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Jose Blanchet

Jose Blanchet appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.PRMachine LearningArtificial Intelligencemath.STStatistics Theorymath.OCComputationComputational Engineering, Finance, and ScienceComputer Science and Game TheoryMethodologyComputation and LanguageComputer VisionData Structures and AlgorithmsDiscrete MathematicsInformation Theorymath.COmath.ITmath.NAq-fin.CPq-fin.STq-fin.TR

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Published work

45 published item(s)

preprint2026arXiv

ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space

Generating continuous-time, continuous-space stochastic processes (e.g., videos, weather forecasts) conditioned on partial observations (e.g., first and last frames) is a fundamental challenge. Existing approaches, (e.g., diffusion models), suffer from key limitations: (1) noise-to-data evolution fails to capture structural similarity between states close in physical time and has unstable integration in low-step regimes; (2) random noise injected is insensitive to the physical process's time elapsed, resulting in incorrect dynamics; (3) they overlook conditioning on arbitrary subsets of states (e.g., irregularly sampled timesteps, future observations). We propose ABC: Any-Subset Autoregressive Models via Non-Markovian Diffusion Bridges in Continuous Time and Space. Crucially, we model the process with one continual SDE whose time variable and intermediate states track the real time and process states. This has provable advantages: (1) the starting point for generating future states is the already-close previous state, rather than uninformative noise; (2) random noise injection scales with physical time elapsed, encouraging physically plausible dynamics with similar time-adjacent states. We derive SDE dynamics via changes-of-measure on path space, yielding another advantage: (3) path-dependent conditioning on arbitrary subsets of the state history and/or future. To learn these dynamics, we derive a path- and time-dependent extension of denoising score matching. Our experiments show ABC's superiority to competing methods on multiple domains, including video generation and weather forecasting.

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preprint2026arXiv

FutureX-Pro: Extending Future Prediction to High-Value Vertical Domains

Building upon FutureX, which established a live benchmark for general-purpose future prediction, this report introduces FutureX-Pro, including FutureX-Finance, FutureX-Retail, FutureX-PublicHealth, FutureX-NaturalDisaster, and FutureX-Search. These together form a specialized framework extending agentic future prediction to high-value vertical domains. While generalist agents demonstrate proficiency in open-domain search, their reliability in capital-intensive and safety-critical sectors remains under-explored. FutureX-Pro targets four economically and socially pivotal verticals: Finance, Retail, Public Health, and Natural Disaster. We benchmark agentic Large Language Models (LLMs) on entry-level yet foundational prediction tasks -- ranging from forecasting market indicators and supply chain demands to tracking epidemic trends and natural disasters. By adapting the contamination-free, live-evaluation pipeline of FutureX, we assess whether current State-of-the-Art (SOTA) agentic LLMs possess the domain grounding necessary for industrial deployment. Our findings reveal the performance gap between generalist reasoning and the precision required for high-value vertical applications.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

On the First Passage Times of Branching Random Walks in $\mathbb R^d$

We study the first passage times of discrete-time branching random walks in ${\mathbb R}^d$ where $d\geq 1$. Here, the genealogy of the particles follows a supercritical Galton-Watson process. We provide asymptotics of the first passage times to a ball of radius one with a distance $x$ from the origin, conditioned upon survival. We provide explicitly the linear dominating term and the logarithmic correction term as a function of $x$. The asymptotics are precise up to an order of $o_{\mathbb P}(\log x)$ for general jump distributions and up to $O_{\mathbb P}(\log\log x)$ for spherically symmetric jumps. A crucial ingredient of both results is the tightness of first passage times. We also discuss an extension of the first passage time analysis to a modified branching random walk model that has been proven to successfully capture shortest path statistics in polymer networks.

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preprint2026arXiv

Simple Approximation and Derivative Free Inference-Time Scaling for Diffusion Models via Sequential Monte Carlo on Path Measures

iffusion-based generative models increasingly rely on inference-time guidance, adding a drift term or reweighting mixture of experts, to improve sample quality on task-specific objectives. However, most existing techniques require repeated score or gradient evaluations, introducing bias, high computational overhead, or both. We introduce \texttt{URGE}, Unbiased Resampling via Girsanov Estimation, a derivative-free inference-time scaling algorithm that performs path-wise importance reweighting via a Girsanov change of measure. Instead of computing gradient-based particle weights in previous work, \texttt{URGE} attaches a simple multiplicative weight to each simulated trajectory and periodically resamples. No score, no Hessian, and no PDE evaluation is required. We establish an equivalence between path-wise and particle-wise SMC: the Girsanov path weight admits a backward conditional expectation that recovers the previous particle-level weights, guaranteeing that both schemes produce the same unbiased terminal law. Empirically, \texttt{URGE} outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks, achieving better generation quality, while being significantly simpler to implement and fully gradient-free.

