Global optimization and stochastic differential equations

Aluffi-Pentini, Filippo; Parisi, Valerio; Zirilli, Francesco

doi:10.1007/bf00941312

Cited by 209 publications

(100 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Straightforward calculations show that the evolution of the bias m t a n d t h e v ariance 2 Note that this behavior is really di erent from the behavior in a xed environment. In a xed environment v = 0 the asymptotic bias is negligible if compared with the variance 4 .…”

Section: Mathematical Approachmentioning

confidence: 96%

“…In practice this means that we may still apply the Fokker-Planck approximation if we study learning with just one minimum, but must suppress the temptation to extend this approach to learning with various minima. From the linear Fokker-Planck equation 12 and the asymptotic evolution equations 20 we conclude that the asymptotic probability distribution for small learning parameters is a simple Gaussian, with its average at the xed point w and a covariance matrix 2 In this section we will point out the di erence between the "intrinsic" noise due to the random presentation of training patterns and the "arti cial" noise in studies on the generalization capabilities of neural networks see e.g. 57, 64 .…”

Section: A Small-uctuations Expansionmentioning

confidence: 99%

“…To linearize around to xed point, we m ust now also require that the rate of change v _ w is much smaller than the typical weight c hange f. Provided these requirements are ful lled, the evolution of the bias m t and the covariance 2 with notation H t def = H w t , and so on. Let us furthermore assume that the changes in the "speed" v, the di usion D, and the curvature H are so slow that they can be considered constant on the local relaxation time local see equation 19 .…”

Section: Nonlinear Learning Rules In Nonstationary Environmentsmentioning

confidence: 99%

See 2 more Smart Citations

On-line learning processes in artificial neural networks

Heskes

Kappen

1993

North-Holland Mathematical Library

View full text Add to dashboard Cite

We study on-line learning processes in arti cial neural networks from a general point of view. On-line learning means that a learning step takes place at each presentation of a randomly drawn training pattern. It can be viewed as a stochastic process governed by a continuous-time master equation.On-line learning is necessary if not all training patterns are available all the time. This occurs in many applications when the training patterns are drawn from a time-dependent environmental distribution. Studying learning in a changing environment, we encounter a con ict between the adaptability and the con dence of the network's representation. Minimization of a criterion incorporating both e ects yields an algorithm for on-line adaptation of the learning parameter.The inherent noise of on-line learning makes it possible to escape from undesired local minima of the error potential on which the learning rule performs stochastic gradient descent. We try to quantify these often made claims by considering the transition times between various minima. We apply our results on the transitions from "twists" in twodimensional self-organizing maps to perfectly ordered con gurations. Finally, w e discuss the capabilities of on-line learning for global optimization.

show abstract

Section: Mathematical Approachmentioning

confidence: 96%

Section: A Small-uctuations Expansionmentioning

confidence: 99%

Section: Nonlinear Learning Rules In Nonstationary Environmentsmentioning

confidence: 99%

See 1 more Smart Citation

On-line learning processes in artificial neural networks

Heskes

Kappen

1993

North-Holland Mathematical Library

View full text Add to dashboard Cite

show abstract

“…For instance, the algorithms of Piyavskii-Shubert [l, 2] and its variants [3], Mladineo [4], Wood [5], Brent [6], and Breiman & Cutler [7], interval methods [8,9]), and "standard" simulated annealing ( [10] for discrete setting, [11,12] for continuous). All of these algorithms share the properties that they require global information in the form of a parameter ( e.g.…”

Section: Introductionmentioning

confidence: 99%

Global Optimization Requires Global Information

Stephens

Baritompa

1998

Journal of Optimization Theory and Applications

View full text Add to dashboard Cite

There are many global optimization algorithms which do not use global information. We broaden previous results, showing limitations on such algorithms, even if allowed to run forever. We show deterministic algorithms must sample a dense set to fi nd the global optimum value and can never be guaranteed to converge only to global optimizers. Further, analogous results show introducing a stochastic element does not overcome these limitations. An example is simulated annealing in practice. Our results show there are functions for which the probability of success is arbitrarily small. Key Words. Global optimization, convergence, stochastic algorithms; deterministic algorithms Global Optimization Requires Global Information William BaritompaChris StephensSeptember 30, 1996 Abstract There are many global optimization algorithms which do not use global information. We broaden previous results, showing limitations on such algorithms, even if allowed to run forever. We show deterministic algorithms must sample a dense set to find the global optimum value and can never be guaranteed to converge only to global optimizers. Further, analogous results show introducing a stochastic element does not overcome these limitations. An example is simulated annealing in practice. Our results show there are functions for which the probability of success is arbitrarily small.

show abstract

“…We adopt the suggestive name of optimotaxis to designate this search procedure. An important feature of optimotaxis is that it can be used with a broad class of signal profiles, including the ones with multiple maxima, a feature that is shared with a few other stochastic optimization algorithms which are not constrained by vehicle kinematics [14,15].…”

Section: Introductionmentioning

confidence: 99%

Optimotaxis: A Stochastic Multi-agent Optimization Procedure with Point Measurements

Mesquita

Hespanha

Åström

Hybrid Systems: Computation and Control

View full text Add to dashboard Cite

Abstract. We consider the problem of seeking the maximum of a scalar signal using a swarm of autonomous vehicles equipped with sensors that can take point measurements of the signal. Vehicles are not able to measure their current position or to communicate with each other. Our approach induces the vehicles to perform a biased random walk inspired by bacterial chemotaxis and controlled by a stochastic hybrid automaton. With such a controller, it is shown that the positions of the vehicles evolve towards a probability density that is a specified function of the spatial profile of the measured signal, granting higher vehicle densities near the signal maxima.

show abstract

Global optimization and stochastic differential equations

Cited by 209 publications

References 3 publications

On-line learning processes in artificial neural networks

On-line learning processes in artificial neural networks

Global Optimization Requires Global Information

Optimotaxis: A Stochastic Multi-agent Optimization Procedure with Point Measurements

Contact Info

Product

Resources

About