This paper deals with the identification of the stochastic Ornstein-Uhlenbeck (OU) process error model, which is characterized by an inverse time constant, and the unknown variances of the process and observation noises. Although the availability of the explicit expression of the log-likelihood function allows one to obtain the maximum likelihood estimator (MLE), this entails evaluating the nontrivial gradient and also often struggles with local optima. To address these limitations, we put forth a sample-efficient global optimization approach based on the Bayesian optimization (BO) framework, which relies on a Gaussian process (GP) surrogate model for the objective function that effectively balances exploration and exploitation to select the query points. Specifically, each evaluation of the objective is implemented efficiently through the Kalman filter (KF) recursion. Comprehensive experiments on various parameter scenarios and sampling intervals corroborate that BO-based estimator consistently outperforms MLE implemented by the steady-state KF approximation and the expectation-maximization algorithm (whose derivation is a side contribution) in terms of root meansquare error (RMSE) and statistical consistency, confirming the effectiveness and robustness of the$\mathbf{B O}$for identification of the stochastic OU process. Notably, the RMSE values produced by the BO-based estimator are smaller than the classical CramérRao lower bound, especially for the inverse time constant, estimating which has been a long-standing challenge. This seemingly counterintuitive result can be explained by the data-driven prior for the learning parameters indirectly injected by BO through the GP prior over the objective function.

The Gaussian process (GP) is a Bayesian non-parametric paradigm that is widely adopted for uncertainty quantification (UQ) in a number of safety-critical applications, including robotics, healthcare, as well as surveillance. The consistency of the resulting uncertainty values however, hinges on the premise that the learning function conforms to the properties specified by the GP model, such as smoothness, periodicity and more, which may not be satisfied in practice, especially with data arriving on the fly. To combat against such model mis-specification, we propose to wed the GP with the prevailing conformal prediction (CP), a distribution-free post-processing framework that produces prediction sets with a provably valid coverage under the sole assumption of data exchangeability. However, this assumption is usually violated in the online setting, where a prediction set is sought before revealing the true label. To ensure long-term coverage guarantee, we will adpatively set the key threshold parameter based on the feedback whether the true label falls inside the prediction set. Numerical results demonstrate the merits of the online GP-CP approach relative to existing alternatives in the long-term coverage performance.

This paper explores how a flying drone can autonomously navigate while constructing a narrowband radio map for signal localization. As flying drones become more ubiquitous, their wireless signals will necessitate new wireless technologies and algorithms to provide robust radio infrastructure while preserving radio spectrum usage. A potential solution for this spectrum-sharing localization challenge is to limit the bandwidth of any transmitter beacon. However, location signaling with a narrow bandwidth necessitates improving a wireless aerial system’s ability to filter a noisy signal, estimate the transmitter’s location, and self-pilot to improve the location estimate. By showing results through simulation, emulation, and a final drone flight experiment, this work provides an algorithm using a Gaussian process for radio signal estimation and Bayesian optimization for drone automatic guidance. This research supports advanced radio and aerial robotics applications in critical areas such as search-and-rescue, last-mile delivery, and large-scale platform digital twin development.

Recent years have witnessed the emergence of mobile edge computing (MEC), on the premise of a cost-effective enhancement in the computational ability of hardware-constrained wireless devices (WDs) comprising the Internet of Things (IoT). In a general multi-server multi-user MEC system, each WD has a computational task to execute and has to select binary (off)loading decisions, along with the analog-amplitude resource allocation variables in an online manner, with the goal of minimizing the overall energy-delay cost (EDC) with dynamic system states. While past works typically rely on the explicit expression of the EDC function, the present contribution considers a practical setting, where in lieu of system state information, the EDC function is not available in analytical form, and instead only the function values at queried points are revealed. Towards tackling such a challenging online combinatorial problem with only bandit information, novel Bayesian optimization (BO) based approaches are put forth by leveraging the multi-armed bandit (MAB) framework. Per time slot, the discrete offloading decisions are first obtained via the MAB method, and the analog resource allocation variables are subsequently optimized using the BO selection rule. By exploiting both temporal and contextual information, two novel BO approaches, termed time-varying BO and contextual time-varying BO, are developed. Numerical tests validate the merits of the proposed BO approaches compared with contemporary benchmarks under different MEC network sizes.

Graph-based inference arises in a gamut of network science-related applications, including smart transportation, climate forecasting, and neuroscience. Given observations over a subset of the nodes due to sampling costs or privacy considerations, extrapolation of time-varying signals over the unobserved nodes can be realized by leveraging their spatio-temporal correlations across the graph. Building on a recently proposed Gaussian process (GP) auto-regressive model to capture spatio-temporal dynamics across slots, the present work further pursues an adaptive framework by ensembling a candidate set of such dynamical models, each representing a unique dynamic pattern of the sought process. With nodal observation arriving on-the-fly, the proposed method simultaneously estimates the missing nodal values and selects the fitted dynamical model via data-adaptive weights. Tests with real data showcase the merits of the proposed method.

