Pre-trained Gaussian processes for Bayesian optimization – Google AI Weblog

Bayesian optimization (BayesOpt) is a robust software broadly used for global optimization duties, resembling hyperparameter tuning, protein engineering, synthetic chemistry, robot learning, and even baking cookies. BayesOpt is a good technique for these issues as a result of all of them contain optimizing black-box functions which might be costly to guage. A black-box operate’s underlying mapping from inputs (configurations of the factor we wish to optimize) to outputs (a measure of efficiency) is unknown. Nevertheless, we are able to try to grasp its inner workings by evaluating the operate for various combos of inputs. As a result of every analysis may be computationally costly, we have to discover the most effective inputs in as few evaluations as attainable. BayesOpt works by repeatedly establishing a surrogate mannequin of the black-box operate and strategically evaluating the operate on the most promising or informative enter location, given the data noticed to date.

Gaussian processes are fashionable surrogate fashions for BayesOpt as a result of they’re straightforward to make use of, may be up to date with new knowledge, and supply a confidence degree about every of their predictions. The Gaussian course of mannequin constructs a probability distribution over attainable features. This distribution is specified by a imply operate (what these attainable features appear like on common) and a kernel operate (how a lot these features can fluctuate throughout inputs). The efficiency of BayesOpt is determined by whether or not the boldness intervals predicted by the surrogate mannequin comprise the black-box operate. Historically, specialists use area information to quantitatively outline the imply and kernel parameters (e.g., the vary or smoothness of the black-box operate) to precise their expectations about what the black-box operate ought to appear like. Nevertheless, for a lot of real-world functions like hyperparameter tuning, it is vitally obscure the landscapes of the tuning targets. Even for specialists with related expertise, it may be difficult to slender down applicable mannequin parameters.

In “Pre-trained Gaussian processes for Bayesian optimization”, we take into account the problem of hyperparameter optimization for deep neural networks utilizing BayesOpt. We suggest Hyper BayesOpt (HyperBO), a extremely customizable interface with an algorithm that removes the necessity for quantifying mannequin parameters for Gaussian processes in BayesOpt. For brand spanking new optimization issues, specialists can merely choose earlier duties which might be related to the present process they’re attempting to resolve. HyperBO pre-trains a Gaussian course of mannequin on knowledge from these chosen duties, and routinely defines the mannequin parameters earlier than working BayesOpt. HyperBO enjoys theoretical ensures on the alignment between the pre-trained mannequin and the bottom fact, in addition to the standard of its options for black-box optimization. We share robust outcomes of HyperBO each on our new tuning benchmarks for near–state-of-the-art deep learning models and traditional multi-task black-box optimization benchmarks (HPO-B). We additionally reveal that HyperBO is powerful to the choice of related duties and has low necessities on the quantity of knowledge and duties for pre-training.

Within the conventional BayesOpt interface, specialists must fastidiously choose the imply and kernel parameters for a Gaussian course of mannequin. HyperBO replaces this guide specification with a choice of associated duties, making Bayesian optimization simpler to make use of. The chosen duties are used for pre-training, the place we optimize a Gaussian course of such that it may possibly progressively generate features which might be just like the features akin to these chosen duties. The similarity manifests in particular person operate values and variations of operate values throughout the inputs.

Loss features for pre-training

We pre-train a Gaussian course of mannequin by minimizing the Kullback–Leibler divergence (a generally used divergence) between the bottom fact mannequin and the pre-trained mannequin. For the reason that floor fact mannequin is unknown, we can’t immediately compute this loss operate. To resolve for this, we introduce two data-driven approximations: (1) Empirical Kullback–Leibler divergence (EKL), which is the divergence between an empirical estimate of the bottom fact mannequin and the pre-trained mannequin; (2) Unfavorable log chance (NLL), which is the the sum of adverse log likelihoods of the pre-trained mannequin for all coaching features. The computational price of EKL or NLL scales linearly with the variety of coaching features. Furthermore, stochastic gradient–primarily based strategies like Adam may be employed to optimize the loss features, which additional lowers the price of computation. In well-controlled environments, optimizing EKL and NLL result in the identical consequence, however their optimization landscapes may be very totally different. For instance, within the easiest case the place the operate solely has one attainable enter, its Gaussian course of mannequin turns into a Gaussian distribution, described by the imply (m) and variance (s). Therefore the loss operate solely has these two parameters, m and s, and we are able to visualize EKL and NLL as follows:

We simulate the loss landscapes of EKL (left) and NLL (proper) for a easy mannequin with parameters m and s. The colours characterize a heatmap of the EKL or NLL values, the place pink corresponds to greater values and blue denotes decrease values. These two loss landscapes are very totally different, however they each goal to match the pre-trained mannequin with the bottom fact mannequin.

