Lecture 4: September 12

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1 36-755: Advanced Statstcal Theory Fall 016 Lecture 4: September 1 Lecturer: Alessandro Rnaldo Scrbe: Xao Hu Ta Note: LaTeX template courtesy of UC Berkeley EECS dept. Dsclamer: These notes have not been subjected to the usual scrutny reserved for formal publcatons. They may be dstrbuted outsde ths class only wth the permsson of the Instructor. 4.1 Last tme Sub-exponental random varables The class of sub-exponental varables s larger than sub-gaussan, and captures varables wth longer tals. For example the χ dstrbuton has a left tal that s Gaussan-lke, but a longer rght tal Bernsten s nequalty We can get better bounds usng Bernsten s nequalty. Hoeffdng s s sharp only f the varance s maxmal. For example, f X E[X] b a.e. and σ = V[X], we get t P[ X E[X] t] e (σ +bt) usng Bernsten s, and P[ X E[X] t] e t b usng Hoeffdng s. If t σ, Bernsten gves sub-gaussan tals wth parameter σ, as opposed to the parameter b usng Hoeffdng s. Snce σ = E[(X E[X]) ] b, we get an mprovement n some range of values of t. In general we use Hoeffdng s because of ts smplcty, unless we need more refned bounds. We should use Bernsten s nequalty especally f V[X] ce[x], and n the lterature ths s sometmes referred to as Bernsten s condton (note that ths s dfferent from the Bernsten s condton ntroduced n the prevous lecture). For an example of Bernsten gvng better bounds than Hoeffdng, refer to Theorem 7.1 n [GKKW0]. Remark If X SG(σ ), then X SE(ν, α), where ν = α = 16σ. Ths s proven n Homework Problem

2 4- Lecture 4: September 1 4. Applcatons 4..1 Maxma Expectaton Let X 1,..., X n be random varables, not necessarly ndependent, such that log(e[e X ]) ψ(), where [0, b), 0 < b. Then log n + ψ() E[max X ] nf. [0,b) Proof: e E[max X] E[e max X ] by Jensen s nequalty = E[max e X ] E[e X ] =1 ne ψ() by assumpton. Takng logs, we have E[max X ] log n+ψ(). We then pck to mnmze the RHS. Example If ψ() = σ (sub-gaussanty), then E[max X ] log n + σ σ log n, where the second nequalty s obtaned by notng that log n s decreasng n and σ s ncreasng, so we balance them by settng log n = σ log n and solvng for. Ths gves = σ. Notce that the bound s on the order of log n, so the maxmum does not grow very fast even f we have many observatons. The followng s one way of obtanng a general result n the case of non-sub-gaussan random varables. Such results can be used, for example, for sub-exponental random varables. Ths result s for reference and the proof s not stated here, but can be found n Secton.5 of [BLM13]. In general, for any u > 0, nf (0,b) { } u + ψ() = nf{t 0 : ψ (t) > u}, where ψ (t) = sup (0,b) {t ψ()}. Usng ths, f ψ() = σ (1 b), where (0, 1 b ), b > 0, then because ψ 1 (u) = σ u + bu, u > 0. E[max X ] σ log n + b log n Example If X 1,..., X n χ p, then E[max X p] p log n + log n.

3 Lecture 4: September Probablty If X 1,..., X n are random varables and X SG(σ ), P[max n X > t] = P[ {X t}] =1 P[X t] =1 ne t σ = e t σ +log n, where the frst nequalty s obtaned usng the unon bound and the second usng the sub-gaussan property of X. If we want a t such that P[max X > t ] δ (0, 1), then settng the RHS of the nequalty above to δ, we get t = σ (log 1 + log n). δ On the other hand, f we consder the X s ndvdually, we get P[X > σ log 1 δ ] δ, and the two only dffer by the log n term. The dmenson of the problem only enters the bound logarthmcally, whch means that dealng wth the maxmum s only almost as dffcult as dealng wth ndvdual random varables. In partcular f we take δ = 1 n, we get t = 4σ log n, where the addtonal log n term only affects the constant. Remark In the above dervaton, the X s do not need to be ndependent. If we want to use ndependence we can do so usng de Morgan s law nstead of the unon bound, but n ths case the exponental decay s so strong that there s not much of a dfference between the two. 4.. Quadratc Forms If X s a random vector of length d, and A s a d x d symmetrc matrx, X T AX s known as a quadratc form n X, and E[X T AX] = tr(aσ) + µ T Aµ, where µ = E[X] and Σ = V[X]. Now, we assume that A s a d x d symmetrc postve defnte matrx, µ = 0, and (wthout loss of generalty 1 ) Σ = I (.e. X N d (0, I d )). We are nterested n the concentraton behavor of the quadratc form X T AX. Now, we have X T AX = X T ΓΛΓ T X d = Z T ΓZ d = Z, where s the th egenvalue of A. =1 1 If X does not have covarance matrx I, we can standardze t by pre-multplyng by Σ 1 : X T AX = X T Σ 1 Σ 1 AΣ 1 Σ 1 X d = Z T BZ, where B = Σ 1 AΣ 1 and Z N(0, I).

