Abstract
Real world experiments are expensive, and thus it is important to reach a target in a minimum number of experiments. Experimental processes often involve control variables that change over time. Such problems can be formulated as functional optimisation problem. We develop a novel Bayesian optimisation framework for such functional optimisation of expensive black-box processes. We represent the control function using Bernstein polynomial basis and optimise in the coefficient space. We derive the theory and practice required to dynamically adjust the order of the polynomial degree, and show how prior information about shape can be integrated. We demonstrate the effectiveness of our approach for short polymer fibre design and optimising learning rate schedules for deep networks.
Highlights
Functional optimisation arises when a time-varying system requires optimal control variable values to change with time
The proposed Bayesian functional optimisation algorithm addresses the optimisation of expensive experimental processes, two of which we present in this paper
We evaluate our proposed functional optimisation method on one synthetic and two real world experiments: optimisation of fibre yield in short polymer fibre production, and learning rate schedule optimisation for neural network training
Summary
Functional optimisation arises when a time-varying system requires optimal control variable values to change with time. An underspecification of the order is detected if the derivative magnitude is close to the theoretical maxima possible within the current order of the polynomial basis based on Figure 2: Short polymer fibre production using recirculation. We derive a key lemma to compute the maximum of the derivative for a function realised using Bernstein polynomial basis of a fixed order. By the same token the theoretical maximum of the magnitude of the derivative is (αmax − αmin), assuming α ∈ [αmax, αmin] Based on this it can be said that an n-th order Bernstein polynomial can only be used to sufficiently approximate a function if the derivate of the function is within the bounds as stated in Lemma 3. If the order of the Bernstein polynomial becomes very high it is possible to use high-dimensional Bayesian optimisation (Rana et al 2017; Oh, Gavves, and Welling 2018)
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