Parameter settings profoundly impact the performance of machine learning algorithms and laboratory experiments. The classical grid search or trial-error methods are exponentially expensive in large parameter spaces, and Bayesian optimization (BO) offers an elegant alternative for global optimization of black box functions. In situations where the black box function can be evaluated at multiple points simultaneously, batch Bayesian optimization is used. Current batch BO approaches are restrictive in that they fix the number of evaluations per batch, and this can be wasteful when the number of specified evaluations is larger than the number of real maxima in the underlying acquisition function. We present the Budgeted Batch Bayesian Optimization (B3O) for hyper-parameter tuning and experimental design - we identify the appropriate batch size for each iteration in an elegant way. To set the batch size flexible, we use the infinite Gaussian mixture model (IGMM) for automatically identifying the number of peaks in the underlying acquisition functions. We solve the intractability of estimating the IGMM directly from the acquisition function by formulating the batch generalized slice sampling to efficiently draw samples from the acquisition function. We perform extensive experiments for both synthetic functions and two real world applications - machine learning hyper-parameter tuning and experimental design for alloy hardening. We show empirically that the proposed B3O outperforms the existing fixed batch BO approaches in finding the optimum whilst requiring a fewer number of evaluations, thus saving cost and time.
The challenge of taking many variables into account in optimization problems may be overcome under the hypothesis of low effective dimensionality. Then, the search of solutions can be reduced to the random embedding of a low dimensional space into the original one, resulting in a more manageable optimization problem. Specifically, in the case of time consuming black-box functions and when the budget of evaluations is severely limited, global optimization with random embeddings appears as a sound alternative to random search. Yet, in the case of box constraints on the native variables, defining suitable bounds on a low dimensional domain appears to be complex. Indeed, a small search domain does not guarantee to find a solution even under restrictive hypotheses about the function, while a larger one may slow down convergence dramatically. Here we tackle the issue of low-dimensional domain selection based on a detailed study of the properties of the random embedding, giving insight on the aforementioned difficulties. In particular, we describe a minimal low-dimensional set in correspondence with the embedded search space. We additionally show that an alternative equivalent embedding procedure yields simultaneously a simpler definition of the low-dimensional minimal set and better properties in practice. Finally, the performance and robustness gains of the proposed enhancements for Bayesian optimization are illustrated on numerical examples.
Solving optimal design problems through crowdsourcing faces a dilemma: On one hand, human beings have been shown to be more effective than algorithms at searching for good solutions of certain real-world problems with high-dimensional or discrete solution spaces; on the other hand, the cost of setting up crowdsourcing environments, the uncertainty in the crowd's domain-specific competence, and the lack of commitment of the crowd, all contribute to the lack of real-world application of design crowdsourcing. We are thus motivated to investigate a solution-searching mechanism where an optimization algorithm is tuned based on human demonstrations on solution searching, so that the search can be continued after human participants abandon the problem. To do so, we model the iterative search process as a Bayesian Optimization (BO) algorithm, and propose an inverse BO (IBO) algorithm to find the maximum likelihood estimators of the BO parameters based on human solutions. We show through a vehicle design and control problem that the search performance of BO can be improved by recovering its parameters based on an effective human search. Thus, IBO has the potential to improve the success rate of design crowdsourcing activities, by requiring only good search strategies instead of good solutions from the crowd.
Bayesian optimization through Gaussian process regression is an effective method of optimizing an unknown function for which every measurement is expensive. It approximates the objective function and then recommends a new measurement point to try out. This recommendation is usually selected by optimizing a given acquisition function. After a sufficient number of measurements, a recommendation about the maximum is made. However, a key realization is that the maximum of a Gaussian process is not a deterministic point, but a random variable with a distribution of its own. This distribution cannot be calculated analytically. Our main contribution is an algorithm, inspired by sequential Monte Carlo samplers, that approximates this maximum distribution. Subsequently, by taking samples from this distribution, we enable Thompson sampling to be applied to (armed-bandit) optimization problems with a continuous input space. All this is done without requiring the optimization of a nonlinear acquisition function. Experiments have shown that the resulting optimization method has a competitive performance at keeping the cumulative regret limited.