Optimizing a black-box function that is expensive to evaluate emerges in a gamut of machine learning and artificial intelligence applications including drug discovery, policy optimization in robotics, and hyperparameter tuning of learning models to list a few. Bayesian optimization (BO) provides a principled framework to find the global optimum of such functions using a limited number of function evaluations. BO relies on a statistical surrogate model to actively select new query points, that is typically captured by a Gaussian process (GP). Unlike most existing approaches that hinge on a single GP surrogate model with a pre-selected kernel function that may confine the expressiveness of the sought function especially under the limited evaluation budget, the present work puts forth a weighted ensemble of GPs as a surrogate model. Building on the advocated Gaussian mixture (GM) posterior, the EGP framework adapts to the most fitted surrogate model as data arrive on-the-fly, offering a richer function space. For the acquisition of next evaluation points, the EGP-based posterior is coupled with an adaptive expected improvement (EI) criterion to balance exploration and exploitation of the search space. Numerical tests on a set of benchmark synthetic functions and two robotic tasks, demonstrate the impressive benefits of the proposed approach.

Belonging to the family of Bayesian nonparametrics, Gaussian process (GP) based approaches have well-documented merits not only in learning over a rich class of nonlinear functions, but also in quantifying the associated uncertainty. However, most GP methods rely on a single preselected kernel function, which may fall short in characterizing data samples that arrive sequentially in time-critical applications. To enable online kernel adaptation, the present work advocates an incremental ensemble (IE-) GP framework, where an EGP assembler employs an ensemble of GP learners, each having a unique kernel belonging to a prescribed kernel dictionary. With each GP expert leveraging the random feature-based approximation to perform online prediction and model update with scalability, the EGP assembler capitalizes on data-adaptive weights to synthesize the per-expert predictions. Further, the novel IE-GP is generalized to accommodate time-varying functions by modeling structured dynamics at the EGP assembler and within each GP learner. To benchmark the performance of IE-GP and its dynamic variant in the adversarial setting where the modeling assumptions are violated, rigorous performance analysis has been conducted via the notion of regret, as the norm in online convex optimization. Last but not the least, online unsupervised learning for dimensionality reduction is explored under the novel IE-GP framework. Synthetic and real data tests demonstrate the effectiveness of the proposed schemes.

Bayesian optimization (BO) has well-documented merits for optimizing black-box functions with an expensive evaluation cost. Such functions emerge in applications as diverse as hyperparameter tuning, drug discovery, and robotics. BO hinges on a Bayesian surrogate model to sequentially select query points so as to balance exploration with exploitation of the search space. Most existing works rely on a single Gaussian process (GP) based surrogate model, where the kernel function form is typically preselected using domain knowledge. To bypass such a design process, this paper leverages an ensemble (E) of GPs to adaptively select the surrogate model fit on-the-fly, yielding a GP mixture posterior with enhanced expressiveness for the sought function. Acquisition of the next evaluation input using this EGP-based function posterior is then enabled by Thompson sampling (TS) that requires no additional design parameters. To endow function sampling with scalability, random feature-based kernel approximation is leveraged per GP model. The novel EGP-TS readily accommodates parallel operation. To further establish convergence of the proposed EGP-TS to the global optimum, analysis is conducted based on the notion of Bayesian regret for both sequential and parallel settings. Tests on synthetic functions and real-world applications showcase the merits of the proposed method.

Value function approximation is a crucial module for policy evaluation in reinforcement learning when the state space is large or continuous. The present paper revisits policy evaluation via temporal-difference (TD) learning from the Gaussian process (GP) perspective. Leveraging random features to approximate the GP prior, an online scalable (OS) approach, termed OS-GPTD, is developed to estimate the value function for a given policy by observing a sequence of state-reward pairs. To benchmark the performance of OS-GPTD even in the adversarial setting, where the modeling assumptions are violated, complementary worst-case analyses are performed. The cumulative Bellman error, as well as the long-term reward prediction error, are upper bounded relative to their counterparts from a fixed value function estimator with the entire state-reward trajectory in hindsight. Performance of the novel OS-GPTD is evaluated on two benchmark problems.

Gaussian process latent variable models (GPLVMs) are powerful, yet computationally heavy tools for nonlinear dimensionality reduction. Existing scalable variants utilize low- rank kernel matrix approximants that in essence subsample the embedding space. This work develops an efficient online approach based on random features by replacing spatial with spectral subsampling. The novel approach bypasses the need for optimizing over spatial samples, without sacrificing performance. Different from GPLVM, whose performance depends on the choice of the kernel, the proposed algorithm relies on an ensemble of kernels - what allows adaptation to a wide range of operating environments. It further allows for initial exploration of a richer function space, relative to methods adhering to a single fixed kernel, followed by sequential contraction of the search space as more data become available. Tests on benchmark datasets demonstrate the effectiveness of the proposed method.

