1 Non-stationaryNoisyFunctionOptimization

2 Non-stationary and Noisy Function Optimisation

• What is a non-stationary problem?
• Effect of uncertainty
• Algorithmic approaches
• Example: Time-varying knapsack problem

3 What is a non-stationary problem?

• Real-world environments contain sources of uncertainty
  • Measuring the fitness more than once will not always result in same fitness
• Not finding single optimum but a sequence of values over time
• Focus on measuring the quality of a solution x, f(x) with different sources of uncertainty:
  • The genotype to phenotype mapping is not exact and one-to-one f observed (x) = f( x+δx )
  • The act of measurement itself is prone to error or uncertainty f observed (x) = f mean ( x ) + f noise ( x ), where noise is _ N(0,σ)_
  • The environment changes over time f observed (x) = f( x,t )

4 Effect of uncertainty (1/2)

• Measuring performance of algorithms dealing with uncertainty is done by:
  • Running them for a fixed period of time
  • Calculating two time averaged metrics
    • On-line measure
    • Offline measure
    • where is time dependent fitness function, the best individual in a population at time t

This figure shows f(x) = 1/(0.1+x2) and the values estimated after 5 samples with two different sorts of uncertainty (uniform between +/- 0.4)

5 Algorithmic Approaches (1/2)

• Increasing robustness/reducing noise by repeatedly re-evaluate solutions and take an average
  • Implicitly via population management
  • Explicitly:
    • Resample when degree of variation present is greater than the range of estimated fitnesses in population
    • Law of diminishing returns
    • Resampling decisions independently for each solution
• Dynamic environments
  • Make sure that there is enough diversity in the population
• Memory based approaches for switching or cyclic environments
  • Expanding memory of EA
  • Example: GA with diploid representation, structured GA
• Explicitly increasing diversity in dynamic environment
  • Examples: GA with a hypermutation operator, random immigrants GA
• Preserving diversity and resampling: modifying selection and replacement policies
  • Steady-state GAs with “delete-oldest” replacement strategy

6 Example: Time-varying knapsack problem

• Number of items having value vit and weight or cost cit
• Select a subset maximising total value meeting time-varying capacity constraint C(t)
• Smith and Vavak did multiple experiments
  • Binary-coded SSGA 100 members
  • Parent selection by binary tournament
  • Uniform crossover
  • Hypermutation operator (triggered if running average drops)
  • Parameter values are decided after initial experiments
• Best performance when combining conservative (binary) tournament with delete-oldest. Using this policy with hypermutation results in finding the global optima in switching environment and continuously moving optima