Practical Bayesian Optimization of Machine Learning Algorithms (1206.2944v2)

Abstract: 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.

Practical Bayesian Optimization of Machine Learning Algorithms

Overview

The paper "Practical Bayesian Optimization of Machine Learning Algorithms," authored by Jasper Snoek, Hugo Larochelle, and Ryan P. Adams, provides a comprehensive paper on the application of Bayesian optimization for the automatic tuning of hyperparameters in ML algorithms. The authors advocate for the use of Gaussian processes (GP) to model the generalization performance of algorithms as a function of their hyperparameters. This method significantly improves the efficiency of choosing which hyperparameter settings to evaluate next, utilizing the information from previous experiments.

Key Contributions

The paper introduces several key contributions that address various practical aspects and challenges in Bayesian optimization:

1. Bayesian Treatment of Hyperparameters: The authors emphasize the importance of a fully Bayesian treatment for GP kernel parameters, rather than simply optimizing hyperparameters. This integrated approach is more robust and tends to yield superior results.
2. Cost-aware Optimization: Recognizing that the time required for function evaluations (training machine learning models) can vary substantially, the paper proposes methods that incorporate the cost (duration) of evaluations. This approach prioritizes both the accuracy and the speed of optimization.
3. Parallel Experiments: To leverage modern multi-core and parallel computing architectures, the authors propose algorithms that support parallel experimentation. This significantly accelerates the optimization process and improves the efficiency of finding optimal hyperparameter settings.

Methodology

Bayesian Optimization Framework

Bayesian optimization is framed as the problem of finding the minimum of an unknown function $f(\mathbf{x})$ over a bounded set $\mathcal{X}$. By assuming that $f$ is sampled from a Gaussian process, the framework constructs a probabilistic model that utilizes all available information from previous evaluations. This approach is especially useful when function evaluations are expensive, such as running deep learning experiments.

Acquisition Functions

Two major steps in Bayesian optimization are choosing the prior (Gaussian process) over functions and selecting an acquisition function. Several acquisition functions are discussed, including:

• Probability of Improvement (PI)
• Expected Improvement (EI)
• Upper Confidence Bound (UCB)

The paper focuses on the expected improvement criterion due to its robust and efficient performance without the need for additional tuning parameters.

Practical Considerations

Covariance Functions

Selecting an appropriate covariance function is critical for the efficacy of Gaussian process models. The authors propose using the ARD Matérn 5/2 kernel as a more flexible and less restrictive alternative to the commonly used squared-exponential kernel.

Cost Modeling

To minimize wallclock time, which is more practical than merely reducing the number of function evaluations, the authors present a method that models the duration of each experiment. This allows the optimization procedure to prefer points that are not only likely to be good but also quick to evaluate.

Parallelization

The paper introduces a method for parallelizing Bayesian optimization where the acquisition function integrates over possible results of pending evaluations. This approach is shown to be highly effective, especially in practical settings with multiple computational resources.

Empirical Analyses

The empirical evaluation of the proposed methods is thorough, spanning various challenging machine learning problems:

• Branin-Hoo Function and Logistic Regression: The proposed GP EI MCMC approach demonstrates superior performance over the Tree Parzen Estimator by requiring fewer evaluations to find the global minimum.
• Online LDA: The approach efficiently optimizes the hyperparameters for online LDA, outperforming exhaustive grid search and significantly reducing the computational time.
• Motif Finding with Structured SVMs: The Bayesian optimization strategies show considerable efficiency improvements over traditional grid search methods, achieving faster and better performance in hyperparameter tuning.
• Convolutional Networks on CIFAR-10: The optimized hyperparameters discovered by the proposed method achieve a test error of 14.98%, improving significantly over the expert-tuned parameters and setting a new state-of-the-art result.

Implications and Future Work

The implications of this work are substantial for both the theoretical development and practical application of machine learning algorithms. By automating the hyperparameter tuning process, Bayesian optimization not only saves time but also often surpasses human expert performance. Future developments may further refine these methods, particularly in areas such as integrating more sophisticated cost models and exploring additional acquisition functions.

Conclusion

This paper successfully addresses several practical challenges in the Bayesian optimization of hyperparameters, offering methodologies that are both theoretically sound and empirically validated. The proposed contributions enhance the robustness, efficiency, and scalability of Bayesian optimization, providing valuable tools for researchers and practitioners in the field of machine learning.