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The lottery ticket hypothesis is IMHO the single most interesting finding for deep learning. It explains why does deep learning works (vs shallow neural nets),
by stared 7y ago
The lottery ticket hypothesis is IMHO the single most interesting finding for deep learning. It explains why does deep learning works (vs shallow neural nets), why is initial over-parametrization is often useful, why deeper is often better than shallow, etc.
I recommend for an overview:
- the original paper "The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks", https://arxiv.org/abs/1803.03635 https://arxiv.org/abs/1803.03635
- "Deconstructing Lottery Tickets: Zeros, Signs, and the Supermask" https://eng.uber.com/deconstructing-lottery-tickets/ https://eng.uber.com/deconstructing-lottery-tickets/ showing that if we remove "non-winning tickets" before the training, the trained network still works well
- jl2718 7y agoThis is really not a new idea for anybody that has studied numerical optimization or done a lot of much simpler linear modeling. It has to do with the gradients, which are basically random steps in high-dimensional spaces unless you align with a ‘large’ eigenvector of the Hessian. This explanation didn’t go over well at all with the ML cargo cultists until someone gave it a name. Interesting how things work.
- stared 7y agoCould you point to concrete papers, showing that empirically? (Many things were rediscovered over and over, sure.) > random steps in high-dimensional spaces unless you align with a ‘large’ eigenvector of the Hessian In this case, the main observation is different than this one, and has much more to do with sparsity (and an exponentially growing number of connections with each layer).
- lsorber 7y agoNocedal has some papers in this direction.
- nograpes 7y agoDo you have any good papers or textbooks where I can learn about this from a numerical optimization perspective?
- adampk 7y agohttps://www.amazon.com/Numerical-Optimization-Operations-Financial-Engineering-ebook/dp/B014P9ICXI https://www.amazon.com/Numerical-Optimization-Operations-Fin... seems to be "the" numerical optimization textbook
- sdenton4 7y agoIt's also fun to consider that most neural network architectures are mostly piecewise linear: relu(Mx + b) for standard feedforward or convolutional layers... So one would expect a lot of knowledge and folklore from (piecewise) linear optimization to carry over nicely. IIRC, ResNet was a good example of that: It's more efficient to learn relu(x + Mx + b) than relu(Mx + b) One difficulty, though, is that proofs generally /don't/ carry over, and IME a lot of 'classical' methods constrain themselves to provable operations... So at times it can be hard to tell the difference between 'useful folklore' and 'tweak that makes the proofs work.' I've been delving into sparse coding literature from about ten years ago, and there's a lot of this kind of difficulty... Interestingly, the best sparse coding architectures ended up being very very similar to shallow neural networks. There's a nice 'deep ksvd denoising' paper from a few months back which improves handily on the older sparse coding architectures by bringing in a smattering of ideas from the DNN age; the rant at the end is makes the case for building these 'blended' architectures to get the best of both worlds. https://arxiv.org/abs/1909.13164 https://arxiv.org/abs/1909.13164 I tend to think that the DNN architectures beat the provable-domain architectures for Good Reasons, but the USEFUL response is to build awesome blended architectures that maybe go a long way towards bringing down model sizes and complexity, while increasing explainability.
- thesz 7y agoIf I am remember correctly, residual layer has form of x+nonlin(Mx+b), not one you provided. I also found (in my "learn DL" experiments) that for ReLU the x - relu(Mx + b) version works better (trains faster and achieves better accuracy).
- sdenton4 7y agoCorrect; the idea is to take Identity + Tweak, so the Relu should hit the 'tweak.'
- srean 7y agoHey sdenton did you work on or publish that harmonic analysis of correlations that you had mentioned long time ago. I am still quite curious about that. If polishing it for ICML etc is too much of a downer, you can upload it to arxiv. I would love to read that.
- moultano 7y agoI don't see any connection between what you've written and the lottery ticket hypothesis. Could you elaborate? (In particular, the core of the lottery ticket hypothesis is that layers make the nets combinatorial, and so I don't see how anything from simple linear modelling should transfer.)
- jl2718 7y agoRELU training is basically the same as active sets. You call it nonlinearity. I call it a constrained sub-problem. The point is that it is always linear for the active variables. If you’re talking only about gradients, you’re not doing anything non-linear.
- moultano 7y agoWhat does that have to do with the lottery ticket hypothesis?
- jl2718 7y agoBecause the opposite hypothesis, that 'winning tickets' can emerge from 'bad tickets' using stochastic gradient descent, implies that randomness doesn't push you back into the null space of the generalization curvature, which you can empirically observe that it almost always will, except for the same eigenvectors which tend to reinforce where there is strong alignment already, iteration after iteration. If you train a lot of small problems you can watch this happen just by looking at the angles between each update. Maybe that's the wrong explanation; I don't know. The practitioner take-away is that some initializations are just lucky/unlucky and you learn this quickly if you start a lot of small problems from scratch. Try stochastic BFGS and the frustration forces some inquiry. The paper is very experimental, which is great, but the conjecture itself is pointing roughly at some feature of SGD (i.e. "SGD seeks out and trains a subset of well-initialized weights.") without offering much direct explanation, for which I'm offering a hand-wavy viewpoint that also comes more from experience/frustration trying to optimize in ill-conditioned local curvatures. There may be a combinatorial way to look at 'lottery tickets', and that may be more productive than looking at it as eigenbasis pursuit, but it seems to me that it certainly might have something to do with that. Either way, the terms are irrelevant, and if it is true now, it was true before it got a name.
- vecter 7y agoCan you summarize and explain those things (why deep learning works, why over-parameterization is useful, and why deeper is better than shallower) assuming an undergraduate-level understanding of mathematics and computer science?
- stared 7y agoDid you try reading these papers? If you want to start from the simpler one, go for the Uber one first. Though, in my understanding, the original paper should be accessible to math/cs undergrads.
- csense 7y agoNeural networks are basically graphs with weighted edges. As you increase number of nodes N in a graph, the number of edges grows at most quadratically, but the number of possible subgraphs grows exponentially. So even if you use totally random edge weights, with high probability some of those O(2^N) subgraphs will be pretty good. OP paper is basically providing formal mathematical definitions for these loose notions, then chasing down a bunch of epsilons and deltas to prove that, in fact, good subgraphs actually do always exist (with high probability for large enough N). From a practical standpoint, the main citation's is pretty informative: - "What's Hidden in a Randomly Weighted Neural Network?" by Ramanujan et al, https://arxiv.org/abs/1911.13299 https://arxiv.org/abs/1911.13299 This paper actually gives an algorithm which empirically does a decent job at finding a "good" subgraph.
- mamp 7y agoI agree. I found Jonathan Frankle’s interview on the NLP highlights podcast to be a great explanation of the findings, and provided some really good intuitions for how deep learning behaves. https://podcasts.apple.com/us/podcast/101-the-lottery-ticket-hypothesis-with-jonathan-frankle/id1235937471?i=1000462500035 https://podcasts.apple.com/us/podcast/101-the-lottery-ticket...