The marginal value of adaptive gradient methods in machine learning
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The document presents findings that adaptive gradient methods may generalize poorly compared to non-adaptive methods. It constructs a dataset where adaptive methods converge to a solution with poor generalization, while non-adaptive methods still find the optimal solution. Experiments on real datasets show adaptive methods achieving lower training loss but higher test error than non-adaptive counterparts. The document thus claims adaptive methods are not advantageous for optimization problems and their popularity may stem from applications like GANs and reinforcement learning where the objective function is unknown.
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The marginal value of adaptive gradient methods in machine learning
- 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
- 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