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The Marginal Value of Adaptive Gradient Methods in Machine Learning

Does deep learning really doing some generalization? 2presented by Jamie Seol

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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

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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

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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?

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Preface

• This holds• [homework: prove it]

• With rewriting by element-wise notation:

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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]

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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

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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

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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

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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?

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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

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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…

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Jamie Seol

Introduction• In summary,

• adaptive gradient descent methods• might be poor

• at generalization

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Preliminaries• Famous non-adaptive gradient descent methods:

• Stochastic Gradient Descent [SGD]

• Heavy-Ball [HB] (Polyak, 1964)

• Nesterov’s Accelerated Gradient [NAG] (Nesterov, 1983)

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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

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Preliminaries• All together

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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

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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

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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

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Jamie Seol

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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]

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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:

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Funny dataset• Let and assume b > 0 (p > 1/2)• Suppose , then

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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!

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Funny dataset• How about non-adaptive method?

• So, when , the solution makes no errors• wow

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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)

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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…

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Experiments• CIFAR-10

• use non-adaptive

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Experiments• low training loss, more test error (Adam vs HB)

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Experiments• Character-level language model

• AdaGrad looks very fast, but indeed, not good• surprisingly, RMSProp closely trails SGD on test

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Experiments• Parsing

• well, it is true that non-adaptive methods are slow

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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

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References• Wilson, Ashia C., et al. "The Marginal Value of Adaptive 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