13.1 Shannon lower bound
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1 ECE598: Iformatio-theoretic methods i high-dimesioal statistics Srig 016 Lecture 13: Shao lower boud, Fao s method Lecturer: Yihog Wu Scribe: Daewo Seo, Mar 8, 016 [Ed Mar 11] I the last class, we leared miimax risk boudig techique by data rocessig iequality of mutual iformatio such that for θ X ˆθ, if Pˆθ θ :E[lθ,ˆθ] R π Iθ; ˆθ Iθ; ˆθ Iθ; X caacity = su Iθ; X 131 P θ Because the exact characterizatio of the LHS is itractable i most cases, we eed a aroriate techique that further lower bouds the LHS, which is called the Shao lower boud Aother techique to get a miimax lower boud, called Fao s method, will be discussed as well 131 Shao lower boud 1311 Shao lower boud Suose that the loss fuctio is rth ower of a arbitrary orm over R, ie, lθ, ˆθ = θ ˆθ r, ad let R π = D The, the LHS ca be writte as if Iθ; ˆθ = if Iθ; ˆθ Pˆθ θ :E[lθ,ˆθ] D = if hθ hθ ˆθ = if hθ hθ ˆθ ˆθ if hθ hθ ˆθ = hθ su hw SLB E W r D where W θ ˆθ ad the very last quatity is called the Shao lower boud To evaluate the suremum term, ay covex otimizatio techique such as Lagrage multilier ca be alied A secial case of the lower boud for Euclidea orm is give by SLB = hθ su hw = hθ h N 0, D E W D P I P = hθ P log πe D P where we used the fact that Gaussia maximizes differetial etroy whe the secod momet is bouded Theorem 131 Shao s Lower Boud Let be a arbitrary orm o R ad r > 0 The { if Iθ; ˆθ Dre r hθ log V Γ 1 + }, Pˆθ θ :E θ ˆθ r D r, 1
2 where V is the volume of the uit radius ball, ie, V volb = vol{x R : x 1} The roof will be give i homework Note: The Shao lower boud is asymtotically tight as D 0 Examle 131 GLM Cosider the -dimesioal -samle GLM, ie, X 1,, X iid N θ, I or equivaletly X N θ, 1 I The the miimax risk with resect to r is R 1 r/ V c r/ Proof Take a rior θ π = N 0, si The the iequality chai 131 is rewritte as log1 + s Iθ, X Iθ; ˆθ if Iθ; ˆθ Pˆθ θ :E[lθ,ˆθ] Rπ SLB = R {V logπes log π re r Γ 1 + } r The, rearragig terms, takig limit s, ad usig the Stirlig s formula we get R π 1 r/ V c r/ R 1 r/ V c r/ 13 Note that for r =, R 1 V /, while the exact boud see Sec 3 is R = E Z for l q orm = I the ext examle, we will see volumes Examle 13 l q -orm Cosider l q -orm, ie, for 1 q 1/q x q = x i q See the volume for several q s i=1 q = Cot d from the revious Note that R = for the quadratic loss The -dimesioal volume of a uit Euclidea ball B is give by V B 1/ = π 1/ Γ 1 + 1/ 1,
3 which follows from the Stirlig s aroximatio, Γ 1 + 1/ / 1/ 1/ 1/ 1/ e e Pluggig i 13 with r =, Hece i this case the SLB is tight R 1 V 1/ = 1 q < Cosider l q orm, where 1 q <, the volume of a uit l q ball is give by [ ] Γ q V B q = Γ 1 + q So usig 13 ad the Stirlig s formula, the miimax boud for a loss fuctio q is give by Aother way to get the same boud is that R /q R 1 E Z q /q Here the roerty that if Z N 0, I, Z q q = Θ P is used q = Recall a uit hyercube i R The, V B =, hece, R 1 the other had, we kow the exact risk, by the SLB O R = 1 E Z log So i this case the SLB is ot tight Here, the equality follows from the fact that if Z N 0, I, Z = Θ P log Note: I the case that we have restrictio o θ such that θ Θ R, where Θ is a covex set with o-emty iterior, the oly thig to be chaged is the SLB art Uer boud by caacity remais uchaged As a examle of uiform rior over some Θ R, caacity SLB = hθ log[ R π ] = log volθ log[ R π ] We get the boud of miimax risk coectig this SLB with caacity formula Also ote that the exact characterizatio of R Θ is oe eve for a covex set Θ 3
