On the Iteration Complexity of Hypergradient Computation (2006.16218v2)

Abstract: We study a general class of bilevel problems, consisting in the minimization of an upper-level objective which depends on the solution to a parametric fixed-point equation. Important instances arising in machine learning include hyperparameter optimization, meta-learning, and certain graph and recurrent neural networks. Typically the gradient of the upper-level objective (hypergradient) is hard or even impossible to compute exactly, which has raised the interest in approximation methods. We investigate some popular approaches to compute the hypergradient, based on reverse mode iterative differentiation and approximate implicit differentiation. Under the hypothesis that the fixed point equation is defined by a contraction mapping, we present a unified analysis which allows for the first time to quantitatively compare these methods, providing explicit bounds for their iteration complexity. This analysis suggests a hierarchy in terms of computational efficiency among the above methods, with approximate implicit differentiation based on conjugate gradient performing best. We present an extensive experimental comparison among the methods which confirm the theoretical findings.

• The paper examines the iteration complexity of Iterative Differentiation (ITD) and Approximate Implicit Differentiation (AID) methods for computing hypergradients in bilevel problems under contraction assumptions.
• Under contraction assumptions, both ITD and AID achieve linear convergence, but experiments show AID, particularly with Conjugate Gradient, proves more efficient in practice.
• The research implies AID methods, especially with Conjugate Gradient, are highly efficient for large-scale ML applications requiring precise hyperparameter tuning with limited resources.

Iteration Complexity of Hypergradient Computation for Bilevel Problems

The paper "On the Iteration Complexity of Hypergradient Computation" provides a meticulous examination of gradient approximation methods for bilevel optimization problems where the upper-level objective function depends on the solution to a parametric fixed-point equation. This class of problems is pertinent to numerous machine learning applications, including hyperparameter optimization, meta-learning, and recurrent neural networks.

Bilevel Framework and Challenges

In bilevel optimization, the lower-level problem often involves a fixed-point equation, and explicitly solving for the fixed-point can be computationally prohibitive or outright infeasible. Consequently, computing exact gradients for the upper-level objective — referred to as hypergradient — can be impractical. This paper focuses on evaluating iterative schemes that approximate hypergradients, examining their iteration complexity under the assumption of contraction mappings for the fixed-point equation.

Methods Examined

Two primary gradient approximation approaches are analyzed:

1. Iterative Differentiation (ITD):
   • ITD calculates hypergradients by differentiating the trajectory of iterates towards the fixed-point using automatic differentiation. The reverse-mode automatic differentiation (RMAD) technique is commonly adopted due to its efficiency on high-dimensional problems.
2. Approximate Implicit Differentiation (AID):
   • AID leverages the implicit function theorem to formulate an equation for the hypergradient, which is solved approximately using methods such as recurrent backpropagation or conjugate gradient methods.

Complexity Results

The paper introduces a unified analysis framework to compare the ITD and AID methods. Under the assumption that the mappings defining the fixed-point equations are contractions, it demonstrates that both methods exhibit linear rates of convergence towards the true hypergradient.

• ITD Complexity:

The authors present non-asymptotic linear error bounds for ITD, highlighting that under contraction assumptions, ITD requires iterating towards the fixed-point with a complexity dependent on the contraction constant and other problem-specific parameters.

• AID Complexity:

For AID methods, approximation errors are evaluated based on the convergence rates of numerical solvers employed for computing the hypergradient. The paper indicates that conjugate gradient methods outperform other solvers due to their superior condition number handling in symmetric systems, providing faster convergence.

Experimental Analysis

The authors validate their theoretical findings empirically across a range of bilevel problems including logistic regression, kernel ridge regression, and hyper-representation. AID, particularly when implemented using conjugate gradient methods, is found to yield more precise gradient estimates and lower computational demands than ITD.

Implications and Future Directions

This paper elucidates the computational hierarchy among hypergradient approximation strategies, indicating conjugate gradient-based AID as highly efficient. The findings carry significant implications for large-scale machine learning scenarios requiring fine hyperparameter tuning, suggesting a preference for AID methods where high precision is needed and computational resources are constrained.

Going forward, further exploration could include scenarios where contraction assumptions are weakened or removed, stochastic approaches for the lower-level problem, and integration with advanced neural architectures like equilibrium models. The paper establishes a robust foundation for such investigations and continues to bridge theoretical advancement with practical applicability in machine learning optimization.