## References

Books, Lecture Notes, Expository Articles
• Concentration Inequalities: A Nonasymptotic Theory of Independencei, by S. Boucheron, G. Lugosi and P. Massart, Oxford University Press, 2013.
• Concentration Inequalities and Model Selection, by P. Massart, Springer Lecture Notes in Mathematics, vol 1605, 2007.
• The Concentration of Measure Phenomenon, by M. Ledoux, 2005, AMS.
• Lecture notes on concentration of measure Inequalities, by G. Lugosi, pdf.
• A Probabilistic Theory of Pattern Recognition, by L. Devroye, L. Gyöfi and G. Lugosi, Springer, 1996 pdf.
• Concentration of Measure Inequalities in Information Theory, Communications and Coding, by M. Raginski and I. Sason, Foundations and Trends in Communications and Information Theory, 2014.
• A New Look at Independence -- Special Invited Paper, by M. Talagrand, the Annals of Applied Probability, 24(1),1--34, 1996.
• Concentration of Measure for the Analysis of Randomized Algorithms, by D.P. Dubhashi and A, Panconesi, Cambridge University Press, 2012.
• Concentration by C. McDiarmid. In M. Habib. C. McDriamid, J.Ramirez-Alfonsin and B. Reed, editors, Probabilistic Methods for Algorithmic Discrete Mathematics, 195--248, Springer, 1998.
• R. Vershynin, Introduction to the non-asymptotic analysis of random matrices. In: Compressed Sensing: Theory and Applications, eds. Yonina Eldar and Gitta Kutyniok. Cambridge University Press,
• Measure Concentration, Lecture Notes for Math 710, by Alexander Barvinok, 2005. pdf
• Probability Theory and Combinatorial Optimization, by M. Steele, CBMS Regionel Conference Series in Applied Mathematics, vol. 69, 1997.

Lecture 1, Tue Sep 1
Some applications of concentration inequalities in high-dimensional statistics:
• Theorem 1 in M. J. Wainwright (2009), Sharp thresholds for noisy and high-dimensional recovery of sparsity using $\ell_1$-constrained quadratic programming (Lasso). IEEE Transactions on Information Theory, 55:2183--2202.
• Theorem 5.2 in A simple proof of the restricted isometry property for random matrices R Baraniuk, M Davenport, R DeVore, M Wakin Constructive Approximation 28 (3), 253-263
• For the covariance estimation example see HW1.
Parameter consistency and central limit theorems for models with increasing dimension d (but still d < n):
• Wasserman, L, Kolar, M. and Rinaldo, A. (2014). Berry-Esseen bounds for estimating undirected graphs, Electronic Journal of Statistics, 8(1), 1188-1224.
• Fan, J. and Peng, H. (2004). Nonconcave penalized likelihood with a diverging number of parameters, the Annals of Statistics, 32(3), 928-961.
• Portnoy, S. (1984). Asymptotic Behavior of M-Estimators of p Regression, Parameters when p^2/n is Large. I. Consistency, tha Annals of Statistics, 12(4), 1298--1309.
• Portnoy, S. (1985). Asymptotic Behavior of M Estimators of p Regression Parameters when p^2/n is Large; II. Normal Approximation, the Annals of Statistics, 13(4), 1403-1417.
• Portnoy, S. (1988). Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity, tha Annals of Statistics, 16(1), 356-366.
Some central limit theorem results in increasing dimension (in the second mini we will see more specialized and stronger results).
• Bentkus, V. (2003). On the dependence of the Berry–Esseen bound on dimension, Journal of Statistical Planning and Inference, 113, 385-402.
• Portnoy, S. (1986). On the central limit theorem in R p when $p \rightarrow \infty$, Probability Theory and Related Fields, 73(4), 571-583.

Lecture 2, Thu Sep 3
For those interested in the mathematical theory of the conecntration of measure, these are good references:
• The Concentration of Measure Phenomenon, by M. Ledoux, 2005, AMS.
• Measure Concentration, Lecture Notes for Math 710, by Alexander Barvinok, 2005. pdf
• Concentration of measure, lecture notes by N. Berestycki and R. Nickl pdf

