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.

翻译：本文提出一种用于求解双层优化问题的新方法——自然超梯度下降（NHGD）。为应对超梯度估计中的计算瓶颈——即需要计算或近似海森矩阵逆——我们利用内层优化问题的统计结构，使用经验费舍尔信息矩阵作为海森矩阵的渐近一致替代项。该设计实现了一种并行优化与近似框架，其中海森逆近似与随机内层优化同步更新，以可忽略的额外代价重用梯度信息。本文的主要理论贡献是建立了NHGD的高概率误差界和样本复杂度保证，其性能与最先进的先优化后近似方法相当，同时显著降低计算时间开销。在代表性双层学习任务上的实证评估进一步展示了NHGD的实际优势，突显了其在大规模机器学习场景中的可扩展性和有效性。