We develop VSD, a method for conditioning a generative model of discrete, combinatorial designs on a rare desired class by efficiently evaluating a black-box (e.g. experiment, simulation) in a batch sequential manner. We call this task active generation; we formalize active generation's requirements and desiderata, and formulate a solution via variational inference. VSD uses off-the-shelf gradient based optimization routines, can learn powerful generative models for desirable designs, and can take advantage of scalable predictive models. We derive asymptotic convergence rates for learning the true conditional generative distribution of designs with certain configurations of our method. After illustrating the generative model on images, we empirically demonstrate that VSD can outperform existing baseline methods on a set of real sequence-design problems in various protein and DNA/RNA engineering tasks.

The expressiveness of flow-based models combined with stochastic variational inference (SVI) has expanded the application of optimization-based Bayesian inference to highly complex problems. However, despite the importance of multi-model Bayesian inference for problems defined on a transdimensional joint model and parameter space, such as Bayesian structure learning, flow-based SVI has been limited to problems defined on a fixed-dimensional parameter space. We introduce CoSMIC normalizing flows (COntextually-Specified Masking for Identity-mapped Components), an extension to neural autoregressive conditional normalizing flow architectures that enables use of a single flow-based variational density for inference over a transdimensional (multi-model) conditional target distribution. We propose a combined stochastic variational transdimensional inference (VTI) approach to training CoSMIC flows using ideas from Bayesian optimization and Monte Carlo gradient estimation. Numerical experiments show the performance of VTI on challenging problems that scale to high-cardinality model spaces.

We propose an extension of Thompson sampling to optimization problems over function spaces where the objective is a known functional of an unknown operator's output. We assume that queries to the operator (such as running a high-fidelity simulator or physical experiment) are costly, while functional evaluations on the operator's output are inexpensive. Our algorithm employs a sample-then-optimize approach using neural operator surrogates. This strategy avoids explicit uncertainty quantification by treating trained neural operators as approximate samples from a Gaussian process (GP) posterior. We derive regret bounds and theoretical results connecting neural operators with GPs in infinite-dimensional settings. Experiments benchmark our method against other Bayesian optimization baselines on functional optimization tasks involving partial differential equations of physical systems, demonstrating better sample efficiency and significant performance gains.

We introduce active generation of Pareto sets (A-GPS), a new framework for online discrete black-box multi-objective optimization (MOO). A-GPS learns a generative model of the Pareto set that supports a-posteriori conditioning on user preferences. The method employs a class probability estimator (CPE) to predict non-dominance relations and to condition the generative model toward high-performing regions of the search space. We also show that this non-dominance CPE implicitly estimates the probability of hypervolume improvement (PHVI). To incorporate subjective trade-offs, A-GPS introduces _preference direction vectors_ that encode user-specified preferences in objective space. At each iteration, the model is updated using both Pareto membership and alignment with these preference directions, producing an amortized generative model capable of sampling across the Pareto front without retraining. The result is a simple yet powerful approach that achieves high-quality Pareto set approximations, avoids explicit hypervolume computation, and flexibly captures user preferences. Empirical results on synthetic benchmarks and protein design tasks demonstrate strong sample efficiency and effective preference incorporation.

The process of calibrating computer models of natural phenomena is essential for applications in the physical sciences, where plenty of domain knowledge can be embedded into simulations and then calibrated against real observations. Current machine learning approaches, however, mostly rely on rerunning simulations over a fixed set of designs available in the observed data, potentially neglecting informative correlations across the design space and requiring a large amount of simulations. Instead, we consider the calibration process from the perspective of Bayesian adaptive experimental design and propose a data-efficient algorithm to run maximally informative simulations within a batch-sequential process. At each round, the algorithm jointly estimates the parameters posterior distribution and optimal designs by maximising a variational lower bound of the expected information gain. The simulator is modelled as a sample from a Gaussian process, which allows us to correlate simulations and real data with the unknown calibration parameters. We show the benefits of our method when compared to related approaches across synthetic and real-data problems.

In large-scale regression problems, random Fourier features (RFFs) have significantly enhanced the computational scalability and flexibility of Gaussian processes (GPs) by defining kernels through their spectral density, from which a finite set of Monte Carlo samples can be used to form an approximate low-rank GP. However, the efficacy of RFFs in kernel approximation and Bayesian kernel learning depends on the ability to tractably sample the kernel spectral measure and the quality of the generated samples. We introduce Stein random features (SRF), leveraging Stein variational gradient descent, which can be used to both generate high-quality RFF samples of known spectral densities as well as flexibly and efficiently approximate traditionally non-analytical spectral measure posteriors. SRFs require only the evaluation of log-probability gradients to perform both kernel approximation and Bayesian kernel learning that results in superior performance over traditional approaches. We empirically validate the effectiveness of SRFs by comparing them to baselines on kernel approximation and well-known GP regression problems.

