Frequency-Scaled Ingredient Fitness

• The paper introduces FFA, a fitness assignment method that uses objective frequency counters to replace direct numerical comparisons in evolutionary algorithms.
• It demonstrates that FFA ensures invariance under injective transformations, eradicating numerical bias and providing consistent selection across diverse problem landscapes.
• Empirical results show that FFA-based EAs convert exponential runtimes on challenging benchmarks to efficient polynomial scaling, enhancing overall optimization performance.

Frequency-Scaled Ingredient Fitness, formalized as Frequency Fitness Assignment (FFA), is a fitness assignment methodology in evolutionary algorithms (EAs) that replaces direct objective function comparison with selection based on the empirical frequency of observed objective values. This approach seeks to eradicate algorithmic bias towards numerically superior solutions and ensures invariance under injective transformations of the objective function, enabling robust optimization across challenging benchmark landscapes such as traps, jumps, plateaus, and random satisfiability problems (Weise et al., 2021).

1. Formal Definition and Computation

Let $f\colon\mathcal{X}\to\mathcal{Y}$ denote the objective function, where $\mathcal{X}$ is the search space and $\mathcal{Y}\subset\mathbb{Z}$ is the image set. In FFA, each encountered objective value $y\in\mathcal{Y}$ is assigned a frequency counter

$$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$

where $x_t$ is the candidate at step $t$. On each selection event (insertion or reevaluation), the counter for the current objective value is incremented by one:

$H[f(x)] \leftarrow H[f(x)] + 1.$

The FFA fitness of a candidate $x$ is defined as

$\varphi(x) := H[f(x)],$

with minimization as the selection criterion. Hence, all individual comparisons for selection are performed with respect to $\mathcal{X}$0. No further normalization or transformation is applied. The entire process relies on maintaining and updating the counter table $\mathcal{X}$1, which encapsulates the historical frequency information of each objective value.

2. Theoretical Properties

FFA possesses distinctive invariance and non-bias properties:

• Invariance Under Injective Transformations: For any injective $\mathcal{X}$2, frequency assignment via $\mathcal{X}$3 or $\mathcal{X}$4 yields an identical frequency histogram up to relabeling:

$\mathcal{X}$5

and thus the search dynamics remain unchanged under any injective scaling, shifting, encryption, or other relabeling of objective values. This ensures algorithmic neutrality to scaling or monotonic coding of fitness landscapes.

• Absence of Numerical Bias: Standard EAs intrinsically prefer "better" (higher or lower, depending on maximization/minimization) objective values. FFA-based EAs, in contrast, are indifferent to magnitude, focusing strictly on the minimization of frequency: only rarely observed $\mathcal{X}$6-values (low $\mathcal{X}$7) are favored. Consequently, selection is invariant to any strict ordering or re-ranking of $\mathcal{X}$8. Pairwise comparison operators relying solely on $\mathcal{X}$9 exhibit complete rank-order independence (Proposition 3.1 of (Weise et al., 2021)), and the procedure is unbiased with respect to the numerical encoding of "goodness."

3. Integration into Evolutionary Algorithms

FFA is integrated by substituting all fitness-based comparisons in established evolutionary algorithms with frequency-based metrics. Two state-of-the-art GA instantiations are examined:

• Greedy (2+1) GA with FFA (GFGA): Maintains two parents, $\mathcal{Y}\subset\mathbb{Z}$0 and $\mathcal{Y}\subset\mathbb{Z}$1. Offspring are generated via uniform crossover and subsequent mutation determined by $\mathcal{Y}\subset\mathbb{Z}$2. After updating $\mathcal{Y}\subset\mathbb{Z}$3, $\mathcal{Y}\subset\mathbb{Z}$4, $\mathcal{Y}\subset\mathbb{Z}$5, selection proceeds by replacing the parent with the highest $\mathcal{Y}\subset\mathbb{Z}$6 value.
• Self-Adjusting (1+($\mathcal{Y}\subset\mathbb{Z}$7,$\mathcal{Y}\subset\mathbb{Z}$8)) GA with FFA (SAFGA): Utilizes a current parent $\mathcal{Y}\subset\mathbb{Z}$9 and mutation parameter $y\in\mathcal{Y}$0. In each generation, $y\in\mathcal{Y}$1 mutants are generated; the best in $y\in\mathcal{Y}$2 among these becomes $y\in\mathcal{Y}$3. After crossover, selection, and $y\in\mathcal{Y}$4 updates, the algorithm adjusts $y\in\mathcal{Y}$5 based on success, only accepting new parent candidates if their frequency-fitness is at least as low as the incumbent.

