Welcome to the probability inequality mines!

When something in your process (measurement, estimation) means that you can be pretty sure that a whole bunch of your stuff is damn likely to be somewhere in particular.

This is basic workhorse stuff in univariate probability, and turns out to be yet more essential in multivariate matrix probability, as seen in matrix factorisation, compressive sensing, PAC-bounds and suchlike.

## Background

Overviews include

- this super simple intro to chaining and controlling maxima by Thomas Lumley
- Dasgupta, Asymptotic Theory of Statistics and Probability (Dasg08) is very easy, and despite its name introduces some nice basic non-asymptotic inequalities
- Raginsky and Sason, Concentration of Measure Inequalities in Information Theory, Communications and Coding (RaSa12)
- Tropp, An Introduction to Matrix Concentration Inequalities (Trop15) high-dimensional data! free!
- Boucheron, Bousquet & Lugosi, Concentration inequalities (BoBl04a) (Clear and brisk but missing some newer stuff)
- Massart, Concentration inequalities and model section (Mass07). Clear, and focussed, but very quick and further, depressingly, by being applied it also demonstrates the limitations of these techniques. Mass00 is an earlier draft.
- Boucheron, Lugosi & Massart, Concentration inequalities: a nonasymptotic theory of independence (BoLM13). Haven’t read it yet.
- Lugosi’s Concentration-of-measure Lecture notes:

The inequalities discussed in these notes bound tail probabilities of general functions of independent random variables.

The taxonomy is interesting:

Several methods have been known to prove such inequalities, including martingale methods (see Milman and Schechtman [1] and the surveys of McDiarmid [2, 3]), information-theoretic methods (see Alhswede, Ga ́cs, and Ko ̈rner [4], Marton [5, 6, 7], Dembo [8], Massart [9] and Rio [10]), Talagrand’s induction method [11, 12, 13] (see also Luczak and McDiarmid [14], McDiarmid [15] and Panchenko [16, 17, 18]), the decoupling method surveyed by de la Pen ̃a and Gin ́e [19], and the so-called “entropy method”, based on logarithmic Sobolev inequalities, developed by Ledoux [20, 21], see also Bobkov and Ledoux [22], Massart [23], Rio [10], Klein [24], Boucheron, Lugosi, and Mas- sart [25, 26], Bousquet [27, 28], and Boucheron, Bousquet, Lugosi, and Massart [29].

(actioned in his Combinatorial statistics notes)

Foundational but impenetrable things I won’t read right now: Talagrand’s opus that is commonly credited with kicking off the modern fad especially with the chaining method. (Tala95)

## Finite sample bounds

These are everywhere in statistics.
Special attention will be given here to finite-sample inequalities.
Asymptotic normality is so last season.
These days we care about finite sample performance, and asymptotic results don’t help us there.
Apparently I can construct useful bounds using concentration inequalities?
One suggested keyword to disambiguate: *Ahlswede-Winterfeld bounds*?

## Basic inequalities

the classics

### Markov

TBD

### Chebychev

TBD

### Hoeffding

TBD

### Chernoff

TBD

### Kolmogorov

TBD.

### Gaussian

For the Gaussian distribution. Filed there.

### Martingale type

TBD.

### Khintchine

Let us copy from wikipedia:

Heuristically: if we pick \(N\) complex numbers \(x_1,\dots,x_N \in\mathbb{C}\)\(, and add them together, each multiplied by jointly independent random signs $\pm 1\), then the expected value of the sum’s magnitude is close to \(\sqrt{|x_1|^{2}+ \cdots + |x_N|^{2}}\).

Let :math:`{varepsilon_n}_{n=1}^N`

i.i.d. random variables
with \(P(\varepsilon_n=\pm1)=\frac12\) for \(n=1,\ldots, N\),
i.e., a sequence with Rademacher distribution.
Let :math:`0<p<infty`

and let :math:`x_1,ldots,x_Nin mathbb{C}`

. Then

for some constants \(A_p,B_p>0 $. It is a simple matter to see that $A_p = 1\) when \(p \ge 2\), and \(B_p = 1\) when \(0 < p \le 2\).

## Empirical process theory

Large deviation inequalities, empirical process inequalities, Talagrand chaining method. Berry-Esseen bound.

## Matrix Chernoff bounds

Nikhil Srivastava’s Discrepancy, Graphs, and the Kadison-Singer Problem has an interesting example of bounds via discrepancy theory (and only indirectly probability). Gros11 is also readable, and gives quantum-mechanical results (i.e. the matrices are complex-valued).

Trop15 summarises:

In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic.

