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Reviewing basic properties of SGD.

About generalization power of adaptive gradient methods (spoiler: they might be arbitrarily poor!).

The paper suggest such case, with related experiments.

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- 1. IDS Lab The Marginal Value of Adaptive Gradient Methods in Machine Learning Does deep learning really doing some generalization? 2 presentedby Jamie Seol
- 2. IDS Lab Jamie Seol Preface • Toy problem: smooth quadratic strong convex optimization • Let object f be as following, and WLOG suppose A to be a symmetric and nonsingular • why WLOG? symmetric because it’s a quadratic form, and singular curvature (curvature of quadratic function is A) is reducible in quadratic function • moreover, strong convex = positive definite curvature • meaning that all eigenvalues are positive
- 3. IDS Lab Jamie Seol Preface • Note that A was a real symmetric matrix, so by the spectral theorem, A has eigendecomposition with unitary basis • In this simple objective function, we can explicitly compute the optima
- 4. IDS Lab Jamie Seol Preface • We’ll apply a gradient descent! let superscript be an iteration: • Will it converge to the optima? let’s check it out! • We use some tricky trick using change of basis • This new sequence x(k) should converge to 0 • But when?
- 5. IDS Lab Jamie Seol Preface • This holds • [homework: prove it] • With rewriting by element-wise notation:
- 6. IDS Lab Jamie Seol Preface • So, the gradient descent converges only if • for all i • In summary, it converges when • And the optimal is • where 𝜎(A) denote a spectral radius of A, meaning the maximal absolute value among eigenvalues [homework: n = 1 case]
- 7. IDS Lab Jamie Seol Preface (appendix) • Actually, this result is rather obvious • Note that A was a curvature of the objective, and the spectral radius or the largest eigenvalue means "stretching" above A’s principal axis • curvature ← see differential geometry • principal axis ← see linear algebra • So, it is vacuous that the learning rate should be in a safe area regarding the "stretching", which can be done with simple normalization
- 8. IDS Lab Jamie Seol Preface • Similarly, the optimal momentum decay can also be induced, using condition number 𝜅 • condition number of a matrix is ratio between maximal and minimal (absolute) eigenvalues • Therefore, if we can control the boundary of the spectral radius of the objective, then we can approximate the optimal parameters for gradient descent • this is the main idea of the YellowFin optimizer
- 9. IDS Lab Jamie Seol Preface • So what? • We pretty much do know well about behaviors of gradient descent • if the objective is smooth quadratic strong convex.. • but the objectives of deep learning is not nice enough! • We just don’t really know about characteristics of deep learning objective functions yet • requires more research
- 10. IDS Lab Jamie Seol Preface 2 • Here’s a typical linear regression problem • If the number of features d is bigger than the number of samples m, than it is underdetermined system • So it has (possibly infinitely) many solutions • Let’s use stochastic gradient descent (SGD) • which solution will SGD find?
- 11. IDS Lab Jamie Seol Preface 2 • Actually, we’ve already discussed about this in the previous seminar • Anyway, even if the system is underdetermined, SGD always converges to some unique solution which belongs to span of X
- 12. IDS Lab Jamie Seol Preface 2 • Moreover, experiments show that SGD’s solution has small norm • We know that the l2-regularization helps generalization • l2-regularization: keeping parameter’s norm small • So, we can say that the SGD has implicit regularization • but there’s evidence that l2-regularization does not help at all… • see previous seminar presented by me • 잘 되지만 사실 잘 안되고, 그래도 좋은 편이지만 그닥 좋지만은 않다…
- 13. IDS Lab Jamie Seol Introduction • In summary, • adaptive gradient descent methods • might be poor • at generalization
- 14. IDS Lab Jamie Seol Preliminaries • Famous non-adaptive gradient descent methods: • Stochastic Gradient Descent [SGD] • Heavy-Ball [HB] (Polyak, 1964) • Nesterov’s Accelerated Gradient [NAG] (Nesterov, 1983)
- 15. IDS Lab Jamie Seol Preliminaries • Adaptive methods can be summarized as: • AdaGrad (Duchi, 2011) • RMSProp (Tieleman and Hinton, 2012, in coursera!) • Adam (Kingma and Ba, 2015) • In short, these methods adaptively changes learning rate and momentum decay
