We consider an atomic congestion game in which each player $i$ either participates in the game with an exogenous and known probability $p_{i}\in(0,1]$, independently of everybody else, or stays out and incurs no cost. We compute the parameterized price of anarchy to characterize the impact of demand uncertainty on the efficiency of selfish behavior, considering two different notions of a social planner. A prophet planner knows the realization of the random participation in the game; the ordinary planner does not. As a consequence, a prophet planner can compute an adaptive social optimum that selects different solutions depending on the players that turn out to be active, whereas an ordinary planner faces the same uncertainty as the players and can only compute social optima with respect to the player participation distribution. For both planners, we derive the precise price of anarchy, which arises from an optimization problem parameterized by the maximum participation probability $q=\max_{i} p_{i}$. For the case of affine costs, we provide an analytic expression for the ordinary and prophet price of anarchy, parameterized as a function of $q$.
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Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that satisfies the standard assumption. Relevant results in this field, such as rates of convergence and confidence sets, depend on the standardness parameter, which in practice may be unknown. In this paper, we review the notion of standardness and its connection to other geometrical restrictions. We prove the almost sure consistency of a plug-in type estimator for the so-called standardness constant, already studied in the literature. We propose a method to correct the bias of the plug-in estimator and corroborate our theoretical findings through a small simulation study. We also show that it is not possible to determine, based on a finite sample, whether a probability measure satisfies the standard assumption.
We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equations driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein-Uhlenbeck processes and illustrate our results with some numerical experiments.
Resiliency plays a critical role in designing future communication networks. How to make edge computing systems resilient against unpredictable failures and fluctuating demand is an important and challenging problem. To this end, this paper investigates a resilient service placement and workload allocation problem for a service provider (SP) who can procure resources from numerous edge nodes to serve its users, considering both resource demand and node failure uncertainties. We introduce a novel two-stage adaptive robust model to capture this problem. The service placement and resource procurement decisions are optimized in the first stage while the workload allocation decision is determined in the second stage after the uncertainty realization. By exploiting the special structure of the uncertainty set, we develop an efficient iterative algorithm that can converge to an exact optimal solution within a finite number of iterations. We further present an affine decision rule approximation approach for solving large-scale problem instances in a reasonable time. Extensive numerical results demonstrate the advantages of the proposed model and approaches, which can help the SP make proactive decisions to mitigate the impacts of the uncertainties.
The separate tasks of denoising, conditional expectation and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that are stable to numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems are also shown.
Computational optimal transport (OT) has recently emerged as a powerful framework with applications in various fields. In this paper we focus on a relaxation of the original OT problem, the entropic OT problem, which allows to implement efficient and practical algorithmic solutions, even in high dimensional settings. This formulation, also known as the Schr\"odinger Bridge problem, notably connects with Stochastic Optimal Control (SOC) and can be solved with the popular Sinkhorn algorithm. In the case of discrete-state spaces, this algorithm is known to have exponential convergence; however, achieving a similar rate of convergence in a more general setting is still an active area of research. In this work, we analyze the convergence of the Sinkhorn algorithm for probability measures defined on the $d$-dimensional torus $\mathbb{T}_L^d$, that admit densities with respect to the Haar measure of $\mathbb{T}_L^d$. In particular, we prove pointwise exponential convergence of Sinkhorn iterates and their gradient. Our proof relies on the connection between these iterates and the evolution along the Hamilton-Jacobi-Bellman equations of value functions obtained from SOC-problems. Our approach is novel in that it is purely probabilistic and relies on coupling by reflection techniques for controlled diffusions on the torus.
