Search

Your search keyword '"Karandikar, Rajeeva L."' showing total 260 results
Start Over Author "Karandikar, Rajeeva L."
260 results on '"Karandikar, Rajeeva L."'

3. Convergence Rates for Stochastic Approximation: Biased Noise with Unbounded Variance, and Applications

Author

Karandikar, Rajeeva L. and Vidyasagar, M.

Subjects
Statistics - Machine Learning, Computer Science - Machine Learning, Mathematics - Optimization and Control, Mathematics - Probability, 62L20, 60G17, 93D05
Abstract

In this paper, we study the convergence properties of the Stochastic Gradient Descent (SGD) method for finding a stationary point of a given objective function $J(\cdot)$. The objective function is not required to be convex. Rather, our results apply to a class of ``invex'' functions, which have the property that every stationary point is also a global minimizer. First, it is assumed that $J(\cdot)$ satisfies a property that is slightly weaker than the Kurdyka-Lojasiewicz (KL) condition, denoted here as (KL'). It is shown that the iterations $J(\boldsymbol{\theta}_t)$ converge almost surely to the global minimum of $J(\cdot)$. Next, the hypothesis on $J(\cdot)$ is strengthened from (KL') to the Polyak-Lojasiewicz (PL) condition. With this stronger hypothesis, we derive estimates on the rate of convergence of $J(\boldsymbol{\theta}_t)$ to its limit. Using these results, we show that for functions satisfying the PL property, the convergence rate of both the objective function and the norm of the gradient with SGD is the same as the best-possible rate for convex functions. While some results along these lines have been published in the past, our contributions contain two distinct improvements. First, the assumptions on the stochastic gradient are more general than elsewhere, and second, our convergence is almost sure, and not in expectation. We also study SGD when only function evaluations are permitted. In this setting, we determine the ``optimal'' increments or the size of the perturbations. Using the same set of ideas, we establish the global convergence of the Stochastic Approximation (SA) algorithm under more general assumptions on the measurement error, compared to the existing literature. We also derive bounds on the rate of convergence of the SA algorithm under appropriate assumptions., Comment: 35 pages, 3 figures

Published
2023

4. Convergence of Batch Asynchronous Stochastic Approximation With Applications to Reinforcement Learning

Author

Karandikar, Rajeeva L. and Vidyasagar, M.

Subjects
Statistics - Machine Learning, Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Electrical Engineering and Systems Science - Systems and Control, Mathematics - Probability, 62L20, 60G17, 93D05
Abstract

We begin by briefly surveying some results on the convergence of the Stochastic Gradient Descent (SGD) Method, proved in a companion paper by the present authors. These results are based on viewing SGD as a version of Stochastic Approximation (SA). Ever since its introduction in the classic paper of Robbins and Monro in 1951, SA has become a standard tool for finding a solution of an equation of the form $f(\theta) = 0$, when only noisy measurements of $f(\cdot)$ are available. In most situations, \textit{every component} of the putative solution $\theta_t$ is updated at each step $t$. In some applications in Reinforcement Learning (RL), \textit{only one component} of $\theta_t$ is updated at each $t$. This is known as \textbf{asynchronous} SA. In this paper, we study \textbf{Block Asynchronous SA (BASA)}, in which, at each step $t$, \textit{some but not necessarily all} components of $\theta_t$ are updated. The theory presented here embraces both conventional (synchronous) SA as well as asynchronous SA, and all in-between possibilities. We provide sufficient conditions for the convergence of BASA, and also prove bounds on the \textit{rate} of convergence of $\theta_t$ to the solution. For the case of conventional SGD, these results reduce to those proved in our companion paper. Then we apply these results to the problem of finding a fixed point of a map with only noisy measurements. This problem arises frequently in RL. We prove sufficient conditions for convergence as well as estimates for the rate of convergence., Comment: 34 pages, 1 figure

Published
2021

5. Stochastic Integrals and Two Filtrations

Author

Karandikar, Rajeeva L. and Rao, B. V.

Subjects
Mathematics - Probability
Abstract

In the definition of the stochastic integral, apart from the integrand and the integrator, there is an underlying filtration that plays a role. Thus, it is natural to ask: {\it Does the stochastic integral depend upon the filtration?} In other words, if we have two filtrations, $({\mathcal F}_\centerdot)$ and $({\mathcal G}_\centerdot)$, a process $X$ that is semimartingale under both the filtrations and a process $f$ that is predictable for both the filtrations, then are the two stochastic integrals - $Y=\int f\,dX$, with filtration $({\mathcal F}_\centerdot)$ and $Z=\int f\,dX$, with filtration $({\mathcal G}_\centerdot)$ the same? When $f$ is left continuous with right limits, then the answer is yes. When one filtration is an enlargement of the other, the two integrals are equal if $f$ is bounded but this may not be the case when $f$ is unbounded. We discuss this and give sufficient conditions under which the two integrals are equal.

