Fast Computation of Superquantile-Constrained Optimization Through Implicit Scenario Reduction

2024-12-31 · Source: JMLR · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, quick

Summary

Jake Roth and Ying Cui introduce a fast, scalable, and robust second-order computational framework for large-scale optimization problems with superquantile-based constraints. Superquantiles are gaining interest as a risk-aware metric for fairness and distribution shifts in statistical learning. The framework utilizes a semismooth-Newton-based augmented Lagrangian method, which benefits from the superquantile operator effectively reducing the dimensions of Newton systems by focusing on fewer tail scenarios. This approach makes the extra cost of second-order information and matrix inversions comparable to, or even less than, gradient computation. The solver excels when the number of scenarios significantly exceeds decision variables. Numerical experiments show it outperforms existing methods, achieving speeds over 750 times faster than OSQP for low-accuracy solutions in synthetic linear and convex diagonal quadratic objectives. It is also up to 70 times faster than Gurobi and 30 times faster than Portfolio Safeguard for high-accuracy solutions, and 20 times faster than Gurobi for quantile regression with over 30 million scenarios.

Key takeaway

For research scientists working on large-scale optimization problems with superquantile constraints, you should investigate this new second-order computational framework. Its demonstrated speed improvements, up to 750x over ADMM and 70x over commercial solvers like Gurobi, suggest it could dramatically reduce computation time for high-accuracy solutions, especially in scenarios where the number of data points far exceeds decision variables.

Key insights

Superquantile-constrained optimization can be significantly accelerated using a specialized second-order computational framework.

Principles

• Superquantiles reduce Newton system dimensions.
• Second-order methods can outperform gradient-based ones.

Method

A semismooth-Newton-based augmented Lagrangian method is used, leveraging the superquantile operator to implicitly reduce scenario dimensions for faster matrix inversions and second-order information computation.

In practice

• Apply to problems with many scenarios, few variables.
• Use for risk-aware statistical learning.
• Consider for quantile regression tasks.

Topics

• Superquantile Optimization
• Second-Order Methods
• Large-Scale Optimization
• Risk-Aware Metrics
• Quantile Regression

Code references

• jacob-roth/superquantile-opt

Best for: Research Scientist, AI Researcher, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by JMLR.