Challenging optimization problems, which elude acceptable solution via conventional calculus methods, arise commonly in different areas of industrial design and practice. Hard optimization problems are those who manifest the following behavior: a) high number of independent input variables; b) very complex or irregular multi-modal fitness; c) computational expensive fitness evaluation. This paper will focus on some theoretical issues that have strong implications for practice. I will stress how an interpretation of the No Free Lunch theorem leads naturally to a general Bayesian optimization framework. The choice of a prior over the space of functions is a critical and inevitable step in every black-box optimization.
We investigate the Student-t process as an alternative to the Gaussian process as a nonparametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the covariance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Student-t process can retain the attractive properties of a Gaussian process -- a nonparametric representation, analytic marginal and predictive distributions, and easy model selection through covariance kernels -- but has enhanced flexibility, and predictive covariances that, unlike a Gaussian process, explicitly depend on the values of training observations. We verify empirically that a Student-t process is especially useful in situations where there are changes in covariance structure, or in applications like Bayesian optimization, where accurate predictive covariances are critical for good performance. These advantages come at no additional computational cost over Gaussian processes.
Bayesian optimization is a powerful global optimization technique for expensive black-box functions. One of its shortcomings is that it requires auxiliary optimization of an acquisition function at each iteration. This auxiliary optimization can be costly and very hard to carry out in practice. Moreover, it creates serious theoretical concerns, as most of the convergence results assume that the exact optimum of the acquisition function can be found. In this paper, we introduce a new technique for efficient global optimization that combines Gaussian process confidence bounds and treed simultaneous optimistic optimization to eliminate the need for auxiliary optimization of acquisition functions. The experiments with global optimization benchmarks and a novel application to automatic information extraction demonstrate that the resulting technique is more efficient than the two approaches from which it draws inspiration. Unlike most theoretical analyses of Bayesian optimization with Gaussian processes, our finite-time convergence rate proofs do not require exact optimization of an acquisition function. That is, our approach eliminates the unsatisfactory assumption that a difficult, potentially NP-hard, problem has to be solved in order to obtain vanishing regret rates.
Recent work on Bayesian optimization has shown its effectiveness in global optimization of difficult black-box objective functions. Many real-world optimization problems of interest also have constraints which are unknown a priori. In this paper, we study Bayesian optimization for constrained problems in the general case that noise may be present in the constraint functions, and the objective and constraints may be evaluated independently. We provide motivating practical examples, and present a general framework to solve such problems. We demonstrate the effectiveness of our approach on optimizing the performance of online latent Dirichlet allocation subject to topic sparsity constraints, tuning a neural network given test-time memory constraints, and optimizing Hamiltonian Monte Carlo to achieve maximal effectiveness in a fixed time, subject to passing standard convergence diagnostics.
BayesOpt is a library with state-of-the-art Bayesian optimization methods to solve nonlinear optimization, stochastic bandits or sequential experimental design problems. Bayesian optimization is sample efficient by building a posterior distribution to capture the evidence and prior knowledge for the target function. Built in standard C++, the library is extremely efficient while being portable and flexible. It includes a common interface for C, C++, Python, Matlab and Octave.
We propose a novel information-theoretic approach for Bayesian optimization called Predictive Entropy Search (PES). At each iteration, PES selects the next evaluation point that maximizes the expected information gained with respect to the global maximum. PES codifies this intractable acquisition function in terms of the expected reduction in the differential entropy of the predictive distribution. This reformulation allows PES to obtain approximations that are both more accurate and efficient than other alternatives such as Entropy Search (ES). Furthermore, PES can easily perform a fully Bayesian treatment of the model hyperparameters while ES cannot. We evaluate PES in both synthetic and real-world applications, including optimization problems in machine learning, finance, biotechnology, and robotics. We show that the increased accuracy of PES leads to significant gains in optimization performance.
In this paper we develop a dynamic form of Bayesian optimization for machine learning models with the goal of rapidly finding good hyperparameter settings. Our method uses the partial information gained during the training of a machine learning model in order to decide whether to pause training and start a new model, or resume the training of a previously-considered model. We specifically tailor our method to machine learning problems by developing a novel positive-definite covariance kernel to capture a variety of training curves. Furthermore, we develop a Gaussian process prior that scales gracefully with additional temporal observations. Finally, we provide an information-theoretic framework to automate the decision process. Experiments on several common machine learning models show that our approach is extremely effective in practice.
