Sample-efficient Bayesian Optimisation Using Known Invariances

TL;DR

This work applies invariant BO to functions that exhibit invariance to a known group of transformations, and provides a method for incorporating group invariances into the kernel of the GP to produce invariance-aware algorithms that achieve significant improvements in sample efficiency.

Abstract

Bayesian optimisation (BO) is a powerful framework for global optimisation of costly functions, using predictions from Gaussian process models (GPs). In this work, we apply BO to functions that exhibit invariance to a known group of transformations. We show that vanilla and constrained BO algorithms are inefficient when optimising such invariant objectives, and provide a method for incorporating group invariances into the kernel of the GP to produce invariance-aware algorithms that achieve significant improvements in sample efficiency. We derive a bound on the maximum information gain of these invariant kernels, and provide novel upper and lower bounds on the number of observations required for invariance-aware BO algorithms to achieve $ε$-optimality. We demonstrate our method's improved performance on a range of synthetic invariant and quasi-invariant functions. We also apply our method in the case where only some of the invariance is incorporated into the kernel, and find that these kernels achieve similar gains in sample efficiency at significantly reduced computational cost. Finally, we use invariant BO to design a current drive system for a nuclear fusion reactor, finding a high-performance solution where non-invariant methods failed.

Figures (6)

• Figure 1: Examples of group-invariant functions, sampled from a Gaussian process with the corresponding invariant kernel. Note that observing a point $x$ (red) provides information about transformed locations $\{\sigma(x): \sigma \in G\}$ (white).
• Figure 2: Regret performance of invariant UCB and MVR algorithms across 3 different tasks; lower is better. Non-invariant kernels (blue) are outperformed by the full group invariant kernel (red) as well as partially specified (subgroup-invariant) kernels (green and yellow). For the permutation invariant function, the search space of standard BO can be constrained by the fundamental domain of the group (purple), but this performs worse than the invariant kernel. Mean $\pm$ s.d., 32 seeds.
• Figure 3: Performance of MVR and UCB on quasi-invariant functions. Regret shown for the noninvariant kernel $k'$ (standard, blue), the invariant kernel $k_G$ (invariant, green), and the quasi-invariant kernel $k_G + \varepsilon k'$ (additive, red). In all cases, the invariant kernel performs almost as well as the true quasi-invariant kernel.
• Figure 4: Nuclear fusion application: optimising safety factor by adjusting current drive actuator. In (a), observe that the order of launchers can be permuted without changing the total profile. Incorporating this invariance into the kernel of the UCB algorithm achieves improved performance (b).
• Figure 5: Effect of group size on cost of data augmentation and invariant kernel methods. Benchmark task is to fit a 6D GP with the given kernel to 100 random datapoints from PermInv-6D. Shown are results from 100 seeds, 64 repeats per seed, performed on one NVIDIA V100-32GB GPU using BoTorch. Invariant kernels are (a) more memory efficient than data augmentation, and (b) can be computed faster. Incorporating full invariance via data augmentation exceeds the GPU memory.

Theorems & Definitions (16)

• Proposition 1: RKHS of invariant functions
• Theorem 1: Upper bound on sample complexity of invariance-aware BO
• proof
• Theorem 2: Lower bound on sample complexity of invariance-aware BO with Matérn kernels
• proof
• Lemma 1
• proof
• Proposition 2
• proof
• Lemma 2: Bounded support function construction cai2021lower