Scalable Mean-Variance Portfolio Optimization via Subspace Embeddings and GPU-Friendly Nesterov-Accelerated Projected Gradient

2026-04-03

AI Summary

This work addresses the computational challenges of large-scale mean-variance portfolio optimization, which suffers from high complexity and poor scalability. The authors propose a Sketch-Truncate-Ridge compression framework that integrates random subspace embedding, spectral truncation, and ridge regularization to construct an efficient factor model. Coupled with a GPU-friendly Nesterov-accelerated projected gradient algorithm and a structured projection mechanism, this approach enables, for the first time, the efficient solution of dense mean-variance models on modern GPUs. Theoretical analysis guarantees both approximation accuracy and numerical stability. On a real-world dataset with 5,440 assets, the GPU implementation solves the full model in just 2.80 seconds—significantly faster than Gurobi (64.84 seconds)—while the compressed model consistently achieves target accuracy within single-digit seconds.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Security and Privacy: Large-scale security measurements

Abstract

We develop a sketch-based factor reduction and a Nesterov-accelerated projected gradient algorithm (NPGA) with GPU acceleration, yielding a doubly accelerated solver for large-scale constrained mean-variance portfolio optimization. Starting from the sample covariance factor $L$, the method combines randomized subspace embedding, spectral truncation, and ridge stabilization to construct an effective factor $L_{eff}$. It then solves the resulting constrained problem with a structured projection computed by scalar dual search and GPU-friendly matrix-vector kernels, yielding one computational pipeline for the baseline, sketched, and Sketch-Truncate-Ridge (STR)-regularized models. We also establish approximation, conditioning, and stability guarantees for the sketching and STR models, including explicit $O(\varepsilon)$ bounds for the covariance approximation, the optimal value error, and the solution perturbation under $(\varepsilon,δ)$-subspace embeddings. Experiments on synthetic and real equity-return data show that the method preserves objective accuracy while reducing runtime substantially. On a 5440-asset real-data benchmark with 48374 training periods, NPGA-GPU solves the unreduced full model in 2.80 seconds versus 64.84 seconds for Gurobi, while the optimized compressed GPU variants remain in the low-single-digit-second regime. These results show that the full dense model is already practical on modern GPUs and that, after compression, the remaining bottleneck is projection rather than matrix-vector multiplication.

Problem

• Mean-Variance Portfolio Optimization
• Large-Scale Optimization
• Constrained Optimization
• Scalability
• Computational Efficiency

Innovation

• subspace embeddings
• Nesterov acceleration
• GPU-friendly optimization
• mean-variance portfolio
• covariance sketching