Comment by LPisGood
1 day ago
It should be noted that optimization of a convex bounded lipschitz function is exactly what most modern statistical learning (AI) models are based on.
1 day ago
It should be noted that optimization of a convex bounded lipschitz function is exactly what most modern statistical learning (AI) models are based on.
Very confused by this comment. The older (poorer) parts of the ML literature focus on models with convex and (gradient-)Lipschitz objectives, but that's not representative of reality, not even close. Modern objectives for AI models are famously nonconvex (catastrophically, from the point of view of classical optimisation theory), and that's where the interesting research is.
I'd push back on this. Most of the core optimization techniques (eg, ADAM, stochastic gradient descent) are straight out of the convex optimization literature. Generally you need to use optimizers that work well on convex objectives because near minimizers, functions tend to be convex. (Proof by contradiction: a non-convex point has a strict descent direction.)
The fact that neural networks are highly nonconvex has encouraged a lot of research, but it's more of the kind aimed at resolving tension: these methods are probably good for convex functions, why do they continue to work for nonconvex problems, and are there tweaks we can make to improve them in that setting? It's not a lot of de novo theory; more standing on the shoulders of giants, etc etc.
No, I have to push back as well, sorry. It takes a very long time to get to the "near-minimizer" stage when training a neural network, and in practice, you never get there (see neural scaling law regimes). What you are saying is the viewpoint from 6-7 years ago. Things have changed.
The reasons why optimizers work well for neural networks in their highly nonconvex landscapes has absolutely nothing to do with their performance in convex landscapes. If that were true, everyone would be using Newton-CG. These optimizers were born in the convex optimization literature as a consequence of the genetic optimization nature of incremental publication (and because that was all we had), but their modern study is through the lens of implicit regularization (their preferences for certain minima) and their stepwise vs. continuous rates for feature learning in multilayer models.
This is completely new theory by the way, and requires painful reinvention of the field. It does not stand on the shoulders of convex optimization. The nonconvex setting is assuredly not a perturbation of the convex setting, and those that do continue to work on deep learning optimization from the convex optimization perspective are well behind the times.
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The optimizers are lifted from convex optimization, but the point above was that they are applied to highly non-convex problems. They work for finding local minima, but a lot of the deeper literature does not translate (e.g. the conjecture being discussed in this post).
ADAM does not work on simple convex problems [1].
[1] refers to [2], which shows that ADAM is not as efficient as gradient descent with line search on some problems, including neural networks.
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Another intuition is that near a minimum you can Taylor expand the function and show that the higher order coefficients (past the square) are negligible.
I'd say it's going to be very hard to come up with a method that works on general nonconvex functions while not working on convex functions
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What do you mean by this? A neural network hypothesis space is not typically strictly convex or a lipschitz function.