Harsha Honnappa

Hemant Gehlot, Harsha Honnappa, Satish V. Ukkusuri

Congestion pricing has become an effective instrument for traffic demand management on road networks. This paper proposes an optimal control approach for congestion pricing for day-to-day timescale that incorporates demand uncertainty and elasticity. Travelers make the decision to travel or not based on the experienced system travel time in the previous day and traffic managers take tolling decisions in order to minimize the average system travel time over a long time horizon. We formulate the problem as a Markov decision process (MDP) and analyze the problem to see if it satisfies conditions for conducting a satisfactory solution analysis. Such an analysis of MDPs is often dependent on the type of state space as well as on the boundedness of travel time functions. We do not constrain the travel time functions to be bounded and present an analysis centered around weighted sup-norm contractions that also holds for unbounded travel time functions. We find that the formulated MDP satisfies a set of assumptions to ensure Bellman's optimality condition. Through this result, the existence of the optimal average cost of the MDP is shown. A method based on approximate dynamic programming is proposed to resolve the implementation and computational issues of solving the control problem. Numerical results suggest that the proposed method efficiently solves the problem and produces accurate solutions.

Harsha Honnappa, Rahul Jain, Amy R. Ward

We introduce the Δ(i)/GI/1 queue, a new queueing model. In this model, customers from a given population independently sample a time to arrive from some given distribution F. Thus, the arrival times are an ordered statistics, and the inter-arrival times are differences of consecutive ordered statistics. They are served by a single server which provides service according to a general distribution G, with independent service times. The exact model is analytically intractable. Thus, we develop fluid and diffusion limits for the various stochastic processes, and performance metrics. The fluid limit of the queue length is observed to be a reflected process, while the diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge process, and is given by a 'netput' process and a directional derivative of the Skorokhod reflected fluid netput in the direction of a diffusion refinement of the netput process. We also observe what may be interpreted as a transient Little's law. Sample path analysis reveals various operating regimes where the diffusion limit switches between a free diffusion, a reflected diffusion process and the zero process, with possible discontinuities during regime switches. The weak convergence is established in the M1 topology, and it is also shown that this is not possible in the J1 topology.

Harsha Honnappa, Rahul Jain

Queueing networks are typically modelled assuming that the arrival process is exogenous, and unaffected by admission control, scheduling policies, etc. In many situations, however, users choose the time of their arrival strategically, taking delay and other metrics into account. In this paper, we develop a framework to study such strategic arrivals into queueing networks. We start by deriving a functional strong law of large numbers (FSLLN) approximation to the queueing network. In the fluid limit derived, we then study the population game wherein users strategically choose when to arrive, and upon arrival which of the K queues to join. The queues start service at given times, which can potentially be different. We characterize the (strategic) arrival process at each of the queues, and the price of anarchy of the ensuing strategic arrival game. We then extend the analysis to multiple populations of users, each with a different cost metric. The equilibrium arrival profile and price of anarchy are derived. Finally, we present the methodology for exact equilibrium analysis. This, however, is tractable for only some simple cases such as two users arriving at a two node queueing network, which we then present.

Imon Banerjee, Harsha Honnappa, Vinayak Rao

In this work, we study a natural nonparametric estimator of the transition probability matrices of a finite controlled Markov chain. We consider an offline setting with a fixed dataset, collected using a so-called logging policy. We develop sample complexity bounds for the estimator and establish conditions for minimaxity. Our statistical bounds depend on the logging policy through its mixing properties. We show that achieving a particular statistical risk bound involves a subtle and interesting trade-off between the strength of the mixing properties and the number of samples. We demonstrate the validity of our results under various examples, such as ergodic Markov chains, weakly ergodic inhomogeneous Markov chains, and controlled Markov chains with non-stationary Markov, episodic, and greedy controls. Lastly, we use these sample complexity bounds to establish concomitant ones for offline evaluation of stationary Markov control policies.

Sharan Srinivasan, Vijay Gupta, Harsha Honnappa

The variational formulation of nonlinear filtering due to Mitter and Newton characterizes the filtering distribution as the unique minimizer of a free energy functional involving the relative entropy with respect to the prior and an expected energy. This formulation rests on an absolute continuity condition between the joint path measure and a product reference measure. We prove that this condition necessarily fails whenever the signal and observation diffusions share a common noise source. Specifically we show that the joint and product measures are mutually singular, so no choice of reference measure can salvage the formulation. We then introduce a conditional variational principle that replaces the prior with a reference measure that preserves the noise correlation structure. This generalization recovers the Mitter--Newton formulation as a special case when the noises are independent, and yields an explicit free energy characterization of the filter in the linear correlated-noise setting.

Yao Ji, Gesualdo Scutari, Ying Sun et al.

