($\boldsymbolθ_l, \boldsymbolθ_u$)-Parametric Multi-Task Optimization: Joint Search in Solution and Infinite Task Spaces

TL;DR

PMTO generalizes multi-task optimization by introducing a bounded, continuous task space and jointly optimizing solutions and tasks. The framework builds a GP-based mapping from solutions and task parameters to objectives, plus a task model mapping tasks to elite solutions, enabling fast online predictions for unseen tasks. A task-evolution mechanism actively samples diverse, informative tasks to improve offline coverage, while an online phase uses the trained task model to rapidly predict near-optimal solutions. Empirical results on synthetic and real-world problems demonstrate faster convergence and improved robustness, with notable gains in adaptive control and minimax design applications.

Abstract

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.

Figures (7)

• Figure 1: Distinguishing multi-task optimization and parametric multi-task optimization. (a) In multi-task optimization, there typically exists a predefined and fixed set of tasks (3 in this example) whose objective functions map points from a unified solution space to the respective objective spaces. (b) In parametric multi-task optimization, continuous task parameters imply a potentially infinite set of tasks. During PMTO's offline optimization mode, a task model $\mathcal{M}: \Theta \rightarrow \mathcal{X}$ is built to actively evolve and optimize tasks with unknown optima. In the online mode, this task model is applied to directly predict optimized solutions for any new task without entailing additional evaluation costs.
• Figure 2: Convergence trends in the offline optimization stage for Ackley-II (top) and Griewank-II (bottom) on four sample tasks each.
• Figure 3: The illustrative examples for the case studies: (a) Parametric Robot Arm Optimization: Optimize the angular position of each joint ($\alpha_1, \alpha_2, \alpha_3$) so that the end effector can approach as close as to the target position. Task parameters include the length of each arm $L$ and the maximum rotation degree $\alpha_{max}$. (b) Parametric Crane-Load System Optimization: Optimize the time intervals $t_1, t_2, t_3$ during which the distinct drive forces $F$ are exerted on the crane-load system so that the system can achieve a goal velocity with minimal operating time and oscillation. Task parameters include Scenario 1: time delays $\Delta t_1, \Delta t_2, \Delta t_3$, and Scenario 2: operating conditions including the length of suspension $l$, the mass of load $m_2$, and the resistance $W$. (c) Plane Truss Design: Optimize the cross-sectional areas of two bars $\theta_1, \theta_2$ and $\theta_3$ (the vertical distance from the second bar) to minimize the overall structural weight and the joint displacement in aware of possible processing errors $x_1, x_2, x_3$. The exact formulation of the (b) and (c) can be found in the supplementary materials.
• Figure 4: Comparative results of the task model's online performance. The results include single-task baseline method, PMTO-FT, and our proposed PMTO method on adaptive control problems. $U=20$ independent trials are considered: (a) Robot arm optimization under distinct operating conditions (b) Crane-load system optimization under distinct environments.
• Figure 5: Comparison of design performance under varying processing errors for the minimax optimization problem. The figure shows the objective values of robust design $\boldsymbol{\theta}$ optimized by Minimax SAEA and our proposed $(\boldsymbol{\theta}_l, \boldsymbol{\theta}_u)$-PMTO algorithm, evaluated across diverse processing error $\mathbf{x}$. To ensure a fair comparison, both methods are provided with the same total evaluation budget and assessed on the same set of processing errors.

Theorems & Definitions (3)

• Theorem
• Remark 1
• Theorem