The expected improvement (EI) algorithm is a popular strategy for information collection in optimization under uncertainty. The algorithm is widely known to be too greedy, but nevertheless enjoys wide use due to its simplicity and ability to handle uncertainty and noise in a coherent decision theoretic framework. To provide rigorous insight into EI, we study its properties in a simple setting of Bayesian optimization where the domain consists of a finite grid of points. This is the so-called best-arm identification problem, where the goal is to allocate measurement effort wisely to confidently identify the best arm using a small number of measurements. In this framework, one can show formally that EI is far from optimal. To overcome this shortcoming, we introduce a simple modification of the expected improvement algorithm. Surprisingly, this simple change results in an algorithm that is asymptotically optimal for Gaussian best-arm identification problems, and provably outperforms standard EI by an order of magnitude.
Chemical space is so large that brute force searches for new interesting molecules are infeasible. High-throughput virtual screening via computer cluster simulations can speed up the discovery process by collecting very large amounts of data in parallel, e.g., up to hundreds or thousands of parallel measurements. Bayesian optimization (BO) can produce additional acceleration by sequentially identifying the most useful simulations or experiments to be performed next. However, current BO methods cannot scale to the large numbers of parallel measurements and the massive libraries of molecules currently used in high-throughput screening. Here, we propose a scalable solution based on a parallel and distributed implementation of Thompson sampling (PDTS). We show that, in small scale problems, PDTS performs similarly as parallel expected improvement (EI), a batch version of the most widely used BO heuristic. Additionally, in settings where parallel EI does not scale, PDTS outperforms other scalable baselines such as a greedy search, $\epsilon$-greedy approaches and a random search method. These results show that PDTS is a successful solution for large-scale parallel BO.
We learn recurrent neural network optimizers trained on simple synthetic functions by gradient descent. We show that these learned optimizers exhibit a remarkable degree of transfer in that they can be used to efficiently optimize a broad range of derivative-free black-box functions, including Gaussian process bandits, simple control objectives, global optimization benchmarks and hyper-parameter tuning tasks. Up to the training horizon, the learned optimizers learn to trade-off exploration and exploitation, and compare favourably with heavily engineered Bayesian optimization packages for hyper-parameter tuning.
Bayesian optimization (BO) methods are useful for optimizing functions that are expensive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the objective function, typically a Gaussian process (GP), upon which an acquisition function is built. This function guides the optimization process and measures the expected utility of performing an evaluation of the objective at a new point. GPs assume continous input variables. When this is not the case, such as when some of the input variables take integer values, one has to introduce extra approximations. A common approach is to round the suggested variable value to the closest integer before doing the evaluation of the objective. We show that this can lead to problems in the optimization process and describe a more principled approach to account for input variables that are integer-valued. We illustrate in both synthetic and a real experiments the utility of our approach, which significantly improves the results of standard BO methods on problems involving integer-valued variables.
We describe a method for searching the optimal hyper-parameters in reservoir computing, which consists of a Gaussian process with Bayesian optimization. It provides an alternative to other frequently used optimization methods such as grid, random, or manual search. In addition to a set of optimal hyper-parameters, the method also provides a probability distribution of the cost function as a function of the hyper-parameters. We apply this method to two types of reservoirs: nonlinear delay nodes and echo state networks. It shows excellent performance on all considered benchmarks, either matching or significantly surpassing results found in the literature. In general, the algorithm achieves optimal results in fewer iterations when compared to other optimization methods. We have optimized up to six hyper-parameters simultaneously, which would have been infeasible using, e.g., grid search. Due to its automated nature, this method significantly reduces the need for expert knowledge when optimizing the hyper-parameters in reservoir computing. Existing software libraries for Bayesian optimization, such as Spearmint, make the implementation of the algorithm straightforward. A fork of the Spearmint framework along with a tutorial on how to use it in practice is available at https://bitbucket.org/uhasseltmachinelearning/spearmint/
Randomized experiments are the gold standard for evaluating the effects of changes to real-world systems. Data in these tests may be difficult to collect and outcomes may have high variance, resulting in potentially large measurement error. Bayesian optimization is a promising technique for efficiently optimizing multiple continuous parameters, but existing approaches degrade in performance when the noise level is high, limiting its applicability to many randomized experiments. We derive an expression for expected improvement under greedy batch optimization with noisy observations and noisy constraints, and develop a quasi-Monte Carlo approximation that allows it to be efficiently optimized. Simulations with synthetic functions show that optimization performance on noisy, constrained problems outperforms existing methods. We further demonstrate the effectiveness of the method with two real-world experiments conducted at Facebook: optimizing a ranking system, and optimizing server compiler flags.