Graph-guided semi-supervised learning (SSL) is a major task emerging in a gamut of network science applications. However, most SSL approaches rely on deterministic similarity metrics for prediction, thus providing only point estimates of the sought function. To allow for uncertainty quantification, which is of utmost importance in safety-critical applications, this work tackles the SSL task in a Gaussian process (GP) based Bayesian framework to propagate the distribution of nonparametric function estimates. Specifically, an incremental learning scenario is considered, where prediction of the desired value of a new node per iteration is followed by processing the corresponding nodal observation. Capitalizing on random features for scalability, an ensemble of GP experts is employed, each associated with a unique kernel from a known dictionary, to choose the fitted kernel combination in a graph- and data-adaptive fashion, thus bypassing the need for offline model training. Experiments with synthetic and real data showcase the merits of the proposed approach.

Combining benefits of kernels with Bayesian models, Gaussian process (GP) based approaches have well-documented merits not only in learning over a rich class of nonlinear functions, but also quantifying the associated uncertainty. While most GP approaches rely on a single preselected prior, the present work employs a weighted ensemble of GP priors, each having a unique covariance (kernel) belonging to a prescribed kernel dictionary – which leads to a richer space of learning functions. Leveraging kernel approximants formed by spectral features for scalability, an online interactive ensemble (OI-E) GP framework is developed to jointly learn the sought function, and for the first time select interactively the EGP kernel on-the-fly. Performance of OI-EGP is benchmarked by the best fixed function estimator via regret analysis. Furthermore, the novel OI-EGP is adapted to accommodate dynamic learning functions. Synthetic and real data tests demonstrate the effectiveness of the proposed schemes.

Semi-supervised learning (SSL) of dynamic processes over graphs is encountered in several applications of network science. Most of the existing approaches are unable to handle graphs with multiple relations, which arise in various real-world networks. This work deals with SSL of dynamic processes over multi-relational graphs (MRGs). Towards this end, a structured dynamical model is introduced to capture the spatio-temporal nature of dynamic graph processes, and incorporate contributions from multiple relations of the graph in a probabilistic fashion. Given nodal samples over a subset of nodes and the MRG, the expectation-maximization (EM) algorithm is adapted to extrapolate nodal features over unobserved nodes, and infer the contributions from the multiple relations in the MRG simultaneously. Experiments with real data showcase the merits of the proposed approach.

One regularity condition for the classical Cramér-Rao lower bound (CRLB) of an unbiased estimator to hold is that the support of the likelihood function (LF) should be independent of the parameter to be estimated. This has been shown to be too stringent and the CRLB has been shown to be valid for the case of parameter-dependent support as long as the LF is continuous at the boundary of its support. For the case where the LF is not continuous at the boundary of its support, a new modified CRLB - designated as the Cramér-Rao-Leibniz lower bound (CRLLB) as it relies on the Leibniz integral rule - has been presented for the scalar parameter and measurement case in [3]. The CRLLB for multidimensional parameter and measurements has been developed in [8]. The present work applies the multidimensional CRLLB to n-dimensional measurement noise with the raised fractional cosine and the truncated Laplace distributions inside an (n - 1)-sphere.

In previous works, it has been shown that the estimation problem of a thrusting/ballistic object in the three-dimensional space can be solved with two-dimensional measurements (azimuth and elevation angles starting from the launch time) assuming the launch point is perfectly known. In this paper, the problem is extended to estimate the target's trajectory with measurements starting after the launch time, i.e., delayed acquisition. Compared to the situation of acquisition at launch time, one has an additional unknown speed (magnitude of the velocity vector) and the unknown acquisition location. The 2D angle measurements are all obtained from a single fixed passive sensor. The parameter vector, in this case, has dimension 8 (velocity vector azimuth angle and elevation angle, drag coefficient, specific thrust, target speed and 3D acquisition position). The invertibility of the Fisher Information Matrix (FIM) of the parameter vector is investigated to test the observability (estimability) of the system. The simulation results prove the statistical efficiency and unbiasedness of the Maximum Likelihood estimator, that is, the Cramer-Rao lower bound (the inverse of the FIM if it is invertible) can be used as the actual covariance.

Parallel combinatory multicarrier (PCMC) modulation is a generalisation of the legacy multicarrier frequency shift keying (FSK) scheme, where for each group of M subcarriers, more than one (say L ) subcarriers are chosen for simultaneous transmission. The PCMC scheme provides an effective way to increase the spectral efficiency while maintaining non‐coherent detection at the receiver. This study provides an in‐depth study of the PCMC scheme with emphasis on how to couple with binary or non‐binary channel coding. One favourable system configuration is identified, having parameters ( L , M ) = (4,8), which increases the spectral efficiency by 50% relative to the most efficient FSK scheme from the FSK family. Coupled with non‐binary low‐density parity‐check (LDPC) coding over the Galois field GF(64), the PCMC scheme with ( L , M ) = (4,8) is shown to have robust performance in multipath fading channels.