Lecture 3, Tue Sep 8
Most of the material on sub-gaussianity was taken from here:
• Omar Rivasplata, Subgaussian random variables: An expository note, Sections 1, 2 and 3. pdf
• Lecture 6 of the course "Machine learning and appplications", by Z. Harchaui, J. Mairal and J. Salmon, pdf
More comprehensive treatment of sub-gaussian variables and processes (and more) are:
• Metric Characterization of Random Variables and Random Processes, by V. V. Buldygin, AMS, 2000.
• Introduction to the non-asymptotic analysis of random matrices, by R. Vershynin, Chapter 5 of: Compressed Sensing, Theory and Applications. Edited by Y. Eldar and G. Kutyniok. Cambridge University Press, 210–268, 2012. pdf
References for Chernoff bounds for Bernoulli (and their multiplicative forms):
• A guided tour of chernoff bounds, by T. Hagerup and C. R\"{u}b, Information and Processing Letters, 33(6), 305--308, 1990.
• Chapter 4 of the book Probability and Computing: Randomized Algorithms and Probabilistic Analysis, by M. Mitzenmacher and E. Upfal, Cambridge University Press, 2005.
• The Probabilistic Method, 3rd Edition, by N. Alon and J. H. Spencer, Wiley, 2008, Appendix A.1.
Improvement of Hoeffding's ineqaulity by Berend and Kantorivich:
• On the concentration of the missing mass, by D. Berend and A. Kntorovich, Electron. Commun. Probab. 18 (3), 1–7, 2013.
• Section 2.2.4 in Raginski's monograph (see references at the top).
Example of how the relative or multiplicative version of Chernoff bounds will lead to substantial improvements:
• Minimax-optimal classification with dyadic decision trees, by C. Scott and R. Nowak, iEEE Transaction on Information Theory, 52(4), 1335-1353.

Lecture 4, Thu Sep 10
For more information on sub-exponential variables, consult:
• Metric Characterization of Random Variables and Random Processes, by V. V. Buldygin, AMS, 2000.

Lecture 5, Tue Sep 15
Empirical Bernstein Inequality:
• Empirical Bernstein Bounds and Sample-Variance Penalization by: Andreas Maurer, Massimiliano Pontil In COLT (2009)
Concentration of chi-squared variables:
• Lemma 1 in Adaptive estimation of a quadratic functional by model selection, by B. Laurent and P. Massart, Annals of Statistics, 2000, 28(5), 1302-1338.
For classic proofs of Hoeffding, Bennet and Bernstein, see, e.g.,
• Chernoff, Hoeffding’s and Bennett’s Inequalities, a write-up by Jimmy Jin, James Wilson and Andrew Nobel. pdf>
For an example of how Bernstein's inequality is preferable to Hoeffding, see Lemma 13 in
• Minimax Rates for Homology Inference, by S. Balakrishnan, A. Rinaldo, D. Sheehy, A. Singh and L. Wasserman, 2011, AISTATS.

Lecture 6, Thu Sep 17
Some references on JL Lemma and random projections:
• An Elementary Proof of a Theorem of Johnson and Lindenstrauss S. Dasgupta and A. Gupta, Random Structures & Algorithms, 22(1), 60-65, 2008.
• D. Achlioptas, Database friendly random projections, Proc 20th ACM Symp Principles of Database Systems, 2001.
• P. Indyk and R. Motwani, Approximate nearest neighbors: Towards removing the curse of dimensionality, Proc 30th Annu ACM Symp Theory of Computing, Dallas, TX, 1998, pp. 604 – 613.
• The Fast Johnson–Lindenstrauss Transform and Approximate Nearest Neighbors, N. Ailon and B. Chazelle, SIAM J. Comput., 39(1), 302–322, 2009.
• Locality-sensitive hashing scheme based on p-stable distributions, by M. Datar, N. Immorlica, P. Indyk, V. Mirrokni, Proceeding SCG '04 Proceedings of the twentieth annual symposium on Computational geometry, 253-262, 2004.
On embedding of of finite metric spaces (of which the JL Lemma is an example):
• Chapter 15 of Lectures on Discrete Geometry, by G. Matousek, 2002, Springer.
A nice, dense monograoh devoted to uses of random projections:
• The Random Projection Method, by S. Vampala. AMS,, 2004.
A sherpening of the JL Lemma, based on empirical process theory:
• Empirical processes and random projection Journal of Functional Analysis, k. Klartag and S. Mendelson, 2005 225(1), 1 August 2005, Pages 229–245.
JL Lemma and the RIP condition in compressed sensing:
• The Restricted Isometry Property and Its Implications for Compressed Sensing, E. Candes, Comptes Rendus Mathematique, Vol. 346, No. 9-10
• Theorem 5.2 in A simple proof of the restricted isometry property for random matrices R. Baraniuk, M. Davenport, R. DeVore, M Wakin, Constructive Approximation 28 (3), 253-263
The JL Lemma and manifolds:
• Distance Preserving Embeddings for General n-Dimensional Manifolds, N. Verma, JMLR, 14 (2013) 2415-2448.
• Tighter bounds for random projections of manifolds, K. Clarkson. Symposium on Computational Geometry (SoCG), 24:39–49, 2008.
• Random projections of smooth manifolds, R. Baraniuk and M. Wakin, Foundations of Computa- tional Mathematics (FoCM), 9(1):51–77, 2009.
And finally, a nice reading course:

