# Optimisation for experiment design

### Bayesian and other surrogate noisy optimisation methods

Closely related is AutoML

## Problem statement

We are interested in solving

\begin{equation*} x^* = \arg \min_x f(x) \end{equation*}

under the constraints that

• $f$ is a black box for which no closed form is known (nor its gradients);
• $f$ is expensive to evaluate;
• evaluations of $y=f(x)$ may be noisy.

This is similar to the typical framing of reinforcement learning problems where there is a similar explore/exploit tradeoff, although I do not know the precise disciplinary boundaries that may transect these areas. They both might be thought of as stochastic optimal control problems.

The most common method seems to the “Bayesian optimisation”, which is based on Gaussian Process regressions. However, this is not a requirement, and many possible wacky regression models can give you the optimisation surrogate.

Of renewed interest for its use in hyperparameter/model selection, in e.g. regularising complex models, which is compactly referred to these days as automl.

You could also obviously use it in industrial process control, which is where I originally saw this kind of thing, in the form of sequential ANOVA design, which is an incredible idea itself, although that is now years old so is not nearly so hip. Since this effectively an attempt at optimal, nonlinear, heteroskedastic sequential ANOVA, I am led to wonder if we can dispense with ANOVA now. Does this stuff actually work well enough? Or is it the same thing, repackaged?

## Refs

FKES15
Feurer, M., Klein, A., Eggensperger, K., Springenberg, J., Blum, M., & Hutter, F. (2015) Efficient and Robust Automated Machine Learning. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, & R. Garnett (Eds.), Advances in Neural Information Processing Systems 28 (pp. 2962–2970). Curran Associates, Inc.
GeSA14
Gelbart, M. A., Snoek, J., & Adams, R. P.(2014) Bayesian Optimization with Unknown Constraints. In Proceedings of the Thirtieth Conference on Uncertainty in Artificial Intelligence (pp. 250–259). Arlington, Virginia, United States: AUAI Press
GAOS10
Grünewälder, S., Audibert, J.-Y., Opper, M., & Shawe-Taylor, J. (2010) Regret Bounds for Gaussian Process Bandit Problems. (Vol. 9, pp. 273–280). Presented at the AISTATS 2010 - Thirteenth International Conference on Artificial Intelligence and Statistics
IaMS00
Ian Dewancker, Michael McCourt, & Scott Clark. (n.d.) Bayesian Optimization Primer.
LJDR16
Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., & Talwalkar, A. (2016) Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization. ArXiv:1603.06560 [Cs, Stat].
Močk75
Močkus, J. (1975) On Bayesian Methods for Seeking the Extremum. In P. D. G. I. Marchuk (Ed.), Optimization Techniques IFIP Technical Conference (pp. 400–404). Springer Berlin Heidelberg DOI.
SnLA12
Snoek, J., Larochelle, H., & Adams, R. P.(2012) Practical bayesian optimization of machine learning algorithms. In Advances in neural information processing systems (pp. 2951–2959). Curran Associates, Inc.
SSZA14
Snoek, J., Swersky, K., Zemel, R., & Adams, R. (2014) Input Warping for Bayesian Optimization of Non-Stationary Functions. In Proceedings of the 31st International Conference on Machine Learning (ICML-14) (pp. 1674–1682).
SKKS12
Srinivas, N., Krause, A., Kakade, S. M., & Seeger, M. (2012) Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design. IEEE Transactions on Information Theory, 58(5), 3250–3265. DOI.
SwSA13
Swersky, K., Snoek, J., & Adams, R. P.(2013) Multi-Task Bayesian Optimization. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, & K. Q. Weinberger (Eds.), Advances in Neural Information Processing Systems 26 (pp. 2004–2012). Curran Associates, Inc.