arXiv subjects

Peter W. Glynn

Publications and source records attributed to Peter W. Glynn.

At least 37 records · Page 2Linked to original sources

arXiv2023

Gaussian Limits for Scheduled Traffic with Super-Heavy Tailed Perturbations

A scheduled arrival model is one in which customers are scheduled to arrive at constant interarrival times, but each customer actual arrival time is perturbed from her scheduled arrival time by a random perturbation. The sequence of perturbations is independent and identically distributed. It has previously been shown that the arrival counting process for scheduled traffic obeys a functional central limit theorem (FCLT) with fractional Brownian motion (fBM) with Hurst parameter H strictly between 0 and 1/2 when the pertubratons have a Pareto-like tail with tail exponent lying in (0,1). Such limit processes exhibit less variability than Brownian motion, because the scheduling feature induces negative correlations in the arrival process. In this paper, we show that when the tail of the perturbations have a super-heavy tail, the FCLT limit process is Brownian motion (i.e. H=1/2), so that the heaviness of the tails eliminates any remaining negative correlations and generates a limit process with independent increments. We further study the case when the perturbations have a Cauchy-like tail, and show that the limit process in this setting is a fBM with H=0. So, this paper shows that the entire range of fBMs with $H \in [0,1/2]$ are possible as limits of scheduled traffic.

math.PR↗

arXiv2022

Unbiased Optimal Stopping via the MUSE

We propose a new unbiased estimator for estimating the utility of the optimal stopping problem. The MUSE, short for Multilevel Unbiased Stopping Estimator, constructs the unbiased Multilevel Monte Carlo (MLMC) estimator at every stage of the optimal stopping problem in a backward recursive way. In contrast to traditional sequential methods, the MUSE can be implemented in parallel. We prove the MUSE has finite variance, finite computational complexity, and achieves $ε$-accuracy with $O(1/ε^2)$ computational cost under mild conditions. We demonstrate MUSE empirically in an option pricing problem involving a high-dimensional input and the use of many parallel processors.

stat.CO↗

arXiv2022

Stability of a Queue Fed by Scheduled Traffic at Critical Loading

Consider the workload process for a single server queue with deterministic service times in which customers arrive according to a scheduled traffic process. A scheduled arrival sequence is one in which customers are scheduled to arrive at constant interarrival times, but each customer actual arrival time is perturbed from her scheduled arrival time by a random perturbation. In this paper, we consider a critically loaded queue in which the service rate equals the arrival rate. Unlike a queue fed by renewal traffic, this queue can be stable even in the presence of critical loading. We identify a necessary and sufficient condition for stability when the perturbations have finite mean. Perhaps surprisingly, the criterion is not reversible, in the sense that such a queue can be stable for a scheduled traffic process in forward time, but unstable for the time-reversal of the same traffic process.

math.PR↗

arXiv2022

Risk-Sensitive Markov Decision Processes with Long-Run CVaR Criterion

CVaR (Conditional Value at Risk) is a risk metric widely used in finance. However, dynamically optimizing CVaR is difficult since it is not a standard Markov decision process (MDP) and the principle of dynamic programming fails. In this paper, we study the infinite-horizon discrete-time MDP with a long-run CVaR criterion, from the view of sensitivity-based optimization. By introducing a pseudo CVaR metric, we derive a CVaR difference formula which quantifies the difference of long-run CVaR under any two policies. The optimality of deterministic policies is derived. We obtain a so-called Bellman local optimality equation for CVaR, which is a necessary and sufficient condition for local optimal policies and only necessary for global optimal policies. A CVaR derivative formula is also derived for providing more sensitivity information. Then we develop a policy iteration type algorithm to efficiently optimize CVaR, which is shown to converge to local optima in the mixed policy space. We further discuss some extensions including the mean-CVaR optimization and the maximization of CVaR. Finally, we conduct numerical experiments relating to portfolio management to demonstrate the main results. Our work may shed light on dynamically optimizing CVaR from a sensitivity viewpoint.

