently small. In the second part of the article numerical tests are performed. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Convergence and Error Bounds for Passive Stochastic Algorithms Using Vanishing Step Size

This work is concerned with passive stochastic approximation (PSA) algorithms having vanishing or decreasing step size and window width. Unlike the traditional stochastic approximation methods, the passive stochastic approximation algorithms utilize passive strategies. Under the framework of PSA, not only the measurement noise is unobservable, but also the “state” {xn} is a randomly generated sequence. In our formulation, both the observation noise and the randomly generated {xn} are correlated random processes. Under rather general conditions, w.p.1. convergence of the algorithms is established. Then upper bounds on estimation errors are obtained. It is shown that the bounds depend on the smoothness of the function under consideration in an essential way, which reveals another distinct feature of the passive algorithms.

Higher-Order Weak Approximation of Ito Diffusions by Markov Chains

This paper proposes a method that allows the construction of discrete-state Markov chains approximating an Ito-diffusion process. The transition probabilities of the Markov chains are chosen in such a way that functionals converge with a desired weak order with respect to vanishing step size under sufficient smoothness assumptions.

Algorithmic alternatives

A large variety of Monte Carlo algorithms are being used for lattice gauge simulations. For purely bosonic theories, present approaches are generally adequate; nevertheless, overrelaxation techniques promise savings by a factor of about three in computer time. For fermionic fields the situation is more difficult and less clear. Algorithms which involve an extrapolation to a vanishing step size are all quite closely related. Methods which do not require such an approximation tend to require computer time which grows as the square of the volume of the system. Recent developments combining global accept/reject stages with Langevin or microcanonical updatings promise to reduce this growth to V 4 3 .

Über Differenzenverfahren zur numerischen Lösung stark gekoppelter Systeme nichtlinearer Diffusionsgleichungen

A class of nonlinear implicit one step difference methods for quasilinear strongly coupled parabolic systems in two space variables is considered. The main part of the paper deals with proving convergence of the discrete approximations for vanishing step sizes. For this purpose, bounds for the inverse difference operators have to be derived previously. This is possible subject to a condition which can be considered as a generalization of the concept “parabolic” to systems. Finally, it is shown that the nonlinear system s arising from the discretizations have one and only one solution for all step sizes being sufficiently small.

Extremum Control in the Presence of Noise†

Abstract An extremum control system is considered in which the output from a plant is minimized by adjusting inputs to the plant in discrete steps of constant size. The direction of stepping is determined by estimates of the true output in the presence of Gaussian noise. The criterion of steady-state performance is the average deviation of the true output from the minimum. Performance of a system with single-input plant is computed when one estimate of output is made after each step; the analysis is supported by experimental results. Analysis is simpler when two independent estimates are made per step; performances of the two types of system are compared. Limiting results for vanishing step size in the two-estimate system are derived for a single-input plant with both output noise and input disturbance, and for a multi-input plant with output noise only.