Adaptive Search

Adaptive (closed-loop) search methods choose each new trial using the results of earlier trials. Unlike grid or random search — where every trial point is fixed before any evaluation runs — an adaptive method runs a feedback loop: evaluate, learn, propose the next point(s), repeat. [L] This buys sample efficiency at the cost of a sequential dependency chain (see evaluation dependency and parallelism).

Bayesian optimization

Bayesian optimization fits a cheap surrogate model of the objective — most commonly a Gaussian process (kriging) — and uses an acquisition function to decide where to sample next, trading off exploration of unknown regions against exploitation of promising ones. [L] Common acquisition functions are probability of improvement, expected improvement, and Thompson sampling. For a current best value f^{*} (minimization), expected improvement scores a candidate x as

\alpha_{\text{EI}}(x) = \mathbb{E}\!\left[\max\!\left(0,\; f^{*} - f(x)\right)\right],

the expected amount by which it would beat the incumbent. [L] Because each evaluation updates the surrogate's posterior, every query depends on all prior results — the method is sequential by construction. It is the method of choice when each evaluation is expensive, since it aims to minimize the number of queries. [C]

Evolutionary and genetic algorithms

Evolutionary algorithms maintain a population of candidate solutions and improve it across generations. In each generation the fitness of every individual is evaluated; fitter individuals are stochastically selected, then combined by crossover and perturbed by mutation to form the next generation. [L] Each generation is therefore a function of the previous generation's fitness values — the dependency is generational. Evolutionary methods make few assumptions about the landscape and handle rugged, discontinuous objectives. [C]

CMA-ES

The Covariance Matrix Adaptation Evolution Strategy is a population-based, derivative-free optimizer for continuous spaces. It samples candidates from a multivariate normal distribution and, each iteration, adapts that distribution's mean, step size, and covariance matrix from the ranking of the candidates — so the search distribution itself learns the local shape of the landscape. [L] It relies only on the ranking of candidates, not on objective values or gradients, which makes it robust on ill-conditioned, non-separable problems. [C]

The trade-off: efficiency vs dependency, and overfitting

Adaptive methods typically reach a good optimum in far fewer evaluations than random search. [C] Two costs come with the feedback loop:

Limited parallelism. A dependency chain cannot be fanned out the way independent evaluations can; parallelism is confined to within a batch or generation, separated by synchronization barriers (details). [L]
Overfitting risk. An optimizer that aggressively maximizes the measured score also fits the noise of the particular evaluation data — the harder it optimizes, the more it can curve-fit a sample. [C] The standard mitigation is to score on held-out data; in trading-strategy validation this is walk forward optimization — optimize on an in-sample window, validate on rolling out-of-sample windows. [C]

A practical corollary: an adaptive method is worth its complexity mainly when evaluations are expensive and a non-adaptive baseline has been measured first. [C]

Claim legend

[L] law / exact — a definition, identity, or theorem that does not vary.
[C] convention — a practitioner rule of thumb that varies by context.

References

Bayesian optimization. Wikipedia. https://en.wikipedia.org/wiki/Bayesian_optimization
Genetic algorithm. Wikipedia. https://en.wikipedia.org/wiki/Genetic_algorithm
CMA-ES. Wikipedia. https://en.wikipedia.org/wiki/CMA-ES
Walk forward optimization. Wikipedia. https://en.wikipedia.org/wiki/Walk_forward_optimization

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