arXiv subjects

Rajesh Sundaresan

Publications and source records attributed to Rajesh Sundaresan.

At least 37 records · Page 2Linked to original sources

arXiv2019

Learning to Detect an Odd Markov Arm

A multi-armed bandit with finitely many arms is studied when each arm is a homogeneous Markov process on an underlying finite state space. The transition law of one of the arms, referred to as the odd arm, is different from the common transition law of all other arms. A learner, who has no knowledge of the above transition laws, has to devise a sequential test to identify the index of the odd arm as quickly as possible, subject to an upper bound on the probability of error. For this problem, we derive an asymptotic lower bound on the expected stopping time of any sequential test of the learner, where the asymptotics is as the probability of error vanishes. Furthermore, we propose a sequential test, and show that the asymptotic behaviour of its expected stopping time comes arbitrarily close to that of the lower bound. Prior works deal with independent and identically distributed arms, whereas our work deals with Markov arms. Our analysis of the rested Markov setting is a key first step in understanding the difficult case of restless Markov setting, which is still open.

cs.IT↗

arXiv2019

On The Equivalence of Projections In Relative $α$-Entropy and Rényi Divergence

The aim of this work is to establish that two recently published projection theorems, one dealing with a parametric generalization of relative entropy and another dealing with Rényi divergence, are equivalent under a correspondence on the space of probability measures. Further, we demonstrate that the associated "Pythagorean" theorems are equivalent under this correspondence. Finally, we apply Eguchi's method of obtaining Riemannian metrics from general divergence functions to show that the geometry arising from the above divergences are equivalent under the aforementioned correspondence.

cs.IT↗

arXiv2019

Reputation-Based Information Design for Inducing Prosocial Behavior

We study the idea of information design for inducing prosocial behavior in the context of electricity consumption. We consider a continuum of agents. Each agent has a different intrinsic motivation to reduce her power consumption. Each agent models the power consumption of the others via a distribution. Using this distribution, agents will anticipate their reputational benefit and choose a power consumption by trading off their own intrinsic motivation to do a prosocial action, the cost of this prosocial action and their reputation. Initially, the service provider can provide two types of quantized feedbacks of the power consumption. We study their advantages and disadvantages. For each feedback, we characterize the corresponding mean field equilibrium, using a fixed point equation. Besides computing the mean field equilibrium, we highlight the need for a systematic study of information design, by showing that revealing less information to the society can lead to more prosociality. In the last part of the paper, we introduce the notion of privacy and provide a new quantized feedback, more flexible than the previous ones, that respects agents' privacy concern but at the same time improves prosociality. The results of this study are also applicable to generic resource sharing problems.

cs.SI↗

arXiv2019

Optimal Mechanisms for Selling Two Items to a Single Buyer Having Uniformly Distributed Valuations

We consider the design of a revenue-optimal mechanism when two items are available to be sold to a single buyer whose valuation is uniformly distributed over an arbitrary rectangle $[c_1,c_1+b_1]\times[c_2,c_2+b_2]$ in the positive quadrant. We provide an explicit, complete solution for arbitrary nonnegative values of $(c_1,c_2,b_1,b_2)$. We identify eight simple structures, each with at most $4$ (possibly stochastic) menu items, and prove that the optimal mechanism has one of these eight structures. We also characterize the optimal mechanism as a function of $(c_1,c_2,b_1,b_2)$. The structures indicate that the optimal mechanism involves (a) an interplay of individual sale and a bundle sale when $c_1$ and $c_2$ are low, (b) a bundle sale when $c_1$ and $c_2$ are high, and (c) an individual sale when one of them is high and the other is low. To the best of our knowledge, our results are the first to show the existence of optimal mechanisms with no exclusion region. We further conjecture, based on promising preliminary results, that our methodology can be extended to a wider class of distributions.

