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I have found that many authors of machine learning papers, that employ the Hebbian learning rule, refer to the biological plausibility of it, as one of the arguments to use it instead of the well known back-propagation algorithm (Whittington & Bogacz, 2017; Pugh, Soltoggio, & Stanley, 2014; Burms, Caluwaerts, & Dambre, 2015) (I imagine the same argument is used for other biologically plausible algorithms).

But why, in actual fact, is biological plausibility important in the first place? Why do so many sources qoute such argument, and what are the advantages of biological plausibility, when related to machine learning applications?

I have searched a lot, but haven't really seen anyone answer this specifically.

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They're mentioning biological plausibility, because they think biological constraints are important for intelligent systems. Also, biologically constrained systems could be easily applied to understanding the brain. Whether biological plausibility is important or not to developing intelligent systems and understanding the brain is a matter of debate.

On one hand, there's the argument that since the brain is the only form of intelligence we know, we should probably imitate it to develop intelligent systems. Also, at the same time, we would learn to understand the brain!

On the other hand, there's the often-repeated argument that we did not get airplanes from imitating birds, but from understanding the principles of aerodynamics. In turn, the principles of aerodynamics were then applied to bird wings and feathers.

For better introduction to this debate applied to understanding the brain and creating scalable intelligence, try reading "Marr's Attacks: On Reductionism and Vagueness" by Eliasmith et al.

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