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preprint2026arXiv

TERMS-Bench: Diagnosing LLM Negotiation Agents Beyond Deal Rate

Negotiation is a central mechanism of economic exchange, shaping markets, procurement, labor agreements, and resource allocation. It is also a canonical testbed for agentic language models, requiring multi-turn interaction under hidden preferences, strategic communication, and binding constraints. These properties make negotiation hard to evaluate: unlike math or code, it has no intrinsic verifier. Existing LLM negotiation evaluations rely on LLM-vs.-LLM interaction or aggregate outcomes such as deal rate, leaving failures opaque. We introduce Terms-Bench, short for Testbed for Economic Reasoning in Multi-turn Strategy, a Bayesian-game framework that makes the environment itself the verifier by specifying the counterpart's latent type, policy, and payoff structure. We instantiate it in bilateral price negotiation, where the counterpart's private state and simulator policy are hidden from the agent but observable to the evaluator. This turns the counterpart from a black-box opponent into a diagnostic instrument, enabling agent-attributable failure analysis and oracle-reference optimality gaps. Evaluating 13 LLM agents spanning frontier systems from major providers, Terms-Bench turns negotiation evaluation from aggregate ranking into actionable diagnosis: where agents fail, why they fail, and what to strengthen. Empirically, frontier models saturate deal rate yet diverge in surplus extraction, cue use, belief calibration, and compliance, revealing agent-specific bargaining bottlenecks masked by prior benchmarks.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Wasserstein Distributionally Robust Regret Optimization for Reinforcement Learning from Human Feedback

Reinforcement learning from human feedback (RLHF) has become a core post-training step for aligning large language models, yet the reward signal used in RLHF is only a learned proxy for true human utility. From an operations research perspective, this creates a decision problem under objective misspecification: the policy is optimized against an estimated reward, while deployment performance is determined by an unobserved objective. The resulting gap leads to reward over-optimization, or Goodharting, where proxy reward continues to improve even after true quality deteriorates. Existing mitigations address this problem through uncertainty penalties, pessimistic rewards, or conservative constraints, but they can be computationally burdensome and overly pessimistic. We propose Wasserstein distributionally robust regret optimization (DRRO) for RLHF. Instead of pessimizing worst-case value as in standard DRO, DRRO pessimizes worst-case regret relative to the best policy under the same plausible reward perturbation. We study the promptwise problem through a simplex allocation model and show that, under an $\ell_1$-ground-cost Wasserstein ambiguity set, the inner worst-case regret admits an exact solution and the optimal policy has a water-filling structure. These results lead to a practical policy-gradient algorithm with a simple sampled-bonus interpretation and only minor changes to GRPO-style RLHF training. The framework also clarifies theoretically why DRRO is less pessimistic than DRO, and our experiments show that DRRO mitigates over-optimization more effectively than existing baselines while standard DRO is systematically over-pessimistic.

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preprint2022arXiv

A Class of Geometric Structures in Transfer Learning: Minimax Bounds and Optimality

We study the problem of transfer learning, observing that previous efforts to understand its information-theoretic limits do not fully exploit the geometric structure of the source and target domains. In contrast, our study first illustrates the benefits of incorporating a natural geometric structure within a linear regression model, which corresponds to the generalized eigenvalue problem formed by the Gram matrices of both domains. We next establish a finite-sample minimax lower bound, propose a refined model interpolation estimator that enjoys a matching upper bound, and then extend our framework to multiple source domains and generalized linear models. Surprisingly, as long as information is available on the distance between the source and target parameters, negative-transfer does not occur. Simulation studies show that our proposed interpolation estimator outperforms state-of-the-art transfer learning methods in both moderate- and high-dimensional settings.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

