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IDS Lab seminar.
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.
The marginal value of adaptive gradient methods in machine learning
The Marginal Value of Adaptive Gradient
Methods in Machine Learning
Does deep learning really doing some generalization? 2
presentedby Jamie Seol
• 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
• 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
• 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?
• This holds
• [homework: prove it]
• With rewriting by element-wise notation:
• 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]
• 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
• 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
• 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
• So what?
• We pretty much do know well about behaviors of gradient
• 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
• 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?
• Actually, we’ve already discussed about this in the previous
• Anyway, even if the system is underdetermined, SGD always
converges to some unique solution which belongs to span of X
• 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…
• In summary,
• adaptive gradient descent methods
• might be poor
• at generalization
• 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
• All together
• 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
• 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
• Let sign(x) denote a function that maps each component of x to its
• 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
• 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]
• Note that for such a dataset, the only discriminative feature is the
• if y = [1, -1, -1, 1, -1] then X becomes:
• Let and assume b > 0 (p > 1/2)
• Suppose , then
• 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!
• How about non-adaptive method?
• So, when , the solution makes no errors
• 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)
• (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
• and it requires as much time as non-adaptive tuning…
• low training loss, more test error (Adam vs HB)
• Character-level language model
• AdaGrad looks very fast, but indeed, not good
• surprisingly, RMSProp closely trails SGD on test
• well, it is true that non-adaptive methods are slow
• 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
• 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/