We propose an adaptive optimisation approach for tuning stochastic model predictive control (MPC) hyper-parameters while jointly estimating probability distributions of the transition model parameters based on performance rewards. In particular, we develop a Bayesian optimisation (BO) algorithm with a heteroscedastic noise model to deal with varying noise across the MPC hyper-parameter and dynamics model parameter spaces. Typical homoscedastic noise models are unrealistic for tuning MPC since stochastic controllers are inherently noisy, and the level of noise is affected by their hyper-parameter settings. We evaluate the proposed optimisation algorithm in simulated control and robotics tasks where we jointly infer control and dynamics parameters. Experimental results demonstrate that our approach leads to higher cumulative rewards and more stable controllers.

Bayesian optimisation (BO) algorithms have shown remarkable success in applications involving expensive black-box functions. Traditionally BO has been set as a sequential decision-making process which estimates the utility of query points via an acquisition function and a prior over functions, such as a Gaussian process. Recently, however, a reformulation of BO via density-ratio estimation (BORE) allowed reinterpreting the acquisition function as a probabilistic binary classifier, removing the need for an explicit prior over functions and increasing scalability. In this paper, we present a theoretical analysis of BORE's regret and an extension of the algorithm with improved uncertainty estimates. We also show that BORE can be naturally extended to a batch optimisation setting by recasting the problem as approximate Bayesian inference. The resulting algorithms come equipped with theoretical performance guarantees and are assessed against other batch and sequential BO baselines in a series of experiments.

Bayesian probabilistic integration, or Bayesian quadrature (BQ), has arisen as a popular means of numerical integral estimation with quantified uncertainty for problems where computational cost limits data availability. BQ leverages flexible Gaussian processes (GPs) to model an integrand which can be subsequently analytically integrated through properties of Gaussian distributions. However, BQ is inherently limited by the fact that the method relies on the use of a strict set of kernels for use in the GP model of the integrand, reducing the flexibility of the method in modeling varied integrand types. In this paper, we present spectral Bayesian quadrature, a form of Bayesian quadrature that allows for the use of any shift-invariant kernel in the integrand GP model while still maintaining the analytical tractability of the integral posterior, increasing the flexibility of BQ methods to address varied problem settings. Additionally our method enables integration with respect to a uniform expectation, effectively computing definite integrals of challenging integrands. We derive the theory and error bounds for this model, as well as demonstrate GBQ’s improved accuracy, flexibility, and data efficiency, compared to traditional BQ and other numerical integration methods, on a variety of quadrature problems.

Volume preservation is usually regarded as a necessary property for the leapfrog transition functions that are used in Hamiltonian Monte Carlo (HMC) and No-U-Turn (NUTS) samplers to guarantee convergence to the target distribution. In this work we rigorously prove that with minimal algorithmic modifications, both HMC and NUTS can be combined with transition functions that are not necessarily volume preserving. In light of these results, we propose a non-volume preserving transition function that conserves the Hamiltonian better than the baseline leapfrog mechanism, on piecewise-continuous distributions. The resulting samplers do not require any assumptions on the geometry of the discontinuity boundaries, and our experimental results show a significant improvement upon traditional HMC and NUTS.

We consider Bayesian inference problems where the likelihood function is either expensive to evaluate or only available via noisy estimates. This setting encompasses application scenarios involving, for example, large datasets or models whose likelihood evaluations require expensive simulations. We formulate this problem within a Bayesian optimisation framework over a space of probability distributions and derive an upper confidence bound (UCB) algorithm to propose non-parametric distribution candidates. The algorithm is designed to minimise regret, which is defined as the Kullback-Leibler divergence with respect to the true posterior in this case. Equipped with a Gaussian process surrogate model, we show that the resulting UCB algorithm achieves asymptotically no regret. The method can be easily implemented as a batch Bayesian optimisation algorithm whose point evaluations are selected via Markov chain Monte Carlo. Experimental results demonstrate the method’s performance on inference problems.