No Weighted-Regret Learning in Adversarial Bandits with Delays

Consider a scenario where a player chooses an action in each round $t$ out of $T$ rounds and observes the incurred cost after a delay of $d_{t}$ rounds. The cost functions and the delay sequence are chosen by an adversary. We show that in a non-cooperative game, the expected weighted ergodic distribution of play converges to the set of coarse correlated equilibria if players use algorithms that have "no weighted-regret" in the above scenario, even if they have linear regret due to too large delays. For a two-player zero-sum game, we show that no weighted-regret is sufficient for the weighted ergodic average of play to converge to the set of Nash equilibria. We prove that the FKM algorithm with $n$ dimensions achieves an expected regret of $O\left(nT^{\frac{3}{4}}+\sqrt{n}T^{\frac{1}{3}}D^{\frac{1}{3}}\right)$ and the EXP3 algorithm with $K$ arms achieves an expected regret of $O\left(\sqrt{\log K\left(KT+D\right)}\right)$ even when $D=\sum_{t=1}^{T}d_{t}$ and $T$ are unknown. These bounds use a novel doubling trick that, under mild assumptions, provably retains the regret bound for when $D$ and $T$ are known. Using these bounds, we show that FKM and EXP3 have no weighted-regret even for $d_{t}=O\left(t\log t\right)$. Therefore, algorithms with no weighted-regret can be used to approximate a CCE of a finite or convex unknown game that can only be simulated with bandit feedback, even if the simulation involves significant delays.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

The Design and Implementation of a Broadly Applicable Algorithm for Optimizing Intra-Day Surgical Scheduling

Surgical scheduling optimization is an active area of research. However, few algorithms to optimize surgical scheduling are implemented and see sustained use. An algorithm is more likely to be implemented, if it allows for surgeon autonomy, i.e., requires only limited scheduling centralization, and functions in the limited technical infrastructure of widely used electronic medical records (EMRs). In order for an algorithm to see sustained use, it must be compatible with changes to hospital capacity, patient volumes, and scheduling practices. To meet these objectives, we developed the BEDS (better elective day of surgery) algorithm, a greedy heuristic for smoothing unit-specific surgical admissions across days. We implemented BEDS in the EMR of a large pediatric academic medical center. The use of BEDS was associated with a reduction in the variability in the number of admissions. BEDS is freely available as a dashboard in Tableau, a commercial software used by numerous hospitals. BEDS is readily implementable with the limited tools available to most hospitals, does not require reductions to surgeon autonomy or centralized scheduling, and is compatible with changes to hospital capacity or patient volumes. We present a general algorithmic framework from which BEDS is derived based on a particular choice of objectives and constraints. We argue that algorithms generated by this framework retain many of the desirable characteristics of BEDS while being compatible with a wide range of objectives and constraints.

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preprint2021arXiv

Confidence Regions in Wasserstein Distributionally Robust Estimation

Wasserstein distributionally robust optimization estimators are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance (in a Wasserstein sense) from the underlying empirical measure. While motivated by the need to identify optimal model parameters or decision choices that are robust to model misspecification, these distributionally robust estimators recover a wide range of regularized estimators, including square-root lasso and support vector machines, among others, as particular cases. This paper studies the asymptotic normality of these distributionally robust estimators as well as the properties of an optimal (in a suitable sense) confidence region induced by the Wasserstein distributionally robust optimization formulation. In addition, key properties of min-max distributionally robust optimization problems are also studied, for example, we show that distributionally robust estimators regularize the loss based on its derivative and we also derive general sufficient conditions which show the equivalence between the min-max distributionally robust optimization problem and the corresponding max-min formulation.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Distributionally Robust Approach to Fair Classification

We propose a distributionally robust logistic regression model with an unfairness penalty that prevents discrimination with respect to sensitive attributes such as gender or ethnicity. This model is equivalent to a tractable convex optimization problem if a Wasserstein ball centered at the empirical distribution on the training data is used to model distributional uncertainty and if a new convex unfairness measure is used to incentivize equalized opportunities. We demonstrate that the resulting classifier improves fairness at a marginal loss of predictive accuracy on both synthetic and real datasets. We also derive linear programming-based confidence bounds on the level of unfairness of any pre-trained classifier by leveraging techniques from optimal uncertainty quantification over Wasserstein balls.