Lecture 7, Thu Sep 22
The DKW inequality with the best constant:
• Massart, P. (1990), x"The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequalityx", The Annals of Probability 18 (3): 1269–1283

Lecture 8, Thu Sep 24
Ale out of town.

Lecture 9, Tue Sep 29
References for martingale methods to get concentration inequalities:
• Colin McDiarmid. Chapter titled Concentration in Probabilistic Methods for Algorithmic Discrete Mathematics Volume 16 of the series Algorithms and Combinatorics pp 195-248.
• Chung F. and L. Lu, Concentration inequalities and martingale inequalities — a survey, 2006, pdf
• Section 2.2 in Raginski's monograph (see references at the top).
Interesting improvements:
• E. Rio, On McDiarmid's concentration inequality Emmanuel Rio, Electronic Communications in Probability, 18(44), 1-11.
• Kontorovich, A. (2014). Concentration in unbounded metric spaces and algorithmic stability. ICML 2014: 28-36
• Sanon, I. (2011). On Refined Versions of the Azuma-Hoeffding Inequality with Applications in Information Theory, pdf

Lecture 10, Thu Oct 1
References the entropy method
• Ledoux, M. (1997). On Talagrand’s deviation inequalities for product measures. ESAIM Probability and Statistics, 1, 63–87.
• Massart, P. (2000a). About the constants in Talagrand’s concentration inequalities for empirical processes. The Annals of Probability, 28, 863–884.
• S. Boucheron, G. Lugosi and P Massart: On concentration of self-bounding functions. Electronic Journal of Probability.14 (2009) 1884-1899.
• Maurer, A. (2006). Concentration inequalities for functions of independent variables. Random Structures & Algorithms, 29, 121–138.
• S. Boucheron and G. Lugosi and P. Massart: Concentration inequalities using the entropy method (2003), Annals of Probability, 31, 1583-1614.
• S Boucheron and G. Lugosi and P. Massart : A sharp concentration inequality with applications Random Structure and Algorithms 16 (2000), 277-292

Lecture 11, Tue Oct 6
References for Max's lectture:
• Arratia, Goldstein, Gordon. Two moments suffice for Poisson approximations: the Chen-Stein method, 1989
• Arratia, Goldstein, Gordon. Poisson Approximation and the Chen-Stein Method, Statistical Science, 1990 Ledoux, M. (1997). On Talagrand’s deviation inequalities for product measures. ESAIM Probability and Statistics, 1, 63–87.

Lecture 12, Thu Oct 8
For today's lecture, the main references are:
• S. Boucheron and G. Lugosi and P. Massart: Concentration inequalities using the entropy method (2003), Annals of Probability, 31, 1583-1614.
• Maurer, A. (2006). Concentration inequalities for functions of independent variables. Random Structures & Algorithms, 29, 121–138.
• A. Maurer (2012). Thermodynamics and concentration, Bernoulli, 18(2), 434-454.

Lecture 13, Tue Oct 13
Same references as the previous lecture.

Lecture 13, Thu Oct 15
The basic references for matrix concentration inequalities are:
• Tropp, J. (2012). User-friendly tail bounds for sums of random matrices, Found. Comput. Math., Vol. 12, num. 4, pp. 389-434, 2012.
• Tropp, J. (2015). An Introduction to Matrix Concentration Inequalities, Found. Trends Mach. Learning, Vol. 8, num. 1-2, pp. 1-230
and references therein.

Lecture 14, Tue Oct 20
Same references as the last time.

Lecture 15, Tue Oct 22
• Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58 (301): 13–30.
• Hoeffding, W. (1948). A Class of Statistics with Asymptotically Normal Distribution, The Annals of Mathematical Statistics, 3(3), 293-325.