Accurate simulation of complex physical systems enables the development, testing, and certification of control strategies before they are deployed into the real systems. As simulators become more advanced, the analytical tractability of the differential equations and associated numerical solvers incorporated in the simulations diminishes, making them difficult to analyse. A potential solution is the use of probabilistic inference to assess the uncertainty of the simulation parameters given real observations of the system. Unfortunately the likelihood function required for inference is generally expensive to compute or totally intractable. In this paper we propose to leverage the power of modern simulators and recent techniques in Bayesian statistics for likelihood-free inference to design a control framework that is efficient and robust with respect to the uncertainty over simulation parameters. The posterior distribution over simulation parameters is propagated through a potentially non-analytical model of the system with the unscented transform, and a variant of the information theoretical model predictive control. This approach provides a more efficient way to evaluate trajectory roll outs than Monte Carlo sampling, reducing the online computation burden. Experiments show that the controller proposed attained superior performance and robustness on classical control and robotics tasks when compared to models not accounting for the uncertainty over model parameters.

Recent advancements in Bayesian likelihood-free inference enables a probabilistic treatment for the problem of estimating simulation parameters and their uncertainty given sequences of observations. Domain randomization can be performed much more effectively when a posterior distribution provides the correct uncertainty over parameters in a simulated environment. In this paper, we study the integration of simulation parameter inference with both model-free reinforcement learning and model-based control in a novel sequential algorithm that alternates between learning a better estimation of parameters and improving the controller. This approach exploits the interdependence between the two problems to generate computational efficiencies and improved reliability when a black-box simulator is available. Experimental results suggest that both control strategies have better performance when compared to traditional domain randomization methods.

We establish a general form of explicit, input-dependent, measure-valued warpings for learning nonstationary kernels. While stationary kernels are uniquitous and simple to use, they struggle to adapt to functions that vary in smoothness with respect to the input. The proposed learning algorithm warps inputs as conditional Gaussian measures that control the smoothness of a standard stationary kernel. This construction allows us to capture non-stationary patterns in the data and provides intuitive inductive bias. The resulting method is based on sparse spectrum Gaussian processes, enabling closed-form solutions, and is extensible to a stacked construction to capture more complex patterns. The method is extensively validated alongside related algorithms on synthetic and real world datasets. We demonstrate a remarkable efficiency in the number of parameters of the warping functions in learning problems with both small and large data regimes.

We consider the problem of sequentially optimising the conditional expectation of an objective function, with both the conditional distribution and the objective function assumed to be fixed but unknown. Assuming that the objective function belongs to a reproducing kernel Hilbert space (RKHS), we provide a novel upper confidence bound (UCB) based algorithm CME-UCB via estimation of the conditional mean embeddings (CME), and derive its regret bound. Along the way, we derive novel approximation guarantees for the CME estimates. Finally, experiments are carried out in a synthetic example and in a likelihood-free inference application that highlight the useful insights of the proposed method.

Bayesian optimisation (BO) has been a successful approach to optimise functions which are expensive to evaluate and whose observations are noisy. Classical BO algorithms, however, do not account for errors about the location where observations are taken, which is a common issue in problems with physical components. In these cases, the estimation of the actual query location is also subject to uncertainty. In this context, we propose an upper confidence bound (UCB) algorithm for BO problems where both the outcome of a query and the true query location are uncertain. The algorithm employs a Gaussian process model that takes probability distributions as inputs. Theoretical results are provided for both the proposed algorithm and a conventional UCB approach within the uncertain-inputs setting. Finally, we evaluate each method’s performance experimentally, comparing them to other input noise aware BO approaches on simulated scenarios involving synthetic and real data.

In the automation of many kinds of processes, the observable outcome can often be described as the combined effect of an entire sequence of actions, or controls, applied throughout the process execution. In these cases, strategies to optimise control policies for individual stages of the process are not applicable, and instead the whole policy needs to be optimised at once. On the other hand, the cost to evaluate the policy's performance might also be high, being desirable that a solution can be found with as few interactions as possible with the real system. We consider the problem of optimising control policies to allow a robot to complete a given race track within a minimum amount of time. We assume that the robot has no prior information about the track or its own dynamical model, just an initial valid driving example. Localisation is only applied to monitor the robot and to provide an indication of its position along the track's centre axis. With that in mind, we propose a method for finding a policy that minimises the time per lap while keeping the vehicle on the track using a Bayesian optimisation (BO) approach over a reproducing kernel Hilbert space. We apply an algorithm to search more efficiently over high-dimensional policy-parameter spaces with BO, by iterating over each dimension individually, in a sequential coordinate descent-like scheme. Experiments demonstrate the performance of the algorithm against other methods in a simulated car racing environment.