Machine Learning (ML) has been successfully applied to a wide range of domains and applications. One of the techniques behind most of these successful applications is Ensemble Learning (EL), the field of ML that gave birth to methods such as Random Forests or Boosting. The complexity of applying these techniques together with the market scarcity on ML experts, has created the need for systems that enable a fast and easy drop-in replacement for ML libraries. Automated machine learning (autoML) is the field of ML that attempts to answers these needs. Typically, these systems rely on optimization techniques such as bayesian optimization to lead the search for the best model. Our approach differs from these systems by making use of the most recent advances on metalearning and a learning to rank approach to learn from metadata. We propose autoBagging, an autoML system that automatically ranks 63 bagging workflows by exploiting past performance and dataset characterization. Results on 140 classification datasets from the OpenML platform show that autoBagging can yield better performance than the Average Rank method and achieve results that are not statistically different from an ideal model that systematically selects the best workflow for each dataset. For the purpose of reproducibility and generalizability, autoBagging is publicly available as an R package on CRAN.
In deterministic optimization, line searches are a standard tool ensuring stability and efficiency. Where only stochastic gradients are available, no direct equivalent has so far been formulated, because uncertain gradients do not allow for a strict sequence of decisions collapsing the search space. We construct a probabilistic line search by combining the structure of existing deterministic methods with notions from Bayesian optimization. Our method retains a Gaussian process surrogate of the univariate optimization objective, and uses a probabilistic belief over the Wolfe conditions to monitor the descent. The algorithm has very low computational cost, and no user-controlled parameters. Experiments show that it effectively removes the need to define a learning rate for stochastic gradient descent.
We consider robust optimization problems, where the goal is to optimize in the worst case over a class of objective functions. We develop a reduction from robust improper optimization to Bayesian optimization: given an oracle that returns $\alpha$-approximate solutions for distributions over objectives, we compute a distribution over solutions that is $\alpha$-approximate in the worst case. We show that de-randomizing this solution is NP-hard in general, but can be done for a broad class of statistical learning tasks. We apply our results to robust neural network training and submodular optimization. We evaluate our approach experimentally on corrupted character classification, and robust influence maximization in networks.
A popular testbed for deep learning has been multimodal recognition of human activity or gesture involving diverse inputs such as video, audio, skeletal pose and depth images. Deep learning architectures have excelled on such problems due to their ability to combine modality representations at different levels of nonlinear feature extraction. However, designing an optimal architecture in which to fuse such learned representations has largely been a non-trivial human engineering effort. We treat fusion structure optimization as a hyper-parameter search and cast it as a discrete optimization problem under the Bayesian optimization framework. We propose a novel graph-induced kernel to compute structural similarities in the search space of tree-structured multimodal architectures and demonstrate its effectiveness using two challenging multimodal human activity recognition datasets.
We present the first general purpose framework for marginal maximum a posteriori estimation of probabilistic program variables. By using a series of code transformations, the evidence of any probabilistic program, and therefore of any graphical model, can be optimized with respect to an arbitrary subset of its sampled variables. To carry out this optimization, we develop the first Bayesian optimization package to directly exploit the source code of its target, leading to innovations in problem-independent hyperpriors, unbounded optimization, and implicit constraint satisfaction; delivering significant performance improvements over prominent existing packages. We present applications of our method to a number of tasks including engineering design and parameter optimization.
