arXiv subjects

Divya Padmanabhan

Publications and source records attributed to Divya Padmanabhan.

9 recordsLinked to original sources

arXiv2023

Distributionally robust optimization through the lens of submodularity

Distributionally robust optimization is used to tackle decision making problems under uncertainty where the distribution of the uncertain data is ambiguous. Many ambiguity sets have been proposed for continuous uncertainty that build on convexity and for which the resulting formulations scale polynomially in the number of random variables. However fewer ambiguity sets have been proposed for discrete uncertainty where the exact formulations scale polynomially in the number of random variables. Towards this, we define a submodular ambiguity set and showcase its expressive power in modeling both discrete and continuous uncertainty. With discrete uncertainty, we show that a class of distributionally robust optimization problems is solvable in polynomial time by viewing it through the lens of submodularity. With continuous uncertainty, we show that it is solvable approximately up to an additive error in pseudo-polynomial time. We then focus on a specific class of submodular ambiguity sets where univariate marginal information and bivariate dependence information on the random vector is specified and provide an exact reformulation as a polynomial sized linear program when the uncertainty is discrete and as a polynomial sized semidefinite program when the uncertainty is continuous. We provide numerical evidence of the modeling flexibility and expressive power of the submodular ambiguity set and demonstrate its applicability in two examples: project networks and multi-newsvendor problems. The paper highlights that the submodular ambiguity set is the natural discrete counterpart of the convex ambiguity set and supplements it for continuous uncertainty, both in modeling and computation.

math.OC↗

arXiv2022

Correlation Robust Influence Maximization

We propose a distributionally robust model for the influence maximization problem. Unlike the classic independent cascade model \citep{kempe2003maximizing}, this model's diffusion process is adversarially adapted to the choice of seed set. Hence, instead of optimizing under the assumption that all influence relationships in the network are independent, we seek a seed set whose expected influence under the worst correlation, i.e. the "worst-case, expected influence", is maximized. We show that this worst-case influence can be efficiently computed, and though the optimization is NP-hard, a ($1 - 1/e$) approximation guarantee holds. We also analyze the structure to the adversary's choice of diffusion process, and contrast with established models. Beyond the key computational advantages, we also highlight the extent to which the independence assumption may cost optimality, and provide insights from numerical experiments comparing the adversarial and independent cascade model.

cs.SI↗

arXiv2021

Extremal Probability Bounds in Combinatorial Optimization

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are ``extremal'' since they are valid across all joint distributions with the given marginals. We analyze the complexity of computing the bounds assuming discrete marginals and identify instances when the bounds are computable in polynomial time. For compact 0/1 V-polytopes, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. On the other hand, the tightest lower bound is shown to be strongly NP-hard to compute for compact 0/1 V-polytopes by restricting attention to Bernoulli random variables. For compact 0/1 H-polytopes, for the special case of PERT networks arising in project management, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. The results in the paper complement existing results in the literature for computing the probability with independent random variables.

math.OC↗

arXiv2020

A Dominant Strategy Truthful, Deterministic Multi-Armed Bandit Mechanism with Logarithmic Regret

Stochastic multi-armed bandit (MAB) mechanisms are widely used in sponsored search auctions, crowdsourcing, online procurement, etc. Existing stochastic MAB mechanisms with a deterministic payment rule, proposed in the literature, necessarily suffer a regret of $Ω(T^{2/3})$, where $T$ is the number of time steps. This happens because the existing mechanisms consider the worst case scenario where the means of the agents' stochastic rewards are separated by a very small amount that depends on $T$. We make, and, exploit the crucial observation that in most scenarios, the separation between the agents' rewards is rarely a function of $T$. Moreover, in the case that the rewards of the arms are arbitrarily close, the regret contributed by such sub-optimal arms is minimal. Our idea is to allow the center to indicate the resolution, $Δ$, with which the agents must be distinguished. This immediately leads us to introduce the notion of $Δ$-Regret. Using sponsored search auctions as a concrete example (the same idea applies for other applications as well), we propose a dominant strategy incentive compatible (DSIC) and individually rational (IR), deterministic MAB mechanism, based on ideas from the Upper Confidence Bound (UCB) family of MAB algorithms. Remarkably, the proposed mechanism $Δ$-UCB achieves a $Δ$-regret of $O(\log T)$ for the case of sponsored search auctions. We first establish the results for single slot sponsored search auctions and then non-trivially extend the results to the case where multiple slots are to be allocated.

