Part 4 · The Maths II: Acquisition Functions
Numerics and batches: LogEI and qEI
A 2023 result embarrassed a quarter-century of default practice. When z is very negative — the model is fairly sure x won’t improve — EI underflows: the value AND its gradient are numerically zero across wide swaths of the domain, so the (gradient-based) inner optimization of the acquisition silently stalls at whatever flat region it started in. Ament et al. (NeurIPS 2023) showed that simply computing and optimizing log EI — with a numerically careful expansion of log(zΦ(z) + φ(z)) — repairs the failure, and that a chunk of EI’s reported underperformance versus fancier acquisitions was this artifact all along. LogEI variants are now the BoTorch default.
Real experiments also run in parallel: you want q candidate points at once, not one. The joint criterion is the expected improvement of the best of the batch:
q the batch size · x₁..xq the points proposed together · the inner max means the batch is rewarded for its best member, not its average
No closed form survives the inner max, so BoTorch estimates it by Monte Carlo: draw posterior samples over the batch jointly (reparameterization trick, so gradients flow), average the improvements, ascend. The joint expectation is what stops the optimizer from proposing q copies of the same argmax — a batch’s members are valued for covering each other’s failure modes.