arXiv subjects
Publications and source records attributed to Rajeeva Laxman Karandikar.
The thrust of this article is to show that on all Banach spaces, stochastic approximation holds when the noise sequence is an i.i.d. sequence with mean 0, without imposing any condition on the geometry of the space. Also, the same is true when the noise is a sequence of independent random variables under appropriate conditions on the moment. In this case, we need to require that the noise sequence is tight.
Impact factor, H-index, citation index, and such other indices have been playing an increasing role in scientific assessment of institutions, researchers, allocation of research funds,... across the globe. These indicies do not have any statistical basis but lots of decisions such as ranking of institutions, ranking of departments, assessment of faculty members for hiring and for promotions as well as selection for various awards are being made by using these indices. Several experts across disciplines have been writing that these indicies should have a marginal role, if any, and judgements should be based on critical assessment of the content by experts. But the dependence on these s steadily increasing. This article cites various such documents being published across the globe and across the disciplies and urging decision makers to ignore or at best give a minimal weight to such indices.
In this paper, we introduce a new approach to proving the convergence of the Stochastic Approximation (SA) and the Stochastic Gradient Descent (SGD) algorithms. The new approach is based on a concept called GSLLN (Generalized Strong Law of Large Numbers), which extends the traditional SLLN. Using this concept, we provide sufficient conditions for convergence, which effectively decouple the properties of the function whose zero we are trying to find, from the properties of the measurement errors (noise sequence). The new approach provides an alternative to the two widely used approaches, namely the ODE approach and the martingale approach, and also permits a wider class of noise signals than either of the two known approaches. In particular, the ``noise'' or measurement error \textit{need not} have a finite second moment, and under suitable conditions, not even a finite mean. By adapting this method of proof, we also derive sufficient conditions for the convergence of zero-order SGD, wherein the stochastic gradient is computed using $2d$ function evaluations, but no gradient computations. The sufficient conditions derived here are the weakest to date, thus leading to a considerable expansion of the applicability of SA and SGD theory.
Stochastic Approximation (SA) was introduced in the early 1950's and has been an active area of research for several decades. While the initial focus was on statistical questions, it was seen to have applications to signal processing, convex optimisation. %Over the last decade, there has been a revival of interest in SA as In later years SA has found application in Reinforced Learning (RL) and led to revival of interest. While bulk of the literature is on SA for the case when the observations are from a finite dimensional Euclidian space, there has been interest in extending the same to infinite dimension. Extension to Hilbert spaces is relatively easier to do, but this is not so when we come to a Banach space - since in the case of a Banach space, even {\em law of large numbers} is not true in general. We consider some cases where approximation works in a Banach space. Our framework includes case when the Banach space $\Bb$ is $\Cb([0,1],\R^d)$, as well as $Ł^1([0,1],\R^d)$, the two cases which do not even have the Radon-Nikodym property.