Traditional methods for black box optimization require a considerable number of evaluations which can be time consuming, unpractical, and often unfeasible for many engineering applications that rely on accurate representations and expensive models to evaluate. Bayesian Optimization (BO) methods search for the global optimum by progressively (actively) learning a surrogate model of the objective function along the search path. Bayesian optimization can be accelerated through multifidelity approaches which leverage multiple black-box approximations of the objective functions that can be computationally cheaper to evaluate, but still provide relevant information to the search task. Further computational benefits are offered by the availability of parallel and distributed computing architectures whose optimal usage is an open opportunity within the context of active learning. This paper introduces the Resource Aware Active Learning (RAAL) strategy, a multifidelity Bayesian scheme to accelerate the optimization of black box functions. At each optimization step, the RAAL procedure computes the set of best sample locations and the associated fidelity sources that maximize the information gain to acquire during the parallel/distributed evaluation of the objective function, while accounting for the limited computational budget. The scheme is demonstrated for a variety of benchmark problems and results are discussed for both single fidelity and multifidelity settings. In particular we observe that the RAAL strategy optimally seeds multiple points at each iteration allowing for a major speed up of the optimization task.
Benchmarking and performance analysis play an important role in understanding the behaviour of iterative optimization heuristics (IOHs) such as local search algorithms, genetic and evolutionary algorithms, Bayesian optimization algorithms, etc. This task, however, involves manual setup, execution, and analysis of the experiment on an individual basis, which is laborious and can be mitigated by a generic and well-designed platform. For this purpose, we propose IOHanalyzer, a new user-friendly tool for the analysis, comparison, and visualization of performance data of IOHs. Implemented in R and C++, IOHanalyzer is fully open source. It is available on CRAN and GitHub. IOHanalyzer provides detailed statistics about fixed-target running times and about fixed-budget performance of the benchmarked algorithms with a real-valued codomain, single-objective optimization tasks. Performance aggregation over several benchmark problems is possible, for example in the form of empirical cumulative distribution functions. Key advantages of IOHanalyzer over other performance analysis packages are its highly interactive design, which allows users to specify the performance measures, ranges, and granularity that are most useful for their experiments, and the possibility to analyze not only performance traces, but also the evolution of dynamic state parameters. IOHanalyzer can directly process performance data from the main benchmarking platforms, including the COCO platform, Nevergrad, the SOS platform, and IOHexperimenter. An R programming interface is provided for users preferring to have a finer control over the implemented functionalities.
Bayesian optimization (BO) is a popular framework to optimize black-box functions. In many applications, the objective function can be evaluated at multiple fidelities to enable a trade-off between the cost and accuracy. To reduce the optimization cost, many multi-fidelity BO methods have been proposed. Despite their success, these methods either ignore or over-simplify the strong, complex correlations across the fidelities, and hence can be inefficient in estimating the objective function. To address this issue, we propose Deep Neural Network Multi-Fidelity Bayesian Optimization (DNN-MFBO) that can flexibly capture all kinds of complicated relationships between the fidelities to improve the objective function estimation and hence the optimization performance. We use sequential, fidelity-wise Gauss-Hermite quadrature and moment-matching to fulfill a mutual information-based acquisition function, which is computationally tractable and efficient. We show the advantages of our method in both synthetic benchmark datasets and real-world applications in engineering design.
Bayesian optimization (BO) is an efficient and flexible global optimization framework that is applicable to a very wide range of engineering applications. To leverage the capability of the classical BO, many extensions, including multi-objective, multi-fidelity, parallelization, latent-variable model, have been proposed to improve the limitation of the classical BO framework. In this work, we propose a novel multi-objective (MO) extension, called srMO-BO-3GP, to solve the MO optimization problems in a sequential setting. Three different Gaussian processes (GPs) are stacked together, where each of the GP is assigned with a different task: the first GP is used to approximate the single-objective function, the second GP is used to learn the unknown constraints, and the third GP is used to learn the uncertain Pareto frontier. At each iteration, a MO augmented Tchebycheff function converting MO to single-objective is adopted and extended with a regularized ridge term, where the regularization is introduced to smoothen the single-objective function. Finally, we couple the third GP along with the classical BO framework to promote the richness and diversity of the Pareto frontier by the exploitation and exploration acquisition function. The proposed framework is demonstrated using several numerical benchmark functions, as well as a thermomechanical finite element model for flip-chip package design optimization.
