[dahart]

> More generally, relative error works only if your computation scales “multiplicatively” from zero; if there’s any additive component, it’s suspect.

IEEE floating point inherently scales from zero, the absolute error in any computation is proportional to the magnitude of the input numbers, whether you’re adding or multiplying or doing something else. It’s the reason that subtracting two large numbers has a higher relative error than subtracting two small numbers, c.f. catastrophic cancellation.

> How does one do this properly?

There’s a little bit of an art to it, but you can start by noting the actual result of any accurate operation has a maximum error of 0.5 LSB (least significant bit) simply as a byproduct of having to store the result in 32 or 64 bits; essentially just think about every single math instruction being required to round the result so it can fit into a register. Now write an expression for your operation in terms of perfect math. If I’m adding two numbers it will look like a[+]b = (a + b)*(1 + e), where e is your 0.5 LSB epsilon value. For 32 bit float, e == +/- 1e^-24. In this case I differentiate between digital addition with a finite precision result, and perfect addition, using [+] for digital addition and + for perfect addition.

This gets hairy and you need more tricks for anything complicated, but multiplying each and every operation by (1+e) is the first step. It quickly becomes apparent that the maximum error is bounded by |e| * (|a|+|b|) for addition or |e| * (|a| * |b|) for multiply… substitute whatever your operation is.

When doing more complicated order-dependent error analysis, it’s helpful to use bounds and to allow error estimates to grow slightly in order to simplify expressions. This way you can prove the error is less than a certain expression, but the expression might be conservative.

A 3d dot product is a good example to work though using (1+e). Typically it’s reasonable to drop e^2 terms, even though it will technically compromise your error bound proof by some minuscule amount.

    a[*]x [+] b[*]y [+] c[*]z = ((ax(1+e) + bx(1+e)) + cx(1+e))(1+e)
    = ((ax+axe + by+bye)(1+e) + cz+cze)(1+e)
    = (ax+by+(ax+by)e + axe+bye+(ax+by)ee + cz+cze)(1+e)
    = (ax + bx + 2e(ax+by) + e^2(ax+by) + cz+cze)(1+e)
    = ax + by + 2e(ax+by) + e^2(ax+by) + cz + cze + axe + bye + 2e^2(ax+by) + e^3(ax+by) + cze + cze^2
    = ax+by+cz + 3e(ax+by) + 2e(cz) + 3e^2(ax+by) + e^2cz + e^3(ax+by)

Now drop all the higher order terms of e.

    = ax+by+cz + 3e(ax+by) + 2e(cz)

Now also notice that 2e|cz| <= 3e|cz|, so we can say the total error bound:

    <= (ax + by + cz) + 3e( |a||x| + |b||y| + |c||z| )

And despite the intermediate mess, this suddenly looks very conceptually simple and doesn’t depend on the order of operations. If the input values are all positive, then we can say the error is proportional to 3 times the magnitude of the dot product. And it’s logical too because we stacked 3 math operations, one multiply for each element of the sum and two adds.

Sorry if that was way too much detail… I got carried away. :P I glossed over some topics, and there could be mistakes but that’s the gist. I’ve had to do this for my work on a much more complicated example, and it took a few tries. There is a good linear algebra book about this, I think called Accuracy and Stability of Numerical Algorithms (Nicholas Higham). The famous PBR graphics book by Pharr et. al. also talks about error estimation techniques.

[tomsmeding]

Right, this is if you know the exact operations that your computation does, and that list is small enough.

My usecase is testing an autodiff algorithm. So I have larger programs (for which doing this process would be quite cumbersome already), and then run them through a code transformation that makes it compute a gradient. What's an appropriate error bound for that gradient?

Ideally I would even want to be able to randomly generate input programs, differentiate them, and test correctness of the computed gradient. I feel like generalising your approach (in particular the dropping of higher-order e terms) smacks of interval arithmetic, but even with a proper error estimate based on interval arithmetic, one would have to incorporate an estimate of the accuracy of their derivatives, too.

And to make life harder, I'd like to do this in a parallel setting where e.g. reductions (sums, products, etc.) have non-deterministic order to improve parallelism. I don't know how to approach this!