cs.GT↗

arXiv2019

On Optimal Mechanisms in the Two-Item Single-Buyer Unit-Demand Setting

We consider the problem of designing a revenue-optimal mechanism in the two-item, single-buyer, unit-demand setting when the buyer's valuations, $(z_1, z_2)$, are uniformly distributed in an arbitrary rectangle $[c,c+b_1]\times[c,c+b_2]$ in the positive quadrant. We provide a complete and explicit solution for arbitrary nonnegative values of $(c,b_1,b_2)$. We identify five simple structures, each with at most five (possibly stochastic) menu items, and prove that the optimal mechanism has one of the five structures. We also characterize the optimal mechanism as a function of $b_1, b_2$, and $c$. When $c$ is low, the optimal mechanism is a posted price mechanism with an exclusion region; when $c$ is high, it is a posted price mechanism without an exclusion region. Our results are the first to show the existence of optimal mechanisms with no exclusion region, to the best of our knowledge.

cs.GT↗

arXiv2019

A Mean-field Approach for Controlling Singularly Perturbed Multi-population SIS Epidemics

We consider a multi-population epidemic model with one or more (almost) isolated communities and one mobile community. Each of the isolated communities has contact within itself and, in addition, contact with the outside world but only through the mobile community. The contact rate between the mobile community and the other communities is assumed to be controlled. We first derive a multidimensional ordinary differential equation (ODE) as a mean-field fluid approximation to the process of the number of infected nodes, after appropriate scaling. We show that the approximation becomes tight as the sizes of the communities grow. We then use a singular perturbation approach to reduce the dimension of the ODE and identify an optimal control policy on this system over a fixed time horizon via Pontryagin's minimum principle. We then show that this policy is close to optimal, within a certain class, on the original problem for large enough communities. From a phenomenological perspective, we show that the epidemic may sustain in time in all communities (and thus the system has a nontrivial metastable regime) even though in the absence of the mobile nodes the epidemic would die out quickly within each of the isolated communities.

math.OC↗

arXiv2018

Network Utility Maximization Revisited: Three Issues and Their Resolution

Distributed and iterative network utility maximization algorithms, such as the primal-dual algorithms or the network-user decomposition algorithms, often involve trajectories where the iterates may be infeasible, convergence to the optimal points of relaxed problems different from the original, or convergence to local maxima. In this paper, we highlight the three issues with iterative algorithms. We then propose a distributed and iterative algorithm that does not suffer from the three issues. In particular, we assert the feasibility of the algorithm's iterates at all times, convergence to the global maximum of the given problem (rather than to global maximum of a relaxed problem), and avoidance of any associated spurious rest points of the dynamics. A benchmark algorithm due to Kelly, Maulloo and Tan (1998) [Rate control for communication networks: shadow prices, proportional fairness and stability, Journal of the Operational Research Society, 49(3), 237-252] involves fast user updates coupled with slow network updates in the form of additive-increase multiplicative-decrease of suggested user flows. The proposed algorithm may be viewed as one with fast user updates and fast network updates that keeps the iterates feasible at all times. Simulations suggest that the convergence rate of the ordinary differential equation (ODE) tracked by our proposed algorithm's iterates is comparable to that of the ODE for the aforementioned benchmark algorithm.

math.OC↗

arXiv2018

Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by \begin{eqnarray*} dX^{\varepsilon}_t &=& b(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \varepsilon^αdB_t, dY^{\varepsilon}_t &=& - \frac{1}{\varepsilon} \nabla_yU(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \frac{s(\varepsilon)}{\sqrt{\varepsilon}} dW_t, \end{eqnarray*} where $B_t, W_t$ are independent Brownian motions on ${\mathbb R}^d$ and ${\mathbb R}^m$ respectively, $b : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow \mathbb{R}$ and $s :(0,\infty) \rightarrow (0,\infty)$. We impose regularity assumptions on $b$, $U$ and let $0 < α< 1.$ When $s(\varepsilon)$ goes to zero slower than a prescribed rate as $\varepsilon \rightarrow 0$, we characterize all weak limit points of $X^{\varepsilon}$, as $\varepsilon \rightarrow 0$, as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of $U(x, \cdot)$ at its global minima we characterize all limit points as Filippov solutions to the differential equation.

math.PR↗

arXiv2018

Augmenting Max-Weight with Explicit Learning for Wireless Scheduling with Switching Costs