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preprint2020arXiv

Delay-Adaptive Learning in Generalized Linear Contextual Bandits

In this paper, we consider online learning in generalized linear contextual bandits where rewards are not immediately observed. Instead, rewards are available to the decision-maker only after some delay, which is unknown and stochastic. We study the performance of two well-known algorithms adapted to this delayed setting: one based on upper confidence bounds, and the other based on Thompson sampling. We describe modifications on how these two algorithms should be adapted to handle delays and give regret characterizations for both algorithms. Our results contribute to the broad landscape of contextual bandits literature by establishing that both algorithms can be made to be robust to delays, thereby helping clarify and reaffirm the empirical success of these two algorithms, which are widely deployed in modern recommendation engines.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Efficient Steady-state Simulation of High-dimensional Stochastic Networks

We propose and study an asymptotically optimal Monte Carlo estimator for steady-state expectations of a d-dimensional reflected Brownian motion. Our estimator is asymptotically optimal in the sense that it requires $\tilde{O}(d)$ (up to logarithmic factors in $d$) i.i.d. Gaussian random variables in order to output an estimate with a controlled error. Our construction is based on the analysis of a suitable multi-level Monte Carlo strategy which, we believe, can be applied widely. This is the first algorithm with linear complexity (under suitable regularity conditions) for steady-state estimation of RBM as the dimension increases.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Exact Simulation for Multivariate Itô Diffusions

We provide the first generic exact simulation algorithm for multivariate diffusions. Current exact sampling algorithms for diffusions require the existence of a transformation which can be used to reduce the sampling problem to the case of a constant diffusion matrix and a drift which is the gradient of some function. Such transformation, called Lamperti transformation, can be applied in general only in one dimension. So, completely different ideas are required for exact sampling of generic multivariate diffusions. The development of these ideas is the main contribution of this paper. Our strategy combines techniques borrowed from the theory of rough paths, on one hand, and multilevel Monte Carlo on the other.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

On distributionally robust extreme value analysis

We study distributional robustness in the context of Extreme Value Theory (EVT). We provide a data-driven method for estimating extreme quantiles in a manner that is robust against incorrect model assumptions underlying the application of the standard Extremal Types Theorem. Typical studies in distributional robustness involve computing worst case estimates over a model uncertainty region expressed in terms of the Kullback-Leibler discrepancy. We go beyond standard distributional robustness in that we investigate different forms of discrepancies, and prove rigorous results which are helpful for understanding the role of a putative model uncertainty region in the context of extreme quantile estimation. Finally, we illustrate our data-driven method in various settings, including examples showing how standard EVT can significantly underestimate quantiles of interest.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Robust Bayesian Classification Using an Optimistic Score Ratio

We build a Bayesian contextual classification model using an optimistic score ratio for robust binary classification when there is limited information on the class-conditional, or contextual, distribution. The optimistic score searches for the distribution that is most plausible to explain the observed outcomes in the testing sample among all distributions belonging to the contextual ambiguity set which is prescribed using a limited structural constraint on the mean vector and the covariance matrix of the underlying contextual distribution. We show that the Bayesian classifier using the optimistic score ratio is conceptually attractive, delivers solid statistical guarantees and is computationally tractable. We showcase the power of the proposed optimistic score ratio classifier on both synthetic and empirical data.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Sequential Batch Learning in Finite-Action Linear Contextual Bandits

We study the sequential batch learning problem in linear contextual bandits with finite action sets, where the decision maker is constrained to split incoming individuals into (at most) a fixed number of batches and can only observe outcomes for the individuals within a batch at the batch's end. Compared to both standard online contextual bandits learning or offline policy learning in contexutal bandits, this sequential batch learning problem provides a finer-grained formulation of many personalized sequential decision making problems in practical applications, including medical treatment in clinical trials, product recommendation in e-commerce and adaptive experiment design in crowdsourcing. We study two settings of the problem: one where the contexts are arbitrarily generated and the other where the contexts are \textit{iid} drawn from some distribution. In each setting, we establish a regret lower bound and provide an algorithm, whose regret upper bound nearly matches the lower bound. As an important insight revealed therefrom, in the former setting, we show that the number of batches required to achieve the fully online performance is polynomial in the time horizon, while for the latter setting, a pure-exploitation algorithm with a judicious batch partition scheme achieves the fully online performance even when the number of batches is less than logarithmic in the time horizon. Together, our results provide a near-complete characterization of sequential decision making in linear contextual bandits when batch constraints are present.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Towards Optimal Running Times for Optimal Transport

