εύʔεਪఆ֓؍ɿϞσϧɾཧ࿦ɾԠ༻ † ླ໦ɹେ࣊ †Tokyo Institute of Technology Department of Mathematical and Computing Sciences 2014 ೥ 9 ݄ 15 ೔ ౷ܭ࿈߹େձ@౦ژେֶ 1 / 56

Outline 1 εύʔεਪఆͷϞσϧ 2 ͍Ζ͍Ζͳεύʔεਖ਼ଇԽ 3 εύʔεਪఆͷཧ࿦ n ≫ p ͷཧ࿦ n ≪ p ͷཧ࿦ 4 ߴ࣍ݩઢܗճؼͷݕఆ 5 εύʔεਪఆͷ࠷దԽख๏ 2 / 56

ߴ࣍ݩσʔλͰͷ໰୊ҙࣝ ήϊϜσʔλ ۚ༥σʔλ ڠௐϑΟϧλϦϯά ίϯϐϡʔλϏδϣϯ Ի੠ೝࣝ ࣍ݩ d = 10000 ͷ࣌ɼαϯϓϧ਺ n = 1000 Ͱਪఆ͕Ͱ͖Δ͔ʁ ͲͷΑ͏ͳ৚͕݅͋Ε͹ਪఆ͕Մೳ͔ʁ ԿΒ͔ͷ௿࣍ݩੑ (εύʔεੑ) Λར༻ɽ 3 / 56

ྺ࢙: εύʔεਪఆͷख๏ͱཧ࿦ 1992 Donoho and Johnstone Wavelet shrinkage (Soft-thresholding) 1996 Tibshirani Lasso ͷఏҊ 2000 Knight and Fu Lasso ͷ઴ۙ෼෍ (n ≫ p) 2006 Candes and Tao, ѹॖηϯγϯά Donoho (੍ݶ౳௕ੑɼ׬શ෮ݩɼp ≫ n) 2009 Bickel et al., Zhang ੍ݶݻ༗஋৚݅ (Lasso ͷϦεΫධՁ, p ≫ n) 2013 van de Geer et al., εύʔεਪఆʹ͓͚Δݕఆ Lockhart et al. (p ≫ n) ͜ΕҎલʹ΋൓ࣹ๏஍਒୳ࠪ΍ը૾ࡶԻআڈɼ๨٫෇͖ߏ଄ֶशʹ L1 ਖ਼ଇԽ͸࢖ΘΕͯ ͍ͨɽৄ͘͠͸ాதར޾ (2010) Λࢀর. 4 / 56

Outline 1 εύʔεਪఆͷϞσϧ 2 ͍Ζ͍Ζͳεύʔεਖ਼ଇԽ 3 εύʔεਪఆͷཧ࿦ n ≫ p ͷཧ࿦ n ≪ p ͷཧ࿦ 4 ߴ࣍ݩઢܗճؼͷݕఆ 5 εύʔεਪఆͷ࠷దԽख๏ 5 / 56

ߴ࣍ݩσʔλղੳ αϯϓϧ਺ ≪ ࣍ݩ όΠΦΠϯϑΥ ςΩετσʔλ ը૾σʔλ 6 / 56

ߴ࣍ݩσʔλղੳ αϯϓϧ਺ ≪ ࣍ݩ ʷ ݹయత਺ཧ౷ܭֶɿαϯϓϧ਺ ≫ ࣍ݩ όΠΦΠϯϑΥ ςΩετσʔλ ը૾σʔλ 6 / 56

εύʔεਪఆ αϯϓϧ਺ ≪ ࣍ݩ ແବͳ৘ใΛ੾Γམͱ͢ˠεύʔεੑ Lasso ਪఆྔ R. Tsibshirani (1996). Regression shrinkage and selection via the lasso. J. Royal. Statist. Soc B., Vol. 58, No. 1, pages 267–288. Ҿ༻਺ɿ10185 (2014 ೥ 5 ݄ 25 ೔) 7 / 56

