Practical Bayesian Optimization of Machine Learning Algorithms

Machine learning algorithms frequently require careful tuning of model hyperparameters, regularization terms, and optimization parameters. Unfortunately, this tuning is often a "black art" that requires expert experience, unwritten rules of thumb, or sometimes brute-force search. Much more appealing is the idea of developing automatic approaches which can optimize the performance of a given learning algorithm to the task at hand. In this work, we consider the automatic tuning problem within the framework of Bayesian optimization, in which a learning algorithm's generalization performance is modeled as a sample from a Gaussian process (GP). The tractable posterior distribution induced by the GP leads to efficient use of the information gathered by previous experiments, enabling optimal choices about what parameters to try next. Here we show how the effects of the Gaussian process prior and the associated inference procedure can have a large impact on the success or failure of Bayesian optimization. We show that thoughtful choices can lead to results that exceed expert-level performance in tuning machine learning algorithms. We also describe new algorithms that take into account the variable cost (duration) of learning experiments and that can leverage the presence of multiple cores for parallel experimentation. We show that these proposed algorithms improve on previous automatic procedures and can reach or surpass human expert-level optimization on a diverse set of contemporary algorithms including latent Dirichlet allocation, structured SVMs and convolutional neural networks

Bayesian Optimization with Unknown Constraints

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.Comment: 14 pages, 3 figure

Raiders of the Lost Architecture: Kernels for Bayesian Optimization in Conditional Parameter Spaces

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.Comment: 6 pages, 3 figures. Appeared in the NIPS 2013 workshop on Bayesian optimizatio

Practical Bayesian Optimization of Machine Learning Algorithms

Machine learning algorithms frequently require careful tuning of model hyperparameters, regularization terms, and optimization parameters. Unfortunately, this tuning is often a "black art" that requires expert experience, unwritten rules of thumb, or sometimes brute-force search. Much more appealing is the idea of developing automatic approaches which can optimize the performance of a given learning algorithm to the task at hand. In this work, we consider the automatic tuning problem within the framework of Bayesian optimization, in which a learning algorithm's generalization performance is modeled as a sample from a Gaussian process (GP). The tractable posterior distribution induced by the GP leads to efficient use of the information gathered by previous experiments, enabling optimal choices about what parameters to try next. Here we show how the effects of the Gaussian process prior and the associated inference procedure can have a large impact on the success or failure of Bayesian optimization. We show that thoughtful choices can lead to results that exceed expert-level performance in tuning machine learning algorithms. We also describe new algorithms that take into account the variable cost (duration) of learning experiments and that can leverage the presence of multiple cores for parallel experimentation. We show that these proposed algorithms improve on previous automatic procedures and can reach or surpass human expert-level optimization on a diverse set of contemporary algorithms including latent Dirichlet allocation, structured SVMs and convolutional neural networks.Engineering and Applied Science

Online Meta-learning by Parallel Algorithm Competition

The efficiency of reinforcement learning algorithms depends critically on a few meta-parameters that modulates the learning updates and the trade-off between exploration and exploitation. The adaptation of the meta-parameters is an open question in reinforcement learning, which arguably has become more of an issue recently with the success of deep reinforcement learning in high-dimensional state spaces. The long learning times in domains such as Atari 2600 video games makes it not feasible to perform comprehensive searches of appropriate meta-parameter values. We propose the Online Meta-learning by Parallel Algorithm Competition (OMPAC) method. In the OMPAC method, several instances of a reinforcement learning algorithm are run in parallel with small differences in the initial values of the meta-parameters. After a fixed number of episodes, the instances are selected based on their performance in the task at hand. Before continuing the learning, Gaussian noise is added to the meta-parameters with a predefined probability. We validate the OMPAC method by improving the state-of-the-art results in stochastic SZ-Tetris and in standard Tetris with a smaller, 10$\times$10, board, by 31% and 84%, respectively, and by improving the results for deep Sarsa($\lambda$) agents in three Atari 2600 games by 62% or more. The experiments also show the ability of the OMPAC method to adapt the meta-parameters according to the learning progress in different tasks.Comment: 15 pages, 10 figures. arXiv admin note: text overlap with arXiv:1702.0311

Multi-Task Bayesian Optimization

Bayesian optimization has recently been proposed as a framework for automatically tuning the hyperparameters of machine learning models and has been shown to yield state-of-the-art performance with impressive ease and efficiency. In this paper, we explore whether it is possible to transfer the knowledge gained from previous optimizations to new tasks in order to find optimal hyperparameter settings more efficiently. Our approach is based on extending multi-task Gaussian processes to the framework of Bayesian optimization. We show that this method significantly speeds up the optimization process when compared to the standard single-task approach. We further propose a straightforward extension of our algorithm in order to jointly minimize the average error across multiple tasks and demonstrate how this can be used to greatly speed up $k$-fold cross-validation. Lastly, our most significant contribution is an adaptation of a recently proposed acquisition function, entropy search, to the cost-sensitive and multi-task settings. We demonstrate the utility of this new acquisition function by utilizing a small dataset in order to explore hyperparameter settings for a large dataset. Our algorithm dynamically chooses which dataset to query in order to yield the most information per unit cost.Engineering and Applied Science