In small-cell wireless networks where users are connected to multiple base stations (BSs), it is often advantageous to switch off dynamically a subset of BSs to minimize energy costs. We consider two types of energy cost: (i) the cost of maintaining a BS in the active state, and (ii) the cost of switching a BS from the active state to inactive state. The problem is to operate the network at the lowest possible energy cost (sum of activation and switching costs) subject to queue stability. In this setting, the traditional approach -- a Max-Weight algorithm along with a Lyapunov-based stability argument -- does not suffice to show queue stability, essentially due to the temporal co-evolution between channel scheduling and the BS activation decisions induced by the switching cost. Instead, we develop a learning and BS activation algorithm with slow temporal dynamics, and a Max-Weight based channel scheduler that has fast temporal dynamics. We show using convergence of time-inhomogeneous Markov chains, that the co-evolving dynamics of learning, BS activation and queue lengths lead to near optimal average energy costs along with queue stability.

eess.SY↗

arXiv2017

Almost Budget Balanced Mechanisms with Scalar Bids For Allocation of a Divisible Good

This paper is about allocation of an infinitely divisible good to several rational and strategic agents. The allocation is done by a social planner who has limited information because the agents' valuation functions are taken to be private information known only to the respective agents. We allow only a scalar signal, called a bid, from each agent to the social planner. Yang and Hajek [Jour. on Selected Areas in Comm., 2007] as well as Johari and Tsitsiklis [Jour. of Oper. Res., 2009] proposed a scalar strategy Vickrey-Clarke-Groves (SSVCG) mechanism with efficient Nash equilibria. We consider a setting where the social planner desires minimal budget surplus. Example situations include fair sharing of Internet resources and auctioning of certain public goods where revenue maximization is not a consideration. Under the SSVCG framework, we propose a mechanism that is efficient and comes close to budget balance by returning much of the payments back to the agents in the form of rebates. We identify a design criterion for {\em almost budget balance}, impose feasibility and voluntary participation constraints, simplify the constraints, and arrive at a convex optimization problem to identify the parameters of the rebate functions. The convex optimization problem has a linear objective function and a continuum of linear constraints. We propose a solution method that involves a finite number of constraints, and identify the number of samples sufficient for a good approximation.

cs.GT↗

arXiv2017

The Power of Side-information in Subgraph Detection

In this work, we tackle the problem of hidden community detection. We consider Belief Propagation (BP) applied to the problem of detecting a hidden Erdős-Rényi (ER) graph embedded in a larger and sparser ER graph, in the presence of side-information. We derive two related algorithms based on BP to perform subgraph detection in the presence of two kinds of side-information. The first variant of side-information consists of a set of nodes, called cues, known to be from the subgraph. The second variant of side-information consists of a set of nodes that are cues with a given probability. It was shown in past works that BP without side-information fails to detect the subgraph correctly when an effective signal-to-noise ratio (SNR) parameter falls below a threshold. In contrast, in the presence of non-trivial side-information, we show that the BP algorithm achieves asymptotically zero error for any value of the SNR parameter. We validate our results through simulations on synthetic datasets as well as on a few real world networks.

cs.LG↗

arXiv2016

A Survey of Algorithms for Separable Convex Optimization with Linear Ascending Constraints

The paper considers the minimization of a separable convex function subject to linear ascending constraints. The problem arises as the core optimization in several resource allocation scenarios, and is a special case of an optimization of a separable convex function over the bases of a polymatroid with a certain structure. The paper presents a survey of state-of-the-art algorithms that solve this optimization problem. The algorithms are applicable to the class of separable convex objective functions that need not be smooth or strictly convex. When the objective function is a so-called $d$-separable function, a simpler linear time algorithm solves the problem.

math.OC↗

arXiv2015

Learning to detect an oddball target

We consider the problem of detecting an odd process among a group of Poisson point processes, all having the same rate except the odd process. The actual rates of the odd and non-odd processes are unknown to the decision maker. We consider a time-slotted sequential detection scenario where, at the beginning of each slot, the decision maker can choose which process to observe during that time slot. We are interested in policies that satisfy a given constraint on the probability of false detection. We propose a generalised likelihood ratio based sequential policy which, via suitable thresholding, can be made to satisfy the given constraint on the probability of false detection. Further, we show that the proposed policy is asymptotically optimal in terms of the conditional expected stopping time among all policies that satisfy the constraint on the probability of false detection. The asymptotic is as the probability of false detection is driven to zero. We apply our results to a particular visual search experiment studied recently by neuroscientists. Our model suggests a neuronal dissimilarity index for the visual search task. The neuronal dissimilarity index, when applied to visual search data from the particular experiment, correlates strongly with the behavioural data. However, the new dissimilarity index performs worse than some previously proposed neuronal dissimilarity indices. We explain why this may be attributed to the experiment conditons.

