Does it need a differentiable objective?

Many optimization uses do not require objective gradients.

Is it guaranteed to find the best solution in a fixed budget?

No general practical guarantee should be assumed.

Can it optimize discrete choices?

Yes, with a suitable sampling distribution and representation.

AI Glossary / AI and machine learning / Advanced

Cross-entropy method

Definition

The cross-entropy method is a sampling-based approach to optimization and rare-event estimation. In optimization, it repeatedly samples candidate solutions and adjusts the sampling distribution toward the better-performing candidates.

What it means

Instead of improving just one candidate at a time, the method maintains a probability distribution over possible candidates. It uses the results of one sampling round to make promising regions more likely in the next round.

The term is related to cross-entropy as a mathematical measure, but it is not simply the cross-entropy loss commonly used to train classifiers. Here it names a broader iterative method with applications in search and simulation.

For a business user, the appeal is a relatively clear search cycle: generate alternatives, evaluate them and concentrate future effort. Its performance still depends on representation, sampling choices and the quality of the objective being evaluated.

How it works

An optimization version starts with a parameterized distribution and draws a population of candidates. It scores them, selects an elite subset and fits updated distribution parameters to favor that subset.

The process repeats until a stopping rule is reached. Smoothing or other safeguards can prevent the distribution from narrowing too quickly around a poor region.

For rare-event estimation, the method adapts a sampling distribution to make informative events easier to observe, with the appropriate probability weighting. That purpose should be distinguished from merely searching for a good solution.

Three practical examples

A simulation team searches for a good parameter setting when objective evaluations are available but useful gradients are not.
A planning researcher samples action sequences and updates the distribution toward high-scoring candidates.
A reliability study investigates rare-event probabilities using an adapted simulation distribution and the required statistical corrections.

Think of it this way

Think of exploring a large menu of design options, then ordering more trials near the options that worked best.

The analogy has a limit: the method updates an explicit probability distribution. Repeatedly choosing personal favorites without controlling exploration or evaluation bias would not provide the same procedure or guarantees.

Worked example

Updating a simple binary sampling choice

Suppose a fictional optimization problem has five yes-or-no decisions. Initially, each is sampled as yes with probability 0.5. The method draws 100 candidate decision vectors and retains the ten with the best valid scores.

For the first decision, eight of the ten elite candidates used yes. A simple unsmoothed update would set that decision's yes probability to 0.8 for the next round. Similar updates apply to the other positions under an independent Bernoulli representation.

This makes successful choices more common but does not prove that the first decision should always be yes. The elite set is small, and interactions among decisions may matter. An independent distribution can also fail to represent those interactions well.

The team can use smoothing and compare repeated runs. It must count all evaluations, including invalid candidates, when assessing cost. These numbers are a teaching illustration rather than a prescription for sample size or elite fraction.

Illustrative case study

A fictional logistics simulator uses the cross-entropy method to search dispatch settings. Early iterations concentrate quickly on one family of schedules, but later tests reveal that a different family performs better under delayed arrivals.

The analysts review how fast the sampling distribution narrowed and whether the objective included realistic disruptions. They broaden the evaluation scenarios and compare several update settings under the same budget.

The method remains a search tool, while the planning team decides whether the resulting schedules are robust enough for use. The case illustrates why concentrating on observed winners can be useful and can also prematurely exclude alternatives.

Common misconceptions and mistakes

Confusing the method with a classification loss: the name refers here to an adaptive sampling procedure.
Assuming elite candidates are globally optimal: they are only the best among the sampled candidates in that round.
Allowing the distribution to collapse too early: exploration can disappear before good regions are found.
Ignoring probability corrections in rare-event estimation: changing the sampling distribution changes how observations must be interpreted.

Business use

The cross-entropy method is relevant to simulation-based optimization, planning research and specialist probability estimation. It can be attractive when candidate solutions can be sampled and scored naturally.

Ask how candidates are represented, how feasibility is handled and how the elite fraction was selected. Compare repeated-run results and total evaluation cost with simpler search methods. Where the method supports operational decisions, validate the selected solution under scenarios beyond those used to shape the search distribution.

Check your understanding

Eight of ten elite candidates use one binary option. Does setting its next sampling probability to 0.8 mean the option has an 80% chance of being globally optimal?

Answer: No. The value controls sampling under that update rule. It is not a calibrated probability that the option belongs to the global optimum.

Related terms

Importance sampling
Cross-entropy
stochastic optimisation
elite sample
Monte Carlo method

Learn next

Read stochastic optimization. Run a small paper exercise with binary candidates and observe how elite selection changes future sampling choices.

Sources

De Boer, Kroese, Mannor and Rubinstein, A Tutorial on the Cross-Entropy Method: https://people.smp.uq.edu.au/DirkKroese/ps/aortut.pdf. Section 1.2 and Algorithm 1.2 were checked for independent Bernoulli sampling, elite selection and probability updates. Rubinstein seminar abstract: https://or.stanford.edu/seminars/rubinstein_112404.html, earlier excerpt consulted. Examples are original.

Previous: Cross-entropy loss Next: Cross-fitting