Asymptotic Optimality Research Articles

Detection and identification of changes of hidden Markov chains: asymptotic theory

This paper revisits a unified framework of sequential change-point detection and hypothesis testing modeled using hidden Markov chains and develops its asymptotic theory. Given a sequence of observations whose distributions are dependent on a hidden Markov chain, the objective is to quickly detect critical events, modeled by the first time the Markov chain leaves a specific set of states, and to accurately identify the class of states that the Markov chain enters. We propose computationally tractable sequential detection and identification strategies and obtain sufficient conditions for the asymptotic optimality in two Bayesian formulations. Numerical examples are provided to confirm the asymptotic optimality.

A Change-Detection-Based Thompson Sampling Framework for Non-Stationary Bandits

We consider a non-stationary two-armed bandit framework and propose a change-detection-based Thompson sampling (TS) algorithm, named TS with change-detection (TS-CD), to keep track of the dynamic environment. The non-stationarity is modeled using a Poisson arrival process, which changes the mean of the rewards on each arrival. The proposed strategy compares the empirical mean of the recent rewards of an arm with the estimate of the mean of the rewards from its history. It detects a change when the empirical mean deviates from the mean estimate by a value larger than a threshold. Then, we characterize the lower bound on the duration of the time-window for which the bandit framework must remain stationary for TS-CD to successfully detect a change when it occurs. Consequently, our results highlight an upper bound on the parameter for the Poisson arrival process, for which the TS-CD achieves asymptotic regret optimality with high probability. Finally, we validate the efficacy of TS-CD by testing it for edge-control of radio access technique (RAT)-selection in a wireless network. Our results show that TS-CD not only outperforms the classical max-power RAT selection strategy but also other actively adaptive and passively adaptive bandit algorithms that are designed for non-stationary environments.

Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure f mathrm {d}x by a discrete probability measure sum _i m_i delta _{z_i}, subject to a constraint on the particle sizes m_i. The locations z_i of the particles, their sizes m_i, and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

Asymptotic Optimality in Byzantine Distributed Quickest Change Detection

The Byzantine distributed quickest change detection (BDQCD) is studied, where a fusion center monitors the occurrence of an abrupt event through a bunch of distributed sensors that may be compromised. We first consider the binary hypothesis case where there is only one post-change hypothesis and prove a novel converse to the first-order asymptotic detection delay in the large mean time to a false alarm regime. This converse is tight in that it coincides with the currently best achievability shown by Fellouris et al.; hence, the optimal asymptotic performance of binary BDQCD is characterized. An important implication of this result is that, even with compromised sensors, a 1-bit link between each sensor and the fusion center suffices to achieve asymptotic optimality. To accommodate multiple post-change hypotheses, we then formulate the multi-hypothesis BDQCD problem and again investigate the optimal first-order performance under different bandwidth constraints. A converse is first obtained by extending our converse from binary to multi-hypothesis BDQCD. Two families of stopping rules, namely the simultaneous d-th alarm and the multi-shot d-th alarm, are then proposed. Under sufficient link bandwidth, the simultaneous d-th alarm, with d being set to the number of honest sensors, can achieve the asymptotic performance that coincides with the derived converse bound; hence, the asymptotically optimal performance of multi-hypothesis BDQCD is again characterized. Moreover, although being shown to be asymptotically optimal only for some special cases, the multi-shot d-th alarm is much more bandwidth-efficient and energy-efficient than the simultaneous d-th alarm. Built upon the above success in characterizing the asymptotic optimality of the BDQCD, a corresponding leader-follower Stackelberg game is formulated and its solution is found.

Robust mean field social control problems with applications in analysis of opinion dynamics

This paper investigates the social optimality of linear quadratic mean field control systems with unmodelled dynamics. The objective of agents is to optimise the social cost, which is the sum of costs of all agents. By variational analysis and direct decoupling methods, the social optimal control problem is analysed, and two equivalent auxiliary robust optimal control problems are obtained for a representative agent. By solving the auxiliary problem with consistent mean field approximations, a set of decentralised strategies is designed, and its asymptotic social optimality is further proved. Next, the results are applied into the study of opinion dynamics in social networks. The evolution of opinions is analysed over finite and infinite horizons, respectively. All opinions are shown to reach agreement with the average opinion in a probabilistic sense. Finally, local interactions among multiple sub-populations are examined via graphon theory.

