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I wonder if this is what Intel is playing at with their comments about beating NVidia in the ML space within a few years without using GPUs?
No it wasn't, they were talking about ASICs.
Bingo!
What do you mean?

I've had a short discussion with a professor at my university about the practical efficiency of circuits. It seems like some tasks are better solved with algorithms and others with circuits.

I think that a proper mix between turing machines and circuits will be important in the future of AI.

Unfortunately they didn't implement the non-linear part in optical form as of yet, though they do at least model a fairly realistic saturable absorber.
It would be cool if they implemented a way of generating true random numbers by measuring a quantum events as part of the system
Is this addressing the learning phase of the neural network, or just the feed-forward phase?
They trained on a computer model of the optical circuit, and only did the feed-forward step on the real thing. The rationale for that is that real-life models spend much more time (and energy) in inference mode, so that is the step you'd most want to optimize.

I can't help but think it would be really cool to automatically produce a circuit that would output the gradient of the error of the actual NN, so you could optimize that directly.

Very interesting paper, optical circuits have always been interesting option for computing whether it is in the form of plasmonics (surface plasmon + electronics) or linear quantum computing (with similar circuits reported in the manuscript). However, the challenge to translate this into any practical application involves solving a lot of engineering challenges....