Published
2020

7. On the Second Fundamental Theorem of Asset Pricing

Author

Karandikar, Rajeeva L and Rao, B V

Subjects
Mathematics - Probability, 91G20, 60G44, 97M30, 62P05
Abstract

Let $X^1,\ldots, X^d$ be sigma-martingales on $(\Omega,{\cal F}, P)$. We show that every bounded martingale (with respect to the underlying filtration) admits an integral representation w.r.t. $X^1,\ldots, X^d$ if and only if there is no equivalent probability measure (other than $P$) under which $X^1,\ldots,X^d$ are sigma-martingales. From this we deduce the second fundamental theorem of asset pricing- that completeness of a market is equivalent to uniqueness of Equivalent Sigma-Martingale Measure (ESMM).

Published
2015

8. Discrete Parameter Martingales

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

9. The Davis Inequality

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

10. SDE Driven by r.c.l.l. Semimartingales

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

11. Pathwise Formula for the Stochastic Integral

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

12. Semimartingales

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

13. Predictable Increasing Processes

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

14. Continuous-Time Processes

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

15. Stochastic Integration

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

16. The Ito’s Integral

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

17. Continuous Semimartingales

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

18. Girsanov Theorem

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

19. Integral Representation of Martingales

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

20. Dominating Process of a Semimartingale

Author

Karandikar, Rajeeva L., Rao, B. V., Biswas, Atanu, Associate Editor, Bhat, B.V. Rajarama, Editor-in-Chief, Bhatt, Abhay G., Editor-in-Chief, Chaudhuri, Arijit, Associate Editor, Daya Sagar, B.S., Associate Editor, Chattopadhyay, Joydeb, Editor-in-Chief, Ponnusamy, S., Editor-in-Chief, Delampady, Mohan, Associate Editor, Ghosh, Ashish, Associate Editor, Neogy, S. K., Associate Editor, Raja, C. R. E., Associate Editor, Rao, T. S. S. R. K., Associate Editor, Sen, Rituparna, Associate Editor, Sury, B., Associate Editor, Karandikar, Rajeeva L., and Rao, B. V.

Published
2018
Full Text
View/download PDF

21. Stochastic approximation in infinite dimensions.

Author

Karandikar, Rajeeva L. and Rao, B. V.

Subjects
*BANACH spaces, *STOCHASTIC approximation, *HILBERT space, *SIGNAL processing, *MARTINGALES (Mathematics)
Abstract

Stochastic approximation (SA) was introduced in the early 1950s 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 optimization. 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 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 픹 is C([0, 1], ℝd), as well as ℒ1([0, 1], ℝd), the two cases which do not even have the Radon–Nikodym property. [ABSTRACT FROM AUTHOR]

Published
2024
Full Text
View/download PDF

22. On characterisation of Markov processes via martingale problems

Author

Bhatt, Abhay G, Karandikar, Rajeeva L, and Rao, B V

Subjects
Mathematics - Probability, 60G17, 60G44
Abstract

It is well-known that well-posedness of a martingale problem in the class of continuous (or r.c.l.l.) solutions enables one to construct the associated transition probability functions. We extend this result to the case when the martingale problem is well-posed in the class of solutions which are continuous in probability. This extension is used to improve on a criterion for a probability measure to be invariant for the semigroup associated with the Markov process. We also give examples of martingale problems that are well-posed in the class of solutions which are continuous in probability but for which no r.c.l.l. solution exists., Comment: 14 pages

Published
2006

23. Measure free martingales

Author

Karandikar, Rajeeva L and Nadkarni, M G

Subjects
Mathematics - Probability, 60G42, 60G48
Abstract

We give a necessary and sufficient condition on a sequence of functions on a set $\Omega$ under which there is a measure on $\Omega$ which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities., Comment: 6 pages

Published
2005

25. Complete Markets

Author

Kallianpur, Gopinath, Karandikar, Rajeeva L., Kallianpur, Gopinath, and Karandikar, Rajeeva L.

Published
2000
Full Text
View/download PDF

44. FOREWORD

Author

Bose, Arup, Chaudhuri, Probal, and Karandikar, Rajeeva L.

Published
2007

Catalog

Books, media, physical & digital resources
See catalog results