High dimensional black-box optimization has broad applications but remains a challenging problem to solve. Given a set of samples $\{\vx_i, y_i\}$, building a global model (like Bayesian Optimization (BO)) suffers from the curse of dimensionality in the high-dimensional search space, while a greedy search may lead to sub-optimality. By recursively splitting the search space into regions with high/low function values, recent works like LaNAS shows good performance in Neural Architecture Search (NAS), reducing the sample complexity empirically. In this paper, we coin LA-MCTS that extends LaNAS to other domains. Unlike previous approaches, LA-MCTS learns the partition of the search space using a few samples and their function values in an online fashion. While LaNAS uses linear partition and performs uniform sampling in each region, our LA-MCTS adopts a nonlinear decision boundary and learns a local model to pick good candidates. If the nonlinear partition function and the local model fits well with ground-truth black-box function, then good partitions and candidates can be reached with much fewer samples. LA-MCTS serves as a \emph{meta-algorithm} by using existing black-box optimizers (e.g., BO, TuRBO) as its local models, achieving strong performance in general black-box optimization and reinforcement learning benchmarks, in particular for high-dimensional problems.
Non-uniform sampling arises when an experimenter does not have full control over the sampling characteristics of the process under investigation. Moreover, it is introduced intentionally in algorithms such as Bayesian optimization and compressive sensing. We argue that Stochastic Differential Equations (SDEs) are especially well-suited for characterizing second order moments of such time series. We introduce new initial estimates for the numerical optimization of the likelihood, based on incremental estimation and initialization from autoregressive models. Furthermore, we introduce model truncation as a purely data-driven method to reduce the order of the estimated model based on the SDE likelihood. We show the increased accuracy achieved with the new estimator in simulation experiments, covering all challenging circumstances that may be encountered in characterizing a non-uniformly sampled time series. Finally, we apply the new estimator to experimental rainfall variability data.
Deployments of Bayesian Optimization (BO) for functions with stochastic evaluations, such as parameter tuning via cross validation and simulation optimization, typically optimize an average of a fixed set of noisy realizations of the objective function. However, disregarding the true objective function in this manner finds a high-precision optimum of the wrong function. To solve this problem, we propose Bayesian Optimization by Sampling Hierarchically (BOSH), a novel BO routine pairing a hierarchical Gaussian process with an information-theoretic framework to generate a growing pool of realizations as the optimization progresses. We demonstrate that BOSH provides more efficient and higher-precision optimization than standard BO across synthetic benchmarks, simulation optimization, reinforcement learning and hyper-parameter tuning tasks.
Bayesian Optimization (BO) is a surrogate-assisted global optimization technique that has been successfully applied in various fields, e.g., automated machine learning and design optimization. Built upon a so-called infill-criterion and Gaussian Process regression (GPR), the BO technique suffers from a substantial computational complexity and hampered convergence rate as the dimension of the search spaces increases. Scaling up BO for high-dimensional optimization problems remains a challenging task. In this paper, we propose to tackle the scalability of BO by hybridizing it with a Principal Component Analysis (PCA), resulting in a novel PCA-assisted BO (PCA-BO) algorithm. Specifically, the PCA procedure learns a linear transformation from all the evaluated points during the run and selects dimensions in the transformed space according to the variability of evaluated points. We then construct the GPR model, and the infill-criterion in the space spanned by the selected dimensions. We assess the performance of our PCA-BO in terms of the empirical convergence rate and CPU time on multi-modal problems from the COCO benchmark framework. The experimental results show that PCA-BO can effectively reduce the CPU time incurred on high-dimensional problems, and maintains the convergence rate on problems with an adequate global structure. PCA-BO therefore provides a satisfactory trade-off between the convergence rate and computational efficiency opening new ways to benefit from the strength of BO approaches in high dimensional numerical optimization.
This paper presents a recursive reasoning formalism of Bayesian optimization (BO) to model the reasoning process in the interactions between boundedly rational, self-interested agents with unknown, complex, and costly-to-evaluate payoff functions in repeated games, which we call Recursive Reasoning-Based BO (R2-B2). Our R2-B2 algorithm is general in that it does not constrain the relationship among the payoff functions of different agents and can thus be applied to various types of games such as constant-sum, general-sum, and common-payoff games. We prove that by reasoning at level 2 or more and at one level higher than the other agents, our R2-B2 agent can achieve faster asymptotic convergence to no regret than that without utilizing recursive reasoning. We also propose a computationally cheaper variant of R2-B2 called R2-B2-Lite at the expense of a weaker convergence guarantee. The performance and generality of our R2-B2 algorithm are empirically demonstrated using synthetic games, adversarial machine learning, and multi-agent reinforcement learning.
