# PAC-Bayesian Inequalities for Martingales

@article{Seldin2012PACBayesianIF,
title={PAC-Bayesian Inequalities for Martingales},
author={Yevgeny Seldin and François Laviolette and Nicol{\o} Cesa-Bianchi and John Shawe-Taylor and Peter Auer},
journal={IEEE Transactions on Information Theory},
year={2012},
volume={58},
pages={7086-7093}
}`
• Published 31 October 2011
• Mathematics
• IEEE Transactions on Information Theory
We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. Our results extend the PAC-Bayesian (probably approximately correct) analysis in learning theory from the i.i.d. setting to martingales opening the way for its application to importance weighted sampling, reinforcement learning, and other interactive learning domains, as well as many other domains in…

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## References

SHOWING 1-10 OF 28 REFERENCES

### A PAC analysis of a Bayesian estimator

• Computer Science
COLT '97
• 1997
The paper uses the techniques to give the first PAC style analysis of a Bayesian inspired estimator of generalisation, the size of a ball which can be placed in the consistent region of parameter space, and the resulting bounds are independent of the complexity of the function class though they depend linearly on the dimensionality of the parameter space.

### Bayesian Gaussian process models : PAC-Bayesian generalisation error bounds and sparse approximations

The tractability and usefulness of simple greedy forward selection with information-theoretic criteria previously used in active learning is demonstrated and generic schemes for automatic model selection with many (hyper)parameters are developed.

### PAC-Bayesian Analysis of Contextual Bandits

• Computer Science
NIPS
• 2011
The analysis allows to provide the algorithm large amount of side information, let the algorithm to decide which side information is relevant for the task, and penalize the algorithm only for the side information that it is using de facto.

### Empirical Bernstein Bounds and Sample-Variance Penalization

• Mathematics, Computer Science
COLT
• 2009
Improved constants for data dependent and variance sensitive confidence bounds are given, called empirical Bernstein bounds, and extended to hold uniformly over classes of functions whose growth function is polynomial in the sample size n, and sample variance penalization is considered.

### PAC-Bayesian Generalisation Error Bounds for Gaussian Process Classification

• M. Seeger
• Computer Science
J. Mach. Learn. Res.
• 2002
By applying the PAC-Bayesian theorem of McAllester (1999a), this paper proves distribution-free generalisation error bounds for a wide range of approximate Bayesian GP classification techniques, giving a strong learning-theoretical justification for the use of these techniques.

### PAC-Bayesian Stochastic Model Selection

A PAC-Bayesian performance guarantee for stochastic model selection that is superior to analogous guarantees for deterministic model selection and shown that the posterior optimizing the performance guarantee is a Gibbs distribution.

### Distribution-Dependent PAC-Bayes Priors

• Mathematics, Computer Science
ALT
• 2010
The idea that the PAC-Bayes prior can be informed by the data-generating distribution is developed, sharp bounds for an existing framework are proved, and insights into function class complexity are developed in this model and means of controlling it with new algorithms are suggested.

### WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES

1. Let be a probability space and,be an increasing family of sub o'-fields of(we put(c) Let (xn)n=1, 2, •c be a sequence of bounded martingale differences on , that is,xn(ƒÖ) is bounded almost surely

### Some PAC-Bayesian Theorems

The PAC-Bayesian theorems given here apply to an arbitrary prior measure on an arbitrary concept space and provide an alternative to the use of VC dimension in proving PAC bounds for parameterized concepts.