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

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.

💡 Research Summary

This paper introduces Natural Hypergradient Descent (NHGD), a novel algorithm for bilevel optimization that eliminates the costly Hessian‑inverse computation traditionally required for hypergradient estimation. The authors focus on bilevel problems where the inner objective is a KL‑divergence loss, i.e., ℓ(v,θ)=E_{ξ∼q}