In this work, we provide faster algorithms for approximating the optimal transport distance, e.g. earth mover's distance, between two discrete probability distributions $μ, ν\in Δ^n$. Given a cost function $C : [n] \times [n] \to \mathbb{R}_{\geq 0}$ where $C(i,j) \leq 1$ quantifies the penalty of transporting a unit of mass from $i$ to $j$, we show how to compute a coupling $X$ between $r$ and $c$ in time $\widetilde{O}\left(n^2 /ε\right)$ whose expected transportation cost is within an additive $ε$ of optimal. This improves upon the previously best known running time for this problem of $\widetilde{O}\left(\text{min}\left\{ n^{9/4}/ε, n^2/ε^2 \right\}\right)$. We achieve our results by providing reductions from optimal transport to canonical optimization problems for which recent algorithmic efforts have provided nearly-linear time algorithms. Leveraging nearly linear time algorithms for solving packing linear programs and for solving the matrix balancing problem, we obtain two separate proofs of our stated running time. Further, one of our algorithms is easily parallelized and can be implemented with depth $\widetilde{O}(1/ε)$. Moreover, we show that further algorithmic improvements to our result would be surprising in the sense that any improvement would yield an $o(n^{2.5})$ algorithm for \textit{maximum cardinality bipartite matching}, for which currently the only known algorithms for achieving such a result are based on fast-matrix multiplication.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

A Distributionally Robust Boosting Algorithm

Distributionally Robust Optimization (DRO) has been shown to provide a flexible framework for decision making under uncertainty and statistical estimation. For example, recent works in DRO have shown that popular statistical estimators can be interpreted as the solutions of suitable formulated data-driven DRO problems. In turn, this connection is used to optimally select tuning parameters in terms of a principled approach informed by robustness considerations. This paper contributes to this growing literature, connecting DRO and statistics, by showing how boosting algorithms can be studied via DRO. We propose a boosting type algorithm, named DRO-Boosting, as a procedure to solve our DRO formulation. Our DRO-Boosting algorithm recovers Adaptive Boosting (AdaBoost) in particular, thus showing that AdaBoost is effectively solving a DRO problem. We apply our algorithm to a financial dataset on credit card default payment prediction. We find that our approach compares favorably to alternative boosting methods which are widely used in practice.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Data-driven Optimal Cost Selection for Distributionally Robust Optimization

Recently, (Blanchet, Kang, and Murhy 2016, and Blanchet, and Kang 2017) showed that several machine learning algorithms, such as square-root Lasso, Support Vector Machines, and regularized logistic regression, among many others, can be represented exactly as distributionally robust optimization (DRO) problems. The distributional uncertainty is defined as a neighborhood centered at the empirical distribution. We propose a methodology which learns such neighborhood in a natural data-driven way. We show rigorously that our framework encompasses adaptive regularization as a particular case. Moreover, we demonstrate empirically that our proposed methodology is able to improve upon a wide range of popular machine learning estimators.

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preprint2019arXiv

Semi-supervised Learning based on Distributionally Robust Optimization

We propose a novel method for semi-supervised learning (SSL) based on data-driven distributionally robust optimization (DRO) using optimal transport metrics. Our proposed method enhances generalization error by using the unlabeled data to restrict the support of the worst case distribution in our DRO formulation. We enable the implementation of our DRO formulation by proposing a stochastic gradient descent algorithm which allows to easily implement the training procedure. We demonstrate that our Semi-supervised DRO method is able to improve the generalization error over natural supervised procedures and state-of-the-art SSL estimators. Finally, we include a discussion on the large sample behavior of the optimal uncertainty region in the DRO formulation. Our discussion exposes important aspects such as the role of dimension reduction in SSL.