ม਺બ୒ͷ໰୊ʢઢܗճؼʣ σβΠϯߦྻ X = (Xij ) ∈ Rn×p. p (࣍ݩ) ≫ n (αϯϓϧ਺). ਅͷϕΫτϧ β∗ ∈ Rp: ඇθϩཁૉͷݸ਺͕͔͔ͨͩ d ݸ (εύʔε). Ϟσϧ : Y = Xβ∗ + ξ. (Y , X) ͔Β β∗ Λਪఆɽ ࣮࣭ਪఆ͠ͳͯ͘͸͍͚ͳ͍ม਺ͷ਺͸ d ݸˠม਺બ୒ɽ 8 / 56

ม਺બ୒ͷ໰୊ʢઢܗճؼʣ σβΠϯߦྻ X = (Xij ) ∈ Rn×p. p (࣍ݩ) ≫ n (αϯϓϧ਺). ਅͷϕΫτϧ β∗ ∈ Rp: ඇθϩཁૉͷݸ਺͕͔͔ͨͩ d ݸ (εύʔε). Ϟσϧ : Y = Xβ∗ + ξ. (Y , X) ͔Β β∗ Λਪఆɽ ࣮࣭ਪఆ͠ͳͯ͘͸͍͚ͳ͍ม਺ͷ਺͸ d ݸˠม਺બ୒ɽ Mallows’ Cp , AIC: ˆ βMC = argmin β∈Rp ∥Y − Xβ∥2 + 2σ2∥β∥0 ͨͩ͠ ∥β∥0 = |{j | βj ̸= 0}|. ˠ 2p ݸͷީิΛ୳ࡧɽNP-ࠔ೉ɽ 8 / 56

Lasso ਪఆྔ Mallows’ Cp ࠷খԽ: ˆ βMC = argmin β∈Rp ∥Y − Xβ∥2 + 2σ2∥β∥0. ໰୊఺: ∥β∥0 ͸ತؔ਺Ͱ͸ͳ͍ɽ࿈ଓͰ΋ͳ͍ɽ୔ࢁͷہॴ࠷దղɽ ˠ ತؔ਺Ͱۙࣅɽ Lasso [L1 ਖ਼ଇԽ] ˆ βLasso = argmin β∈Rp ∥Y − Xβ∥2 + λ∥β∥1 ͨͩ͠ ∥β∥1 = ∑ p j=1 |βj |. ˠ ತ࠷దԽʂ L1 ϊϧϜ͸ L0 ϊϧϜͷ [−1, 1]p ʹ͓ ͚Δತแ (Լ͔Β཈͑Δ࠷େͷತؔ਺) L1 ϊϧϜ͸ཁૉ਺ؔ਺ͷ Lov´ asz ֦ு 9 / 56

Lasso ਪఆྔͷεύʔεੑ p = n, X = I ͷ৔߹ɽ ˆ βLasso = argmin β∈Rp 1 2 ∥Y − β∥2 + C∥β∥1 ⇒ ˆ βLasso,i = argmin b∈R 1 2 (yi − b)2 + C|b| = { sign(yi )(yi − C) (|yi | > C) 0 (|yi | ≤ C). খ͍͞γάφϧ͸ 0 ʹॖখ͞ΕΔˠεύʔεʂ 10 / 56

Lasso ਪఆྔͷεύʔεੑ ˆ β = arg min β∈Rp 1 n ∥Xβ − Y ∥2 2 + λn p ∑ j=1 |βj |. 11 / 56

εύʔεੑͷԸܙ ˆ β = arg min β∈Rp 1 n ∥Xβ − Y ∥2 2 + λn p ∑ j=1 |βj |. Theorem (Lasso ͷऩଋϨʔτ) ͋Δ৚݅ͷ΋ͱɼఆ਺ C ͕ଘࡏͯ͠ߴ͍֬཰Ͱ࣍ͷෆ౳͕ࣜ੒Γཱͭɿ ∥ˆ β − β∗∥2 2 ≤ C dlog(p) n . ˞࣍ݩ͕ߴͯ͘΋ɼ͔͔ͨͩ log(p) Ͱ͔͠ޮ͍ͯ͜ͳ͍ɽ࣮࣭తͳ࣍ݩ d ͕ࢧ഑తɽ ʢ ʮ͋Δ৚݅ʯʹ͍ͭͯ͸ޙͰৄࡉΛઆ໌ʣ 12 / 56

