On-Line Learning in Neural Networks by David Saad

By David Saad

Online studying is likely one of the most typically used thoughts for education neural networks. even though it's been used effectively in lots of real-world functions, such a lot education equipment are in accordance with heuristic observations. the shortcoming of theoretical aid damages the credibility in addition to the potency of neural networks education, making it tough to settle on trustworthy or optimum tools. This booklet offers a coherent photograph of the cutting-edge within the theoretical research of online studying. An advent relates the topic to different advancements in neural networks and explains the general photo. Surveys via major specialists within the box mix new and verified fabric and let nonexperts to benefit extra concerning the ideas and strategies used. This ebook, the 1st within the sector, presents a complete view of the topic and should be welcomed via mathematicians, scientists and engineers, either in and academia.

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Let us assume that the algorithm converges. 42). 43) t—>oo This result means that the quantities 7*varzff (z, w*) and (wt — w)VwC(w) keep the same order of magnitude during the convergence. Since the latter quantity is related to the distance to the optimum (cf. 2) the convergence speed depends on how fast the learning rates 7^ decrease. 39). This analysis can be repeated with non scalar learning rates approximating the inverse of the Hessian. This algorithm converges faster than using a scalar learning rate equal to the inverse of the largest eigenvalue of the Hessian.

The convergence of ft means that the norm w\ of the parameter vector wt is bounded. 3) guarantees that the parameters will be confined in a bounded region containing the origin. This confinement property means that all continuous functions of wt are bounded (we assume of course that the parameter space has finite dimension). This include w%, E z (iJ(z,w)) 2 and all the derivatives of the cost function C(wt). In the rest of this section, positive constants K\, K2, etc.. are introduced whenever such a bound is used.

We can then bound the variations of the criterion ht using a first order Taylor expansion and bounding the second derivatives with K\. | ht+l -ht + 2ytH(x, wt)VwC(wt) | < 7 ? s. s. 30) implies that ht = C(wt) converges almost surely. 17) x Step c. The last step of the proof departs from the convex case. Proving that C(wt) converges to zero would be a very strong result, equivalent to proving the convergence to the global minimum. We can however prove that the gradient VwC(wt) converges to zero almost surely.

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