cs.GT↗

arXiv2019

Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of $n$ dependent Bernoulli random variables exceeds an integer $k$. Under knowledge of all pairs of bivariate distributions denoted by a complete graph, the bounds are NP-hard to compute. When the bivariate distributions are specified on a tree graph, we show that tight bounds are computable in polynomial time using linear optimization. These bounds provide robust probability estimates when the assumption of conditional independence in a tree structured graphical model is violated. Generalization of the result to finding probability bounds of order statistic for more general random variables and instances where the bounds provide the most significant improvements over univariate bounds is also discussed in the paper.

math.OC↗

arXiv2018

Exploiting Partial Correlations in Distributionally Robust Optimization

In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set $\{(\bold{x}, \bold{x}\bold{x}'): \bold{x} \in \mathcal{X}\}$ where $\mathcal{X}$ is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in Project Evaluation and Review Technique (PERT) networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled.

math.OC↗

arXiv2016

Topic Model Based Multi-Label Classification from the Crowd

Multi-label classification is a common supervised machine learning problem where each instance is associated with multiple classes. The key challenge in this problem is learning the correlations between the classes. An additional challenge arises when the labels of the training instances are provided by noisy, heterogeneous crowdworkers with unknown qualities. We first assume labels from a perfect source and propose a novel topic model where the present as well as the absent classes generate the latent topics and hence the words. We non-trivially extend our topic model to the scenario where the labels are provided by noisy crowdworkers. Extensive experimentation on real world datasets reveals the superior performance of the proposed model. The proposed model learns the qualities of the annotators as well, even with minimal training data.

cs.LG↗

arXiv2016

A Truthful Mechanism with Biparameter Learning for Online Crowdsourcing

We study a problem of allocating divisible jobs, arriving online, to workers in a crowdsourcing setting which involves learning two parameters of strategically behaving workers. Each job is split into a certain number of tasks that are then allocated to workers. Each arriving job has to be completed within a deadline and each task has to be completed satisfying an upper bound on probability of failure. The job population is homogeneous while the workers are heterogeneous in terms of costs, completion times, and times to failure. The job completion time and time to failure of each worker are stochastic with fixed but unknown means. The requester is faced with the challenge of learning two separate parameters of each (strategically behaving) worker simultaneously, namely, the mean job completion time and the mean time to failure. The time to failure of a worker depends on the duration of the task handled by the worker. Assuming non-strategic workers to start with, we solve this biparameter learning problem by applying the Robust UCB algorithm. Then, we non-trivially extend this algorithm to the setting where the workers are strategic about their costs. Our proposed mechanism is dominant strategy incentive compatible and ex-post individually rational with asymptotically optimal regret performance.

cs.AI↗

arXiv2016

A Robust UCB Scheme for Active Learning in Regression from Strategic Crowds

We study the problem of training an accurate linear regression model by procuring labels from multiple noisy crowd annotators, under a budget constraint. We propose a Bayesian model for linear regression in crowdsourcing and use variational inference for parameter estimation. To minimize the number of labels crowdsourced from the annotators, we adopt an active learning approach. In this specific context, we prove the equivalence of well-studied criteria of active learning like entropy minimization and expected error reduction. Interestingly, we observe that we can decouple the problems of identifying an optimal unlabeled instance and identifying an annotator to label it. We observe a useful connection between the multi-armed bandit framework and the annotator selection in active learning. Due to the nature of the distribution of the rewards on the arms, we use the Robust Upper Confidence Bound (UCB) scheme with truncated empirical mean estimator to solve the annotator selection problem. This yields provable guarantees on the regret. We further apply our model to the scenario where annotators are strategic and design suitable incentives to induce them to put in their best efforts.

cs.LG↗