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About this paper
Louis Claeys, Artur Goldman, Zebang Shen, Niao He
A new Schrödinger eigenfunction method addresses long-horizon stochastic optimal control in high dimensions, demonstrating up to an order-of-magnitude improvement in control accuracy and reducing computational complexity from O(Td) to O(d) compared to existing deep learning solvers. This approach also provides a closed-form analytic solution for symmetric linear-quadratic regulators with arbitrary terminal costs.
Key insights
- Stochastic Optimal Control problems with gradient-drift dynamics can be formally mapped to the spectral problem of a Schrödinger operator, enabling the use of spectral theory for analysis.
- For long planning horizons, the optimal control policy is primarily determined by the principal eigenfunction of the associated Schrödinger operator, with contributions from higher eigenfunctions decaying exponentially.
- Standard deep learning loss functions for eigenfunction approximation implicitly reweight spatial regions, making them insensitive to errors in crucial high-value function areas, which must be corrected for accurate control learning.
Summary by alphaXiv.
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