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preprint2016arXiv

$ε$-Strong Simulation for Multidimensional Stochastic Differential Equations via Rough Path Analysis

Consider a multidimensional diffusion process $X=\{X\left(t\right) :t\in\lbrack0,1]\}$. Let $\varepsilon>0$ be a \textit{deterministic}, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of $X$, we construct a probability space, supporting both $X$ and an explicit, piecewise constant, fully simulatable process $X_{\varepsilon}$ such that \[ \sup_{0\leq t\leq1}\left\Vert X_{\varepsilon}\left(t\right) -X\left(t\right) \right\Vert_{\infty}<\varepsilon \] with probability one. Moreover, the user can adaptively choose $\varepsilon^{\prime}\in\left(0,\varepsilon\right) $ so that $X_{\varepsilon^{\prime}}$ (also piecewise constant and fully simulatable) can be constructed conditional on $X_{\varepsilon}$ to ensure an error smaller than $\varepsilon^{\prime}>0$ with probability one. Our construction requires a detailed study of continuity estimates of the Ito map using Lyon's theory of rough paths. We approximate the underlying Brownian motion, jointly with the Lévy areas with a deterministic $\varepsilon$ error in the underlying rough path metric.

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preprint2016arXiv

Exact Sampling of the Infinite Horizon Maximum of a Random Walk Over a Non-linear Boundary

We present the first algorithm that samples $\max_{n\geq0}\{S_{n}-n^α\},$ where $S_n$ is a mean zero random walk, and $n^α$ with $α\in(1/2,1)$ defines a nonliner boundary. We show that our algorithm has finite expected running time. We also apply the algorithm to construct the first exact simulation method for the steady-state departure process of a $GI/GI/\infty$ queue where the service time distribution has infinite mean.

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preprint2016arXiv

Perfect Sampling of Generalized Jackson Network

We provide the first perfect sampling algorithm for a Generalized Jackson Network of FIFO queues under arbitrary topology and non-Markovian assumptions on the input of the network. We assume (in addition to stability) that the interarrival and service times of customers have finite moment generating function in a neighborhood of the origin, and the interarrival times have unbounded support.

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preprint2016arXiv

Rates of Convergence to Stationarity for Multidimensional RBM

We provide the first rate of convergence analysis for RBM as the dimension grows under natural uniformity conditions. In particular, if the underlying routing matrix is uniformly contractive, uniform stability of the drift vector holds, and the variances of the underlying Brownian Motion (BM) are bounded, then we show that the RBM converges exponentially fast to stationarity with a relaxation time of order $O(d^4\log(d)^2)$ as $d\to\infty$.

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preprint2015arXiv

Exact Sampling of Stationary and Time-Reversed Queues

We provide the first algorithm that under minimal assumptions allows to simulate the stationary waiting-time sequence of a single-server queue backwards in time, jointly with the input processes of the queue (inter-arrival and service times). The single-server queue is useful in applications of DCFTP (Dominated Coupling From The Past), which is a well known protocol for simulation without bias from steady-state distributions. Our algorithm terminates in finite time assuming only finite mean of the inter-arrival and service times. In order to simulate the single-server queue in stationarity until the first idle period in finite expected termination time we require the existence of finite variance. This requirement is also necessary for such idle time (which is a natural coalescence time in DCFTP applications) to have finite mean. Thus, in this sense, our algorithm is applicable under minimal assumptions.

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preprint2015arXiv

On the Limiting Ratio of Current Age to Total Life for Null Recurrent Renewal Processes

If the inter-arrival time distribution of a renewal process is regularly varying with index $α\in\left( 0,1\right) $ (i.e. the inter-arrival times have infinite mean) and if $A\left( t\right) $ is the associated age process at time $t$. Then we show that if $C\left( t\right) $ is the length of the current cycle at time $t$, \[ A\left( t\right) /C\left( t\right) \Rightarrow U^{1/α}, \] where $U$ is $U\left( 0,1\right) $. This extends a classical result in renewal theory in the finite mean case which indicates that the limit is $U\left( 0,1\right) $.

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preprint2015arXiv

Perfect Sampling of GI/GI/c Queues

We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backwards in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.

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preprint2015arXiv

Steady-state simulation of reflected Brownian motion and related stochastic networks

This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first consider the case of compound Poisson (possibly Markov modulated) input. In this case, we analyze the complexity of our procedure as the dimension of the network increases and show that, under certain assumptions, the algorithm has polynomial-expected termination time. Our methodology includes procedures that are of interest beyond steady-state simulation and reflected processes. For instance, we use wavelets to construct a piecewise linear function that can be guaranteed to be within $\varepsilon$ distance (deterministic) in the uniform norm to Brownian motion in any compact time interval.

