Differential Evolution

How DE optimizes Bayesian Network parameters in dplbnDE

Differential Evolution (DE) is a population-based optimizer for continuous parameter spaces. dplbnDE applies DE to the space of CPT parameters, using CLL as the fitness function.

The DE cycle

Each generation, every individual goes through mutation, crossover, constraint repair, and selection.

1. Mutation

   A mutant vector is formed by combining existing population members. For DE/best/1:

   mutant <- vec(best$.params) + F * (vec(r1$.params) - vec(r2$.params))
   

   best is the current best individual, r1 and r2 are randomly selected, and F is the mutation factor.

2. Crossover

   A trial vector mixes the mutant with the current individual. Binomial crossover:

   cross_points <- runif(dim) > CR
   trial[cross_points] <- mutant[cross_points]
   
3. Constraint repair

   Two repairs enforce the probability constraints:

   # Fold out-of-bounds values back into [0, 1]
   trial <- reflect(trial)
   # Renormalize CPT rows to sum to 1
   trial <- keepSumFast(trial, group_indices)
   
4. Selection

   The trial replaces the current individual only if its CLL is strictly better:

   if (f > fitness[j]) {
     pop[[j]] <- trial
   }
   

Mutation strategies

Strategy	Formula	Used by
DE/best/1	best + F×(r1 − r2)	DEbest (pairs=1)
DE/best/2	best + F×(r1−r2+r3−r4)	DEbest (pairs=2)
DE/rand/1	r0 + F×(r1 − r2)	DErand (pairs=1)
DE/rand/2	r0 + F×(r1−r2+r3−r4)	DErand (pairs=2)
current-to-pbest	xi + F×(xp−xi) + F×(r1−r2)	JADE, L-SHADE, jSO, NL-SHADE-RSP

Key parameters

NP — Population size

Number of candidate solutions. Minimum enforced is 6. Larger NP explores more broadly but costs more evaluations per generation. Default: 40.

G — Generations

Maximum number of generations. Total evaluations ≈ NP × G for classic variants. Default: 100.

F — Mutation factor

Controls mutation step size. Range [0, 2]. Typical values 0.4–0.9. Adaptive variants update F automatically using a Cauchy distribution. Default: 0.5.

CR — Crossover rate

Probability of inheriting a dimension from the mutant. Range [0, 1]. Adaptive variants update CR using a Normal distribution. Default: 0.7.

Adaptive variants

JADE, L-SHADE, jSO, and NL-SHADE-RSP automatically tune F and CR each generation using success histories:

• F is sampled from a Cauchy distribution centered on the historical mean of successful F values
• CR is sampled from a Normal distribution centered on the historical mean of successful CR values
• Successful values are those that produced an improvement — they are stored in a memory of size H (default 6)

L-SHADE and jSO additionally apply Linear Population Size Reduction (LPSR): the population shrinks linearly from NP to a minimum of 4 over the course of evolution, concentrating evaluations on the most promising individuals.