4 131 Gaussia width of a covex body K Suose Z N0, I ad a set K is covex ad symmetric Defie the Gaussia width of K [ ] wk E su x, Z x K Lemma 131 Urysoh volk 1/ wk Urysoh s lemma hels us characterize the boud of miimax risk I our case, K = B, the [ ] [ ] wk = E su x, Z x K = E su x 1 x, Z = E Z, which is i fact the exected dual orm of Z From the lemma, we have V 1/ R 1 V / 1 E Z E Z Therefore, 13 Fao s method Recall the iequality chai, if Pˆθ θ :E ˆθ θ R π Iθ; ˆθ Iθ; ˆθ Iθ; X caacity I this sectio, we discuss Fao s method that reduces the LHS to multile hyothesis testig roblem, which is easier to comute The stes are followigs: 1 Discretize Istead of Θ, cosider a discrete subset Θ = {θ 1,, θ } Θ Poits are icked to satisfy θ i θ j for all i j Figure 131 visualizes this discretizatio Reduce to multile hyothesis testig Assume uiform rior such that θ π = uif{θ 1,, θ } ad let f be a quatizer that mas θ Θ to θ i Θ, the closest oit to θ Note that fθ = θ because θ is draw over Θ So by data rocessig iequality for θ X ˆθ fˆθ, Iθ; ˆθ Iθ; fˆθ Note that Iθ; fˆθ is a fuctio of joit robability mass over discrete sace Θ Θ Let s see the error evets {θ fˆθ} Let say the true source is θ = θ k If error haes, it imlies our estimate ˆθ closer to θ j = fˆθ tha θ k for some j I other words, if error haes, ˆθ fˆθ ˆθ θ k 4
5 Figure 131: Discretizatio So due to triagular iequality, the error evet imlies Hece, fˆθ θ k = fˆθ ˆθ + ˆθ θ k fˆθ ˆθ + ˆθ θ k ˆθ θ k, ˆθ θ k P e Prθ k fˆθ Pr ˆθ θ k E ˆθ θ k R π / / = R π if Pˆθ θ :E ˆθ θ R π Iθ; ˆθ if Iθ; ˆθ if Iθ; fˆθ So, we reduce the LHS to a multile hyothesis test roblem where θ, fˆθ are both discrete 3 Aly Fao s iequality Recall the data rocessig iequality for KL divergece by Figure 13 Here our rocessor is 1{θ ˆθ}, ad we ca further lower boud as Iθ; fˆθ = D P θ,fˆθ P θ P fˆθ D BerP e Ber 1 1 = P e log P e P e log 1 P e 1 = hp e + log P e log 1 log + log P e log, P e 1 Iθ; fˆθ + log, log where h is a biary etroy fuctio So fially we reach the boud R R π P e 1 Iθ; fˆθ + log log R 1 Iθ; fˆθ + log log 5
6 Figure 13: Data rocessig kerel for Fao s iequality Note: The situatio of the Fao s iequality i class is that 1 θ uiformly takes M values Markov chai θ X ˆθ holds The, the Fao s iequality says that Iθ; X log + log M P e logm 1 P e 1 log + 1 P e log M, Iθ; X + log log M The Fao s iequality ituitively meas that whe the mutual iformatio is fixed, P e caot be less tha a certai value O the other had, whe P e is fixed, the mutual iformatio must be greater tha a certai value Note: We ca also use the Fao iequality as followig: ad similarly as above, Iθ; X mi Iθ; X, R R π Iθ; X + log P e 1 log R Iθ; X + log 1 log M Note: If the loss fuctio is, mi Iθ; X R 1 Iθ; X + log log M 6
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