Bayesian optimization has proven to be a highly effective methodology for the global optimization of unknown, expensive and multimodal functions. The ability to accurately model distributions over functions is critical to the effectiveness of Bayesian optimization. Although Gaussian processes provide a flexible prior over functions which can be queried efficiently, there are various classes of functions that remain difficult to model. One of the most frequently occurring of these is the class of non-stationary functions. The optimization of the hyperparameters of machine learning algorithms is a problem domain in which parameters are often manually transformed a priori, for example by optimizing in "log-space," to mitigate the effects of spatially-varying length scale. We develop a methodology for automatically learning a wide family of bijective transformations or warpings of the input space using the Beta cumulative distribution function. We further extend the warping framework to multi-task Bayesian optimization so that multiple tasks can be warped into a jointly stationary space. On a set of challenging benchmark optimization tasks, we observe that the inclusion of warping greatly improves on the state-of-the-art, producing better results faster and more reliably.
We consider the problem of maximizing a real-valued continuous function $f$ using a Bayesian approach. Since the early work of Jonas Mockus and Antanas \v{Z}ilinskas in the 70's, the problem of optimization is usually formulated by considering the loss function $\max f - M_n$ (where $M_n$ denotes the best function value observed after $n$ evaluations of $f$). This loss function puts emphasis on the value of the maximum, at the expense of the location of the maximizer. In the special case of a one-step Bayes-optimal strategy, it leads to the classical Expected Improvement (EI) sampling criterion. This is a special case of a Stepwise Uncertainty Reduction (SUR) strategy, where the risk associated to a certain uncertainty measure (here, the expected loss) on the quantity of interest is minimized at each step of the algorithm. In this article, assuming that $f$ is defined over a measure space $(\mathbb{X}, \lambda)$, we propose to consider instead the integral loss function $\int_{\mathbb{X}} (f - M_n)_{+}\, d\lambda$, and we show that this leads, in the case of a Gaussian process prior, to a new numerically tractable sampling criterion that we call $\rm EI^2$ (for Expected Integrated Expected Improvement). A numerical experiment illustrates that a SUR strategy based on this new sampling criterion reduces the error on both the value and the location of the maximizer faster than the EI-based strategy.
In practical Bayesian optimization, we must often search over structures with differing numbers of parameters. For instance, we may wish to search over neural network architectures with an unknown number of layers. To relate performance data gathered for different architectures, we define a new kernel for conditional parameter spaces that explicitly includes information about which parameters are relevant in a given structure. We show that this kernel improves model quality and Bayesian optimization results over several simpler baseline kernels.
The Higgs boson is thought to provide the interaction that imparts mass to the fundamental fermions, but while measurements at the Large Hadron Collider (LHC) are consistent with this hypothesis, current analysis techniques lack the statistical power to cross the traditional 5$\sigma$ significance barrier without more data. \emph{Deep learning} techniques have the potential to increase the statistical power of this analysis by \emph{automatically} learning complex, high-level data representations. In this work, deep neural networks are used to detect the decay of the Higgs to a pair of tau leptons. A Bayesian optimization algorithm is used to tune the network architecture and training algorithm hyperparameters, resulting in a deep network of eight non-linear processing layers that improves upon the performance of shallow classifiers even without the use of features specifically engineered by physicists for this application. The improvement in discovery significance is equivalent to an increase in the accumulated dataset of 25\%.
Estimation of Distribution Algorithms (EDAs) require flexible probability models that can be efficiently learned and sampled. Restricted Boltzmann Machines (RBMs) are generative neural networks with these desired properties. We integrate an RBM into an EDA and evaluate the performance of this system in solving combinatorial optimization problems with a single objective. We assess how the number of fitness evaluations and the CPU time scale with problem size and with problem complexity. The results are compared to the Bayesian Optimization Algorithm, a state-of-the-art EDA. Although RBM-EDA requires larger population sizes and a larger number of fitness evaluations, it outperforms BOA in terms of CPU times, in particular if the problem is large or complex. RBM-EDA requires less time for model building than BOA. These results highlight the potential of using generative neural networks for combinatorial optimization.
Some statistical models are specified via a data generating process for which the likelihood function cannot be computed in closed form. Standard likelihood-based inference is then not feasible but the model parameters can be inferred by finding the values which yield simulated data that resemble the observed data. This approach faces at least two major difficulties: The first difficulty is the choice of the discrepancy measure which is used to judge whether the simulated data resemble the observed data. The second difficulty is the computationally efficient identification of regions in the parameter space where the discrepancy is low. We give here an introduction to our recent work where we tackle the two difficulties through classification and Bayesian optimization.