[dahart]

The Higham book does work toward error analysis of matrix multiplies, it would be useful to see how that’s done.

In the case of autodiff, you do presumably know the exact computations that are done, there just might be so many of them that it’s infeasible to work it out analytically.

It depends on your requirements, so I’m not sure if this suggestion will work for you, but one strategy to consider would be to build the error bound computation as a function into your math operations. It’s relatively much easier to compute error bounds than it is to write an expression for them or to prove them. That strategy won’t give you conservative bounds and if your input is non-deterministic, the answer will vary on every run. But you could sample your error bounds enough times to have some confidence in the statistical answer.

I’m assuming in both paragraphs above that you have control over the autodiff implementation and can modify it. If that’s not true, if it’s not yours and not open source, then the only alternative is to ask the maintainer.

IIRC this is what the PBR book does, it weaves an error bound function into the base class of a math operation and then you can query the error from a parse tree of different math ops, or something like that.

[tomsmeding]

Context: I'm writing an autodiff algorithm, and that's the thing I want to test.

> one strategy to consider would be to build the error bound computation as a function into your math operations

That's autodiff! :D The error (change) in a function's output given an error (change) in its input is the definition of its derivative.

So I guess I can use my autodiff algorithm to compute the error bounds for testing my autodiff algorithm! ... Uhm.

Actually, though, I want _one_ error bound, not one that varies per run. But I guess you can do the same thing symbolically: you just end up with

1. a slightly-too-optimistic bound because you will (for practicality) discard higher-order e terms;

2. a conservative bound because you'll be pessimistic in the face of control flow, array sizes, etc. where the precise operations performed depend on the input.

I guess this symbolic approach works until you have unbounded sequential loops (which pessimistically have unbounded error, because it may accumulate indefinitely). Or perhaps it breaks down already with arbitrary-size arrays; what is rhe error bound on `lambda x. sum([x]*n)`, assuming n is unknown? (Using python syntax as "universal syntax".)

> then you can query the error from a parse tree of different math ops, or something like that.

If there is no dynamic control flow nor variable-size data structures etc. in that parse tree, I suspect they do kind of what I hand-wavingly described above.

It's not perfect enough that I'm going to implement this approach immediately (variable-size arrays are kind of core to what I want to do). But perhaps there's some trick I can pull out of this. Thanks for the ideas!

[dahart]

> That's autodiff! :D The error (change) in a function's output given an error (change) in its input is the definition of its derivative

Ah but the error of your autodiff math isn’t the same thing as the error of the function you’re autodiffing! And the math error isn’t even a derivative. Specifically, the absolute value handling of rounding error analysis will cause the error bounds to diverge from being proportional to the function and/or derivative values.

I assume your unknown sum sizes are eventually finite… otherwise you won’t get an answer? Evaluating the rounding error will just take the same time as evaluating your sum. Long sums, btw, are bad for precision. Typically people try to sort them or compute them in hierarchical/ butterfly reduce fashion, or use other tricks.

[tomsmeding]

> Specifically, the absolute value handling of rounding error analysis will cause the error bounds to diverge from being proportional to the function and/or derivative values.

Oh, I see! Not quite autodiff then, indeed.

> Long sums, btw, are bad for precision. Typically people try to sort them or compute them in hierarchical/ butterfly reduce fashion, or use other tricks.

Typical parallel reduction algorithms do precisely this, but for different reasons (parallelisation). I guess I'd have to compute a partially symbolic expression for the error bound depending on the array size, and hope that I'm not going to change the parallel reduction algorithm without changing the error bounds in tests.

It's possible, but finicky. I see why people slap simple relative bounds on things and call it a day. :)

[dahart]

There’s probably a way to get a stable conservative error bound for a parallel sum reduction that’s non-deterministic.

Here’s the PBR book’s chapter on error analysis. It’s way easier to implement error bound tracking than it is to come up with symbolic bounds, and it’s way better than having nothing or even a guess pulled out of thin air. It’s probably worth trying, just to see what happens, it might be surprising. Plus if you implement the error bounds, you may find very interesting and/or enlightening ways to visualize them.

https://pbr-book.org/4ed/Shapes/Managing_Rounding_Error#