TuRBO: the trust-region algebra

TuRBO abandons global modeling: it keeps a hyperrectangle centered on the incumbent and only models/acquires inside it. The region has base side L, shaped per-dimension by the fitted lengthscales (so the box is wide where the function is flat):

λ_i = ℓ_i · L / ( ∏_j ℓ_j )^1/d

λ_i the trust region's side length along dimension i · ℓ_i that dimension's fitted lengthscale · L the base side length · d the dimensionality; the product term renormalizes so volume is preserved

The side length then breathes by simple counters: τ_succ consecutive improving evaluations double it (capped at L_max); τ_fail consecutive failures halve it; and when L < L_min the region has collapsed onto a local optimum — declare victory locally, restart elsewhere, and let an implicit multi-armed bandit (Thompson sampling across regions, in the full algorithm) allocate the budget. Here is that dynamic, computed live for this page:

Our simplified single-region TuRBO (numpy, one run on negated Branin): the side length halves through dry spells, and a dotted line marks a restart after collapse — the algorithm concluding a local search and spending its remaining budget elsewhere. Local + restart beats confidently-wrong global modeling in high dimensions; that is TuRBO’s entire bet.

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