math.OC↗

arXiv2022

The Typical Behavior of Bandit Algorithms

We establish strong laws of large numbers and central limit theorems for the regret of two of the most popular bandit algorithms: Thompson sampling and UCB. Here, our characterizations of the regret distribution complement the characterizations of the tail of the regret distribution recently developed by Fan and Glynn (2021) (arXiv:2109.13595). The tail characterizations there are associated with atypical bandit behavior on trajectories where the optimal arm mean is under-estimated, leading to mis-identification of the optimal arm and large regret. In contrast, our SLLN's and CLT's here describe the typical behavior and fluctuation of regret on trajectories where the optimal arm mean is properly estimated. We find that Thompson sampling and UCB satisfy the same SLLN and CLT, with the asymptotics of both the SLLN and the (mean) centering sequence in the CLT matching the asymptotics of expected regret. Both the mean and variance in the CLT grow at $\log(T)$ rates with the time horizon $T$. Asymptotically as $T \to \infty$, the variability in the number of plays of each sub-optimal arm depends only on the rewards received for that arm, which indicates that each sub-optimal arm contributes independently to the overall CLT variance.

cs.LG↗

arXiv2022

Smoothed Variable Sample-size Accelerated Proximal Methods for Nonsmooth Stochastic Convex Programs

We consider minimizing $f(x) = \mathbb{E}[f(x,ω)]$ when $f(x,ω)$ is possibly nonsmooth and either strongly convex or convex in $x$. (I) Strongly convex. When $f(x,ω)$ is $μ-$strongly convex in $x$, we propose a variable sample-size accelerated proximal scheme (VS-APM) and apply it on $f_η(x)$, the ($η$-)Moreau smoothed variant of $\mathbb{E}[f(x,ω)]$; we term such a scheme as (m-VS-APM). We consider three settings. (a) Bounded domains. In this setting, VS-APM displays linear convergence in inexact gradient steps, each of which requires utilizing an inner (SSG) scheme. Specifically, mVS-APM achieves an optimal oracle complexity in SSG steps; (b) Unbounded domains. In this regime, under a weaker assumption of suitable state-dependent bounds on subgradients, an unaccelerated variant mVS-PM is linearly convergent; (c) Smooth ill-conditioned $f$. When $f$ is $L$-smooth and $κ= L/μ\ggg 1$, we employ mVS-APM where increasingly accurate gradients $\nabla_x f_η(x)$ are obtained by VS-APM. Notably, mVS-APM displays linear convergence and near-optimal complexity in inner proximal evaluations (upto a log factor) compared to VS-APM. But, unlike a direct application of VS-APM, this scheme is characterized by larger steplengths and better empirical behavior; (II) Convex. When $f(x,ω)$ is merely convex but smoothable, by suitable choices of the smoothing, steplength, and batch-size sequences, smoothed VS-APM (or sVS-APM) produces sequences for which expected sub-optimality diminishes at the rate of $\mathcal{O}(1/k)$ with an optimal oracle complexity of $\mathcal{O}(1/ε^2)$. Finally, sVS-APM and VS-APM produce sequences that converge almost surely to a solution of the original problem.

math.OC↗

arXiv2022

A New Truncation Algorithm for Markov Chain Equilibrium Distributions with Computable Error Bounds

This paper introduces a new algorithm for numerically computing equilibrium (i.e. stationary) distributions for Markov chains and Markov jump processes with either a very large finite state space or a countably infinite state space. The algorithm is based on a ratio representation for equilibrium expectations in which the numerator and denominator correspond to expectations defined over paths that start and end within a given return set $K$. When $K$ is a singleton, this representation is a well-known consequence of regenerative process theory. For computational tractability, we ignore contributions to the path expectations corresponding to excursions out of a given truncation set $A$. This yields a truncation algorithm that is provably convergent as $A$ gets large. Furthermore, in the presence of a suitable Lyapunov function, we can bound the path expectations, thereby providing computable and convergent error bounds for our numerical procedure. Our paper also provides a computational comparison with two other truncation methods that come with computable error bounds. The results are in alignment with the observation that our bounds have associated computational complexities that typically scale better as the truncation set gets bigger.

math.PR↗

arXiv2022

A Short Proof of a Convex Representation for Stationary Distributions of Markov Chains with an Application to State Space Truncation

