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- 1. IDS Lab Understanding deep learning requires rethinking generalization Does deep learning really doing some generalization? presentedby Jamie Seol
- 2. IDS Lab Jamie Seol Motivation • Normally, we measure a generalization by: • generalization error = |training error - test error| • if we overfit, the training error should be low, while test error becomes large = high generalization error! • However, a complex neural network is fragile to be overfitted! • for example, let’s train some human baby by randomly labeled CIFAR-10 dataset • then, give’em some sample in the training set (2nd+ epoch) • they will say "what the…" to any question • because it’s impossible to generalize some kind of abtracted concept! • what about in neural network?
- 3. IDS Lab Jamie Seol CIFAR-10 • This is the CIFAR-10 dataset • The goal of this task is to classify given image into one of 10 classes • CNNs that we know well will solve this rather easily
- 4. IDS Lab Jamie Seol Randomized CIFAR-10 • When we randomize information of CIFAR-10’s training set, the result of accuracy becomes:
- 5. IDS Lab Jamie Seol Randomized CIFAR-10 • This is just nothing more than over-overfit! • What’s the problem than? • neural networks memorized datasets • even if it should have no meaning! • it’s random! raaaaandddddddommm!!! • aaaaarrrrrrrr!!! • it did not generalize some concepts • it just memorized!!!!
- 6. IDS Lab Jamie Seol Randomized CIFAR-10 • Even if you didn’t intend to, neural nets can just memorize thing rather than generalizing! • According to the experiment, • the effective capacity of neural network is sufficient for memorizing the entire data set • randomizing (corrupting) data set makes task harder just by small constant factor compared to the origial task! • Again, even if you didn’t want to!! neural network is fragile to overfit in natural sense!! • "You don’t have to explain the meanings. I’ll just memorize it" - Chatur, from the movie "3 Idiots"
- 7. IDS Lab Jamie Seol Regularization • However, we do know that there are a lot of techniques for regularization, which supports generalizations! • dropout, batch norm, early stopping, weigh decay… • It does seem help, but wait…. • can someone prove that regularizations fundamentally improves generalization? • does this works really really well? really???
- 8. IDS Lab Jamie Seol Regularization • Isn’t data augmentation significantly more important than weight decay? • Even with regulizations, neural networks are good memorizers • Just changing the model increased test accuracy
- 9. IDS Lab Jamie Seol Regularization • Early stopping helps • but not necessarily…
- 10. IDS Lab Jamie Seol Regularization • Well… these techniques seem does helpful, but suspicion remains…
- 11. IDS Lab Jamie Seol Rademacher complexity • By the way, what’s so big deal about memorizing everything? • The following measurement is called Rademacher complexity • Detailed math is omitted here • The thing is, if some model can memorize everything (actually, if the hypothesis have power to fit randomized dataset), then theoritical upper bound of generalization error is just 1 • which is useless!!!! • actually, using regularization scheme lowers the bound, but this is not true in ReLU, and we’ll show that there is some situation that regularization helps nothing
- 12. IDS Lab Jamie Seol Finite-sample expressivity • Remember Universal Approximation Theorem? • finite-sample expressivity theorem is more practical version of it • note that this statement shows that UAT does not guarantees generalization! • Theorem 1: there exists a 2-layer NN by ReLU with 2n+d weights that can represent any function given by n samples in d dimensions • This is not a hard theorem to prove, so let’s do it
- 13. IDS Lab Jamie Seol Lemma 1 • Lemma 1: for b1 < x1 < b2 < … < bn < xn, matrix A = [ReLU(xi - bj)]ij has full rank • Proof: obvious
- 14. IDS Lab Jamie Seol Theorem 1 • Theorem 1: there exists a 2-layer NN by ReLU with 2n+d weights that can represent any function given by n samples in d dimensions • Proof: Note that 2-layered neural network with ReLU can be expressed as • where w, b ∈ ℝn and a ∈ ℝd • for data S = {z1, …, zn} and label y ∈ ℝn where zi ∈ ℝd, WTS yi = NN2(zi) for all i from 1 to n • choose a, b so that xi = ⟨a, zi⟩ meets the condition for Lemma 1 • Then, this becomes y = Aw, while Lemma 1 says that A is invertable • done
- 15. IDS Lab Jamie Seol Finite-sample expressivity • What does it mean? • It means that once you have more than about 2n + d parameters, your model already possesses a willingful power to super-overfit and just to remember everything instead of generalizing some concept, therefore it gains trivial bound for generalization error and is exposed to sudden-death-danger of doing nothing more than a memorizer • long story short: we can’t speak formally about generalization in deep learning yet • a snake’s leg: for deeper network, use intermediate layers to choose splitted interval rather than target, resulting similar O(n + k) parameters required
- 16. IDS Lab Jamie Seol Stochastic Gradient Descent • Let’s think about linear optimization • If we have large d, which is a underdetermined problem, then we can have multiple globla minima • But hey, can we determine which optima gives best generalization? • in non-linear systems, peeking curvature helped • but there’s no such thing as a curvature in linear system!
- 17. IDS Lab Jamie Seol Stochastic Gradient Descent • Funny thing about SGD is, it gives optima for l2 loss for underdetermined system, and known to be a regularizer itself
- 18. IDS Lab Jamie Seol Stochastic Gradient Descent • However… the result shows minimum l2 norm wasn’t always the global optima in sense of generalization • furthermore, it is possible to generate some dataset that minimum l2 norm is not optima! a constructive counter example! • adding l2 regularization to parameters didn’t help a bit (not shown in the table) norm = 220 norm = 390
- 19. IDS Lab Jamie Seol Conclusion • "Be careful whenever you speak 'generalization' in deep learning" • Contributions of this paper: • experimental framework for suspecting suspicious activities of generalization techniques • proof for lack of theoritical boundary of generalization error in deep learning (since it can just memorize it all with small effective capacity) • optimization does not necessarily means generalization • "beware of the light" - Caliban, from the movie "Logan"
- 20. IDS Lab Jamie Seol References • Zhang, Chiyuan, et al. "Understanding deep learning requires rethinking generalization." arXiv preprint arXiv:1611.03530 (2016). • https://www.slideshare.net/JungHoonSeo2/understanding-deep-learning- requires-rethinking-generalization-2017-12 • https://www.slideshare.net/JungHoonSeo2/understanding-deep-learning- requires-rethinking-generalization-2017-2-22

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