Optimization

Conclusion and Research Directions

We have introduced Bayesian optimization, first discussing GP regression, then the expected improve- ment, knowledge gradient, entropy search, and predictive entropy search acquisition functions. We then discussed a variety of exotic Bayesian optimization problems: those with noisy measurements; paral- lel evaluations; constraints; multiple fidelities and multiple information sources; random environmental conditions and multi-task BayesOpt; and derivative observations.
Many research directions present themselves in this exciting field. First, there is substantial room for developing a deeper theoretical understanding of Bayesian optimization. As described in Section 4.4, settings where we can compute multi-step optimal algorithms are extremely limited. Moreover, while the acquisition functions we currently use in practice seem to perform almost as well as optimal multi- step algorithms when we can compute them, we do not currently have finite-time bounds that explain their near-optimal empirical performance, nor do we know whether multi-step optimal algorithms can provide substantial practical benefit in yet-to-be-understood settings. Even in the asymptotic regime, relatively little is known about rates of convergence for Bayesian optimization algorithms: while Bull (2011) establishes a rate of convergence for expected improvement when it is combined with periodic uniform sampling, it is unknown whether removing uniform sampling results in the same or different rate.
Second, there is room to build Bayesian optimization methods that leverage novel statistical ap- proaches. Gaussian processes (or variants thereof such as Snoek et al. (2014) and Kersting et al. (2007)) are used in most work on Bayesian optimization, but it seems likely that classes of problems exist where the objective could be better modeled through other approaches. It is both of interest to develop new statistical models that are broadly useful, and to develop models that are specifically designed for appli- cations of interest.
Third, developing Bayesian optimization methods that work well in high dimensions is of great prac- tical and theoretical interest. Directions for research include developing statistical methods that identify and leverage structure present in high-dimensional objectives arising in practice, which has been pur- sued by recent work including Wang et al. (2013, 2016b); Kandasamy et al. (2015). See also Shan and Wang (2010). It is also possible that new acquisition functions may provide substantial value in high dimensional problems.
Fourth, it is of interest to develop methods that leverage exotic problem structure unconsidered by today’s methods, in the spirit of the Section 5. It may be particularly fruitful to combine such methodological development with applying Bayesian optimization to important real-world problems, as using methods in the real world tends to reveal unanticipated difficulties and spur creativity.
Fifth, substantial impact in a variety of fields seems possible through application of Bayesian opti- mization. One set of application areas where Bayesian optimization seems particularly well-positioned to offer impact is in chemistry, chemical engineering, materials design, and drug discovery, where prac- titioners undertake design efforts involving repeated physical experiments consuming years of effort and substantial monetary expense. While there is some early work in these areas (Ueno et al., 2016; Frazier and Wang, 2016; Negoescu et al., 2011; Seko et al., 2015; Ju et al., 2017) the number of researchers working in these fields aware of the power and applicability of Bayesian optimization is still relatively small.