Domain similarity measures can be used to gauge adaptability and select suitable data for transfer learning, but existing approaches define ad hoc measures that are deemed suitable for respective tasks. Inspired by work on curriculum learning, we propose to \emph{learn} data selection measures using Bayesian Optimization and evaluate them across models, domains and tasks. Our learned measures outperform existing domain similarity measures significantly on three tasks: sentiment analysis, part-of-speech tagging, and parsing. We show the importance of complementing similarity with diversity, and that learned measures are -- to some degree -- transferable across models, domains, and even tasks.
Bayesian optimization has recently attracted the attention of the automatic machine learning community for its excellent results in hyperparameter tuning. BO is characterized by the sample efficiency with which it can optimize expensive black-box functions. The efficiency is achieved in a similar fashion to the learning to learn methods: surrogate models (typically in the form of Gaussian processes) learn the target function and perform intelligent sampling. This surrogate model can be applied even in the presence of noise; however, as with most regression methods, it is very sensitive to outlier data. This can result in erroneous predictions and, in the case of BO, biased and inefficient exploration. In this work, we present a GP model that is robust to outliers which uses a Student-t likelihood to segregate outliers and robustly conduct Bayesian optimization. We present numerical results evaluating the proposed method in both artificial functions and real problems.
This paper studies Bayesian ranking and selection (R&S) problems with correlated prior beliefs and continuous domains, i.e. Bayesian optimization (BO). Knowledge gradient methods [Frazier et al., 2008, 2009] have been widely studied for discrete R&S problems, which sample the one-step Bayes-optimal point. When used over continuous domains, previous work on the knowledge gradient [Scott et al., 2011, Wu and Frazier, 2016, Wu et al., 2017] often rely on a discretized finite approximation. However, the discretization introduces error and scales poorly as the dimension of domain grows. In this paper, we develop a fast discretization-free knowledge gradient method for Bayesian optimization. Our method is not restricted to the fully sequential setting, but useful in all settings where knowledge gradient can be used over continuous domains. We show how our method can be generalized to handle (i) batch of points suggestion (parallel knowledge gradient); (ii) the setting where derivative information is available in the optimization process (derivative-enabled knowledge gradient). In numerical experiments, we demonstrate that the discretization-free knowledge gradient method finds global optima significantly faster than previous Bayesian optimization algorithms on both synthetic test functions and real-world applications, especially when function evaluations are noisy; and derivative-enabled knowledge gradient can further improve the performances, even outperforming the gradient-based optimizer such as BFGS when derivative information is available.
When approaching a novel visual recognition problem in a specialized image domain, a common strategy is to start with a pre-trained deep neural network and fine-tune it to the specialized domain. If the target domain covers a smaller visual space than the source domain used for pre-training (e.g. ImageNet), the fine-tuned network is likely to be over-parameterized. However, applying network pruning as a post-processing step to reduce the memory requirements has drawbacks: fine-tuning and pruning are performed independently; pruning parameters are set once and cannot adapt over time; and the highly parameterized nature of state-of-the-art pruning methods make it prohibitive to manually search the pruning parameter space for deep networks, leading to coarse approximations. We propose a principled method for jointly fine-tuning and compressing a pre-trained convolutional network that overcomes these limitations. Experiments on two specialized image domains (remote sensing images and describable textures) demonstrate the validity of the proposed approach.
This work studies how an AI-controlled dog-fighting agent with tunable decision-making parameters can learn to optimize performance against an intelligent adversary, as measured by a stochastic objective function evaluated on simulated combat engagements. Gaussian process Bayesian optimization (GPBO) techniques are developed to automatically learn global Gaussian Process (GP) surrogate models, which provide statistical performance predictions in both explored and unexplored areas of the parameter space. This allows a learning engine to sample full-combat simulations at parameter values that are most likely to optimize performance and also provide highly informative data points for improving future predictions. However, standard GPBO methods do not provide a reliable surrogate model for the highly volatile objective functions found in aerial combat, and thus do not reliably identify global maxima. These issues are addressed by novel Repeat Sampling (RS) and Hybrid Repeat/Multi-point Sampling (HRMS) techniques. Simulation studies show that HRMS improves the accuracy of GP surrogate models, allowing AI decision-makers to more accurately predict performance and efficiently tune parameters.