Enzyme Compilation Errors: Reverse Derivatives

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Enzyme Compilation Errors: Unraveling the Mystery of Reverse Derivatives

Enzyme Compilation Errors: Unraveling the Mystery of Reverse Derivatives

Hey guys! Let's dive into a fascinating challenge in the world of automatic differentiation (AD) using Enzyme.jl. We're going to explore a specific error that pops up when dealing with "reverse-over-reverse" derivatives, particularly when matrices are involved. This can be super tricky, but don't worry, we'll break it down step by step and make it understandable. This is a common problem, so understanding this can save you a ton of headaches.

The Problem: Reverse-over-Reverse Derivatives with Matrices

In the realm of AD, we often use the concept of derivatives. In this case, we have the num_to_vec and num_to_mat functions. Enzyme.jl is a powerful tool to compute these derivatives efficiently. When we try to calculate the second derivative of num_to_mat, something unexpected happens: Enzyme fails to compile, and we get an error! The error message isn't the most descriptive, but it hints at an internal issue within Enzyme itself. Basically, Enzyme chokes on the reverse-over-reverse structure when matrices are involved. This is where the derivative of a derivative is calculated, and it is a known issue.

Let's get into some code: the functions num_to_vec converts a number to a vector and num_to_mat builds a matrix using that vector. The derivative and second_derivative functions are then used to calculate the first and second derivatives respectively. The issue is when you try to calculate the second derivative of num_to_mat. It's a tricky problem, but that's what we're going to address now.

This behavior is documented in the bug report and we will give more information about it.

Reproducing the Error: A Minimal Working Example (MWE)

The good thing is that we have a Minimal Working Example (MWE) that shows exactly how to reproduce the error. MWEs are awesome for debugging. They're the smallest pieces of code needed to trigger a bug. The code in the original report is small and concise and the error is easy to reproduce. Let's take a look at it:

import Enzyme

num_to_vec(x::Number) = sin.([1, 2] .* x)
num_to_mat(x::Number) = hcat(num_to_vec(x), num_to_vec(3x))

derivative(f, x) = Enzyme.jacobian(Enzyme.Reverse, f, x)[1]
second_derivative(f, x) = derivative(_x -> derivative(f, _x), x)

derivative(num_to_mat, 1.0)  # works
second_derivative(num_to_mat, 1.0)  # errors

In this example, num_to_mat creates a matrix based on the input x. The derivative function uses Enzyme's jacobian to compute the derivative. Then, second_derivative calculates the derivative of the derivative. When you run this code, the first derivative calculation works fine. But when we try to compute the second derivative, it fails. This is a critical problem for anyone doing complex numerical calculations.

The Error Message: Decoding the Jargon

Let's go over the error message and see if we can understand what is happening here. Error messages can be tricky. Here's a snippet of what we get:

ERROR: Enzyme compilation failed due to an internal error.
Please open an issue with the code to reproduce and full error log on github.com/EnzymeAD/Enzyme.jl

The message says that the Enzyme compilation has failed. This means that the AD engine, which transforms your code to compute derivatives, couldn't finish its job. It suggests opening an issue on the Enzyme.jl GitHub page. This is important: if you encounter this, definitely report it so the developers can fix it. The message also suggests setting Enzyme.Compiler.VERBOSE_ERRORS[] = true to get more detailed information, which is useful when reporting the bug.

Diving Deeper: The Stacktrace

Let's go through the Stacktrace. A stacktrace shows the sequence of function calls that led to the error. This is critical for understanding where the error originated. The stacktrace is a series of function calls, and the error originates in the num_to_mat function, which is then called by second_derivative. Then it goes deeper into Enzyme's internals. It's difficult to understand the exact root cause of the error. It's safe to say it's related to how Enzyme handles reverse-mode differentiation when calculating the second derivative of a function involving matrices.

Technical Deep Dive: The Heart of the Problem

To really understand what's happening, you need to understand how Enzyme works. Enzyme is a source-to-source transformation tool. It works by transforming the original code to compute the derivatives. When the reverse-over-reverse derivative comes into play, especially with matrices, the complexity increases dramatically.

  • Reverse Mode Differentiation: Enzyme uses reverse-mode differentiation. This is usually efficient for calculating gradients, but it can be memory-intensive. Calculating the second derivative compounds this complexity. The engine tries to create an augmented primal that causes the compilation error.
  • Matrix Operations: Matrices bring another layer of complexity. The way Enzyme handles matrix operations, especially with the repeated application of reverse-mode AD, is not ideal.
  • Internal Errors: The error message indicates an internal error. It means that something went wrong during Enzyme's internal operations. This usually means a bug or a limitation in the library.

Potential Workarounds and Solutions

  • Simplify the Function: Try to simplify the function you're differentiating. This may include breaking down num_to_mat into simpler functions or avoiding matrix operations if possible. See if this will solve the compilation error.
  • Alternative AD Libraries: If the issue persists, consider using a different AD library in Julia, such as ForwardDiff.jl, especially if the problem is specific to reverse-mode differentiation. See if this solves the problem, if it does, it's a good alternative.
  • Contribute to Enzyme: The best solution is to contribute to Enzyme. You can help by providing clear bug reports, minimal working examples, and potentially even helping fix the code. Open a GitHub issue and provide detailed information.

The Future of Enzyme and AD

AD is a rapidly developing field. Enzyme is a very promising library. With the help of the community, issues like these will be resolved. Remember to stay updated with the latest versions and to report any issues you find.

Conclusion: Navigating the AD Maze

We've explored a tricky issue where reverse-over-reverse derivatives with matrices cause Enzyme compilation failures. By understanding the error, its origins, and potential workarounds, you can navigate the AD maze more effectively. Remember to report issues, contribute, and stay patient – the world of AD is constantly improving!