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.

## Follow-the-regularized leader

TBD

## Parameter-free

## 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 \[s_k^2 = S_k/(k — 1).\]

## Anomaly detection

See also anomaly detection.

## References

*PMLR*, 126–35.

*arXiv:1703.00573 [Cs]*, March.

*Annual Review of Statistics and Its Application*8 (1): 165–90.

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*Learning Theory*, edited by Nader H. Bshouty and Claudio Gentile, 4539:278–92. Berlin, Heidelberg: Springer Berlin Heidelberg.

*arXiv:1701.00251 [Cs, Stat]*, January.

*Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation - GECCO ’06*, 453. Seattle, Washington, USA: ACM Press.

*arXiv:1612.04111 [Cs, Stat]*, December.

*Journal of the American Statistical Association*69 (348): 859–66.

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*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.

*Proceedings of the Tenth ACM International Conference on Web Search and Data Mining*, 51–60. WSDM ’17. New York, NY, USA: ACM Press.

*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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