browniansearch

Introduction

The literature on search over correlated landscapes, organized by how many evaluations the searcher gets.

Search on an unknown landscape

A firm chooses a strategy, a designer a parameter, an engineer a location. Each choice has a performance, and performance varies with the choice in a way nobody knows in advance. Nearby choices perform similarly, and distant ones may perform very differently. That unknown function is the landscape.

Trying a choice reveals its performance at that point only. A searcher who can afford a few trials must decide where to try next and, at the end, where to commit.

Correlated learning

Bardhi and Callander (2026) survey this area under the name correlated learning: how an observed outcome changes beliefs about other choices when outcomes are correlated. The modelling device is to treat the unknown map from choices to outcomes as the path of a stochastic process, indexed by choices rather than time, most often Brownian motion.

Brownian motion gives learning a simple shape. Beliefs about an untried choice depend only on its nearest observed neighbours. Uncertainty grows with distance from what is known, and learning one region reveals little about the rest.

The classic index results for search and bandits (Gittins 1979; Weitzman 1979) assume independent outcomes. Correlated extensions had made little progress beyond special cases such as negatively correlated bandits (Klein and Rady 2011).

The Brownian framework has been applied to four kinds of problem:

One shot

In the workhorse search model (Callander 2011a) each agent in a sequence lives one period, tries one choice and cares only about its outcome. Each decision is therefore a one-shot problem, solved afresh with everything learned so far.

The solution is a risk–return threshold set by the drift and volatility of the landscape. An agent keeps the status quo when its outcome is close enough to ideal, and otherwise searches, more boldly the worse the status quo. Learning can get stuck: after a bad outcome, reaching a good one looks too risky, and search stops even though better choices exist.

The same one-shot logic appears in elections (Callander 2011b), judicial precedent (Callander and Clark 2017), search with non-satiated preferences (Callander and Matouschek 2019) and agenda control (Callander and McCarty 2024).

In communication, Callander (2008) introduced the Brownian landscape. An expert who knows the landscape recommends one action, and the recommendation is followed when her bias is below the same threshold. Aybas and Callander (2025) extend the one-shot cheap-talk game to Itô-diffusion landscapes, including mean-reverting ones.

Two shots

With two evaluations the first can be spent on information. Callander and Hummel (2014) give each of two policymakers one experiment in a two-period game. An incumbent who expects to lose power experiments early to shape what its successor learns.

Garfagnini and Strulovici (2016) study agents who live two periods and search twice, with a cost of novelty. Trying something new has option value because the second period exploits the best outcome found. Ganz (2020) allows a manager one offline trial before an online choice, so only the second evaluation is paid.

Bayesian optimization has the same horizon. With one informative trial before a final recommendation, the knowledge-gradient policy (Frazier, Powell and Dayanik 2009) is optimal by construction. Wu and Frazier (2019) implement a practical two-step lookahead.

Three shots

A searcher knows an incumbent's performance, runs one trial, and then commits to a final position, which is the third evaluation and the only one paid. When the Grass Is Greener solves this problem exactly on the exponential of an Ornstein–Uhlenbeck landscape.

The final placement is in closed form, with a phase diagram for equal observations: abandon a below-median incumbent, revert after a disappointing trial, commit between two good positions, and stay put once a position is strong enough. The opening trial reduces to a one-dimensional optimization.

Commitment between positions rests on a variance-ratio identity. An interior position has the same conditional mean as an explicit exterior one and a conditional variance larger by the factor

$$\frac{1+\rho}{1-\rho},$$

where $\rho$ is the correlation between the two observed positions.

Many shots and unbounded search

Farsighted search by a long-lived agent with a free choice of location is largely open (Bardhi and Callander 2026). Progress has come under restrictions:

An early hybrid is Jovanovic and Rob (1990), with correlated learning inside an interval and independence across intervals.

Optimization studies the same object with large budgets. Calvin (1997) derives average-case convergence rates under Wiener measure. Grill, Valko and Munos (2018) approximate the maximum of a Brownian path, and Grosse, Zhang and Hennig (2023) extend this to Gaussian-process samples.

Bayesian optimization mostly uses one-step rules: probability of improvement (Kushner 1964), expected improvement (Mockus et al. 1978; Jones et al. 1998), upper confidence bounds (Srinivas et al. 2010) and entropy search (Hennig and Schuler 2012; Wang and Jegelka 2017). Multi-step lookahead exists in approximate form (Ginsbourger and Le Riche 2010; Lam, Willcox and Wolpert 2016; Jiang et al. 2020). Shahriari et al. (2016) review the field.

A line search is a few evaluations along one direction. The probabilistic line search of Mahsereci and Hennig (2017) uses gradient observations and a Gaussian-process model. The grass rule used as a line search is on the line-search page.

Mean-reverting landscapes

A Brownian landscape drifts without bound away from known territory. An Ornstein–Uhlenbeck landscape reverts, so performance far from what is known regresses to an average, and every choice is equally uncertain before any is tried.

Bardhi (2024) introduces the powered-exponential covariances $\exp(-(|x-x'|/\ell)^p)$. The case $p = 1$ is the Ornstein–Uhlenbeck covariance, the only member with the nearest-neighbour property. The case $p = 2$ is the smooth squared-exponential kernel, which loses it. In machine learning $p = 1$ is the Matérn kernel with $\nu = 1/2$ (Rasmussen and Williams 2006).

The survey calls for testing such distance-based covariances more widely. The same kernel already appears elsewhere:

Practice and experiment

Across four randomized trials covering 759 firms, 59% never pivoted radically during the study, 24% pivoted once and 17% more than once (Camuffo et al. 2024; Camuffo et al. 2020). Practitioners call the fork pivot or persevere (Ries 2011). Rugged-landscape models in management study search by heuristics such as hill climbing (Levinthal 1997).

In the lab, subjects respond to Brownian uncertainty but struggle when it is highly complex (Glick and Myers 2015). In a dynamic search experiment, participants move away from low prizes, stay near high ones, and concentrate around their best discoveries (Malladi, Martínez-Marquina and Morozov 2025).