Notes here iff I need 'em.
Second order optimisation that does not require the hessian matrix to be given explicitly.
Andre Gibiansky's example for coders.
LiSSA attempts to make 2nd order gradient descent methods practical (AgBH16):
linear time stochastic second order algorithm that achieves linear convergence for typical problems in machine learning while still maintaining run-times theoretically comparable to state-of-the-art first order algorithms. This relies heavily on the special structure of the optimization problem that allows our unbiased hessian estimator to be implemented efficiently, using only vector-vector products.
David McAllester observes:
Since can be computed efficiently whenever we can run backpropagation, the conditions under which the LiSSA algorithm can be run are actually much more general than the paper suggests. Backpropagation can be run on essentially any natural loss function.
What is Francis Bach's new baby? Finite sample guarantees for certain Newton-like treatments of SGD for certain problems: (BaMo11, BaMo13)
Beyond stochastic gradient descent for large-scale machine learning
Many machine learning and signal processing problems are traditionally cast as convex optimization problems. A common difficulty in solving these problems is the size of the data, where there are many observations ('large n') and each of these is large ('large p'). In this setting, online algorithms such as stochastic gradient descent which pass over the data only once, are usually preferred over batch algorithms, which require multiple passes over the data. In this talk, I will show how the smoothness of loss functions may be used to design novel algorithms with improved behavior, both in theory and practice: in the ideal infinite-data setting, an efficient novel Newton-based stochastic approximation algorithm leads to a convergence rate of O(1/n) without strong convexity assumptions, while in the practical finite-data setting, an appropriate combination of batch and online algorithms leads to unexpected behaviors, such as a linear convergence rate for strongly convex problems, with an iteration cost similar to stochastic gradient descent. (joint work with Nicolas Le Roux, Eric Moulines and Mark Schmidt).
Secant conditions and update designs
See eg. vendenberghe
the BFGS update satisfies the secant condition i.e.
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- BMAS14: Nicolas Boumal, Bamdev Mishra, P.-A. Absil, Rodolphe Sepulchre (2014) Manopt, a Matlab Toolbox for Optimization on Manifolds. Journal of Machine Learning Research, 15, 1455–1459.
- MaNT04: K Madsen, H.B. Nielsen, O. Tingleff (2004) Methods for non-linear least squares problems
- BaMo11: Francis Bach, Eric Moulines (2011) Non-Asymptotic Analysis of Stochastic Approximation Algorithms for Machine Learning. In Advances in Neural Information Processing Systems (NIPS) (p. ). Spain
- BaMo13: Francis R. Bach, Eric Moulines (2013) Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n). In arXiv:1306.2119 [cs, math, stat] (pp. 773–781).
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- AgBH16: Naman Agarwal, Brian Bullins, Elad Hazan (2016) Second Order Stochastic Optimization in Linear Time. ArXiv:1602.03943 [Cs, Stat].
- MaSu12: James Martens, Ilya Sutskever (2012) Training deep and recurrent networks with hessian-free optimization. In Neural networks: Tricks of the trade (pp. 479–535). Springer