Outline 1 εύʔεਪఆͷϞσϧ 2 ͍Ζ͍Ζͳεύʔεਖ਼ଇԽ 3 εύʔεਪఆͷཧ࿦ n ≫ p ͷཧ࿦ n ≪ p ͷཧ࿦ 4 ߴ࣍ݩઢܗճؼͷݕఆ 5 εύʔεਪఆͷ࠷దԽख๏ 13 / 56

Lasso ΛҰൠԽ Lasso: min β∈Rp 1 n n ∑ i=1 (yi − x⊤ i β)2 + ∥β∥1 ਖ਼ଇԽ߲ . 14 / 56

Lasso ΛҰൠԽ Lasso: min β∈Rp 1 n n ∑ i=1 (yi − x⊤ i β)2 + ∥β∥1 ਖ਼ଇԽ߲ . ҰൠԽͨ͠εύʔεਖ਼ଇԽਪఆ๏: min w∈Rp 1 n n ∑ i=1 ℓ(zi , β) + ψ(β). L1 ਖ਼ଇԽ߲Ҏ֎ʹͲͷΑ͏ͳਖ਼ଇԽ߲͕༗༻Ͱ͋Ζ͏͔ʁ 14 / 56

L1 ਖ਼ଇԽʹΑͬͯεύʔεʹͳΔཧ༝ɿ ࠲ඪ࣠ʹԊͬͯ ઑ͍ͬͯΔɽ ਖ਼ଇԽ߲ͷ ઑΓํ Λ޻෉͢Δ͜ͱͰ༷ʑͳεύʔεੑ͕ಘΒΕΔɽ 15 / 56

άϧʔϓਖ਼ଇԽ C ∑ g∈G ∥βg ∥ ॏෳͳ͠ ॏෳ͋Γ άϧʔϓ಺͢΂ͯͷม਺͕ಉ࣌ʹ 0 ʹͳΓ΍͍͢ɽ ΑΓੵۃతʹεύʔεʹͰ͖Δɽ Ԡ༻ྫɿήϊϜϫΠυ૬ؔղੳ 16 / 56

άϧʔϓਖ਼ଇԽͷԠ༻ྫ ϚϧνλεΫֶश Lounici et al. (2009) T ݸͷλεΫͰಉ࣌ʹਪఆ: y(t) i ≈ x(t)⊤ i β(t) (i = 1, . . . , n(t), t = 1, . . . , T). min β(t) T ∑ t=1 n(t) ∑ i=1 (yi − x(t)⊤ i β(t))2 + C p ∑ k=1 ∥(β(1) k , . . . , β(T) k )∥ άϧʔϓਖ਼ଇԽ . β(1) β(2) β(T) *URXS *URXS *URXS ؞؞؞ ؞؞؞ λεΫؒڞ௨Ͱඇθϩͳม਺Λબ୒ 17 / 56

άϧʔϓਖ਼ଇԽͷԠ༻ྫ ϚϧνλεΫֶश Lounici et al. (2009) T ݸͷλεΫͰಉ࣌ʹਪఆ: y(t) i ≈ x(t)⊤ i β(t) (i = 1, . . . , n(t), t = 1, . . . , T). min β(t) T ∑ t=1 n(t) ∑ i=1 (yi − x(t)⊤ i β(t))2 + C p ∑ k=1 ∥(β(1) k , . . . , β(T) k )∥ άϧʔϓਖ਼ଇԽ . β(1) β(2) β(T) *URXS *URXS *URXS ؞؞؞ ؞؞؞ λεΫؒڞ௨Ͱඇθϩͳม਺Λબ୒ 17 / 56

τϨʔεϊϧϜਖ਼ଇԽ W : M × N ߦྻɽ ∥W ∥Tr = Tr[(WW ⊤)1 2 ] = min{M,N} ∑ j=1 σj (W ) σj (W ) ͸ W ͷ j ൪໨ͷಛҟ஋ (ඇෛͱ͢Δ)ɽ ಛҟ஋ͷ࿨ = ಛҟ஋΁ͷ L1 ਖ਼ଇԽ ˠ ಛҟ஋͕εύʔε ಛҟ஋͕εύʔε = ௿ϥϯΫ 18 / 56

