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Thank you for sharing your work. I confess that I am not that versed in mathematics, but I'm working on something for my PhD in the same line as to what you propose here and was wondering if my initial understanding of your proposal is correct. IMO, my proposal would go under the "How to do this quickly, efficiently, effectively?" category. My proposal is that I want to approach the problem in an axiomatic way. I am under the impression that the term "axiomatic" is very controversial, so allow me to elaborate.
So what do I mean by axioms? I certainly am not talking about axioms in the sense of "facts". For example, "the thigh bone is connected to the hip bone" is not an axiom in my mind. Another non-example would be the "Resolution" law in logic. What I am looking for is the axioms by which we define what constitutes the reactions of an "intelligent agent". For example, "An intelligent agent should be stable in its decision making" is an axiom in my view. Actually, this is more than an example and I want to propose this as the first axiom of an "intelligent agent". To put it in more precise terms, "there should not be adversarial examples for an intelligent agent". Think of a mosquito, it is a very simple intelligent agent that follows the sources of light (or heat, I'm not sure). But it is not possible to produce an adversarial pattern of light (e.g. rapidly blinking light) to convince the mosquito to NOT follow the light source.At the first glance, this axiom is not that helpful. But if we get mathematical, it turns out that this axiom has profound consequences. I am planning on uploading a paper on this topic in arxiv in the coming days. To be able to continue the discussion though, I will put some of the math here and am looking forward to getting your opinion on it.To put the axiom in mathematical terms, I propose a concept that I call "local robustness". An intelligent agent is locally robust if the magnitude of the change in its decision is independent of the direction of an infinitesimal change in the input. To get a feel of the rationale behind the definition, consider an image and its label. If I introduce some small perturbation to the image, the change in the confidence of the label an intelligent agent assigns to the perturbed image should be independent of the "content" of the perturbation. In math form:The definition could be readily generalized to complex vectors. From complex analysis we know that a function would satisfy the local robustness condition if and only if it satisfies the Cauchy-Riemman equations. I should mention that CR equations are defined for a scalar input, but it is straightforward to generalize this to complex vectors. The functions that satisfy the homogeneous CR equations are very special and they are called "holomorphic" functions. These functions have other desirable properties as well. A complex-valued function is analytic and complex differentiable if and only if it is holomorphic (it's more nuanced than that, but I will omit the discussion for brevity). It is proved that only complex-valued functions of a complex vector could be holomorphic. In other words, it is impossible to construct a holomorphic function of a real vector. So the first consequence of the axiom would be "an intelligent agent uses a complex-valued function of a complex vector for decision making". I can go on about the other consequences of the axiom, but bringing them up would be futile if you find the approach not satisfactory or relevant.
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It seems naively that the magnitude of change in an intelligent system's decision SHOULD depend on the direction of change of its input (even if the change of input is small) ... I can't see why not...
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Right now I am trying to provide for myself a stable job and to set my foot on a firm ground financially. I think that I am living that part of a man's life in which one needs to bear the fruits of his first endeavors. The geopolitical situation in the middle-east and especially Iran is of no help though. Nevertheless, I am always interested in the discussions in the mailing list and try to follow them as much as possible.
On another note, I also have been reading about quantum probability recently. While the subject is certainly out of my reach right now, I think that I have found something interesting that I would like to share with you and Linas and ask if you see any potential there. Before that I would like to give my thanks to Linas for introducing the reading materials on these subjects and to tell you that I would surely look them up. On the subject of Riemannian surfaces, I had a hunch that the subject is important but I lack the math to read the literature. I figured that I need to get a better understanding of vector fields and geometric algebra and I am reading a book called "Geometric Algebra for Computer Science". I would be glad if you could suggest an introductory book on the subject of Riemannian surfaces itself.
The idea is that the output of a classifier is a quantum probability distribution. So a classifier is something like a Dirichlet process but for quantum probability distributions. The output of a k-class classifier is a pure complex antisymmetric k-by-k matrix and using matrix exponential we can map that matrix to a matrix in SU(k).
Hi Linas,
We have the INLP workshop approved as part of AGI-2021 conference so looking forward to hear your keynote there!
https://aigents.github.io/inlp/
Best regards,
-Anton
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