Written by: Stephen Hsu
Primary Source: Information Processing, 07/17/2018.
It’s never been a better time to work on AI/ML. Vast resources are being deployed in this direction, by corporations and governments alike. In addition to the marvelous practical applications in development, a theoretical understanding of Deep Learning may emerge in the next few years.
The notes below are to keep track of some interesting things I encountered at the meeting.
Some ML learning resources:
I heard a more polished version of this talk by Elad at the Theory of Deep Learning workshop. He is trying to connect results in sparse learning (e.g., performance guarantees for L1 or threshold algos) to Deep Learning. (Video is from UCLA IPAM.)
It may turn out that the problems on which DL works well are precisely those in which the training data (and underlying generative processes) have a hierarchical structure which is sparse, level by level. Layered networks perform a kind of coarse graining (renormalization group flow): first layers filter by feature, subsequent layers by combinations of features, etc. But the whole thing can be understood as products of sparse filters, and the performance under training is described by sparse performance guarantees (ReLU = thresholded penalization?). Given the inherent locality of physics (atoms, molecules, cells, tissue; atoms, words, sentences, …) it is not surprising that natural phenomena generate data with this kind of hierarchical structure.
Off-topic: At dinner with one of my former students and his colleague (both researchers at an AI lab in Germany), the subject of Finitism came up due to a throwaway remark about the Continuum Hypothesis.
The reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic, cannot be found in mathematics or logic. The reason is that the laws of physics “happen” to permit the existence of physical models for the operations of arithmetic such as addition, subtraction and multiplication.
My perspective: We experience the physical world directly, so the highest confidence belief we have is in its reality. Mathematics is an invention of our brains, and cannot help but be inspired by the objects we find in the physical world. Our idealizations (such as “infinity”) may or may not be well-founded. In fact, mathematics with infinity included may be very sick, as evidenced by Godel’s results, or paradoxes in set theory. There is no reason that infinity is needed (as far as we know) to do physics. It is entirely possible that there are only a (large but) finite number of degrees of freedom in the physical universe.
I will ascribe to Skolem a view, not explicitly stated by him, that there is a reality to mathematics, but axioms cannot describe it. Indeed one goes further and says that there is no reason to think that any axiom system can adequately describe it.
This “it” (mathematics) that Cohen describes may be the set of idealizations constructed by our brains extrapolating from physical reality. But there is no guarantee that these idealizations have a strong kind of internal consistency and indeed they cannot be adequately described by any axiom system.