Ever since the control bug bit me in my EE undergrad years I am happy to see how useful the knowledge remains. Of course the underlying math of optimization remains general but the direct applications of control theory made it much more appetizing for me to struggle through.
lain98 34 minutes ago [-]
I find myself completely outclassed by mathematicians in my own field.
I tried to learn a little math on the side after my regular software engineer gig but I'm completely outclassed by phd's.
I am unsure of the next course of action or if software will survive another 5 years and how my career will look like in the future. Seems like I am engaged in the ice trade and they are about to invent the refrigerator.
measurablefunc 4 hours ago [-]
It's not clear or obvious why continuous semantics should be applicable on a digital computer. This might seem like nitpicking but it's not, there is a fundamental issue that is always swept under the rug in these kinds of analysis which is about reconciling finitary arithmetic over bit strings & the analytical equations which only work w/ infinite precision over the real or complex numbers as they are usually defined (equivalence classes of cauchy sequences or dedekind cuts).
There are no dedekind cuts or cauchy sequences on digital computers so the fact that the analytical equations map to algorithms at all is very non-obvious.
sfpotter 42 minutes ago [-]
This is what the field of numerical analysis exists for. These details definitely have been treated, but this was done mainly early in the field's history; for example, by people like Wilkinson and Kahan...
jampekka 3 hours ago [-]
Continuous formulations are used with digital computers all the time. Limited precision of floats sometimes causes numerical instability for some algorithms, but usually these are fixable with different (sometimes less efficient) implementations.
Discretizing e.g. time or space is perhaps a bigger issue, but the issues are usually well understood and mitigated by e.g. advanced numerical integration schemes, discrete-continuous formulations or just cranking up the discretization resolution.
Analytical tools for discrete formulations are usually a lot less developed and don't as easily admit closed-form solutions.
phreeza 3 hours ago [-]
Doesn't continuous time basically mean "this is what we expect for sufficiently small time steps"? Very similar to how one would for example take the first order Taylor dynamics and use them for "sufficiently small" perturbations from equilibrium. Is there any other magic to continuous time systems that one would not expect to be solved by sufficiently small time steps?
measurablefunc 2 hours ago [-]
You should look into condition numbers & how that applies to numerical stability of discretized optimization. If you take a continuous formulation & naively discretize you might get lucky & get a convergent & stable implementation but more often than not you will end up w/ subtle bugs & instabilities for ill-conditioned initial conditions.
phreeza 2 hours ago [-]
I understand that much, but it seems like "your naive timestep may need to be smaller than you think or you need to do some extra work" rather than the more fundamental objection from OP?
measurablefunc 2 hours ago [-]
The translation from continuous to discrete is not automatic. There is a missing verification in the linked analysis. The mapping must be verified for stability for the proper class of initial/boundary conditions. Increasing the resolution from 64 bit floats to 128 bit floats doesn't automatically give you a stable discretized optimizer from a continuous formulation.
phyalow 2 hours ago [-]
Or you can just try stuff and see if it works
measurablefunc 2 hours ago [-]
Point still stands, translation from continuous to discrete is not as simple as people think.
phreeza 1 hours ago [-]
Numerical issues totally exist but the reason has nothing to do with the fact that Cauchy sequences don't exist on a computer imo.
heyethan 1 hours ago [-]
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cubefox 1 hours ago [-]
Real numbers mostly appear in calculus (e.g. the chain rule in gradient descent/backpropagation), but "discrete calculus" is then used as an approximation of infinitesimal calculus. It uses "finite differences" rather than derivatives, which doesn't require real numbers:
I am unsure of the next course of action or if software will survive another 5 years and how my career will look like in the future. Seems like I am engaged in the ice trade and they are about to invent the refrigerator.
There are no dedekind cuts or cauchy sequences on digital computers so the fact that the analytical equations map to algorithms at all is very non-obvious.
Discretizing e.g. time or space is perhaps a bigger issue, but the issues are usually well understood and mitigated by e.g. advanced numerical integration schemes, discrete-continuous formulations or just cranking up the discretization resolution.
Analytical tools for discrete formulations are usually a lot less developed and don't as easily admit closed-form solutions.
https://en.wikipedia.org/wiki/Finite_difference
I'm not sure about applications of real numbers outside of calculus, and how to replace them there.