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Probabilistic Numerics for Computer Scientists

By Michael Schober, 2016-05-18 12:00:00 +0200

This post is the first of a series originating in a PhD student meeting in Tübingen in April 2016.

It has been two years since the inaugural probabilistic numerics roundtable meeting. While the field has started to generate some momentum, it is growing more important to explain the concept of probabilistic numerics to all relevant parties. This post will attempt to explain the basic ideas, research goals and obstacles to a trained computer scientist.

Probabilistic numerics is an emerging research area with goals, problems and tools from both applied mathematics – numerical analysis and probability theory – and theoretical computer science – mostly algorithms and data structures. Numerical algorithms are tools for solving equations that either are too big to be solved manually or don’t possess a closed-form solution at all. In both cases, an iterative procedure is implemented in a computer which is then proven to converge to the correct solution. Common problems are finding optimal values of some function, evaluating tricky integrals or solving differential equations.

However, in many cases even these algorithms are too expensive to be evaluated to very high precision. For instance, when solving integrals of many variables, the approximation quality depends exponentially on the number of function evaluations – the so-called curse of dimensionality. Or, the function to be optimized might be too costly to be evaluated accurately which is the case in deep learning and big data (just to throw in some more buzz words). In these situations, the (slightly modified) algorithms might still work, but guarantees are harder to come by.

To draw an analogy to a classical computer science problem, I will use the A* search algorithm. Imagine a huge weighted graph. The task is to compute the cost from node X to node Y, but you are only given a small computational time budget, a tiny fraction of the guaranteed worst case running time. Since the A* algorithm uses an heuristic to estimate the remaining cost, the algorithm can be terminated any time to produce an approximate output.

Now, the second key ingredient comes into play: probability theory. In probabilistic numerics, probabilities can represent the uncertainties stemming from approximations in the algorithm or finite run time. The calculus of probability theory, most importantly Bayes’ rule, allows to incorporate and extend algorithms in many ways in a consistent and well-studied framework. E.g., the early stopping A* algorithm might not only return an expected value for the cost, but also give a standard deviation. Or the algorithm could spend some time comparing the estimated remaining cost with the observed values from the graph weights, adjusting the overall estimation if, for instance, all previous estimates have been too high. Another possible extension: the weights of the graph might be represented by probability distributions themselves. If the graph represents a road network, there might be traffic jams with some probability.

Note, however, that this does not necessarily mean that there is randomness or stochastic elements in a probabilistic numerical algorithm. While some researchers also use sampling based methods in their algorithms, other methods are completely deterministic working on deterministic problems to produce deterministic results.

While many problems in probabilistic numerics are of mathematical nature, there are also good reasons to get interested in this area when your focus is more on computer science. One common problem is to represent the necessary probability distributions with low memory cost. What are good programming interfaces when dealing with probability distributions? There are also challenges when dealing with specialized hardware, e.g., GPUs for large-scale optimization.

In conclusion, probabilistic numerics is a young and active research area using probability theory to describe iterative approximative algorithms. There are a plethora of open problems and potential applications and lots of interesting challenges covering the whole spectrum of applied mathematics and (theoretical) computer science.

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