Diagnosing Optimizer-Landscape Interaction via Representation in Subset Selection Multitasking: Evidence from Piano Fingering

Ananda Phan Iman; Tingyang Wei; Chang Wook Ahn

cs.NEOct 29, 2026

Diagnosing Optimizer-Landscape Interaction via Representation in Subset Selection Multitasking: Evidence from Piano Fingering

Authors: Ananda Phan Iman, Tingyang Wei, Chang Wook Ahn

Organizations: Department of AI Convergence, Gwangju Inst. of Sci. & Tech. (GIST), South Korea · College of Computing and Data Science, Nanyang Technological University, Singapore

Abstract

Continuous relaxations are commonly used as an encoding scheme for evolutionary multitasking optimization (EMTO), allowing heterogeneous tasks to be optimized within a unified search space. However, these continuous representations fundamentally reshape the induced fitness landscape, and their impact on optimization behavior and knowledge transfer remains underexplored. In this work, we bridge the gap by analyzing the interaction between the representation-induced fitness landscape and optimizer behavior through fitness landscape analysis. Using piano fingering estimation as a representative case of the subset selection problem, we introduce several encoding variants and investigate how their induced landscapes influence search dynamics and solution quality under EMTO. Our analysis indicates that transfer efficacy is strongly associated with changes in representation-induced landscape characteristics, rather than algorithmic factors alone, highlighting the representation design as the key factor of effective knowledge transfer in multitasking optimization.

Figures & tables

Fig. 1 : Illustration of how different unified continuous representations of the same discrete piano fingering selection $\mathbf{x}$ induce distinct search space. Top: A short piano passage with an example of fingering selection. Bottom: Two alternative unified continuous search spaces, within which optimization is performed; color indicates fitness values after decoding.

Panel of Figure 2 — Fig. 2 : Illustration of five representation variants(R1–R5) grouped by their underlying representation mechanisms. (1) R1 (Direct): each continuous variable is deterministically mapped to a discrete fingering selection. (2) R2–R3 (Modular): Fingering selections are controlled by blocks of continuous variables. (3) R4–R5 (Indirect): discrete fingerings are embedded in a low-dimensional latent space learned offline, with a fixed latent-to-discrete mapping during optimization.

Panel of Figure 3 — Fig. 3 : Example of region-based mapping for homogeneous and heterogeneous blocks. (a) homogeneous block with top- $k_{j}$ truncation mapping, (b) heterogeneous block with sequential selection without replacement.

ID	Type	Mapping
R1	Direct	Partition-based pointwise mapping
R2	Modular (hom.)	top- $k_{j}$ truncation mapping
R3	Modular (het.)	Sequential selection without replacement
R4	Indirect	Fixed VAE decoder, $p=2$
R5	Indirect	Fixed VAE decoder, $p=1$

TABLE I : Representation Variant Summary

Fig. 4 : Distribution of representation-induced landscape features of R1-R5 across tasks, including lag-1 autocorrelation $\rho(1)$ , autocorrelation length $\tau$ , and neutrality $\nu$ .

		$\Delta\ (\%)$						$\Delta\ (\%)$			$\Delta\ (\%)$
Rep.	GA	AT-MFEA	MFEA-AKT	MFEA-DGD	MFMP	MTEA-AD	DE	BLKT-DE	MKTDE	CMA-ES	MTES-KG
R1	—	$+0.78\%$	$-6.54\%$	$-136.34\%$	$-9.80\%$	$+1.06\%$	—	$+8.11\%$	$-3.82\%$	—	$-26.27\%$
R2	—	$+7.01\%$	$-26.35\%$	$-106.73\%$	$-12.35\%$	$+7.26\%$	—	$+11.56\%$	$+2.74\%$	—	$-106.03\%$
R3	—	$+5.62\%$	$-15.02\%$	$-98.69\%$	$-56.29\%$	$\mathbf{+9.26\%}$	—	$\mathbf{+27.27\%}$	$+13.70\%$	—	$-43.83\%$
R4	—	$+3.50\%$	$-13.11\%$	$-101.10\%$	$-6.34\%$	$+4.82\%$	—	$+5.15\%$	$+0.89\%$	—	$-20.93\%$
R5	—	$+0.23\%$	$-7.64\%$	$-116.24\%$	$-11.27\%$	$+1.71\%$	—	$+5.69\%$	$-9.33\%$	—	$-6.06\%$

TABLE II : Median percent transfer effect

\Delta\ (\%)

across 8 tasks,

\Delta\ (\%)=(f^{\text{STO}}-f^{\text{MTO}})/f^{\text{STO}}\times 100

(minimization, matched STO per optimizer family).

\Delta<-100

indicates

f^{\text{MTO}}>2f^{\text{STO}}

. Bold indicates the largest

\Delta

(best transfer effect) per family.

Fig. 5 : Performance rank distributions across representations and optimizer. Boxplots show the median and interquartile range (IQR) across all tasks and runs. Collapsed boxes indicate zero IQR (stable central performance). Optimizers are grouped by their family; lower ranks indicate better performance.

Fig. 6 : Heatmap of Spearman correlation ( $r_{s}$ ) between algorithm performance and landscape, including lag-1 autocorrelation $\rho(1)$ , autocorrelation length $\tau$ , and neutrality $\nu$ . Columns grouped by optimizer (STO and MTO), and only statistically significant with Holm-corrected ( $\alpha=0.05$ ) are shown; others are blanks.

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Evolutionary Optimization Multi-Task Optimization

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Multi-task optimization is typically characterized by a fixed and finite set of tasks. The present paper relaxes this condition by considering a non-fixed and potentially infinite set of optimization tasks defined in a parameterized, continuous and bounded task space. We refer to this unique problem setting as parametric multi-task optimization (PMTO). Assuming the bounds of the task parameters to be (

\boldsymbolθ_l

\boldsymbolθ_u

), a novel (

\boldsymbolθ_l

\boldsymbolθ_u

)-PMTO algorithm is crafted to operate in two complementary modes. In an offline optimization mode, a joint search over solution and task spaces is carried out with the creation of two approximation models: (1) for mapping points in a unified solution space to the objective spaces of all tasks, which provably accelerates convergence by acting as a conduit for inter-task knowledge transfers, and (2) for probabilistically mapping tasks to their corresponding solutions, which facilitates evolutionary exploration of under-explored regions of the task space. In the online mode, the derived models enable direct optimization of any task within the bounds without the need to search from scratch. This outcome is validated on both synthetic test problems and practical case studies, with the significant real-world applicability of PMTO shown towards fast reconfiguration of robot controllers under changing task conditions. The potential of PMTO to vastly speedup the search for solutions to minimax optimization problems is also demonstrated through an example in robust engineering design.

Abstract

Evolutionary Multitask Optimization

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