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

Deyi Kong, Zaiwei Chen, Shuzhong Zhang, Shancong Mou

TL;DR

NHGD reformulates bilevel optimization by replacing the Hessian inverse with the inverse EFIM, computed in parallel with inner SGD. This natural gradient-style preconditioning yields a parallel, optimize-and-approximate scheme that preserves high-probability convergence guarantees while reducing runtime overhead. Theoretical results provide high-probability bounds for the Hessian-inverse approximation and a finite-sample convergence rate matching state-of-the-art methods, and empirical experiments on hyper-data cleaning, data distillation, and PDE-constrained optimization demonstrate practical scalability and robustness. The approach enables efficient, scalable bilevel learning and suggests avenues for acceleration with blockwise/K-FAC-like structures and extensions to broader inner-objective families.

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.

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

TL;DR

NHGD reformulates bilevel optimization by replacing the Hessian inverse with the inverse EFIM, computed in parallel with inner SGD. This natural gradient-style preconditioning yields a parallel, optimize-and-approximate scheme that preserves high-probability convergence guarantees while reducing runtime overhead. Theoretical results provide high-probability bounds for the Hessian-inverse approximation and a finite-sample convergence rate matching state-of-the-art methods, and empirical experiments on hyper-data cleaning, data distillation, and PDE-constrained optimization demonstrate practical scalability and robustness. The approach enables efficient, scalable bilevel learning and suggests avenues for acceleration with blockwise/K-FAC-like structures and extensions to broader inner-objective families.

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.
Paper Structure (44 sections, 12 theorems, 141 equations, 7 figures, 5 tables, 2 algorithms)

This paper contains 44 sections, 12 theorems, 141 equations, 7 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Works
  4. Hypergradient Descent
  5. Other Hessian Inverse Approximation Methods
  6. Natural Gradient Descent
  7. Natural Hypergradient Descent Algorithm
  8. SGD algorithm for inner problem:
  9. Iterative approximation of the Hessian inverse:
  10. Iterative approximation of the cross-partial derivative:
  11. Practical Considerations
  12. Convergence Analysis
  13. Analysis for Hessian Inverse Approximation Bound
  14. Hessian Inverse Approximation Bound
  15. Proof Sketch of Theorem \ref{['theorem:hessian_inv_approx']}
  16. ...and 29 more sections

Key Result

Theorem 4.7

Suppose Assumptions as:l_inner, as:unbias_grad, as:bound_grad_inner and as:inner_dist hold. For any outer iteration $k$, given the outer variable $v_k$ and set the inner stepsize as $\eta_t = 4/(\mu (t+\frac{8L^2}{\mu^2}))$. For any $\delta \in (0,1)$ and $T\geq T_0(\delta)$, the following holds wit where $T_0(\delta)$ is defined in eq:condition_T0.

Figures (7)

  • Figure 1: Overview of NHGD. The inner problem is solved using SGD on Device 1, while gradient information is sent to Device 2 for iterative rank-one updates of the EFIM inverse $A_t$ and the cross-derivatives $L_t$. After the inner loop, Device 1 sends gradient information to Device 2, which approximates the hypergradient and updates the outer variable. The updated $v_{k+1}$ is then returned to Device 1 to resume the inner optimization. This design enables synchronous hypergradient estimation alongside inner-loop optimization.
  • Figure 2: Test accuracy for the Hyper-data Cleaning task.
  • Figure 3: Test Accuracy for Data Distillation task.
  • Figure 4: Outer loss for PDE Constrained Optimization task
  • Figure 5: Outer loss for the Hyper-data Cleaning task.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Remark 3.1
  • Remark 4.6
  • Theorem 4.7
  • Lemma 4.8
  • Remark 4.9
  • Theorem 4.10
  • Remark 4.11
  • Proposition 4.12
  • Lemma B.1
  • proof
  • ...and 18 more