ྫ: ਪનγεςϜ өը A өը B өը C · · · өը X Ϣʔβ 1 4 8 * · · · 2 Ϣʔβ 2 2 * 2 · · · * Ϣʔβ 3 2 4 * · · · * . . . (e.g., Srebro et al. (2005), NetFlix Bennett and Lanning (2007)) 19 / 56

ྫ: ਪનγεςϜ ϥϯΫ 1 ͱԾఆ͢Δ өը A өը B өը C · · · өը X Ϣʔβ 1 4 8 4 · · · 2 Ϣʔβ 2 2 4 2 · · · 1 Ϣʔβ 3 2 4 2 · · · 1 . . . (e.g., Srebro et al. (2005), NetFlix Bennett and Lanning (2007)) 19 / 56

ྫ: ਪનγεςϜ W*= N M өը Ϣʔβ ˠ ௿ϥϯΫߦྻิ׬: ௿ϥϯΫߦྻͷ Rademacher Complexity: Srebro et al. (2005). Compressed sensing: Cand` es and Tao (2009), Cand` es and Recht (2009). 20 / 56

ྫ: ॖখϥϯΫճؼ ॖখϥϯΫճؼ (Anderson, 1951, Burket, 1964, Izenman, 1975) ϚϧνλεΫֶश (Argyriou et al., 2008) ॖখϥϯΫճؼ = W* n Y X N M N W + W ∗ ͸ ௿ϥϯΫ. 21 / 56

εύʔεڞ෼ࢄબ୒ xk ∼ N(0, Σ) (i.i.d., Σ ∈ Rp×p), Σ = 1 n ∑ n k=1 xkx⊤ k . ˆ S = argmin S:൒ਖ਼ఆରশ { − log(det(S)) + Tr[SΣ] + λ p ∑ i,j=1 |Si,j | } . (Meinshausen and B uhlmann, 2006, Yuan and Lin, 2007, Banerjee et al., 2008) Σ ͷٯߦྻ S Λਪఆɽ Si,j = 0 ⇔ X(i) , X(j) ͕৚݅෇͖ಠཱɽ Ψ΢γΞϯάϥϑΟΧϧϞσϧ͕ತ࠷దԽͰਪఆͰ͖Δɽ 22 / 56

(ҰൠԽ) Fused Lasso ψ(β) = C ∑ (i,j)∈E |βi − βj |. (Tibshirani et al. (2005), Jacob et al. (2009)) Fused lasso ʹΑΔҨ఻ࢠσʔλղੳ (Tibshirani and Taylor ‘11) TV σϊΠδϯά (Chambolle ‘04) 23 / 56

ඇತਖ਼ଇԽ SCAD (Smoothly Clipped Absolute Deviation) (Fan and Li, 2001) MCP (Minimax Concave Penalty) (Zhang, 2010) Lq ਖ਼ଇԽ (q < 1), Bridge ਖ਼ଇԽ (Frank and Friedman, 1993) ΑΓεύʔεͳղɽͦͷ୅ΘΓ࠷దԽ͸೉͘͠ͳΔɽ 24 / 56

ͦͷଞ L1 ਖ਼ଇԽͷ֦ு Adaptive Lasso (Zou, 2006) ͋ΔҰகਪఆྔ ˜ β ͕͋Δͱͯ͠ɼͦΕΛར༻ɽ ψ(β) = C p ∑ j=1 |βj | |˜ βj |γ Lasso ΑΓ΋খ͍͞όΠΞε (઴ۙෆภ)ɽ ΦϥΫϧϓϩύςΟɽ εύʔεՃ๏Ϟσϧ (Hastie and Tibshirani, 1999, Ravikumar et al., 2009) f (x) = ∑ p j=1 fj (xj ) ͳΔඇઢܗؔ਺Λਪఆɽ fj ∈ Hj (Hj : ࠶ੜ֩ώϧϕϧτۭؒ) ͱ͢Δɽ ψ(f ) = C p ∑ j=1 ∥fj ∥Hj Group Lasso ͷҰൠԽɽ Multiple Kernel Learning ͱ΋ݺ͹ΕΔɽ 25 / 56

