The Curse of Unrolling: Rate of Differentiating Through Optimization (2209.13271v3)

Abstract: Computing the Jacobian of the solution of an optimization problem is a central problem in machine learning, with applications in hyperparameter optimization, meta-learning, optimization as a layer, and dataset distillation, to name a few. Unrolled differentiation is a popular heuristic that approximates the solution using an iterative solver and differentiates it through the computational path. This work provides a non-asymptotic convergence-rate analysis of this approach on quadratic objectives for gradient descent and the Chebyshev method. We show that to ensure convergence of the Jacobian, we can either 1) choose a large learning rate leading to a fast asymptotic convergence but accept that the algorithm may have an arbitrarily long burn-in phase or 2) choose a smaller learning rate leading to an immediate but slower convergence. We refer to this phenomenon as the curse of unrolling. Finally, we discuss open problems relative to this approach, such as deriving a practical update rule for the optimal unrolling strategy and making novel connections with the field of Sobolev orthogonal polynomials.

• The paper presents a non-asymptotic convergence analysis of unrolled differentiation for quadratic problems using gradient descent and Chebyshev methods.
• It identifies a two-phase dynamic where a larger step size triggers an initial burn-in period before exponential convergence is achieved.
• Empirical findings validate the trade-offs in step size choices, offering actionable insights for hyperparameter optimization and meta-learning tasks.

An Analysis of Unrolled Differentiation in Optimization

The paper "The Curse of Unrolling: Rate of Differentiating Through Optimization" investigates the convergence rates of unrolled differentiation, a widely used heuristic in machine learning for estimating the Jacobian of implicit functions defined by optimization problems. The context of this paper is rooted in scenarios such as hyperparameter optimization, meta-learning, and similar domains within machine learning where the calculation of such Jacobians is crucial but challenging due to the implicit nature of the functions involved.

Central Contribution and Methodological Approach

The paper's principal contribution is a non-asymptotic convergence rate analysis of unrolled differentiation when applied to quadratic optimization problems using gradient descent and the Chebyshev method. The authors delineate an inherent trade-off, termed the "curse of unrolling," which manifests in a two-phase convergence dynamic: an initial burn-in phase marked by potential increases in suboptimality, followed by a later phase characterized by exponential convergence. This dichotomy poses a critical choice between a rapid asymptotic convergence with a prolonged burn-in period induced by a larger step size, or a slower yet immediate convergence facilitated by a smaller step size.

Formally, the paper introduces a master identity that links the rate of convergence of an unrolled Jacobian to both the polynomial associated with the iterative optimizer and its derivative. The convergence bounds derived are applicable specifically to quadratic objectives, and the analysis extends to provide explicit rates for both gradient descent and the Chebyshev method.

Numerical Insights and Implications

Empirical results presented in the paper illustrate the burn-in phenomenon and its dependency on the chosen step size. The findings echo the theoretical predictions, reinforcing the identified trade-offs. In practical terms, this suggests that practitioners employing unrolled differentiation must carefully calibrate the step size to balance initial convergence behavior against long-term efficiency. The derived rates also reveal the impact of step size choices on the maximum suboptimality encountered during the optimization process.

Theoretical Implications and Future Directions

From a theoretical standpoint, this analysis contributes to a deeper understanding of the limitations and capabilities of unrolled differentiation. The emphasis on quadratic objectives provides a tractable yet informative setting, illuminating the potential for extending these insights to more general non-quadratic settings. The paper also opens avenues for exploring novel optimization algorithms that are specifically tuned for unrolled differentiation, especially in the context of mitigating the burn-in phase.

Conclusion

This paper offers a significant contribution to the understanding of the dynamics of unrolled differentiation in optimization. The curse of unrolling, as articulated through this paper, not only sheds light on the intrinsic trade-offs in current practices but also sets the stage for advancements in both algorithm design and theoretical exploration. Moving forward, addressing the identified challenges could lead to more robust and efficient methodologies for machine learning tasks where differentiating through optimization layers is essential.