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LibMOON: A Gradient-based MultiObjective OptimizatioN Library in PyTorch

Xiaoyuan Zhang, Liang Zhao, Yingying Yu, Xi Lin, Yifan Chen, Han Zhao, Qingfu Zhang

TL;DR

LibMOON is introduced, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.

Abstract

Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with thousands / millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order / meta-heuristic methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with thousands / millions of parameters. In light of the above gap, this paper introduces LibMOON, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.

LibMOON: A Gradient-based MultiObjective OptimizatioN Library in PyTorch

TL;DR

LibMOON is introduced, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.

Abstract

Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with thousands / millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order / meta-heuristic methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with thousands / millions of parameters. In light of the above gap, this paper introduces LibMOON, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.
Paper Structure (31 sections, 2 theorems, 2 equations, 8 figures, 13 tables)

This paper contains 31 sections, 2 theorems, 2 equations, 8 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Notation.
  3. Related works
  4. Gradient-based multiobjective optimization
  5. Multiobjective optimization libraries
  6. LibMOON: A gradient-based MOO library in PyTorch
  7. Framework
  8. Supported problems.
  9. Supported metrics.
  10. MOO solvers
  11. Aggregation-based methods.
  12. Gradient manipulation-based methods.
  13. Zero-order optimization.
  14. Pareto set learning solvers
  15. Multiobjective Bayesian optimization solvers
  16. ...and 16 more sections

Key Result

Theorem 1

If $g_{\bm{\lambda}}(\cdot)$ is strictly decreasing w.r.t vector ${\bm{L}}({\bm{\theta}})$, i.e., $g_{\bm{\lambda}}({\bm{L}}({\bm{\theta}}^{(a)})) < g_{\bm{\lambda}}({\bm{L}}({\bm{\theta}}^{(b)}))$ when ${\bm{L}}_i({\bm{\theta}}^{(a)}) \preceq_\mathrm{strict} {\bm{L}}_i({\bm{\theta}}^{(b)})$, then t

Figures (8)

  • Figure 1: Supported solvers and problems in LibMOON: LibMOON addresses synthetic, real-world and MTL problems with three categories of solvers: MOO, PSL and MOBO solvers.
  • Figure 2: Architecture of Pareto models.
  • Figure 3: Finite Pareto solutions by ten MOO solvers on VLMOP2 problem.
  • Figure 4: Predicted Pareto solutions by different PSL solvers on VLMOP2 problem.
  • Figure 5: Finite Pareto solutions by different solvers on Adult problem.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Theorem 1: Adapted from Theorem 2.6.2 miettinen1999nonlinear
  • Theorem 2