Outline 1 εύʔεਪఆͷϞσϧ 2 ͍Ζ͍Ζͳεύʔεਖ਼ଇԽ 3 εύʔεਪఆͷཧ࿦ n ≫ p ͷཧ࿦ n ≪ p ͷཧ࿦ 4 ߴ࣍ݩઢܗճؼͷݕఆ 5 εύʔεਪఆͷ࠷దԽख๏ 26 / 56

໰୊ઃఆ ؆୯ͷͨΊઢܗճؼΛߟ͑Δɽ Y = Xβ∗ + ϵ. Y ∈ Rn : Ԡ౴ม਺, X ∈ Rn×p : આ໌ม਺, ϵ = [ϵ1, . . . , ϵn]⊤ ∈ Rn. ҰൠԽઢܗճؼ΁ͷҰൠԽ΋Մೳɽ 27 / 56

n ≫ p ͷཧ࿦ 28 / 56

Lasso ͷ઴ۙ෼෍ p ͸ݻఆɼn → ∞ ͷ઴ۙతৼΔ෣͍Λߟ͑Δɽ 1 n X⊤X p −→ C ≻ O. ϊΠζ ϵi ͸ฏۉ 0 ෼ࢄ σ2 ͱ͢Δ. Theorem (Lasso ͷ઴ۙ෼෍ (Knight and Fu, 2000)) λn √ n → λ0 ≥ 0 ͳΒ √ n(ˆ β − β∗) d −→ argmin u V (u), ͨͩ͠ɼ V (u) = u⊤Cu − 2u⊤W + λ0 ∑ p j=1 [uj sign(β∗ j )1(β∗ j ̸= 0) + |uj |1(β∗ j = 0)], W ∼ N(0, σ2C). ˆ β ͸ √ n-consistent Ͱ͋Δɽ β∗ j = 0 ͳΔ੒෼Ͱ ˆ βj ͸ ਖ਼ͷ֬཰Ͱ 0 ͱͳΔɽ ୈࡾ߲ͷ͍ͤͰɼ઴ۙతʹόΠΞε͕࢒Δɽ ˆ β = argminβ 1 n ∥Y − Xβ∥2 + λn ∑ p j=1 |βj |. 29 / 56

Adaptive Lasso ͷΦϥΫϧϓϩύςΟ ˜ β ͸͋ΔҰகਪఆྔɽ ˆ β = argmin β 1 n ∥Y − Xβ∥2 + λn p ∑ j=1 |βj | |˜ βj |γ . Theorem (Adaptive Lasso ͷΦϥΫϧϓϩύςΟ (Zou, 2006)) λn √ n → 0, λnn1+γ 2 → ∞ ͷͱ͖ɼ 1 limn→∞ P(ˆ J = J) → 1 (ˆ J := {j | |ˆ βj | ̸= 0}, J := {j | |β∗ j | ̸= 0}), 2 √ n(ˆ βJ − β∗ J ) d −→ N(0, σ2C−1 JJ ). ม਺બ୒ͷҰகੑ͋Γɽ ઴ۙෆภɼ઴ۙਖ਼نੑ͋Γɽ ͨͩ͠ɼβ∗ ͷ͋Δ੒෼͕ݪ఺ʹۙͮ͘Α͏ͳہॴతͳٞ࿦ (β∗ j = O(1/ √ n) ͳΔঢ়گ) ʹ͍ͭͯ͸Կ΋ݴ͍ͬͯͳ͍͜ͱʹ஫ҙɽ 30 / 56

