Hidden Limits of Machine Learning
Note: Machine learning(ML) is a subset of Artificial Intelligence(AI), AI includes, for example, evolutional programming, swamp intelligence etc. In other words Machine learning is a collection of algorithms and techniques used to design systems that learn from data.
The lure of Machine Learning methods has been overwhelming. The range of Machine Learning applications is wide from Detection of Credit Card Fraud to Classification of Biological Particles in Electron-microscopy Images. People are getting to believe that Machine Learning is Universal Tool capable to solve any classification problem (or at least with 99% reliability)
The purpose of the given note is to add a new argument against of Machine Learning’s omnipotence. It’s worth mentioning that there are theoretical limitations on Machine Learning, e.g. Cybenko’s Theorem for Neural Networks.
To warm up let’s consider a classification (or separation) of two-dimensional classes (objects) with the following restriction on the objects: the ‘amount’ of boundary points is negligible compared with the ‘amount’ of interior points (generally, one can easily create a weird object where boundary points are 99% of all points, we omit these cases).
So, for two-dimensional classes even if some of the classes touch each another and even if class boundaries are totally fractal then a non-linear separation curve (or set of curves) can be found with low percentage of false classifications. False classifications are expected to happen if some classes touch each another (however due to the restriction the percentage of false classifications tends to be low since boundary samples consists a small percent).
Let’s for the sake of simplicity consider Two Classes Classification Problem. Moreover, let’s assume that each class (object) represents a region in N-dimensional space (where N is much more one). Let’s also assume that each object is close to a N-dimensional sphere of radius R and the two objects touch each another at some boundary segment.
Volume of N-dimensional sphere of the radius R:
Vol(R) = πN/2 RN / Γ(N/2+1)
where Γ is the gamma function.
Then the volume of near-surface “peel” of thickness ε (ε<R):
Vol(R) – Vol(R- ε) = πN/2 RN (1 – (R – ε)N/RN) / Γ(N/2+1)
For ε=0.1 R the ε-peel’s volume equal to πN/2 RN (1 – 0.9N) / Γ(N/2+1). Hence, when N is much greater 1 then the volume of 0.1-peel is close to Vol(R). In other words, almost all mass of N-dimensional sphere (for N much greater than 1) is concentrated at the thin 0.1R peel. Actually the interior of this sphere is almost ‘empty’.
Consequently, if the two classes touch each another along some boundary, for example touching boundary is 10% of the whole boundary) then the separation curve would mismatch roughly 50% of boundary points. Hence, the total number of false classifications is about 10%.
This is an interesting property of high-dimensional objects is that most of their volume is near the surface.
Expert Systems
Fundamental problem in development of expert systems is not in techniques of inference but in experts themselves. Because human experts are not able to articulate their thinking process to codify it.
Note:
In Machine Learning the concept “machine” is not necessarily a collection of physical parts (e.g. bearings, shafts), but any entity which does a specific job.
23+ years’ programming and theoretical experience in the computer science fields such as video compression, media streaming and artificial intelligence (co-author of several papers and patents).
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