Natural Hypergradient Descent: Algorithm Design, Convergence Analysis, and Parallel Implementation

• URL: http://arxiv.org/abs/2602.10905v1
• Date: Wed, 11 Feb 2026 14:31:33 GMT
• Title: Natural Hypergradient Descent: Algorithm Design, Convergence Analysis, and Parallel Implementation
• Authors: Deyi Kong, Zaiwei Chen, Shuzhong Zhang, Shancong Mou,
• Abstract summary: Natural Hypergradient Descent (NHGD) is a new method for solving bilevel optimization problems.<n>Our main theoretical contribution establishes high-probability error bounds and sample complexity guarantees for NHGD.<n> Empirical evaluations on representative bilevel learning tasks demonstrate the practical advantages of NHGD.
• Score: 5.754044493040163
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we propose Natural Hypergradient Descent (NHGD), a new method for solving bilevel optimization problems. To address the computational bottleneck in hypergradient estimation--namely, the need to compute or approximate Hessian inverse--we exploit the statistical structure of the inner optimization problem and use the empirical Fisher information matrix as an asymptotically consistent surrogate for the Hessian. This design enables a parallel optimize-and-approximate framework in which the Hessian-inverse approximation is updated synchronously with the stochastic inner optimization, reusing gradient information at negligible additional cost. Our main theoretical contribution establishes high-probability error bounds and sample complexity guarantees for NHGD that match those of state-of-the-art optimize-then-approximate methods, while significantly reducing computational time overhead. Empirical evaluations on representative bilevel learning tasks further demonstrate the practical advantages of NHGD, highlighting its scalability and effectiveness in large-scale machine learning settings.

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