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Sequences of linear systems arise in the predictor- corrector method when computing the Pareto front for multi- objective optimization. Rather than discarding information gen- erated when solving one system, it may be advantageous to recycle information for subsequent systems. To accomplish this, we seek to reduce the overall cost of computation when solving linear systems using common recycling methods. In this work, we assessed the performance of recycling minimum residual (RMIN- RES) method along with a map between coefficient matrices. For these methods to be fully integrated into the software used in Enouen et al. (2022), there must be working version of each in both Python and PyTorch. Herein, we discuss the challenges we encountered and solutions undertaken (and some ongoing) when computing efficient Python implementations of these recycling strategies. The goal of this project was to implement RMINRES in Python and PyTorch and add it to the established Pareto front code to reduce computational cost. Additionally, we wanted to implement the sparse approximate maps code in Python and PyTorch, so that it can be parallelized in future work. 

Issues of fairness often arise in graphical neural networks used for misinformation detection. However, improving fairness can often come at the cost of reducing accuracy and vice versa. Therefore, we formulate the task of balancing accuracy and fairness as a multi-objective optimization (MOO) problem where we seek to find a set of Pareto optimal solutions. Traditional first-order approaches to solving MOO problems such as multigradient descent can be costly, especially with large neural networks. Instead, we describe a more efficient approach using the predictor-corrector method. Given an initial Pareto optimal point, this approach predicts the direction of a neighboring solution and refines this prediction using a few steps of multigradient descent. We show experimentally that this approach allows for the generation of high-quality Pareto fronts faster than baseline optimization methods. 

Multiple-objective optimization (MOO) aims to simultaneously optimize multiple conflicting objectives and has found important applications in machine learning, such as simultaneously minimizing classification and fairness losses. At an optimum, further optimizing one objective will necessarily increase at least another objective, and decision-makers need to comprehensively explore multiple optima to pin-point one final solution. We address the efficiency of exploring the Pareto front that contains all optima. First, stochastic multi-gradient descent (SMGD) takes time to converge to the Pareto front with large neural networks and datasets. Instead, we explore the Pareto front as a manifold from a few initial optima, based on a predictor-corrector method. Second, for each exploration step, the predictor iteratively solves a large-scale linear system that scales quadratically in the number of model parameters, and requires one backpropagation to evaluate a second-order Hessian-vector product per iteration of the solver. We propose a Gauss-Newton approximation that scales linearly, and that requires only first-order inner-product per iteration. T hird, we explore different linear system solvers, including the MINRES and conjugate gradient methods for approximately solving the linear systems. The innovations make predictor-corrector efficient for large networks and datasets. Experiments on a fair misinformation detection task show that 1) the predictor-corrector method can find Pareto fronts better than or similar to SMGD with less time, and 2) the proposed first-order method does not harm the quality of the Pareto front identified by the second-order method, while further reducing running time.