In an influential paper, Courtois and Semal (1984) establish that when $G$ is an irreducible substochastic matrix for which $\sum_{n=0}^{\infty}G^n <\infty$, then the stationary distribution of any stochastic matrix $P\ge G$ can be expressed as a convex combination of the normalized rows of $(I-G)^{-1} = \sum_{n=0}^{\infty} G^n$. In this note, we give a short proof of this result that extends the theory to the countably infinite and continuous state space settings. This result plays an important role in obtaining error bounds in algorithms involving nearly decomposable Markov chains, and also in state truncations for Markov chains. We also use the representation to establish a new total variation distance error bound for truncated Markov chains.

math.PR↗

arXiv2022

On Convergence of a Truncation Scheme for Approximating Stationary Distributions of Continuous State Space Markov Chains and Processes

In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior of the truncated chain or process is close to that of the untruncated system. For example, such questions arise naturally when considering numerical methods for computing stationary distributions on unbounded state space. In this paper, we use the principle of "regeneration" to show that the stationary distributions of "fixed state" truncations converge in great generality (in total variation norm) to the stationary distribution of the untruncated limit, when the untruncated chain is positive Harris recurrent. Even in countable state space, our theory extends known results by showing that the augmentation can correspond to an $r$-regular measure. In addition, we extend our theory to cover an important subclass of Harris recurrent Markov processes that include non-explosive Markov jump processes on countable state space.

math.PR↗

arXiv2022

On Convergence of General Truncation-Augmentation Schemes for Approximating Stationary Distributions of Markov Chains

In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior of the truncated chain or process is close to that of the untruncated system. For example, such questions arise naturally when considering numerical methods for computing stationary distributions on unbounded state space. In this paper, we study general truncation-augmentation schemes, in which the substochastic truncated "northwest corner" of the transition matrix or kernel is stochasticized (or augmented) arbitrarily. In the presence of a Lyapunov condition involving a coercive function, we show that such schemes are generally convergent in countable state space, provided that the truncation is chosen as a sublevel set of the Lyapunov function. For stochastically monotone Markov chains on $\mathbb Z_+$, we prove that we can always choose the truncation sets to be of the form $\{0,1,...,n\}$. We then provide sufficient conditions for weakly continuous Markov chains under which general truncation-augmentation schemes converge weakly in continuous state space. Finally, we briefly discuss the extension of the theory to continuous time Markov jump processes.

math.PR↗

arXiv2022

Solving Poisson's Equation: Existence, Uniqueness, Martingale Structure, and CLT

The solution of Poisson's equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study two different representations for the solution in countable state space, one based on regenerative structure and the other based on an infinite sum of expectations. We also consider integrability and related uniqueness issues associated with solutions to Poisson's equation, and provide verifiable Lyapunov conditions to support our theory. Our key results include a central limit theorem and law of the iterated logarithm for Markov dependent sums, under Lyapunov conditions weaker than have previously appeared in the literature.

math.PR↗

arXiv2022

Solutions of Poisson's Equation for Stochastically Monotone Markov Chains

Stochastically monotone Markov chains arise in many applied domains, especially in the setting of queues and storage systems. Poisson's equation is a key tool for analyzing additive functionals of such models, such as cumulative sums of waiting times or sums of rewards. In this paper, we show that when the reward function for such a Markov chain is monotone, the solution of Poisson's equation is monotone. This implies that the value function associated with infinite horizon average reward is monotone in the state when the reward is monotone.

math.PR↗

arXiv2021

Distributionally Robust Martingale Optimal Transport

We study the problem of bounding path-dependent expectations (within any finite time horizon $d$) over the class of discrete-time martingales whose marginal distributions lie within a prescribed tolerance of a given collection of benchmark marginal distributions. This problem is a relaxation of the martingale optimal transport (MOT) problem and is motivated by applications to super-hedging in financial markets. We show that the empirical version of our relaxed MOT problem can be approximated within $O\left( n^{-1/2}\right)$ error where $n$ is the number of samples of each of the individual marginal distributions (generated independently) and using a suitably constructed finite-dimensional linear programming problem.