Bayesian optimization is a powerful tool for fine-tuning the hyper-parameters of a wide variety of machine learning models. The success of machine learning has led practitioners in diverse real-world settings to learn classifiers for practical problems. As machine learning becomes commonplace, Bayesian optimization becomes an attractive method for practitioners to automate the process of classifier hyper-parameter tuning. A key observation is that the data used for tuning models in these settings is often sensitive. Certain data such as genetic predisposition, personal email statistics, and car accident history, if not properly private, may be at risk of being inferred from Bayesian optimization outputs. To address this, we introduce methods for releasing the best hyper-parameters and classifier accuracy privately. Leveraging the strong theoretical guarantees of differential privacy and known Bayesian optimization convergence bounds, we prove that under a GP assumption these private quantities are also near-optimal. Finally, even if this assumption is not satisfied, we can use different smoothness guarantees to protect privacy.
When applying machine learning to problems in NLP, there are many choices to make about how to represent input texts. These choices can have a big effect on performance, but they are often uninteresting to researchers or practitioners who simply need a module that performs well. We propose an approach to optimizing over this space of choices, formulating the problem as global optimization. We apply a sequential model-based optimization technique and show that our method makes standard linear models competitive with more sophisticated, expensive state-of-the-art methods based on latent variable models or neural networks on various topic classification and sentiment analysis problems. Our approach is a first step towards black-box NLP systems that work with raw text and do not require manual tuning.
Bayesian optimization is a sample-efficient method for black-box global optimization. How- ever, the performance of a Bayesian optimization method very much depends on its exploration strategy, i.e. the choice of acquisition function, and it is not clear a priori which choice will result in superior performance. While portfolio methods provide an effective, principled way of combining a collection of acquisition functions, they are often based on measures of past performance which can be misleading. To address this issue, we introduce the Entropy Search Portfolio (ESP): a novel approach to portfolio construction which is motivated by information theoretic considerations. We show that ESP outperforms existing portfolio methods on several real and synthetic problems, including geostatistical datasets and simulated control tasks. We not only show that ESP is able to offer performance as good as the best, but unknown, acquisition function, but surprisingly it often gives better performance. Finally, over a wide range of conditions we find that ESP is robust to the inclusion of poor acquisition functions.
This works extends the Random Embedding Bayesian Optimization approach by integrating a warping of the high dimensional subspace within the covariance kernel. The proposed warping, that relies on elementary geometric considerations, allows mitigating the drawbacks of the high extrinsic dimensionality while avoiding the algorithm to evaluate points giving redundant information. It also alleviates constraints on bound selection for the embedded domain, thus improving the robustness, as illustrated with a test case with 25 variables and intrinsic dimension 6.
We address the problem of synthetic gene design using Bayesian optimization. The main issue when designing a gene is that the design space is defined in terms of long strings of characters of different lengths, which renders the optimization intractable. We propose a three-step approach to deal with this issue. First, we use a Gaussian process model to emulate the behavior of the cell. As inputs of the model, we use a set of biologically meaningful gene features, which allows us to define optimal gene designs rules. Based on the model outputs we define a multi-task acquisition function to optimize simultaneously severals aspects of interest. Finally, we define an evaluation function, which allow us to rank sets of candidate gene sequences that are coherent with the optimal design strategy. We illustrate the performance of this approach in a real gene design experiment with mammalian cells.
We introduce Bayesian optimization, a technique developed for optimizing time-consuming engineering simulations and for fitting machine learning models on large datasets. Bayesian optimization guides the choice of experiments during materials design and discovery to find good material designs in as few experiments as possible. We focus on the case when materials designs are parameterized by a low-dimensional vector. Bayesian optimization is built on a statistical technique called Gaussian process regression, which allows predicting the performance of a new design based on previously tested designs. After providing a detailed introduction to Gaussian process regression, we introduce two Bayesian optimization methods: expected improvement, for design problems with noise-free evaluations; and the knowledge-gradient method, which generalizes expected improvement and may be used in design problems with noisy evaluations. Both methods are derived using a value-of-information analysis, and enjoy one-step Bayes-optimality.
Bayesian optimization has shown to be a fundamental global optimization algorithm in many applications: ranging from automatic machine learning, robotics, reinforcement learning, experimental design, simulations, etc. The most popular and effective Bayesian optimization relies on a surrogate model in the form of a Gaussian process due to its flexibility to represent a prior over function. However, many algorithms and setups relies on the stationarity assumption of the Gaussian process. In this paper, we present a novel nonstationary strategy for Bayesian optimization that is able to outperform the state of the art in Bayesian optimization both in stationary and nonstationary problems.