LDPC-coded 50-Gbaud IM/DD PAM-4 transmission employing probabilistic amplitude shaping based on prefix-free code distribution matching

We propose and experimentally demonstrate the intensity-modulation direct-detection (IM/DD) transmission employing low-density parity-check (LDPC) coded probabilistic shaping 4-level pulse amplitude modulation (PS-PAM4), assist with pairwise-distributed probabilistic amplitude shaping (PAS) structure based on prefix-free code distribution matching (PCDM). Unlike traditional PAS, the probability of adjacent symbols in the pairwise-distributed PAS architecture is consistent, which facilitates the filling of check bits generated by LDPC in IM/DD systems. On the other hand, PCDM has attracted increasing attention in recent years because of its asymptotic optimality similar to constant composition distribution matching (CCDM) in block length, and its high parallelism in hardware implementation. This is the first time that PCDM is applied in PAM4 signal transmission of PAS architecture in IM/DD system together with LDPC. Experiments of different code rates of LDPC and PCDM are performed. The experimental results show that the receiver sensitivity can be improved by more than 1 dB compared with the uniform-distributed signal transmission.

Towards robust, fast solutions of elliptic equations on complex domains through hybrid high‐order discretizations and non‐nested multigrid methods

AbstractThe use of modern discretization technologies such as hybrid high‐order (HHO) methods, coupled with appropriate linear solvers, allow for the robust and fast solution of partial differential equations (PDEs). Although efficient linear solvers have recently been made available for simpler cases, complex geometries remain a challenge for large scale problems. To address this problem, we propose in this work a geometric multigrid algorithm for unstructured non‐nested meshes. The non‐nestedness is handled in the prolongation operator through the use of the ‐orthogonal projection from the coarse elements onto the fine ones. However, as the exact evaluation of this projection can be computationally expensive, we develop a cheaper approximate implementation that globally preserves the approximation properties of the ‐orthogonal projection. Additionally, as the multigrid method requires not only the coarsening of the elements, but also that of the faces, we leverage the geometric flexibility of polytopal elements to define an abstract non‐nested coarsening strategy based on element agglomeration and face collapsing. Finally, the multigrid method is tested on homogeneous and heterogeneous diffusion problems in two and three space dimensions. The solver exhibits near‐perfect asymptotic optimality for moderate degrees of approximation.

COMPLETE SUBSET AVERAGING FOR QUANTILE REGRESSIONS

We propose a novel conditional quantile prediction method based on complete subset averaging (CSA) for quantile regressions. All models under consideration are potentially misspecified, and the dimension of regressors goes to infinity as the sample size increases. Since we average over the complete subsets, the number of models is much larger than the usual model averaging method which adopts sophisticated weighting schemes. We propose to use an equal weight but select the proper size of the complete subset based on the leave-one-out cross-validation method. Building upon the theory of Lu and Su (2015, Journal of Econometrics 188, 40–58), we investigate the large sample properties of CSA and show the asymptotic optimality in the sense of Li (1987, Annals of Statistics 15, 958–975) We check the finite sample performance via Monte Carlo simulations and empirical applications.

Two-timescale online learning of joint user association and resource scheduling in dynamic mobile edge computing

For the mobile edge computing network consisting of multiple base stations and resource-constrained user devices, network cost in terms of energy and delay will incur during task offoading from the user to the edge server. With the limitations imposed on transmission capacity, computing resource, and connection capacity, the perslot online learning algorithm is first proposed to minimize the time-averaged network cost. In particular, by leveraging the theories of stochastic gradient descent and minimum cost maximum flow, the user association is jointly optimized with resource scheduling in each time slot. The theoretical analysis proves that the proposed approach can achieve asymptotic optimality without any prior knowledge of the network environment. Moreover, to alleviate the high network overhead incurred during user handover and task migration, a two-timescale optimization approach is proposed to avoid frequent changes in user association. With user association executed on a large time scale and the resource scheduling decided on the single time slot, the a symptotic optimality is preserved. Simulation results verify the effectiveness of the proposed online learning algorithms.

Mind your own customers and ignore the others: Asymptotic optimality of a local policy in multi-class queueing systems with customer feedback

This work contributes to the investigation of optimal routing and scheduling policies in multi-class multi-server queueing systems with customer feedback. We propose a new policy, dubbed local policy that requires access to only local queue information. Our new local policy specifies how an idle server chooses the next customer by using the queue length information of not all queues, but only those this server is eligible to serve. To gain useful insights and mathematical tractability, we consider a simple W model with customer feedback, and we establish limit theorems to show that our local policy is asymptotically optimal among all policies that may use the global system information, with the objective of minimizing the cumulative queueing costs measured by convex functions of the queue lengths. Numerical experiments provide convincing engineering confirmations of the effectiveness of our local policy for both W model and a more general non-W model.

