Online learning



An online learning perspective gives bounds on the regret: the gap between in performance between online estimation and the optimal estimator when we have access to the entire data.

A lot of things are sort-of online learning; stochastic gradient descent, for example, is closely related. However, if you meet someone who claims to study “online learning” they usually mean to emphasis particular things. Frequently seen in the context of bandit problems; connection TBD.

Hazan’s Introduction to online convex optimization looks fresh.

Mirror descent

TBD

Follow-the-regularized leader

TBD

Covariance

Learning covariance online is a much more basic application than the other fancy things considered here, but I guess it still fits. John D Cook:

This better way of computing variance goes back to a 1962 paper by B. P. Welford and is presented in Donald Knuth’s Art of Computer Programming, Vol 2, page 232, 3rd edition. […]

  • Initialize \(M_1 = x_1\) and \(S_1 = 0.\)
  • For subsequent \(x\)s, use the recurrence formulas \[M_k = M_{k-1} + (x_k — M_{k-1})/k\] \[S_k = S_{k-1} + (x_k — M_{k-1})(x_k — M_k).\]
  • For \(2 \leq k \leq n\), the \(k\)th estimate of the variance is \[s_k^2 = S_k/(k — 1).\]

References

Abernethy, Jacob, Peter L Bartlett, and Elad Hazan. 2011. “Blackwell Approachability and No-Regret Learning Are Equivalent.” In, 20.
Allen-Zhu, Zeyuan, Yuanzhi Li, Aarti Singh, and Yining Wang. 2017. Near-Optimal Design of Experiments via Regret Minimization.” In PMLR, 126–35.
Arora, Sanjeev, Rong Ge, Yingyu Liang, Tengyu Ma, and Yi Zhang. 2017. Generalization and Equilibrium in Generative Adversarial Nets (GANs).” arXiv:1703.00573 [Cs], March.
Cesa-Bianchi, Nicolò, and Francesco Orabona. 2021. Online Learning Algorithms.” Annual Review of Statistics and Its Application 8 (1): 165–90.
Chan, Tony F., Gene H. Golub, and Randall J. Leveque. 1983. Algorithms for Computing the Sample Variance: Analysis and Recommendations.” The American Statistician 37 (3): 242–47.
Dasgupta, Sanjoy, and Daniel Hsu. 2007. On-Line Estimation with the Multivariate Gaussian Distribution.” In Learning Theory, edited by Nader H. Bshouty and Claudio Gentile, 4539:278–92. Berlin, Heidelberg: Springer Berlin Heidelberg.
Feng, Jiashi, Huan Xu, and Shie Mannor. 2017. Outlier Robust Online Learning.” arXiv:1701.00251 [Cs, Stat], January.
Igel, Christian, Thorsten Suttorp, and Nikolaus Hansen. 2006. A Computational Efficient Covariance Matrix Update and a (1+1)-CMA for Evolution Strategies.” In Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation - GECCO ’06, 453. Seattle, Washington, USA: ACM Press.
Koppel, Alec, Garrett Warnell, Ethan Stump, and Alejandro Ribeiro. 2016. Parsimonious Online Learning with Kernels via Sparse Projections in Function Space.” arXiv:1612.04111 [Cs, Stat], December.
Ling, Robert F. 1974. Comparison of Several Algorithms for Computing Sample Means and Variances.” Journal of the American Statistical Association 69 (348): 859–66.
Lyu, Hanbaek, Deanna Needell, and Laura Balzano. 2020. Online Matrix Factorization for Markovian Data and Applications to Network Dictionary Learning.” Journal of Machine Learning Research 21 (251): 1–49.
Orabona, Francesco, David Pal, Orabona Com, and Yahoo-Inc Com. n.d. “Open Problem: Parameter-Free and Scale-Free Online Algorithms,” 6.
Vervoort, Marco R. 1996. Blackwell Games.” In Statistics, Probability and Game Theory: Papers in Honor of David Blackwell, edited by T.S. Ferguson, L.S. Shapley, and J.B. MacQueen, 369–90. Institute of Mathematical Statistics.
Zarezade, Ali, Utkarsh Upadhyay, Hamid R. Rabiee, and Manuel Gomez-Rodriguez. 2017. RedQueen: An Online Algorithm for Smart Broadcasting in Social Networks.” In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining, 51–60. WSDM ’17. New York, NY, USA: ACM Press.
Zinkevich, Martin. 2003. Online Convex Programming and Generalized Infinitesimal Gradient Ascent.” In Proceedings of the Twentieth International Conference on International Conference on Machine Learning, 928–35. ICML’03. Washington, DC, USA: AAAI Press.

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