n ≪ p ͷཧ࿦ 31 / 56

Lass ͷϦεΫͷ্ք ˆ β = arg min β∈Rp 1 n ∥Xβ − Y ∥2 2 + λn p ∑ j=1 |βj |. Theorem (Lasso ͷऩଋϨʔτ (Bickel et al., 2009, Zhang, 2009)) σβΠϯߦྻ͕ Restricted eigenvalue condition (Bickel et al., 2009) ͔ͭ maxi,j |Xij | ≤ 1 Λຬͨ͠ɼϊΠζ͕ E[eτξi ] ≤ eσ2τ2/2 (∀τ > 0) Λຬͨ͢ ͳΒ, ֬཰ 1 − δ Ͱ ∥ˆ β − β∗∥2 2 ≤ C d log(p/δ) n . ࣍ݩ͕ߴͯ͘΋ɼ͔͔ͨͩ log(p) Ͱ͔͠ޮ͍ͯ͜ͳ͍ɽ࣮࣭తͳ࣍ ݩ d ͕ࢧ഑తɽ ϊΠζͷ৚݅͸αϒΨ΢γΞϯͷඞཁे෼৚݅ɽ 32 / 56

Lasso ͷ minimax ࠷దੑ Theorem (ϛχϚΫε࠷దϨʔτ (Raskutti and Wainwright, 2011)) ͋Δ৚݅ͷ΋ͱɼ֬཰ 1/2 Ҏ্Ͱɼ min ˆ β:ਪఆྔ max β∗:d-εύʔε ∥ˆ β − β∗∥2 ≥ C d log(p/d) n . Lasso ͸ minimax ϨʔτΛୡ੒͢Δ (d log(d) n ͷ߲Λআ͍ͯ)ɽ ͜ͷ݁ՌΛ Multiple Kernel Learning ʹ֦ுͨ݁͠Ռ: Raskutti et al. (2012), Suzuki and Sugiyama (2013). 33 / 56

੍ݶݻ༗஋৚݅ (Restricted eigenvalue condition) A = 1 n X⊤X ͱ͢Δɽ Definition (੍ݶݻ༗஋৚݅ (RE(k′, C))) ϕRE (k′, C) = ϕRE (k′, C, A) := inf J⊆{1,...,n},v∈Rp: |J|≤k′,C∥vJ ∥1≥∥vJc ∥1 v⊤Av ∥vJ ∥2 2 ʹର͠ɼϕRE > 0 ͕੒Γཱͭɽ ΄΅εύʔεͳϕΫτϧʹ੍ݶͯ͠ఆٛͨ͠࠷খݻ༗஋ɽ J ৼঢ়ऋ৵औः ইঝছথॡ 34 / 56

ద߹ੑ৚݅ (Compatibility condition) A = 1 n X⊤X ͱ͢Δɽ Definition (ద߹ੑ৚݅ (COM(J, C))) ϕCOM(J, C) = ϕCOM(J, C, A) := inf v∈Rp: C∥vJ ∥1≥∥vJc ∥1 k v⊤Av ∥vJ∥2 1 ʹର͠ɼϕCOM > 0 ͕੒Γཱͭɽ |J| ≤ k′ ͳΒɼRE ΑΓ΋ऑ͍৚݅ɽ 35 / 56

੍ݶ౳௕ੑ৚݅ (Restricted isometory condition) Definition (੍ݶ౳௕ੑ৚݅ (RI(k′, δ))) ͋Δීวఆ਺ c ͕ଘࡏͯ͠ɼ͋Δ δ > 0 ʹର͠ɼ (1 − δ)∥β∥2 ≤ ∥Xβ∥2 ≤ (1 + δ)∥β∥2 ͕શͯͷ k′-εύʔεͳϕΫτϧ β ∈ Rp ʹରͯ͠੒Γཱͭɽ RE, COM ΑΓ΋ڧ͍৚݅ɽ Johnson-Lindenstrauss ͷิ୊. ѹॖηϯγϯάʹ͓͚Δ׬શ෮ݩͷจ຺ͰΑ͘༻͍ΒΕΔɽ 36 / 56