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preprint2015arXiv

Tail Asymptotics for Delay in a Half-loaded GI/GI/2 Queue with Heavy-tailed Job Sizes

We obtain asymptotic bounds for the tail distribution of steady-state waiting time in a two server queue where each server processes incoming jobs at a rate equal to the rate of their arrivals (that is, the half-loaded regime). The job sizes are taken to be regularly varying. When the incoming jobs have finite variance, there are basically two types of effects that dominate the tail asymptotics. While the quantitative distinction between these two manifests itself only in the slowly varying components, the two effects arise from qualitatively very different phenomena (arrival of one extremely big job (or) two big jobs). Then there is a phase transition that occurs when the incoming jobs have infinite variance. In that case, only one of these effects dominate the tail asymptotics, the one involving arrival of one extremely big job.

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preprint2014arXiv

A Weak Convergence Criterion Constructing Changes of Measure

Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and changes of measures for jump processes.

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preprint2014arXiv

State-independent Importance Sampling for Random Walks with Regularly Varying Increments

We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and elegant state-dependent efficient simulation algorithms have been developed in the literature over the last few years. We propose that by suitably decomposing these rare event probabilities into a dominant and further residual components, simpler state-independent importance sampling algorithms can be devised for each component resulting in composite unbiased estimators with desirable efficiency properties. When the increments have infinite variance, there is an added complexity in estimating the level crossing probabilities as even the well known zero-variance measures have an infinite expected termination time. We adapt our algorithms so that this expectation is finite while the estimators remain strongly efficient. Numerically, the proposed estimators perform at least as well, and sometimes substantially better than the existing state-dependent estimators in the literature.

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preprint2014arXiv

Theoretical analysis of a Stochastic Approximation approach for computing Quasi-Stationary distributions

This paper studies a method, which has been proposed in the Physics literature by [8, 7, 10], for estimating the quasi-stationary distribution. In contrast to existing methods in eigenvector estimation, the method eliminates the need for explicit transition matrix manipulation to extract the principal eigenvector. Our paper analyzes the algorithm by casting it as a stochastic approximation algorithm (Robbins-Monro) [23, 16]. In doing so, we prove its convergence and obtain its rate of convergence. Based on this insight, we also give an example where the rate of convergence is very slow. This problem can be alleviated by using an improved version of the algorithm that is given in this paper. Numerical experiments are described that demonstrate the effectiveness of this improved method.

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preprint2014arXiv

Total variation approximations and conditional limit theorems for multivariate regularly varying random walks conditioned on ruin

We study a new technique for the asymptotic analysis of heavy-tailed systems conditioned on large deviations events. We illustrate our approach in the context of ruin events of multidimensional regularly varying random walks. Our approach is to study the Markov process described by the random walk conditioned on hitting a rare target set. We construct a Markov chain whose transition kernel can be evaluated directly from the increment distribution of the associated random walk. This process is shown to approximate the conditional process of interest in total variation. Then, by analyzing the approximating process, we are able to obtain asymptotic conditional joint distributions and a conditional functional central limit theorem of several objects such as the time until ruin, the whole random walk prior to ruin, and the overshoot on the target set. These types of joint conditional limit theorems have been obtained previously in the literature only in the one dimensional case. In addition to using different techniques, our results include features that are qualitatively different from the one dimensional case. For instance, the asymptotic conditional law of the time to ruin is no longer purely Pareto as in the multidimensional case.

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preprint2013arXiv

Continuous-time Modeling of Bid-Ask Spread and Price Dynamics in Limit Order Books

We derive a continuous time model for the joint evolution of the mid price and the bid-ask spread from a multiscale analysis of the whole limit order book (LOB) dynamics. We model the LOB as a multiclass queueing system and perform our asymptotic analysis using stylized features observed empirically. We argue that in the asymptotic regime supported by empirical observations the mid price and bid-ask-spread can be described using only certain parameters of the book (not the whole book itself). Our limit process is characterized by reflecting behavior and state-dependent jumps. Our analysis allows to explain certain characteristics observed in practice such as: the connection between power-law decaying tails in the volumes of the order book and the returns, as well as statistical properties of the long-run spread distribution.

