From ee8375f47495072c93e1f000ff2b82ec44c132fb Mon Sep 17 00:00:00 2001 From: Brummel Date: Wed, 17 Jun 2026 10:12:20 +0200 Subject: [PATCH] Add 'Adaptive Search' (parameter-search cluster) --- Adaptive-Search.md | 83 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 83 insertions(+) create mode 100644 Adaptive-Search.md diff --git a/Adaptive-Search.md b/Adaptive-Search.md new file mode 100644 index 0000000..4ecf16e --- /dev/null +++ b/Adaptive-Search.md @@ -0,0 +1,83 @@ +# Adaptive Search + +**Adaptive** (closed-loop) search methods choose each new trial *using the +results of earlier trials*. Unlike [grid](Grid-Search) or +[random](Random-Search) 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](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: + +1. **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](Evaluation-Dependency-and-Parallelism)). [L] +2. **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](https://en.wikipedia.org/wiki/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. + +- *Genetic algorithm.* Wikipedia. + +- *CMA-ES.* Wikipedia. + +- *Walk forward optimization.* Wikipedia. +