We propose a numerical method for the Vlasov-Poisson-Fokker-Planck model written as an hyperbolic system thanks to a spectral decomposition in the basis of Hermite functions with respect to the velocity variable and a structure preserving finite volume scheme for the space variable. On the one hand, we show that this scheme naturally preserves both stationary solutions and linearized free-energy estimate. On the other hand, we adapt previous arguments based on hypocoercivity methods to get quantitative estimates ensuring the exponential relaxation to equilibrium of the discrete solution for the linearized Vlasov-Poisson-Fokker-Planck system, uniformly with respect to both scaling and discretization parameters. Finally, we perform substantial numerical simulations for the nonlinear system to illustrate the efficiency of this approach for a large variety of collisional regimes (plasma echos for weakly collisional regimes and trend to equilibrium for collisional plasmas) and to highlight its robustness (unconditional stability, asymptotic preserving properties).
The distributed computation of a Nash equilibrium in aggregative games is gaining increased traction in recent years. Of particular interest is the mediator-free scenario where individual players only access or observe the decisions of their neighbors due to practical constraints. Given the competitive rivalry among participating players, protecting the privacy of individual players becomes imperative when sensitive information is involved. We propose a fully distributed equilibrium-computation approach for aggregative games that can achieve both rigorous differential privacy and guaranteed computation accuracy of the Nash equilibrium. This is in sharp contrast to existing differential-privacy solutions for aggregative games that have to either sacrifice the accuracy of equilibrium computation to gain rigorous privacy guarantees, or allow the cumulative privacy budget to grow unbounded, hence losing privacy guarantees, as iteration proceeds. Our approach uses independent noises across players, thus making it effective even when adversaries have access to all shared messages as well as the underlying algorithm structure. The encryption-free nature of the proposed approach, also ensures efficiency in computation and communication. The approach is also applicable in stochastic aggregative games, able to ensure both rigorous differential privacy and guaranteed computation accuracy of the Nash equilibrium when individual players only have stochastic estimates of their pseudo-gradient mappings. Numerical comparisons with existing counterparts confirm the effectiveness of the proposed approach.
We study stochastic online resource allocation: a decision maker needs to allocate limited resources to stochastically-generated sequentially-arriving requests in order to maximize reward. At each time step, requests are drawn independently from a distribution that is unknown to the decision maker. Online resource allocation and its special cases have been studied extensively in the past, but prior results crucially and universally rely on the strong assumption that the total number of requests (the horizon) is known to the decision maker in advance. In many applications, such as revenue management and online advertising, the number of requests can vary widely because of fluctuations in demand or user traffic intensity. In this work, we develop online algorithms that are robust to horizon uncertainty. In sharp contrast to the known-horizon setting, no algorithm can achieve even a constant asymptotic competitive ratio that is independent of the horizon uncertainty. We introduce a novel generalization of dual mirror descent which allows the decision maker to specify a schedule of time-varying target consumption rates, and prove corresponding performance guarantees. We go on to give a fast algorithm for computing a schedule of target consumption rates that leads to near-optimal performance in the unknown-horizon setting. In particular, our competitive ratio attains the optimal rate of growth (up to logarithmic factors) as the horizon uncertainty grows large. Finally, we also provide a way to incorporate machine-learned predictions about the horizon which interpolates between the known and unknown horizon settings.
The congestion game is a powerful model that encompasses a range of engineering systems such as traffic networks and resource allocation. It describes the behavior of a group of agents who share a common set of $F$ facilities and take actions as subsets with $k$ facilities. In this work, we study the online formulation of congestion games, where agents participate in the game repeatedly and observe feedback with randomness. We propose CongestEXP, a decentralized algorithm that applies the classic exponential weights method. By maintaining weights on the facility level, the regret bound of CongestEXP avoids the exponential dependence on the size of possible facility sets, i.e., $\binom{F}{k} \approx F^k$, and scales only linearly with $F$. Specifically, we show that CongestEXP attains a regret upper bound of $O(kF\sqrt{T})$ for every individual player, where $T$ is the time horizon. On the other hand, exploiting the exponential growth of weights enables CongestEXP to achieve a fast convergence rate. If a strict Nash equilibrium exists, we show that CongestEXP can converge to the strict Nash policy almost exponentially fast in $O(F\exp(-t^{1-\alpha}))$, where $t$ is the number of iterations and $\alpha \in (1/2, 1)$.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
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