Abstract
In Computer Science there is a strong consensus that it is highly desirable to combine the versatility of Machine Learning (ML) with the assurances formal verification can provide. However, it is unclear what such ‘verified ML’ should look like.
This paper is the first to formalise the concepts of classifiers and learners in ML in terms of computable analysis. It provides results about which properties of classifiers and learners are computable. By doing this we establish a bridge between the continuous mathematics underpinning ML and the discrete setting of most of computer science.
We define the computational tasks underlying the newly suggested verified ML in a model-agnostic way, i.e., they work for all machine learning approaches including, e.g., random forests, support vector machines, and Neural Networks. We show that they are in principle computable.
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Notes
- 1.
Arguing informally, continuity means that sufficiently good approximations of the input specify approximations of the output to desired precision. Computability means that we can actually compute the desired approximations of the output from sufficiently good approximations of the input. The latter cannot be possible for discontinuous function.
- 2.
By uncurrying, we can move from a learner L to the function \(\ell : (\textbf{X} \times k)^* \times \textbf{X} \rightarrow \textbf{k}_\bot \) such that \(L((x_i,n_i)_{i \le j})(x) = \ell ((x_i,n_i)_{i \le j}, x)\). One of them is continuous iff the other is. Now we can see that if \(\ell \) returns a colour, it returns the same colour on an open neighbourhood of both training sample and test point.
- 3.
Our overall algorithms are theoretical and not easily put into code. However, the individual concepts can be, in order to help the understanding of mathematical concepts such as robustness/sparsity and density.
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Crook, T., Morgan, J., Pauly, A., Roggenbach, M. (2023). A Computability Perspective on (Verified) Machine Learning. In: Madeira, A., Martins, M.A. (eds) Recent Trends in Algebraic Development Techniques. WADT 2022. Lecture Notes in Computer Science, vol 13710. Springer, Cham. https://doi.org/10.1007/978-3-031-43345-0_3
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