Scalable Min-Max Optimization via Primal-Dual Exact Pareto Optimization

TL;DR

The proposed Exact Pareto Optimization via Augmented Lagrangian (EPO-AL) algorithm scales better with the number of objectives than subgradient-based strategies, while exhibiting lower per-iteration complexity than recent smoothing-based counterparts.

Abstract

In multi-objective optimization, minimizing the worst objective can be preferable to minimizing the average objective, as this ensures improved fairness across objectives. Due to the non-smooth nature of the resultant min-max optimization problem, classical subgradient-based approaches typically exhibit slow convergence. Motivated by primal-dual consensus techniques in multi-agent optimization and learning, we formulate a smooth variant of the min-max problem based on the augmented Lagrangian. The resultant Exact Pareto Optimization via Augmented Lagrangian (EPO-AL) algorithm scales better with the number of objectives than subgradient-based strategies, while exhibiting lower per-iteration complexity than recent smoothing-based counterparts. We establish that every fixed-point of the proposed algorithm is both Pareto and min-max optimal under mild assumptions and demonstrate its effectiveness in numerical simulations.

Figures (2)

• Figure 1: Multi-objective optimization trajectories (top) for subgradient algorithm (\ref{['eq:subgrad']}) in (a), EPO Search mahapatra2020multiin (b), and the proposed approach via augmented Lagrangian in (c), referred to as EPO-AL; the table shows the per-iteration computational complexities (bottom). Optimization trajectories are obtained for $K=2$ non-convex objectives $J_1(w) = 1 - e^{-\|w-1/\sqrt{d}\|^2}$ and $J_2(w) = 1 - e^{-\|w+1/\sqrt{d}\|^2}$lin2019pareto with $w \in \mathds{R}^3$ (see Sec. \ref{['subsec:vis']} for details). The intersection between the Pareto front (red arc, see Def. \ref{['def:weak_PO']}) and the fair solutions (blue line, see Def. \ref{['def:FO']}) is an exact Pareto optimal (EPO) mahapatra2020multi solution (white cross, see Def. \ref{['def:epo']}), which satisfies the min-max optimality (\ref{['eq:obj']}) under mild assumptions (see Prop. \ref{['prop:epo_minmax']}). Observe that the proposed strategy first finds the Pareto front, and then searches for the Pareto solution that is min-max optimal according to \ref{['eq:obj']}. The subgradient algorithm (\ref{['eq:subgrad']}) exhibits oscillations around the min-max optimal solution Pareto front due to the non-smooth behavior of maximum operator in (\ref{['eq:obj']}), unlike both EPO-based approaches that smoothly converge to the min-max optimal solution.
• Figure 2: Iteration complexity $i^o$ (a,b) and wall-clock time complexity $t^o$ (c,d) as a function of number of objectives $K$. The results are averaged over $30$ independent experiments after removing the minimum and maximum, where each experiment assumes different preference vector $r$ and different initial model $w_0$. Shaded area corresponds to $99 \%$ confidence interval.

Theorems & Definitions (11)

• Definition 1: Weak Pareto optimality miettinen1999nonlinear
• Definition 2: Fairness li2019fairhamidi2025over
• Definition 3: Exact Pareto optimality mahapatra2020multi
• Proposition 1: Exact Pareto optimality implies min-max optimality
• proof
• Remark 1: Per-iteration computational complexity
• Definition 4: Pareto stationarity
• Theorem 1: Fixed point analysis
• proof
• Corollary 1: Convex objectives