We study sparse linear regression over a network of agents, modeled as an undirected graph (with no centralized node). The estimation problem is formulated as the minimization of the sum of the local LASSO loss functions plus a quadratic penalty of the consensus constraint -- the latter being instrumental to obtain distributed solution methods. While penalty-based consensus methods have been extensively studied in the optimization literature, their statistical and computational guarantees in the high dimensional setting remain unclear. This work provides an answer to this open problem. Our contribution is two-fold. First, we establish statistical consistency of the estimator: under a suitable choice of the penalty parameter, the optimal solution of the penalized problem achieves near optimal minimax rate $\mathcal{O}(s \log d/N)$ in $\ell_2$-loss, where $s$ is the sparsity value, $d$ is the ambient dimension, and $N$ is the total sample size in the network -- this matches centralized sample rates. Second, we show that the proximal-gradient algorithm applied to the penalized problem, which naturally leads to distributed implementations, converges linearly up to a tolerance of the order of the centralized statistical error -- the rate scales as $\mathcal{O}(d)$, revealing an unavoidable speed-accuracy dilemma.Numerical results demonstrate the tightness of the derived sample rate and convergence rate scalings.

Prateek Jaiswal, Harsha Honnappa, Vinayak A. Rao

This paper considers data-driven chance-constrained stochastic optimization problems in a Bayesian framework. Bayesian posteriors afford a principled mechanism to incorporate data and prior knowledge into stochastic optimization problems. However, the computation of Bayesian posteriors is typically an intractable problem, and has spawned a large literature on approximate Bayesian computation. Here, in the context of chance-constrained optimization, we focus on the question of statistical consistency (in an appropriate sense) of the optimal value, computed using an approximate posterior distribution. To this end, we rigorously prove a frequentist consistency result demonstrating the convergence of the optimal value to the optimal value of a fixed, parameterized constrained optimization problem. We augment this by also establishing a probabilistic rate of convergence of the optimal value. We also prove the convex feasibility of the approximate Bayesian stochastic optimization problem. Finally, we demonstrate the utility of our approach on an optimal staffing problem for an M/M/c queueing model.

Ruixin Wang, Prateek Jaiwal, Harsha Honnappa

We present methodology for estimating the stochastic intensity of a doubly stochastic Poisson process. Statistical and theoretical analyses of traffic traces show that these processes are appropriate models of high intensity traffic arriving at an array of service systems. The statistical estimation of the underlying latent stochastic intensity process driving the traffic model involves a rather complicated nonlinear filtering problem. We develop a novel simulation methodology, using deep neural networks to approximate the path measures induced by the stochastic intensity process, for solving this nonlinear filtering problem. Our simulation studies demonstrate that the method is quite accurate on both in-sample estimation and on an out-of-sample performance prediction task for an infinite server queue.

Prateek Jaiswal, Harsha Honnappa, Vinayak A. Rao

We study system design problems stated as parameterized stochastic programs with a chance-constraint set. We adopt a Bayesian approach that requires the computation of a posterior predictive integral which is usually intractable. In addition, for the problem to be a well-defined convex program, we must retain the convexity of the feasible set. Consequently, we propose a variational Bayes-based method to approximately compute the posterior predictive integral that ensures tractability and retains the convexity of the feasible set. Under certain regularity conditions, we also show that the solution set obtained using variational Bayes converges to the true solution set as the number of observations tends to infinity. We also provide bounds on the probability of qualifying a true infeasible point (with respect to the true constraints) as feasible under the VB approximation for a given number of samples.

Prateek Jaiswal, Harsha Honnappa, Vinayak A. Rao

This paper establishes the asymptotic consistency of the {\it loss-calibrated variational Bayes} (LCVB) method. LCVB was proposed in~\cite{LaSiGh2011} as a method for approximately computing Bayesian posteriors in a `loss aware' manner. This methodology is also highly relevant in general data-driven decision-making contexts. Here, we not only establish the asymptotic consistency of the calibrated approximate posterior, but also the asymptotic consistency of decision rules. We also establish the asymptotic consistency of decision rules obtained from a `naive' variational Bayesian procedure.

Prateek Jaiswal, Vinayak A. Rao, Harsha Honnappa

We study the asymptotic consistency properties of $α$-Rényi approximate posteriors, a class of variational Bayesian methods that approximate an intractable Bayesian posterior with a member of a tractable family of distributions, the member chosen to minimize the $α$-Rényi divergence from the true posterior. Unique to our work is that we consider settings with $α> 1$, resulting in approximations that upperbound the log-likelihood, and consequently have wider spread than traditional variational approaches that minimize the Kullback-Liebler (KL) divergence from the posterior. Our primary result identifies sufficient conditions under which consistency holds, centering around the existence of a 'good' sequence of distributions in the approximating family that possesses, among other properties, the right rate of convergence to a limit distribution. We further characterize the good sequence by demonstrating that a sequence of distributions that converges too quickly cannot be a good sequence. We also extend our analysis to the setting where $α$ equals one, corresponding to the minimizer of the reverse KL divergence, and to models with local latent variables. We also illustrate the existence of good sequence with a number of examples. Our results complement a growing body of work focused on the frequentist properties of variational Bayesian methods.