4 4-4 Lecture 4: September 1 The frst equalty s obtaned usng the spectral theorem, whch tells us that f A s symmetrc postve defnte, we can wrte A = ΓΛΓ T, where Λ s a dagonal matrx wth the egenvalues of A, and Γ s an orthogonal matrx. The second equalty uses the property of rotatonal nvarance of standard normal varables: f a standard normal random varable s pre-multpled by an orthogonal matrx, the result s stll standard normal,.e. Γ T X = d Z. In D, we can thnk of ths property as rotatng a contour plot of a bvarate standard normal densty, where after rotaton the result s stll standard normal. Now, we note that Z χ 1, so we have reduced a potentally complcated quadratc form to a weghted sum of χ 1 random varables. Hence we look at concentraton bounds for W = d =1 (Y 1), where Y χ 1. Then, t > 0, P[W t + t] e t and P[W t] e t, where = = A F, the Frobenus norm, and = max = A op, the operator norm. The operator norm s also equal to sup {x: x =1} x T Ax. For detals on the dervaton see Example.1 n [BLM13] and Lemma 1 n [LM00]. The followng result s an extenson to sub-gaussan random varables, and s known as the Hanson-Wrght nequalty. The proof s n [RV13] and Homework Problem 5. If X 1,..., X d are ndependent and X SG(σ ), where c 1 and c depend on σ. { c P[ X T AX E[X T c mn 1 t AX] t] e } c, t A F A op, 4.3 Bounded Dfference Inequalty The bounded dfference nequalty allows us to get bounds on a functon of ndependent random varables, where the functon could be more complcated than just a sum. Let Z = f(x 1,..., X n ), where X 1,..., X n are ndependent. Let Y 0 = E[Z] and for k = 1,..., n let Y k = E[Z X 1,..., X k ]. In partcular, Y n = Z. Then, by telescopng, Z E[Z] = Y n Y 0 = (Y k Y k 1 ) = k=1 D k, k=1 where D k = Y k Y k 1. We have expressed Z, a possbly complcated functon of the X s, as a sum of k random varables. However, these D k s are not ndependent, so we wll need several more tools. Remark The orderng of X s s not mportant! Martngales Let (Ω, F) be a probablty space. F 0 = {φ, Ω} s the trval σ-feld. A fltraton s a sequence F 0 F 1 F... of sub-σ-felds of F.

5 Lecture 4: September The sequence {Y k } k=1,,... s adapted to the fltraton f Y k s F k -measurable k. Ths sequence s a martngale f E[ Y k ] < and E[Y k+1 F k ] = Y k k Doob Constructon Let Z = f(x 1,..., X n ) be ntegrable, and F k = σ(x 1,..., X k ) for k = 0,..., n. Then Y k = E[Z F k ] s a martngale. Ths s known as the Doob martngale or Levy martngale. If (Y k, F k ) k=0,1,... s a martngale, then (D k, F k ) k=1,,..., where D k = Y k Y k 1, s a martngale dfference sequence. It s adapted to the fltraton, and E[D k ] = 0 k Concentraton bounds for martngale dfference sequences Theorem 4.1 Let (D k, F k ) k=1,,... be a martngale dfference such that E[e D k F k 1 ] e ν k < 1, α k where ν k, α k > 0. Then 1. n SE(ν, α ), where ν = n k=1 ν k and α = max k α k.. P[ n k=1 D e t ν f 0 t ν k t] α. e t α otherwse Proof: E[e n ] = E[E[e n F n 1 ]] = E[e n 1 E[e Dn F n 1 ]] E[e n 1 ]e ν n, f < 1 α n e n k=1 ν k, f < 1 max k α k, where the second equalty s because e n 1 s F n 1 -measurable, and the last nequalty s obtaned by teratng ths procedure. Pont follows drectly from pont 1. Remark Ths result s the same as for ndependent sub-exponental varables. References [GKKW0] László Györf, Mchael Kohler, Adam Krzyżak and Harro Walk, A Dstrbuton- Free Theory of Nonparametrc Regresson, Sprnger, 00.

6 4-6 Lecture 4: September 1 [BLM13] [LM00] [RV13] S. Boucheron, G. Lugos and P. Massart, Concentraton Inequaltes: A Nonasymptotc Theory of Independence, Oxford Unversty Press, 013. B. Laurent and P. Massart, Adaptve estmaton of a quadratc functonal by model selecton, Annals of Statstcs, 000, 8(5), M. Rudelson and R. Vershynn, Hanson-Wrght nequalty and sub-gaussan concentraton, Electron. Commun. Probab., 013, 18(8), 1 9.

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