The frequency table is updated for every new or surviving individual, ensuring up-to-date frequency-based selection throughout the search process.

4. Empirical Performance on Benchmark Problems

Empirical studies demonstrate marked improvements in runtime scaling on multiple benchmark classes:

Benchmark	Pure EA Runtime (Expected)	FFA-based EA Mean Runtime
Trap, Jump, Twomax	Exponential in $y\in\mathcal{Y}$6	Polynomial (e.g. $y\in\mathcal{Y}$7-$y\in\mathcal{Y}$8)
Plateau (width$y\in\mathcal{Y}$91)	Exponential	Sub-$$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$0 (for SAFGA/SAFGAP)
MaxSAT (Random 3-SAT, $$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$1)	$$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$2 FEs (fails)	$$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$3 FEs (virtually all solved)

FFA-based variants (OFEA, GFGA) outperform standard EAs by transforming exponential scaling to empirical polynomial runtimes in problems where pure EAs encounter hard fitness landscapes such as traps, jumps, and plateaus. For uniform random 3-SAT at the phase transition ($$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$4), FFA-EAs solve all tested instances while pure EAs fail after upwards of $$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$5 evaluations. The log-mean expected runtime under FFA is reduced by $$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$6–$$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$7 orders of magnitude.

5. Hybrid Selection Mechanisms

FFA is further utilized in hybrid algorithms that alternate or adapt between direct objective-based and frequency-based optimization:

• EAFEA (Alternating): Alternates every fitness evaluation between the standard (1+1) EA and objective-frequency-based OFEA, with cross-communication via a global best-so-far value. Empirically, this combines fast performance on simple landscapes and polynomial behavior on hard benchmarks.
• SAFGAP (Adaptive Switch): Commences as self-adjusting (1+($$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$8,$$H[y] = |\{t \leq \text{current step} : f(x_t) = y\}|,$$9)) GA, switching to SAFGA once $x_t$0 reaches $x_t$1 (mutation of all bits). Reverts to the base algorithm upon objective improvement in FFA mode. This strategy matches or surpasses pure and FFA-only methods across all benchmarks by leveraging both standard and FFA-based search strengths.

These hybrid methods demonstrate that hybridizing FFA with standard selection enhances robustness and preserves asymptotic performance on both easy and hard problem instances.

6. Key Equations and Benchmark Definitions

• FFA Assignment: $x_t$2, $x_t$3, minimize $x_t$4.
• Jump Function ($x_t$5):

$x_t$6

• Plateau Function (width $x_t$7):

$x_t$8

A summary of the empirical exponents (largest $x_t$9 such that all runs complete in at most $t$0 FEs) shows that FFA-variants never exceeded $t$1 over all benchmarks, while pure EA approaches required exponential time.

7. Context, Limitations, and Implications

FFA is a general-purpose "plug-in" mechanism for EAs, requiring minimal modification—namely, the substitution of objective-based comparison with frequency-based selection and the maintenance of frequency counters. It eradicates bias toward numerically "good" solutions and achieves strict invariance to all injective renamings of objective values. Empirical results indicate that this methodology can transform the empirical runtime complexity on hard benchmark classes from exponential to polynomial, broadening the class of problems efficiently solvable by black-box EAs (Weise et al., 2021). A plausible implication is that the absence of objective numerical bias facilitates traversal of complex fitness landscapes, particularly those engineered to deceive or plateau standard selection schemes.

No explicit disadvantages, pathologies, or scenarios in which FFA worsens performance relative to standard selection are identified in the referenced empirical studies, but the non-monotonic relationship between objective value and frequency-fitness suggests potential trade-offs on unimodal or non-deceptive landscapes. Future research could clarify the generality of FFA improvements, especially on application-specific or noisy objective functions.