Bayesian optimization is a sequential decision making framework for optimizing expensive-to-evaluate black-box functions. Computing a full lookahead policy amounts to solving a highly intractable stochastic dynamic program. Myopic approaches, such as expected improvement, are often adopted in practice, but they ignore the long-term impact of the immediate decision. Existing nonmyopic approaches are mostly heuristic and/or computationally expensive. In this paper, we provide the first efficient implementation of general multi-step lookahead Bayesian optimization, formulated as a sequence of nested optimization problems within a multi-step scenario tree. Instead of solving these problems in a nested way, we equivalently optimize all decision variables in the full tree jointly, in a ``one-shot'' fashion. Combining this with an efficient method for implementing multi-step Gaussian process ``fantasization,'' we demonstrate that multi-step expected improvement is computationally tractable and exhibits performance superior to existing methods on a wide range of benchmarks.
Active search refers to the problem of efficiently locating targets in an unknown environment by actively making data-collection decisions, and has many applications including detecting gas leaks, radiation sources or human survivors of disasters using aerial and/or ground robots (agents). Existing active search methods are in general only amenable to a single agent, or if they extend to multi agent they require a central control system to coordinate the actions of all agents. However, such control systems are often impractical in robotics applications. In this paper, we propose two distinct active search algorithms called SPATS (Sparse Parallel Asynchronous Thompson Sampling) and LATSI (LAplace Thompson Sampling with Information gain) that allow for multiple agents to independently make data-collection decisions without a central coordinator. Throughout we consider that targets are sparsely located around the environment in keeping with compressive sensing assumptions and its applicability in real world scenarios. Additionally, while most common search algorithms assume that agents can sense the entire environment (e.g. compressive sensing) or sense point-wise (e.g. Bayesian Optimization) at all times, we make a realistic assumption that each agent can only sense a contiguous region of space at a time. We provide simulation results as well as theoretical analysis to demonstrate the efficacy of our proposed algorithms.
Searching for accurate Machine and Deep Learning models is a computationally expensive and awfully energivorous process. A strategy which has been gaining recently importance to drastically reduce computational time and energy consumed is to exploit the availability of different information sources, with different computational costs and different "fidelity", typically smaller portions of a large dataset. The multi-source optimization strategy fits into the scheme of Gaussian Process based Bayesian Optimization. An Augmented Gaussian Process method exploiting multiple information sources (namely, AGP-MISO) is proposed. The Augmented Gaussian Process is trained using only "reliable" information among available sources. A novel acquisition function is defined according to the Augmented Gaussian Process. Computational results are reported related to the optimization of the hyperparameters of a Support Vector Machine (SVM) classifier using two sources: a large dataset - the most expensive one - and a smaller portion of it. A comparison with a traditional Bayesian Optimization approach to optimize the hyperparameters of the SVM classifier on the large dataset only is reported.
We consider the problem of optimizing a vector-valued objective function $\boldsymbol{f}$ sampled from a Gaussian Process (GP) whose index set is a well-behaved, compact metric space $({\cal X},d)$ of designs. We assume that $\boldsymbol{f}$ is not known beforehand and that evaluating $\boldsymbol{f}$ at design $x$ results in a noisy observation of $\boldsymbol{f}(x)$. Since identifying the Pareto optimal designs via exhaustive search is infeasible when the cardinality of ${\cal X}$ is large, we propose an algorithm, called Adaptive $\boldsymbol{\epsilon}$-PAL, that exploits the smoothness of the GP-sampled function and the structure of $({\cal X},d)$ to learn fast. In essence, Adaptive $\boldsymbol{\epsilon}$-PAL employs a tree-based adaptive discretization technique to identify an $\boldsymbol{\epsilon}$-accurate Pareto set of designs in as few evaluations as possible. We provide both information-type and metric dimension-type bounds on the sample complexity of $\boldsymbol{\epsilon}$-accurate Pareto set identification. We also experimentally show that our algorithm outperforms other Pareto set identification methods.
While Bayesian Optimization (BO) is a very popular method for optimizing expensive black-box functions, it fails to leverage the experience of domain experts. This causes BO to waste function evaluations on bad design choices (e.g., machine learning hyperparameters) that the expert already knows to work poorly. To address this issue, we introduce Bayesian Optimization with a Prior for the Optimum (BOPrO). BOPrO allows users to inject their knowledge into the optimization process in the form of priors about which parts of the input space will yield the best performance, rather than BO's standard priors over functions, which are much less intuitive for users. BOPrO then combines these priors with BO's standard probabilistic model to form a pseudo-posterior used to select which points to evaluate next. We show that BOPrO is around 6.67x faster than state-of-the-art methods on a common suite of benchmarks, and achieves a new state-of-the-art performance on a real-world hardware design application. We also show that BOPrO converges faster even if the priors for the optimum are not entirely accurate and that it robustly recovers from misleading priors.
