Content deleted Content added
m Task 18 (cosmetic): eval 20 templates: hyphenate params (1×); |
Ritalinowitz (talk | contribs) |
||
Line 14:
::<math>\theta_{n+1}=\theta_n - a_n(N(\theta_n) - \alpha)</math>
Here, <math>a_1, a_2, \dots</math> is a sequence of positive step sizes. [[Herbert Robbins|Robbins]] and Monro proved<ref name="rm" /><sup>, Theorem 2</sup> that <math>\theta_n</math> [[convergence of random variables|converges]] in <math>L^2</math> (and hence also in probability) to <math>\theta^*</math>, and Blum<ref name=":0">{{Cite journal|last=Blum|first=Julius R.|date=1954-06-01|title=Approximation Methods which Converge with Probability one|journal=The Annals of Mathematical Statistics|language=EN|volume=25|issue=2|pages=382–386|doi=10.1214/aoms/1177728794|issn=0003-4851|doi-access=free}}</ref> later proved the convergence is actually with probability one, provided that:
* <math display="inline">N(\theta)</math> is uniformly bounded,
* <math display="inline">M(\theta)</math> is nondecreasing,
| |||