Classification by likelihood accordance functions

In this paper, we introduce the likelihood accordance function (LA function for short), which is defined to characterize the accordance of a new observation to be classified with training samples. The LA classifier is then constructed using the ratio of LA functions. It is shown that, the LA functions are invariant under orthogonal linear transformations, while LA classifier is invariant under non-degenerate linear transformations. Moreover, the asymptotic optimality of LA classifier is obtained. At last, several simulations illustrated that the new LA classifier performs much better than the traditional classifiers.

F-RRT*: An improved path planning algorithm with improved initial solution and convergence rate

During the last decades, sampling-based algorithms have been used to solve the problem of motion planning. RRT*, as an optimal variant of RRT, provides asymptotic optimality. However, the slow convergence rate and costly initial solution make it inefficient. To overcome these limitations, this paper proposes a modified RRT* algorithm, F-RRT*, which generates a better initial solution and converges faster than RRT*. F-RRT* optimizes the cost of paths by creating a parent node for the random point, instead of selecting it among the existing vertices. The creation process can be divided into two steps, the FindReachest, and CreatNode procedures, which require few calculations, and the triangle inequality is used repeatedly throughout the process, thus, resulting in paths with higher performance than those of RRT*. Since the algorithm proposed in this paper is a tree extending algorithm, its performance can be further enhanced when combined with other sampling strategies. The advantages of the proposed algorithm in the initial solution and fast convergence rate are demonstrated by comparing with RRT*, RRT*-Smart, and Q-RRT* through numerical simulations in this paper.

Optimal Load Balancing with Locality Constraints

Applications in cloud platforms motivate the study of efficient load balancing under job-server constraints and server heterogeneity. In this paper, we study load balancing on a bipartite graph where left nodes correspond to job types and right nodes correspond to servers, with each edge indicating that a job type can be served by a server. Thus edges represent locality constraints, i.e., an arbitrary job can only be served at servers which contain certain data and/or machine learning (ML) models. Servers in this system can have heterogeneous service rates. In this setting, we investigate the performance of two policies named Join-the-Fastest-of-the-Shortest-Queue (JFSQ) and Join-the-Fastest-of-the-Idle-Queue (JFIQ), which are simple variants of Join-the-Shortest-Queue and Join-the-Idle-Queue, where ties are broken in favor of the fastest servers. Under a "well-connected'' graph condition, we show that JFSQ and JFIQ are asymptotically optimal in the mean response time when the number of servers goes to infinity. In addition to asymptotic optimality, we also obtain upper bounds on the mean response time for finite-size systems. We further show that the well-connectedness condition can be satisfied by a random bipartite graph construction with relatively sparse connectivity.

Robust Change Detection via Information Projection

We study the robust transient and quickest change detection problems with unknown post-change distributions under finite alphabets. When the distribution after the change point is unknown, the change in the distribution of observations may occur in multiple ways without much structure on the observations, whereas, before the change point, a false alarm is highly structured, following a particular sample path with respect to the distribution of observations and the detection scheme. We first characterize these likely events for the deviation and propose a method to test the empirical distribution, relative to the most likely way for it to occur as an outlier. We show performance guarantees and prove asymptotic optimality for a robust quickest change detection problem up to a multiplicative constant before benchmarking our method with the finite moving average (FMA) method, generalized likelihood ratio test (GLRT) and M-statistic kernel (MSK) change point detection method under 4 different performance criteria including the run time complexity. Finally, we apply our method on economic market indicators and climate data. Our method successfully captures the regime shifts during times of historical significance for the markets and identifies the current climate change phenomenon to be a highly likely regime shift rather than a random event.

An overview on recent progress of extended state observers for uncertain systems: Methods, theory, and applications

AbstractExtended state observer (ESO) is featured with online estimating both the system's states and the total effect of external disturbance as well as nonlinear uncertain dynamics. The original ESO design, proposed for the cascade‐of‐integrator system without measurement noise, has been well studied with theoretical foundation and many successful applications. In the last years, multifarious modified ESOs have also been developed to handle many engineering systems captured by the models with general nonlinear structure, time‐delay, the noises with stochastic properties, and etc. This paper aims to comprehensively investigate the representative modified ESOs in recent years. Firstly, the ESO's design and tuning law are illustrated for general uncertain systems with multiple control inputs, multiple measured outputs and various uncertainties. The quantitative results regarding the estimation errors of ESO are also presented. Next, the extended state‐based Kalman‐typed filters to handle state estimation problem against both stochastic noises and uncertain dynamics are introduced. The consistency, stability, and asymptotic optimization of the proposed filters are rigorously shown. Furthermore, some novel designs of ESO to deal with the estimation problem of uncertain nonlinear systems with time‐delay and measurement biases are discussed. We believe this overview will be helpful for practitioners in applications of recent modified ESOs.

