Empirical Comparison of Low-Level and High-Level Differentiation Strategies for Linear Solvers

Concetti Chiave

Empirical evaluation of the tradeoffs between low-level and high-level differentiation approaches for linear solvers, demonstrating that high-level differentiation is generally preferable but low-level differentiation can be effective for certain solvers.

Sintesi

The article examines the tradeoffs between low-level and high-level differentiation strategies when applying automatic differentiation (AD) to computer programs containing calls to linear solvers.
The key highlights are:

Previous publications have advised against differentiating through the low-level solver implementation, and instead advocated for high-level approaches that express the derivative in terms of a modified linear system. However, the accuracy of both approaches has not been empirically compared.

The authors implemented low-level differentiation by applying the Tapenade AD tool to the SPARSKIT implementation of GMRES, TFQMR and BiCGStab solvers. They also implemented high-level differentiation at the matrix calculus level.

Experiments were conducted on 65 matrices from the SuiteSparse collection, comparing the performance of the original, undifferentiated solvers with the low-level and high-level differentiation strategies.

The results show that high-level differentiation generally performs nearly as well as the original solver, but there are typically a few problems that require more iterations to achieve similar levels of accuracy.

The effectiveness of low-level differentiation is highly solver-dependent. For TFQMR and restarted GMRES, the low-level differentiation strategy is nearly as effective as high-level differentiation. However, for BiCGStab, there is a significant gap in performance between high-level and low-level differentiation.

The authors conclude that the common advice to use high-level differentiation is justified, but a careful solver choice may lead to useful gradients even with low-level approaches in certain situations.

Statistiche

The L2 norm of the difference between the computed x (or ∂x/∂u) and the reference value is less than 10^-2 or 10^-4.

Citazioni

"Despite this ubiquitous advice, we are not aware of prior work comparing the accuracy of both approaches."
"We demonstrate with this article that the common advice is justified, and that high-level differentiation is indeed usually preferable to low-level differentiation."

Approfondimenti chiave tratti da

Differentiating Through Linear Solvers

by Paul... alle arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17039.pdf

Domande più approfondite

What are the theoretical underpinnings that explain the observed differences in performance between low-level and high-level differentiation for different linear solvers

The observed differences in performance between low-level and high-level differentiation for different linear solvers can be attributed to the theoretical foundations of each approach.
In low-level differentiation, the differentiation process involves directly differentiating through the solver's implementation details, which can lead to issues with stability and accuracy. The theoretical underpinnings suggest that for certain linear solvers, especially those with complex iterative methods like Krylov solvers, low-level differentiation may not be well-understood or supported. This lack of theoretical development for low-level differentiation of Krylov solvers, particularly for nonsymmetric systems, can result in erratic behavior and divergence, as seen in the experimental results for BiCGStab and GMRES.
On the other hand, high-level differentiation treats the linear solver as an elementary function and applies matrix calculus rules to compute derivatives directly. The theoretical framework for high-level differentiation is more established and straightforward, especially in the forward mode. By expressing the derivative in terms of a modified linear system that can be solved separately, high-level differentiation can often achieve better stability and accuracy compared to low-level differentiation. However, the manual effort involved in developing high-level differentiation strategies may be a drawback in some cases.
Therefore, the differences in performance between low-level and high-level differentiation for different linear solvers can be explained by the level of theoretical understanding and support for each approach, as well as the complexity of the iterative methods used in the solvers.

How would the results change if the experiments were conducted using reverse mode differentiation instead of forward mode

If the experiments were conducted using reverse mode differentiation instead of forward mode, the results would likely show different performance characteristics for the low-level and high-level differentiation strategies.
In reverse mode differentiation, the chain rule is applied starting with the dependent variables and propagating backward to the independent variables. This mode is particularly useful when the number of independent variables is much larger than the number of dependent variables, as is often the case in machine learning and optimization problems.
When applied to linear solvers, reverse mode differentiation may offer advantages in terms of computational efficiency and memory usage compared to forward mode. However, the effectiveness of reverse mode differentiation through linear solvers would depend on the specific characteristics of the solvers and the nature of the problem being solved.
Overall, conducting the experiments using reverse mode differentiation could provide insights into how the performance of low-level and high-level strategies varies in this mode and whether one approach is more suitable than the other for different types of linear solvers.

How do roundoff errors and other numerical considerations influence the relative performance of the differentiation strategies, and can these effects be quantified

Roundoff errors and other numerical considerations can significantly influence the relative performance of differentiation strategies, and these effects can be quantified to some extent.
In the context of differentiating through linear solvers, roundoff errors can accumulate during the iterative solution process, affecting the accuracy of the computed derivatives. These errors can be amplified in iterative methods like Krylov solvers, leading to numerical instability and divergence. By quantifying the impact of roundoff errors on the convergence behavior and accuracy of the computed derivatives, researchers can assess the robustness of different differentiation strategies.
Additionally, the choice of numerical precision, preconditioners, and convergence criteria can also impact the performance of differentiation strategies. High-level differentiation may be more sensitive to numerical considerations due to the direct computation of derivatives, while low-level differentiation may be affected by the stability of the solver implementation.
To quantify the influence of roundoff errors and other numerical considerations, researchers can analyze the convergence behavior of differentiation strategies under varying conditions, such as different matrix sizes, condition numbers, and solver configurations. By measuring the sensitivity of the differentiation results to numerical parameters, it is possible to assess the reliability and accuracy of the differentiation process in the presence of numerical uncertainties.