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preprint2013arXiv

Perfect sampling for infinite server and loss systems

We present the first class of perfect sampling (also known as exact simulation) algorithms for the steady-state distribution of non-Markovian loss networks. We use a variation of Dominated Coupling From The Past for which we simulate a stationary infinite server queue backwards in time and analyze the running time in heavy traffic. In particular, we are able to simulate stationary renewal marked point processes in unbounded regions. We use the infinite server queue as an upper bound process to simulate loss systems. The running time analysis of our perfect sampling algorithm for loss systems is performed in the Quality-Driven (QD) and the Quality-and-Efficiency-Driven regimes. In both cases, we show that our algorithm achieves sub-exponential complexity as both the number of servers and the arrival rate increase. Moreover, in the QD regime, our algorithm achieves a nearly optimal rate of convergence.

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preprint2012arXiv

Efficient Rare-event Simulation for Perpetuities

We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) + ... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the Y_j's satisfy the so-called Cramer condition with associated root theta_{ast} in (0,infty) and that the tails of the B_j's are appropriately behaved so that D is regularly varying with index theta_{ast}. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the Y_j's according to theta_{ast} fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.

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preprint2012arXiv

Rare-Event Simulation for Many-Server Queues

We develop rare-event simulation methodology for the analysis of loss events in a many-server loss system under quality-driven regime, focusing on the steady-state loss probability (i.e. fraction of lost customers over arrivals) and the behavior of the whole system leading to loss events. The analysis of these events requires working with the full measure-valued process describing the system. This is the first algorithm that is shown to be asymptotically optimal, in the rare-event simulation context, under the setting of many-server queues involving a full measure-valued descriptor.

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preprint2012arXiv

Sampling Point Processes on Stable Unbounded Regions and Exam Simulation of Queues

Given a marked renewal point process (assuming that the marks are i.i.d.) we say that an unbounded region is stable if it contains finitely many points of the point process with probability one. In this paper we provide algorithms that allow to sample these finitely many points efficiently. We explain how exact simulation of the steady-state measure valued state descriptor of the infinite server queue follows as a simple corollary of our algorithms. We provide numerical evidence supporting that our algorithms are not only theoretically sound but also practical. Finally, having simulation optimization in mind, we also apply our results to gradient estimation of steady-state performance measures.

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preprint2011arXiv

Characterizing Optimal Sampling of Binary Contingency Tables via the Configuration Model

A binary contingency table is an m x n array of binary entries with prescribed row sums r=(r_1,...,r_m) and column sums c=(c_1,...,c_n). The configuration model for uniformly sampling binary contingency tables proceeds as follows. First, label N=\sum_{i=1}^{m} r_i tokens of type 1, arrange them in m cells, and let the i-th cell contain r_i tokens. Next, label another set of tokens of type 2 containing N=\sum_{j=1}^{n}c_j elements arranged in n cells, and let the j-th cell contain c_j tokens. Finally, pair the type-1 tokens with the type-2 tokens by generating a random permutation until the total pairing corresponds to a binary contingency table. Generating one random permutation takes O(N) time, which is optimal up to constant factors. A fundamental question is whether a constant number of permutations is sufficient to obtain a binary contingency table. In the current paper, we solve this problem by showing a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as N goes to \infty. Our finding shows surprising differences from recent results for binary symmetric contingency tables.

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preprint2011arXiv

Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson's integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.

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preprint2011arXiv

On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case

Two of the most popular approximations for the distribution of the steady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, $ρ$, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of $W_{\infty}$ is roughly exponential at scale $O((1-ρ)^{-1})$, while the heavy tailed asymptotic describes power law decay in the tail of the distribution of $W_{\infty}$ for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of $(1-ρ)$, that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold.

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preprint2010arXiv

Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2β+1}) function evaluations suffice to achieve a given relative precision, where β is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.

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preprint2010arXiv

Efficient Simulation and Conditional Functional Limit Theorems for Ruinous Heavy-tailed Random Walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed stochastic processes. Our changes-of-measure are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity -- a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain, as a corollary, functional conditional central limit theorems that extend classical results in the literature. We illustrate our methodology in the context of the ruin probability $P(\sup_n S_n >b)$, where $S_n$ is a random walk with heavy-tailed increments that have negative drift. Our techniques are based on the use of Lyapunov inequalities for variance control and termination time. The conditional limit theorems combine the application of Lyapunov bounds with coupling arguments.

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