In recent years, leveraging parallel and distributed computational resources has become essential to solve problems of high computational cost. Bayesian optimization (BO) has shown attractive results in those expensive-to-evaluate problems such as hyperparameter optimization of machine learning algorithms. While many parallel BO methods have been developed to search efficiently utilizing these computational resources, these methods assumed synchronous settings or were not scalable. In this paper, we propose a simple and scalable BO method for asynchronous parallel settings. Experiments are carried out with a benchmark function and hyperparameter optimization of multi-layer perceptrons, which demonstrate the promising performance of the proposed method.
Finding methods for making generalizable predictions is a fundamental problem of machine learning. By looking into similarities between the prediction problem for unknown data and the lossless compression we have found an approach that gives a solution. In this paper we propose a compression principle that states that an optimal predictive model is the one that minimizes a total compressed message length of all data and model definition while guarantees decodability. Following the compression principle we use Bayesian approach to build probabilistic models of data and network definitions. A method to approximate Bayesian integrals using a sequence of variational approximations is implemented as an optimizer for hyper-parameters: Bayesian Stochastic Gradient Descent (BSGD). Training with BSGD is completely defined by setting only three parameters: number of epochs, the size of the dataset and the size of the minibatch, which define a learning rate and a number of iterations. We show that dropout can be used for a continuous dimensionality reduction that allows to find optimal network dimensions as required by the compression principle.
Bayesian optimization (BO) is a sample-efficient global optimization algorithm for black-box functions which are expensive to evaluate. Existing literature on model based optimization in conditional parameter spaces are usually built on trees. In this work, we generalize the additive assumption to tree-structured functions and propose an additive tree-structured covariance function, showing improved sample-efficiency, wider applicability and greater flexibility. Furthermore, by incorporating the structure information of parameter spaces and the additive assumption in the BO loop, we develop a parallel algorithm to optimize the acquisition function and this optimization can be performed in a low dimensional space. We demonstrate our method on an optimization benchmark function, as well as on a neural network model compression problem, and experimental results show our approach significantly outperforms the current state of the art for conditional parameter optimization including SMAC, TPE and Jenatton et al. (2017).
We propose MUMBO, the first high-performing yet computationally efficient acquisition function for multi-task Bayesian optimization. Here, the challenge is to perform efficient optimization by evaluating low-cost functions somehow related to our true target function. This is a broad class of problems including the popular task of multi-fidelity optimization. However, while information-theoretic acquisition functions are known to provide state-of-the-art Bayesian optimization, existing implementations for multi-task scenarios have prohibitive computational requirements. Previous acquisition functions have therefore been suitable only for problems with both low-dimensional parameter spaces and function query costs sufficiently large to overshadow very significant optimization overheads. In this work, we derive a novel multi-task version of entropy search, delivering robust performance with low computational overheads across classic optimization challenges and multi-task hyper-parameter tuning. MUMBO is scalable and efficient, allowing multi-task Bayesian optimization to be deployed in problems with rich parameter and fidelity spaces.
Bayesian optimization (BO) is an efficient method for optimizing expensive black-box functions. In real-world applications, BO often faces a major problem of missing values in inputs. The missing inputs can happen in two cases. First, the historical data for training BO often contain missing values. Second, when performing the function evaluation (e.g. computing alloy strength in a heat treatment process), errors may occur (e.g. a thermostat stops working) leading to an erroneous situation where the function is computed at a random unknown value instead of the suggested value. To deal with this problem, a common approach just simply skips data points where missing values happen. Clearly, this naive method cannot utilize data efficiently and often leads to poor performance. In this paper, we propose a novel BO method to handle missing inputs. We first find a probability distribution of each missing value so that we can impute the missing value by drawing a sample from its distribution. We then develop a new acquisition function based on the well-known Upper Confidence Bound (UCB) acquisition function, which considers the uncertainty of imputed values when suggesting the next point for function evaluation. We conduct comprehensive experiments on both synthetic and real-world applications to show the usefulness of our method.
Matrix square roots and their inverses arise frequently in machine learning, e.g., when sampling from high-dimensional Gaussians $\mathcal{N}(\mathbf 0, \mathbf K)$ or whitening a vector $\mathbf b$ against covariance matrix $\mathbf K$. While existing methods typically require $O(N^3)$ computation, we introduce a highly-efficient quadratic-time algorithm for computing $\mathbf K^{1/2} \mathbf b$, $\mathbf K^{-1/2} \mathbf b$, and their derivatives through matrix-vector multiplication (MVMs). Our method combines Krylov subspace methods with a rational approximation and typically achieves $4$ decimal places of accuracy with fewer than $100$ MVMs. Moreover, the backward pass requires little additional computation. We demonstrate our method's applicability on matrices as large as $50,\!000 \times 50,\!000$ - well beyond traditional methods - with little approximation error. Applying this increased scalability to variational Gaussian processes, Bayesian optimization, and Gibbs sampling results in more powerful models with higher accuracy.