SMOP: Stochastic trust region method for multi-objective problems

TL;DR

SMOP introduces a stochastic trust-region framework for multi-objective optimization under noisy evaluations by employing probabilistically fully linear models. It proves almost sure convergence to a Pareto stationary point and demonstrates practical effectiveness on noisy benchmarks and fairness-aware logistic regression tasks, aided by adaptive subsampling. A Pareto-front finding routine extends SMOP to approximate the entire Pareto set, enabling practical exploration of trade-offs with reduced computational cost. This approach advances scalable, inexact-model optimization for multiobjective problems with finite-sum structure and noisy information.

Abstract

The problem considered is a multi-objective optimization problem, in which the goal is to find an optimal value of a vector function representing various criteria. The aim of this work is to develop an algorithm which utilizes the trust region framework with probabilistic model functions, able to cope with noisy problems, using inaccurate functions and gradients. We prove the almost sure convergence of the proposed algorithm to a Pareto critical point if the model functions are good approximations in probabilistic sense. Numerical results demonstrate effectiveness of the probabilistic trust region by comparing it to competitive stochastic multi-objective solvers. The application in supervised machine learning is showcased by training non discriminatory Logistic Regression models on different size data groups. Additionally, we use several test examples with irregularly shaped fronts to exhibit the efficiency of the algorithm.

Figures (5)

• Figure 1: Pareto front for problem \ref{['tp1']} for different levels of noise $\sigma\in\{0.01,0.1,1\}$. "Start" indicates function values for starting point $f(x_0)$. "Last" indicates all values $f(x^*)$ obtained in the last iterations $x^*$ for 10 simulations. "Mean" indicates $f(x^*_{mean})$, where $x^*_{mean}$ is the mean of last iterations of simulations. Model parameters are: $x_0=(9,9), k_{max}=500, \theta=10^{-4}, \gamma_1=0.5, \gamma_2=2, \eta_1=10^{-4}$.
• Figure 2: Pareto front for problem \ref{['tp2']} for different levels of noise $\sigma\in\{0.01,0.1,1\}$. "Start" indicates function values for starting point $f(x_0)$. "Last" indicates all values $f(x^*)$ obtained in the last iterations $x^*$ for 10 simulations. "Mean" indicates $f(x^*_{mean})$, where $x^*_{mean}$ is the mean of last iterations of simulations. Model parameters: $x_0=(-0.5,1), k_{max}=500, \theta=0.4, \gamma_1=0.5, \gamma_2=2, \eta_1=0.4$.
• Figure 3: Comparison of SMOP with DMOP VOS and SMG LV, in terms of $\omega(x_k)$, $\phi(x_k)$ and number of scalar products. Model parameters: $x_0=(0,...,0), k_{max}=150, \theta=0.01, \gamma_1=0.5, \gamma_2=2, \eta_1=0.25$
• Figure 4: Algorithm comparison. Model parameters: $x_0=(0,..,0), k_{max}=150, \theta=0.01,\gamma_1=0.5,\gamma_2=2, \eta_1=0.25$
• Figure 5: Pareto front for logistic regression (left), and test example 2 (right)

Theorems & Definitions (29)

• Definition 1
• Lemma 1
• Definition 2
• Definition 3
• Lemma 2
• proof
• Theorem 1
• Theorem 2
• Lemma 3
• proof