cs.IT↗

arXiv2015

Minimization Problems Based on Relative $α$-Entropy I: Forward Projection

Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative $α$-entropies (denoted $\mathscr{I}_α$), arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $α$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative $α$-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Rényi or Tsallis entropy principle. The minimizing probability distribution (termed forward $\mathscr{I}_α$-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse $\mathscr{I}_α$-projection is studied.

cs.IT↗

arXiv2015

Minimization Problems Based on Relative $α$-Entropy II: Reverse Projection

In part I of this two-part work, certain minimization problems based on a parametric family of relative entropies (denoted $\mathscr{I}_α$) were studied. Such minimizers were called forward $\mathscr{I}_α$-projections. Here, a complementary class of minimization problems leading to the so-called reverse $\mathscr{I}_α$-projections are studied. Reverse $\mathscr{I}_α$-projections, particularly on log-convex or power-law families, are of interest in robust estimation problems ($α>1$) and in constrained compression settings ($α<1$). Orthogonality of the power-law family with an associated linear family is first established and is then exploited to turn a reverse $\mathscr{I}_α$-projection into a forward $\mathscr{I}_α$-projection. The transformed problem is a simpler quasiconvex minimization subject to linear constraints.

cs.IT↗

arXiv2015

Active Search with a Cost for Switching Actions

Active Sequential Hypothesis Testing (ASHT) is an extension of the classical sequential hypothesis testing problem with controls. Chernoff (Ann. Math. Statist., 1959) proposed a policy called Procedure A and showed its asymptotic optimality as the cost of sampling was driven to zero. In this paper we study a further extension where we introduce costs for switching of actions. We show that a modification of Chernoff's Procedure A, one that we call Sluggish Procedure A, is asymptotically optimal even with switching costs. The growth rate of the total cost, as the probability of false detection is driven to zero, and as a switching parameter of the Sluggish Procedure A is driven down to zero, is the same as that without switching costs.

cs.IT↗

arXiv2015

Neural dissimilarity indices that predict oddball detection in behaviour

Neuroscientists have recently shown that images that are difficult to find in visual search elicit similar patterns of firing across a population of recorded neurons. The $L^{1}$ distance between firing rate vectors associated with two images was strongly correlated with the inverse of decision time in behaviour. But why should decision times be correlated with $L^{1}$ distance? What is the decision-theoretic basis? In our decision theoretic formulation, we modeled visual search as an active sequential hypothesis testing problem with switching costs. Our analysis suggests an appropriate neuronal dissimilarity index which correlates equally strongly with the inverse of decision time as the $L^{1}$ distance. We also consider a number of other possibilities such as the relative entropy (Kullback-Leibler divergence) and the Chernoff entropy of the firing rate distributions. A more stringent test of equality of means, which would have provided a strong backing for our modeling fails for our proposed as well as the other already discussed dissimilarity indices. However, test statistics from the equality of means test, when used to rank the indices in terms of their ability to explain the observed results, places our proposed dissimilarity index at the top followed by relative entropy, Chernoff entropy and the $L^{1}$ indices. Computations of the different indices requires an estimate of the relative entropy between two Poisson point processes. An estimator is developed and is shown to have near unbiased performance for almost all operating regions.

cs.IT↗

arXiv2014

Relative $α$-Entropy Minimizers Subject to Linear Statistical Constraints

We study minimization of a parametric family of relative entropies, termed relative $α$-entropies (denoted $\mathscr{I}_α(P,Q)$). These arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $α$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimization of $\mathscr{I}_α(P,Q)$ over the first argument on a set of probability distributions that constitutes a linear family is studied. Such a minimization generalizes the maximum Rényi or Tsallis entropy principle. The minimizing probability distribution (termed $\mathscr{I}_α$-projection) for a linear family is shown to have a power-law.

cs.IT↗