Can any existing Machine Learning structures perfectly emulate recursive functions like the Fibonacci sequence?

Question

To be clear I don't mean, provided the last two numbers in the sequence provide the next one:

(2, 3, -> 5)

But rather given any index provide the Fibonacci number:

(0 -> 1) or (7 -> 21) or (11 -> 144) 

Adding two numbers is a very simple task for any machine learning structure, and by extension counting by ones, twos or any fixed number is a simple addition rule. Recursive calculations however...

To my understanding, most learning networks rely on forwards only evaluation, whereas most programming languages have loops, jumps, or circular flow patterns (all of which are usually ASM jumps of some kind), thus allowing recursion.

Sure some networks aren't forwards only; But can processing weights using the hyperbolic tangent or sigmoid function enter any computationally complete state?

i.e. conditional statements, conditional jumps, forced jumps, simple loops, complex loops with multiple conditions, providing sort order, actual reordering of elements, assignments, allocating extra registers, etc?

It would seem that even a non-forwards only network would only find a polynomial of best fit, reducing errors across the expanse of the training set and no further.

Am I missing something obvious, or did most of Machine Learning just look at recursion and pretend like those problems don't exist?

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Update

Technically any programming language can be considered the DNA of a genetic algorithm, where the compiler (and possibly console out measurement) would be the fitness function. The issue is that programming (so far) cannot be expressed in a hill climbing way - literally, the fitness is 0, until the fitness is 1. Things don't half work in programming, and if they do, there is no way of measuring how 'working' a program is for unknown situations. Even an off by one error could appear to be a totally different and chaotic system with no output. This is exactly the reason learning to code in the first place is so difficult, the learning curve is almost vertical.

Some might argue that you just need to provide stronger foundation rules for the system to exploit - but that just leads to attempting to generalize all programming problems, which circles right back to designing a programming language and loses all notion of some learning machine at all. Following this road brings you to a close variant of LISP with mutate-able code and virtually meaningless fitness functions that brute force the 'nice' and 'simple' looking code-space in attempt to follow human coding best practices.

Others might argue that we simply aren't using enough population or momentum to gain footing on the error surface, or make a meaningful step towards a solution. But as your population approaches the number of DNA permutations, you are really just brute forcing (and very inefficiently at that). Brute forcing code permutations is nothing new, and definitely not machine learning - it's actually quite common in regex golf, I think there's even an xkcd about it...

The real problem isn't finding a solution that works for some specific recursive function, but finding a solution space that can encompass the recursive domain in some useful way.

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So other than Neural Networks trained using Backpropagation hypothetically finding the closed form of a recursive function (if a closed form even exists, and they don't in most real cases where recursion is useful), or a non-forwards only network acting like a pseudo-programming language with awful fitness prospects in the best case scenario, plus the virtually impossible task of tuning exit constraints to prevent infinite recursion... That's really it so far for machine learning and recursion?

Answer 1

According to Kolmogorov et al's On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, a three layer neural network can model arbitrary function with the linear and logistic functions, including f(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n) / (2^n * sqrt(5)), which is the close form solution of Fibonacci sequence.

If you would like to treat the problem as a recursive sequence without a closed-form solution, I would view it as a special sliding window approach (I called it special because your window size seems fixed as 2). There are more general studies on the proper window size for your interest. See these two posts:

Time Series Prediction via Neural Networks

Proper way of using recurrent neural network for time series analysis

Answer 2

Ok, where to start...

Firstly, you talk about 'machine learning' and 'perfectly emulate'. This is not generally the purpose of machine learning algorithms. They make informed guesses given some evidence and some general notions about structures that exist in the world. That typically means an approximate answer is better than an 'exact' one that is wrong. So, no, most existing machine learning approaches aren't the right tools to answer your question.

Second, you talk of 'recursive structures' as some sort of magic bullet. Yet they are merely convenient ways to represent functions, somewhat analogous to higher order differential equations. Because of the feedbacks they tend to introduce, the functions tend to be non-linear. Some machine learning approaches will have trouble with this, but many (neural networks for example) should be able to approximate you function quite well, given sufficient evidence.

As an aside, having or not having closed form solutions is somewhat irrelevant here. What matters is how well the function at hand fits with the assumptions embodied in the machine learning algorithm. That relationship may be complex (eg: try approximating fibbonacci with a support vector machine), but that's the essence.

Now, if you want a machine learning algorithm tailored to the search for exact representations of recursive structures, you could set up some assumptions and have your algorithm produce the most likely 'exact' recursive structure that fits your data. There are probably real world problems in which such a thing would be useful. Indeed the field of optimisation approaches similar problems.

The genetic algorithms mentioned in other answers could be an example of this, especially if you provided a 'genome' that matches the sort of recursive function you think you may be dealing with. Closed form primitives could form part of that space too, if you believe they are more likely to be 'exact' than more complex genetically generated algorithms.

Regarding your assertion that programming cannot be expressed in a hill climbing way, that doesn't prevent a learning algorithm from scoring possible solutions by how many much of your evidence it's able to reproduce and how complex they are. In many cases (most? though counting cases here isn't really possible) such an approach will find a correct answer. Sure, you can come up with pathological cases, but with those, there's little hope anyway.

Summing up, machine learning algorithms are not usually designed to tackle finding 'exact' solutions, so aren't the right tools as they stand. But, by embedding some prior assumptions that exact solutions are best, and perhaps the sort of exact solution you're after, you'll probably do pretty well with genetic algorithms, and likely also with algorithms like support vector machines.

I think you also sum things up nicely with this:

The real problem isn't finding a solution that works for some specific recursive function, but finding a solution space that can encompass the recursive domain in some useful way.

The other answers go a long way to telling you where the state of the art is. If you want more, a bright new research path lies ahead!