Pre-training improves Bayesian optimization

Within the BayesOpt algorithm, selections on the place to guage the black-box operate are made iteratively. The choice standards are primarily based on the boldness ranges offered by the Gaussian course of, that are up to date in every iteration by conditioning on earlier knowledge factors acquired by BayesOpt. Intuitively, the up to date confidence ranges must be good: not overly assured or too uncertain, since in both of those two circumstances, BayesOpt can’t make the choices that may match what an knowledgeable would do.

In HyperBO, we change the hand-specified mannequin in conventional BayesOpt with the pre-trained Gaussian course of. Below gentle situations and with sufficient coaching features, we are able to mathematically confirm good theoretical properties of HyperBO: (1) Alignment: the pre-trained Gaussian course of ensures to be near the bottom fact mannequin when each are conditioned on noticed knowledge factors; (2) Optimality: HyperBO ensures to discover a near-optimal answer to the black-box optimization drawback for any features distributed in accordance with the unknown floor fact Gaussian course of.

We visualize the Gaussian course of (areas shaded in purple are 95% and 99% confidence intervals) conditional on observations (black dots) from an unknown check operate (orange line). In comparison with the standard BayesOpt with out pre-training, the expected confidence ranges in HyperBO captures the unknown check operate a lot better, which is a vital prerequisite for Bayesian optimization.

Empirically, to outline the construction of pre-trained Gaussian processes, we select to make use of very expressive imply features modeled by neural networks, and apply well-defined kernel functions on inputs encoded to a better dimensional house with neural networks.

To guage HyperBO on difficult and real looking black-box optimization issues, we created the PD1 benchmark, which accommodates a dataset for multi-task hyperparameter optimization for deep neural networks. PD1 was developed by coaching tens of hundreds of configurations of near–state-of-the-art deep learning models on fashionable picture and textual content datasets, in addition to a protein sequence dataset. PD1 accommodates roughly 50,000 hyperparameter evaluations from 24 totally different duties (e.g., tuning Wide ResNet on CIFAR100) with roughly 12,000 machine days of computation.

We reveal that when pre-training for only some hours on a single CPU, HyperBO can considerably outperform BayesOpt with fastidiously hand-tuned fashions on unseen difficult duties, together with tuning ResNet50 on ImageNet. Even with solely ~100 knowledge factors per coaching operate, HyperBO can carry out competitively in opposition to baselines.

Tuning validation error charges of ResNet50 on ImageNet and Broad ResNet (WRN) on the Street View House Numbers (SVHN) dataset and CIFAR100. By pre-training on solely ~20 duties and ~100 knowledge factors per process, HyperBO can considerably outperform conventional BayesOpt (with a fastidiously hand-tuned Gaussian course of) on beforehand unseen duties.

Conclusion and future work

HyperBO is a framework that pre-trains a Gaussian course of and subsequently performs Bayesian optimization with a pre-trained mannequin. With HyperBO, we not should hand-specify the precise quantitative parameters in a Gaussian course of. As an alternative, we solely must establish associated duties and their corresponding knowledge for pre-training. This makes BayesOpt each extra accessible and more practical. An essential future course is to allow HyperBO to generalize over heterogeneous search areas, for which we’re creating new algorithms by pre-training a hierarchical probabilistic model.

Acknowledgements

The next members of the Google Analysis Mind Group performed this analysis: Zi Wang, George E. Dahl, Kevin Swersky, Chansoo Lee, Zachary Nado, Justin Gilmer, Jasper Snoek, and Zoubin Ghahramani. We would prefer to thank Zelda Mariet and Matthias Feurer for assist and session on switch studying baselines. We would additionally prefer to thank Rif A. Saurous for constructive suggestions, and Rodolphe Jenatton and David Belanger for suggestions on earlier variations of the manuscript. As well as, we thank Sharat Chikkerur, Ben Adlam, Balaji Lakshminarayanan, Fei Sha and Eytan Bakshy for feedback, and Setareh Ariafar and Alexander Terenin for conversations on animation. Lastly, we thank Tom Small for designing the animation for this publish.