- 16. IDS Lab Jamie Seol Preliminaries • All together
- 17. IDS Lab Jamie Seol Synopsis • For a system with multiple solution, what solution does an algorithm find and how well does it generalize to unseen data? • Claim: there exists a constructive problem(dataset) in which • non-adaptive methods work well and • finds a solution with good generalization power • adaptive methods work poor • finds a solution with poor generalization power • we even can make this arbitrarily poor, while the non- adaptive solution still working
- 18. IDS Lab Jamie Seol Problem settings • Think of a simple binary least-squares classification problem • When d > n, if there is a optima with loss 0 then there are infinite number of optima • But as shown in preface 2, SGD converges to the unique solution • with known to be the minimum norm solution • which generalizes well • why? becase in here, it’s also the largest margin solution • All other non-adaptive methods also converges to the same
- 19. IDS Lab Jamie Seol Lemma • Let sign(x) denote a function that maps each component of x to its sign • ex) sign([2, -3]) = [1, -1] • If there exists a solution proportional to sign(XTy), this is precisely the unique solution where all adaptive methods converge • quite interesting lemma! • pf) use induction • Note that this solution is just: • mean of positive labeled vectors - mean of negative labeled vectors
- 20. IDS Lab Jamie Seol
- 21. IDS Lab Jamie Seol Funny dataset • Let’s fool adaptive methods • first, assign yi to 1 with probability p > 1/2 • when y = [-1, -1, -1, -1] • when y = [1, 1, 1, 1]
- 22. IDS Lab Jamie Seol Funny dataset • Note that for such a dataset, the only discriminative feature is the first one! • if y = [1, -1, -1, 1, -1] then X becomes:
- 23. IDS Lab Jamie Seol Funny dataset • Let and assume b > 0 (p > 1/2) • Suppose , then
- 24. IDS Lab Jamie Seol Funny dataset • So, holds! • Take a closer look • all first three are 1, and rest is 0 for new data • this solution is bad! • it will classify every new data to positive class!!! • what a horrible generalization!
- 25. IDS Lab Jamie Seol Funny dataset • How about non-adaptive method? • So, when , the solution makes no errors • wow
- 26. IDS Lab Jamie Seol Funny dataset • Think this is too extreme? • Well, even in the real dataset, the following are rather common: • a few frequent feature (j = 2, 3) • some are good indicators, but hard to identify (j = 1) • many other sparse feature (other)
- 27. IDS Lab Jamie Seol Experiments • (authors said that they downloaded models from internet…) • Results in summary: • adaptive makes poor generalization • even if it had lower loss than the non-adaptive ones!!! • adaptive looks fast, but that’s it • adaptive says "no more tuning" but tuning initial values were still significant • and it requires as much time as non-adaptive tuning…
- 28. IDS Lab Jamie Seol Experiments • CIFAR-10 • use non-adaptive
- 29. IDS Lab Jamie Seol Experiments • low training loss, more test error (Adam vs HB)
- 30. IDS Lab Jamie Seol Experiments • Character-level language model • AdaGrad looks very fast, but indeed, not good • surprisingly, RMSProp closely trails SGD on test
- 31. IDS Lab Jamie Seol Experiments • Parsing • well, it is true that non-adaptive methods are slow
- 32. IDS Lab Jamie Seol Conclusion • Adaptive methods are not advantageous for optimization • It might be fast, but poor generalization • then why is Adam so popular? • because it’s popular…? • specially, known to be popular in GAN and Q-learning • these are not exactly optimization problems • we don’t know any nature of objectives in those two yet
- 33. IDS Lab Jamie Seol References • Wilson,Ashia C., et al. "The Marginal Value ofAdaptive Gradient Methods in Machine Learning." arXiv preprint arXiv:1705.08292 (2017). • Zhang, Jian, Ioannis Mitliagkas, and Christopher Ré. "YellowFin and the Art of Momentum Tuning." arXiv preprint arXiv:1706.03471 (2017). • Zhang, Chiyuan, et al. "Understanding deep learning requires rethinking generalization." arXiv preprint arXiv:1611.03530 (2016). • Polyak, Boris T. "Some methods of speeding up the convergence of iteration methods." USSR Computational Mathematics and Mathematical Physics 4.5 (1964): 1-17. • Goh, "Why Momentum Really Works", Distill, 2017. http://doi.org/ 10.23915/distill.00006

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