֤৚݅ͷؔ܎ͱऩଋϨʔτ ˆ β : Lasso ਪఆྔ. J := {j | β∗ j ̸= 0}. d := |J|. ڧ͍ 1 n ∥X(ˆ β − β∗)∥2 2 ∥ˆ β − β∗∥2 2 ∥ˆ β − β∗∥2 1 RI(2d, δ) ˠ ѹॖηϯγϯάʹ͓͚Δ׬શ෮ݩ ⇓ RE(2d, 3) ˠ d log(p) n d log(p) n d2 log(p) n ⇓ COM(J, 3) ˠ d log(p) n d2 log(p) n d2 log(p) n ऑ͍ ؔ࿈ࣄ߲ͷৄࡉ͸ B¨ uhlmann and van de Geer (2011) ʹ໢ཏ͞Ε͍ͯΔɽ 37 / 56

੍ݶݻ༗஋৚݅ (RE) ͕੒ཱ͢Δ֬཰ ੍ݶݻ༗஋৚݅͸ͲΕ͚ͩ੒Γཱͪ΍͍͔͢ʁ p ࣍ݩ֬཰ม਺ Z ͕౳ํత: E[⟨Z, z⟩2] = ∥z∥2 2 (∀z ∈ Rp). αϒΨ΢γΞϯϊϧϜ ∥Z∥ψ2 Λ࣍ͷΑ͏ʹఆٛ͢Δ: ∥Z∥ψ2 = supz∈Rp,∥z∥=1 inft{t | E[exp(⟨Z, z⟩)2/t2] ≤ 2}. 1. Z = [Z1, Z2, . . . , Zn]⊤ ∈ Rn×p ͷ֤ߦ Zi ∈ Rp ͕ಠཱͳ౳ํతαϒΨ ΢γΞϯ֬཰ม਺ͱ͢Δɽ 2. ͋Δ൒ਖ਼ఆରশߦྻ Σ ∈ Rp×p Λ༻͍ͯ X = ZΣ Ͱ͋Δͱ͢Δɽ Theorem (Rudelson and Zhou (2013)) ∥Zi ∥ψ2 ≤ κ (∀i) ͱ͢Δɽ͋Δීวఆ਺ c0 ͕ଘࡏ͠ m = c0 maxi (Σi,i )2 ϕ2 RE (k,9,Σ) ʹର ͠ɼn ≥ 4c0mκ4 log(60ep/(mκ)) ͳΒ͹ P ( ϕRE(k, 3, Σ) ≥ 1 2 ϕRE(k, 9, Σ) ) ≥ 1 − 2 exp(−n/(4c0κ4)). ͭ·Γɼਅͷ෼ࢄڞ෼ࢄߦྻ੍͕ݶݻ༗஋৚݅Λຬͨ͢ ⇒ ܦݧత෼ࢄڞ෼ࢄߦྻ΋ߴ͍֬཰Ͱಉ৚݅Λຬͨ͢ɽ 38 / 56

ತ࠷దԽΛ༻͍ͳ͍εύʔεਪఆ๏ͷੑ࣭ ৘ใྔن४ܕਪఆྔ: Massart (2003), Bunea et al. (2007), Rigollet and Tsybakov (2011). min β∈Rp ∥Y − Xβ∥2 + Cσ2∥β∥0 { 1 + log ( p ∥β∥0 )} . Bayes ਪఆྔ: Dalalyan and Tsybakov (2008), Alquier and Lounici (2011), Suzuki (2012). ΦϥΫϧෆ౳ࣜ: X ʹ Կ΋৚݅Λ՝ͣ͞ʹ ࣍ͷෆ౳͕ࣜ੒Γཱͭ: 1 n ∥Xβ∗ − X ˆ β∥2 ≤ Cσ2 d n log ( 1 + p d ) . ■ ϛχϚοΫε࠷దɽ ■ ತਖ਼ଇԽਪఆ๏ͱେ͖ͳΪϟ οϓɽ ■ ܭࢉྔͱ౷ܭతੑ࣭ͷτϨʔυΦϑɽ 39 / 56

Outline 1 εύʔεਪఆͷϞσϧ 2 ͍Ζ͍Ζͳεύʔεਖ਼ଇԽ 3 εύʔεਪఆͷཧ࿦ n ≫ p ͷཧ࿦ n ≪ p ͷཧ࿦ 4 ߴ࣍ݩઢܗճؼͷݕఆ 5 εύʔεਪఆͷ࠷దԽख๏ 40 / 56