math.PR↗

arXiv2021

Distributed stochastic optimization with large delays

One of the most widely used methods for solving large-scale stochastic optimization problems is distributed asynchronous stochastic gradient descent (DASGD), a family of algorithms that result from parallelizing stochastic gradient descent on distributed computing architectures (possibly) asychronously. However, a key obstacle in the efficient implementation of DASGD is the issue of delays: when a computing node contributes a gradient update, the global model parameter may have already been updated by other nodes several times over, thereby rendering this gradient information stale. These delays can quickly add up if the computational throughput of a node is saturated, so the convergence of DASGD may be compromised in the presence of large delays. Our first contribution is that, by carefully tuning the algorithm's step-size, convergence to the critical set is still achieved in mean square, even if the delays grow unbounded at a polynomial rate. We also establish finer results in a broad class of structured optimization problems (called variationally coherent), where we show that DASGD converges to a global optimum with probability $1$ under the same delay assumptions. Together, these results contribute to the broad landscape of large-scale non-convex stochastic optimization by offering state-of-the-art theoretical guarantees and providing insights for algorithm design.

math.OC↗

arXiv2021

On Incorporating Forecasts into Linear State Space Model Markov Decision Processes

Weather forecast information will very likely find increasing application in the control of future energy systems. In this paper, we introduce an augmented state space model formulation with linear dynamics, within which one can incorporate forecast information that is dynamically revealed alongside the evolution of the underlying state variable. We use the martingale model for forecast evolution (MMFE) to enforce the necessary consistency properties that must govern the joint evolution of forecasts with the underlying state. The formulation also generates jointly Markovian dynamics that give rise to Markov decision processes (MDPs) that remain computationally tractable. This paper is the first to enforce MMFE consistency requirements within an MDP formulation that preserves tractability.

math.OC↗

arXiv2020

Multivariate Distributionally Robust Convex Regression under Absolute Error Loss

This paper proposes a novel non-parametric multidimensional convex regression estimator which is designed to be robust to adversarial perturbations in the empirical measure. We minimize over convex functions the maximum (over Wasserstein perturbations of the empirical measure) of the absolute regression errors. The inner maximization is solved in closed form resulting in a regularization penalty involves the norm of the gradient. We show consistency of our estimator and a rate of convergence of order $ \widetilde{O}\left( n^{-1/d}\right) $, matching the bounds of alternative estimators based on square-loss minimization. Contrary to all of the existing results, our convergence rates hold without imposing compactness on the underlying domain and with no a priori bounds on the underlying convex function or its gradient norm.

math.ST↗

arXiv2019

Unbiased Multilevel Monte Carlo: Stochastic Optimization, Steady-state Simulation, Quantiles, and Other Applications

We present general principles for the design and analysis of unbiased Monte Carlo estimators in a wide range of settings. Our estimators posses finite work-normalized variance under mild regularity conditions. We apply our estimators to various settings of interest, including unbiased optimization in Sample Average Approximations, unbiased steady-state simulation of regenerative processes, quantile estimation and nested simulation problems.

math.ST↗

arXiv2018

The stochastic counterpart of conservation laws with heterogeneous conductivity fields: application to deterministic problems and uncertainty quantification

Conservation laws in the form of elliptic and parabolic partial differential equations (PDEs) are fundamental to the modeling of many problems such as heat transfer and flow in porous media. Many of such PDEs are stochastic due to the presence of uncertainty in the conductivity field. Based on the relation between stochastic diffusion processes and PDEs, Monte Carlo (MC) methods are available to solve these PDEs. These methods are especially relevant for cases where we are interested in the solution in a small subset of the domain. The existing MC methods based on the stochastic formulation require restrictively small time steps for high variance conductivity fields. Moreover, in many applications the conductivity is piecewise constant and the existing methods are not readily applicable in these cases. Here we provide an algorithm to solve one-dimensional elliptic problems that bypasses these two limitations. The methodology is demonstrated using problems governed by deterministic and stochastic PDEs. It is shown that the method provides an efficient alternative to compute the statistical moments of the solution to a stochastic PDE at any point in the domain. A variance reduction scheme is proposed for applying the method for efficient mean calculations.

physics.comp-ph↗