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>What do you mean by “gravitational theory” apart from “correctly accounts for predicting the gravitational effects that we observe”? F = G (m1 * m2) / r^2 Ne

>What do you mean by “gravitational theory” apart from “correctly accounts for predicting the gravitational effects that we observe”? F = G (m1 * m2) / r^2 Newton's theory of gravitation identified the relevant variables and their relationships. It had no junk variables or parameters. You can do algebraic transformations on it and use the formula outside of the original context. It was not fully correct, but it made exact predictions. People were able to notice the discrepancies it had with orbits of some celestial bodies, which helped in identifying general theory of relativity. All of this would have been lost of the formula was a state vector of a large neural network.

mlthoughts2018 8y ago
> “Newton's theory of gravitation identified the relevant variables and their relationships.” No, it introduced a useful low-dimensional fiction that approximately works, with high accuracy at coarse scales, and doesn’t account for relativistic or quantum effects (and so, in an absolute sense, is only an approximation, not an interpretation of what is real). Besides, what you glibly type out as a few characters of a formula unpackages into a deeply hard-to-explain theory in natural language. It’s vastly less interpretable than simpler (and less correct) models because it introduces much more complexity of the meaning and physical concepts behind the formula terms. Just as quantum or relativistic extensions add even more complexity and counter-intuitive concepts and become even harder to interpret. > “All of this would have been lost of the formula was a state vector of a large neural network.” Why? I see no reason why that would be true. If there is a very low-dimensional manifold in parameter space that accounts for all the predictive power, that would be a very easy thing to diagnose from a neural network model. Lots of things have been discovered that way, like graph centrality metrics coming from eigen decompositions, style transfer, glove vectors, eigenfaces. Getting the high-dimensional representation that yields accurate predictions is often step 1, great scientific progress and often a huge leap forward. Then step 2 is often asking if there is some natural low-dimensional compression of the parameters that is approximately as good, and if so, does it correspond to any known quantities. Though sometimes this isn’t possible or even desirable, such as with non-parametric models, and yet such models can still represent “interpretable” scientific progress in the sense of allowing us to predict how states of affairs will develop when we previously couldn’t and solving inference goals.
- psyklic 8y ago
  Science (and especially mathematics) is all about increasing human understanding. A neural network that makes correct predictions would be very interesting to both engineers and scientists. From a scientific perspective, the fact something can be modeled well using a neural network would provide interesting constraints on the system, which may eventually yield breakthroughs in understanding the original system. However, the neural network's parameterized equation by itself would never be accepted as good science, since its complexity would hardly contribute anything to our understanding.
- romaniv 8y ago
  > If there is a very low-dimensional manifold in parameter space that accounts for all the predictive power, that would be a very easy thing to diagnose from a neural network model. People wouldn't waste time researching neural network visualization tools, coming up with adversarial examples and developing alternative (more explainable) machine learning models if that was the case.

mlthoughts2018 8y ago

> “Newton's theory of gravitation identified the relevant variables and their relationships.” No, it introduced a useful low-dimensional fiction that approximately works, with high accuracy at coarse scales, and doesn’t account for relativistic or quantum effects (and so, in an absolute sense, is only an approximation, not an interpretation of what is real). Besides, what you glibly type out as a few characters of a formula unpackages into a deeply hard-to-explain theory in natural language. It’s vastly less interpretable than simpler (and less correct) models because it introduces much more complexity of the meaning and physical concepts behind the formula terms. Just as quantum or relativistic extensions add even more complexity and counter-intuitive concepts and become even harder to interpret. > “All of this would have been lost of the formula was a state vector of a large neural network.” Why? I see no reason why that would be true. If there is a very low-dimensional manifold in parameter space that accounts for all the predictive power, that would be a very easy thing to diagnose from a neural network model. Lots of things have been discovered that way, like graph centrality metrics coming from eigen decompositions, style transfer, glove vectors, eigenfaces. Getting the high-dimensional representation that yields accurate predictions is often step 1, great scientific progress and often a huge leap forward. Then step 2 is often asking if there is some natural low-dimensional compression of the parameters that is approximately as good, and if so, does it correspond to any known quantities. Though sometimes this isn’t possible or even desirable, such as with non-parametric models, and yet such models can still represent “interpretable” scientific progress in the sense of allowing us to predict how states of affairs will develop when we previously couldn’t and solving inference goals.

psyklic 8y ago
Science (and especially mathematics) is all about increasing human understanding. A neural network that makes correct predictions would be very interesting to both engineers and scientists. From a scientific perspective, the fact something can be modeled well using a neural network would provide interesting constraints on the system, which may eventually yield breakthroughs in understanding the original system. However, the neural network's parameterized equation by itself would never be accepted as good science, since its complexity would hardly contribute anything to our understanding.
romaniv 8y ago
> If there is a very low-dimensional manifold in parameter space that accounts for all the predictive power, that would be a very easy thing to diagnose from a neural network model. People wouldn't waste time researching neural network visualization tools, coming up with adversarial examples and developing alternative (more explainable) machine learning models if that was the case.