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We propose MIVABO, a simple yet effective method for efficient optimization of expensive-toevaluate mixed-variable black-box objective functions, combining a linear model of expressive features with Thompson sampling
Mixed-Variable Bayesian Optimization
IJCAI 2020, pp.2633-2639, (2020)
The optimization of expensive to evaluate, black-box, mixed-variable functions, i.e. functions that have continuous and discrete inputs, is a difficult and yet pervasive problem in science and engineering. In Bayesian optimization (BO), special cases of this problem that consider fully continuous or fully discrete domains have been wide...More
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- Bayesian optimization (BO)  is a well-established paradigm to optimize costly-to-evaluate, black-box objectives that has been successfully applied to multiple scientific domains.
- Since evaluating f is costly, the goal is to query inputs based on past observations to find a global minimizer x∗ ∈ arg minx∈X f (x) as efficiently and accurately as possible
- To this end, BO algorithms leverage two components: (i) a probabilistic function model, known as surrogate, that encodes the belief about f based on the observations available, and an acquisition function α : X → R that expresses the informativeness of input x about the location of x∗, given the surrogate of f.
- The goal of the acquisition function is to simultaneously learn about the inputs that are likely to be optimal and about poorly explored regions of the input space, i.e. to trade-off exploitation against exploration
- Bayesian optimization (BO)  is a well-established paradigm to optimize costly-to-evaluate, black-box objectives that has been successfully applied to multiple scientific domains
- We show how to use Thompson sampling  to suggest informative inputs to query (Sec. 3.3) and, provide a bound on the regret incurred by MIVABO. (Sec. 3.4)
- For the continuous model part, we employ Random Fourier Features (RFFs) approximating a Gaussian process with a squared exponential (SE) kernel, as we found Random Fourier Features to provide the best tradeoff between complexity and accuracy in practice
- We consider one of the most popular algorithms from OpenML, namely XGBoost, which is an efficient implementation of the extreme gradient boosting framework from 
- We propose MIVABO, a simple yet effective method for efficient optimization of expensive-toevaluate mixed-variable black-box objective functions, combining a linear model of expressive features with Thompson sampling
- Our method is characterized by a high degree of flexibility due to the modularity of its components, i.e., the feature mapping used to model the mixed-input objective, and the optimization oracles used as subroutines for the acquisition procedure
- The authors here present experimental results on tuning the hyperparameters of two machine learning algorithms, namely gradient boosting and a deep generative model.
- Random TPE SMAC GPyOpt MiVaBO.
- 0.0 1 10 20 30 40 50 60 70 80 90 100 # of sampled hyperparameter configurations validation error Random TPE.
- 0.00 1 10 20 30 40 50 60 70 80 90 100 # of sampled hyperparameter configurations negative test log-likelihood MiVaBo. 80 1 4 8 12 16 20 24 28 32 # of sampled hyperparameter configurations
- The authors use the publicly available OpenML database , which contains evaluations for various machine learning methods trained on several datasets with many hyperparameter settings.
- The authors consider one of the most popular algorithms from OpenML, namely XGBoost, which is an efficient implementation of the extreme gradient boosting framework from .
- The results in Fig. 1 show that MIVABO achieves performance which is either significantly stronger than or competitive with the state-of-the-art mixed-variable BO algorithms on this challenging task.
- As compared to TPE and SMAC, the method likely benefits from more sophisticated uncertainty estimation
- The authors propose MIVABO, a simple yet effective method for efficient optimization of expensive-toevaluate mixed-variable black-box objective functions, combining a linear model of expressive features with Thompson sampling.
- The authors' method is characterized by a high degree of flexibility due to the modularity of its components, i.e., the feature mapping used to model the mixed-input objective, and the optimization oracles used as subroutines for the acquisition procedure.
- This allows practitioners to tailor MIVABO to specific objectives, e.g. by incorporating prior knowledge in the feature design or by exploiting optimization oracles that can handle specific types of constraints.
- The authors empirically demonstrate that MIVABO significantly improves optimization performance as compared to state-of-the-art data driven methods for mixed-variable optimization
- Table1: Hyperparameters of the VAE. The architecture of the VAE (if all layers are enabled) is C1-C2-F1-F2-z-F3-F4-D1-D2, with C denoting a convolutional (conv.) layer, F a fullyconnected (fc.) layer, D a deconvolutional (deconv.) layer and z the latent space. Layers F2 and F3 have fixed sizes of 2dz and dz units respectively, where dz denotes the dimensionality of the latent space z. The domain of the number of units of the fc. layers F1 and F4 is discretized with a step size of 64, i.e. [0, 64, 128, . . . , 832, 896, 960], denoted by [0 . . . 960] in the table for brevity. For dz, the domain [16 . . . 64] refers to all integers within that interval
- Table2: Mean plus/minus one standard deviation of the negative test log-likelihood over 8 random initializations, achieved by the best VAE configuration found by SMAC, TPE and GPyOpt after 16 BO iterations, for constraint violation penalties of 500, 250 and 125 nats. Performance values of MIVABO and random search (which are not affected by the penalty) are included for reference
- Table3: Mean plus/minus one standard deviation of the number of constraint violations by SMAC, TPE, GPyOpt and random search within 16 BO iterations over 8 random initializations, for constraint violation penalties of 500, 250 and 125 nats
- Table4: Hyperparameters of the XGBoost algorithm. 10 parameters, 3 of which are discrete
- The efficient optimization of black-box functions over continuous domains has been extensively studied in the BO literature [58, 62, 20]. However, to adapt these methods to the mixed-variable setting, it is necessary to use ad-hoc relaxation techniques to map the problem to a fully continuous one and rounding methods to map the resulting solution to the original domain. This procedure ignores the original structure of the domain and makes the quality of the solution dependent on the choice of relaxation and rounding methods. Moreover, in this setting, it is hard to
Preprint. Under review.
incorporate constraints over the discrete input variables. More recently, BO algorithms for discrete domains have been proposed [4, 42]. However, the application of these methods to the mixedvariable setting requires discretizing the continuous part of the domain, where the discretization granularity plays a crucial role: If it is too small, it makes the input space prohibitively large; if it is too large, the resulting domain may contain only poorly performing values of the continuous inputs.
- This research has been partially supported by SNSF NFP75 grant 407540 167189
- Matteo Turchetta was supported through the ETH-MPI Center for Learning Systems
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