## To read

## Refs

- DaHV12: (2012) A concentration theorem for projections.
*ArXiv Preprint ArXiv:1206.6813*. - Tala96: (1996) A new look at independence.
*The Annals of Probability*, 24(1), 1–34. - MaMi11: (2011) A non asymptotic penalized criterion for Gaussian mixture model selection.
*ESAIM: Probability and Statistics*, 15, 41–68. DOI - Trop15: (2015) An Introduction to Matrix Concentration Inequalities.
*ArXiv:1501.01571 [Cs, Math, Stat]*. - Dasg08: (2008)
*Asymptotic Theory of Statistics and Probability*. New York: Springer New York - HoPr02: (2002) Concentration and deviation inequalities in infinite dimensions via covariance representations.
*Bernoulli*, 8(6), 697–720. - BoBL04: (2004) Concentration inequalities. In Advanced Lectures in Machine Learning.
- BoLM13: (2013)
*Concentration inequalities: a nonasymptotic theory of independence*. Oxford: Oxford University Press - Mass07: (2007)
*Concentration inequalities and model selection: Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003*. Berlin ; New York: Springer-Verlag - Krol16: (2016) Concentration inequalities for Poisson point processes with application to adaptive intensity estimation.
*ArXiv:1612.07901 [Math, Stat]*. - BoLM03: (2003) Concentration inequalities using the entropy method. , 31(3), 1583–1614. DOI
- Tala95: (1995) Concentration of measure and isoperimetric inequalities in product spaces.
*Publications Mathématiques de L’IHÉS*, 81(1), 73–205. DOI - RaSa12: (2012) Concentration of Measure Inequalities in Information Theory, Communications and Coding.
*Foundations and Trends in Communications and Information Theory*. - WuZh16: (2016) Distribution-dependent concentration inequalities for tighter generalization bounds.
*ArXiv:1607.05506 [Stat]*. - CaRe09: (2009) Exact matrix completion via convex optimization.
*Foundations of Computational Mathematics*, 9(6), 717–772. DOI - Dasg00: (2000) Experiments with Random Projection. In Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (pp. 143–151). San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.
- Geer95: (1995) Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes.
*The Annals of Statistics*, 23(5), 1779–1801. DOI - KuMo16: (2016) Generalization Bounds for Non-stationary Mixing Processes. In Machine Learning Journal.
- KuMo14: (2014) Generalization Bounds for Time Series Prediction with Non-stationary Processes. In Algorithmic Learning Theory (pp. 260–274). Bled, Slovenia: Springer International Publishing DOI
- BoBL04: (2004) Introduction to Statistical Learning Theory. In Advanced Lectures on Machine Learning (pp. 169–207). Springer Berlin Heidelberg
- HaRR15: (2015) Lasso and probabilistic inequalities for multivariate point processes.
*Bernoulli*, 21(1), 83–143. DOI - KuMo15: (2015) Learning Theory and Algorithms for Forecasting Non-Stationary Time Series. In Advances in Neural Information Processing Systems (pp. 541–549). Curran Associates, Inc.
- BeHK12: (2012) Minimum KL-divergence on complements of \(L_1\) balls.
*ArXiv:1206.6544 [Cs, Math]*. - LeGe14: (2014) New concentration inequalities for suprema of empirical processes.
*Bernoulli*, 20(4), 2020–2038. DOI - Geer02: (2002) On Hoeffdoing’s inequality for dependent random variables. In Empirical Process Techniques for Dependent Data. Birkhhäuser
- DeHW11: (2011) On the concentration properties of Interacting particle processes.
*Foundations and Trends® in Machine Learning*, 3(3–4), 225–389. DOI - BePo05: (2005) Optimal Inequalities in Probability Theory: A Convex Optimization Approach.
*SIAM Journal on Optimization*, 15(3), 780–804. DOI - Kolt11: (2011)
*Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems*. Springer Berlin Heidelberg DOI - GLFB10: (2010) Quantum state tomography via compressed sensing.
*Physical Review Letters*, 105(15). DOI - FGLE12: (2012) Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators.
*New Journal of Physics*, 14(9), 095022. DOI - LaGG16: (2016) Random projections of random manifolds.
*ArXiv:1607.04331 [Cs, q-Bio, Stat]*. - Gros11: (2011) Recovering Low-Rank Matrices From Few Coefficients in Any Basis.
*IEEE Transactions on Information Theory*, 57(3), 1548–1566. DOI - Houd02: (2002) Remarks on deviation inequalities for functions of infinitely divisible random vectors.
*The Annals of Probability*, 30(3), 1223–1237. DOI - BaBM99: (1999) Risk bounds for model selection via penalization.
*Probability Theory and Related Fields*, 113(3), 301–413. - CaRT06: (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information.
*IEEE Transactions on Information Theory*, 52(2), 489–509. DOI - Kenn15: (2015) Semiparametric theory and empirical processes in causal inference.
*ArXiv Preprint ArXiv:1510.04740*. - Bach13: (2013) Sharp analysis of low-rank kernel matrix approximations. In COLT (Vol. 30, pp. 185–209).
- DSBN13: (2013) Sketching Sparse Matrices.
*ArXiv:1303.6544 [Cs, Math]*. - Mass00: (2000) Some applications of concentration inequalities to statistics. In Annales de la Faculté des sciences de Toulouse: Mathématiques (Vol. 9, pp. 245–303).
- Horn79: (1979) Some inequalities for the expectation of a product of functions of a random variable and for the multivariate distribution function at a random point.
*Biometrical Journal*, 21(3), 243–245. DOI - ReRo07: (2007) Some non asymptotic tail estimates for Hawkes processes.
*Bulletin of the Belgian Mathematical Society - Simon Stevin*, 13(5), 883–896. - Geer14: (2014) Statistical Theory for High-Dimensional Models.
*ArXiv:1409.8557 [Math, Stat]*. - BüGe11: (2011)
*Statistics for High-Dimensional Data: Methods, Theory and Applications*. Heidelberg ; New York: Springer - Ligg10: (2010) Stochastic models for large interacting systems and related correlation inequalities.
*Proceedings of the National Academy of Sciences of the United States of America*, 107(38), 16413–16419. DOI - GeLe11: (2011) The Lasso, correlated design, and improved oracle inequalities.
*ArXiv:1107.0189 [Stat]*. - AuNe11: (2011) The multiplicative property characterizes \(\ell_p\) and \(L_p\) norms.
*Confluentes Mathematici*, 03(04), 637–647. DOI - BeLT17: (2017) Towards the study of least squares estimators with convex penalty.
*ArXiv:1701.09120 [Math, Stat]*. - GiNi09: (2009) Uniform limit theorems for wavelet density estimators.
*The Annals of Probability*, 37(4), 1605–1646. DOI - RaRe09: (2009) Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning. In Advances in neural information processing systems (pp. 1313–1320). Curran Associates, Inc.