Lenient Regret for Multi-Armed Bandits

We consider the Multi-Armed Bandit (MAB) problem, where an agent sequentially chooses actions and observes rewards for the actions it took. While the majority of algorithms try to minimize the regret, i.e., the cumulative difference between the reward of the best action and the agent's action, this criterion might lead to undesirable results. For example, in large problems, or when the interaction with the environment is brief, finding an optimal arm is infeasible, and regret-minimizing algorithms tend to over-explore. To overcome this issue, algorithms for such settings should instead focus on playing near-optimal arms. To this end, we suggest a new, more lenient, regret criterion that ignores suboptimality gaps smaller than some ε. We then present a variant of the Thompson Sampling (TS) algorithm, called ε-TS, and prove its asymptotic optimality in terms of the lenient regret. Importantly, we show that when the mean of the optimal arm is high enough, the lenient regret of ε-TS is bounded by a constant. Finally, we show that ε-TS can be applied to improve the performance when the agent knows a lower bound of the suboptimality gaps.

Social Optima of Backward Linear-Quadratic-Gaussian Mean-Field Teams

This paper studies a class of stochastic linear-quadratic-Gaussian (LQG) dynamic optimization problems involving a large number of weakly-coupled heterogeneous agents. By “heterogeneous,” we mean agents are endowed with different types of parameters thus they are not statistically identical. Specifically, discrete-type heterogeneous agents are considered here which are more practical than homogeneous-type agents, and at the same time, more tractable than continuum-type heterogeneous agents. Unlike well-studied mean-field-game, these agents formalize a team with cooperation to minimize some social cost functional. Moreover, unlike standard social optima literature, the state here evolves by some backward stochastic differential equation (BSDE) in which the terminal instead initial condition is specified. Accordingly, the related social cost is represented by some recursive functional for which the initial state is considered. Applying a backward version of person-by-person optimality, we construct an auxiliary control problem for each agent based on decentralized information. The decentralized social strategy is derived by a class of new consistency condition (CC) systems, which are mean-field-type forward-backward stochastic differential equations (FBSDEs). The well-posedness of such consistency condition system is obtained via Riccati decoupling method. The related asymptotic social optimality is also verified.

Mallows criterion for heteroskedastic linear regressions with many regressors

We present a feasible generalized Mallows criterion for model selection for a linear regression setup with conditional heteroskedasticity and possibly numerous explanatory variables. The feasible version exploits unbiased individual variance estimates from recent literature. The property of asymptotic optimality of the feasible criterion is shown. A simulation experiment shows large discrepancies between model selection outcomes and those yielded by the classical Mallows criterion or other available alternatives.

An Optimal Control Approach for Consensus of General Linear Time-Invariant Multi-Agent Systems

Abstract In this paper, the consensus problem for general linear time-invariant (LTI) multi-agent systems (MASs) with a single input is studied in a new optimal control framework. The optimal cooperative control law is designed from a modified linear quadratic regulator (LQR) method and an inverse optimal control formulation. Three cost function terms are constructed to address the consensus, control effort, and cooperative tracking, respectively. Three distinct features of this approach can be achieved. First, the optimal feedback control law is derived analytically without involving any numerical solution. Second, this formulation guarantees both asymptotic stability and optimality. Third, the cooperative control law is distributed and only requires local information based on the communication topology to enable the agents to achieve consensus and track a desired trajectory. The performance of this optimal cooperative control method is demonstrated through an example of attitude synchronization of multiple satellites.

Model Averaging Estimation for Varying-Coefficient Single-Index Models

The varying-coefficient single-index model (VCSIM) is widely used in economics, statistics and biology. A model averaging method for VCSIM based on a Mallows-type criterion is proposed to improve prodictive capacity, which allows the number of candidate models to diverge with sample size. Under model misspecification, the asymptotic optimality is derived in the sense of achieving the lowest possible squared errors. The authors compare the proposed model averaging method with several other classical model selection methods by simulations and the corresponding results show that the model averaging estimation has a outstanding performance. The authors also apply the method to a real dataset.