όΠΞεআڈʹΑΔํ๏ ΞΠσΟΞɿLasso ਪఆྔ ˆ β ͔ΒόΠΞεΛআڈ. (van de Geer et al., 2014, Javanmard and Montanari, 2014) ˜ β = ˆ β + MX⊤(Y − X ˆ β) M ͕ (X⊤X)−1 ͳΒɼ ˜ β = β∗ + (X⊤X)−1X⊤ϵ. ˠ όΠΞεͳ͠ɼ (઴ۙ) ਖ਼نɽ ໰୊఺: n ≫ p ͷͱ͖ɼX⊤X ͸ඇՄٯɽ 41 / 56

M ͷٻΊํ: min M∈Rp×p |ΣM⊤ − I|∞. (| · |∞ ͸த਎ΛϕΫτϧͱΈͳͨ͠ແݶେϊϧϜ) Theorem (Javanmard and Montanari (2014)) ϵi ∼ N(0, σ2) (i.i.d.) ͱ͢Δɽ √ n(˜ β−β∗) = Z +∆, Z ∼ N(0, σ2MΣM⊤), ∆ = √ n(MΣ−I)(β∗− ˆ β). ·ͨɼX ͕ϥϯμϜͰ෼ࢄڞ෼ࢄߦྻ͕ਖ਼ఆͳ࣌ɼλn = cσ √ log(p)/n ͱ ͢Δͱɼ ∥∆∥∞ = Op ( d log(p) √ n ) . ˞ n ≫ d2 log2(p) ͳΒ͹ ∆ ≈ 0 Ͱɼ √ n(˜ β − β∗) ͸΄΅ਖ਼ن෼෍ʹै͏ɽ ˠ ৴པ۠ؒͷߏங΍ݕఆ͕Ͱ͖Δɽ 42 / 56

M ͷٻΊํ: min M∈Rp×p |ΣM⊤ − I|∞. (| · |∞ ͸த਎ΛϕΫτϧͱΈͳͨ͠ແݶେϊϧϜ) Theorem (Javanmard and Montanari (2014)) √ n(˜ β − β∗) = Z ਖ਼ن෼෍ + ∆ X ͕ඇՄٯͰ͋Δ͜ͱʹΑΔ࢒ΓΧε ৚͕݅ྑ͍࣌ 0 ΁ऩଋ ˞ n ≫ d2 log2(p) ͳΒ͹ ∆ ≈ 0 Ͱɼ √ n(˜ β − β∗) ͸΄΅ਖ਼ن෼෍ʹै͏ɽ ˠ ৴པ۠ؒͷߏங΍ݕఆ͕Ͱ͖Δɽ 42 / 56

਺஋࣮ݧ (a) ਓޱσʔλʹ͓͚Δ 95%৴པ۠ؒ. (n, p, d) = (1000, 600, 10). (b) ਓޱσʔλʹ͓͚Δ p ஋ͷ CDF. (n, p, d) = (1000, 600, 10). ਤ͸ Javanmard and Montanari (2014) ͔ΒҾ༻ɽ 43 / 56

ڞ෼ࢄݕఆ౷ܭྔ (Lockhart et al., 2014) 0 500 1000 1500 2000 2500 3000 −600 −400 −200 0 200 400 600 L1 Norm Coe cients 0 2 4 6 8 10 10 44 / 56

ڞ෼ࢄݕఆ౷ܭྔ (Lockhart et al., 2014) 0 500 1000 1500 2000 2500 3000 −600 −400 −200 0 200 400 600 L1 Norm Coe cients 0 2 4 6 8 10 10 J = supp(ˆ β(λk)), J∗ = supp(β∗), ˜ β(λk+1) := argmin β:βJ ∈R|J|,βJc =0 ∥Y − XJβJ∥2 + λk+1∥βJ∥1. J∗ ⊆ J ͳΒ͹ (β∗ j = 0 Ͱ͋Δ)ɼ Tk = ( ⟨Y , X ˆ β(λk+1)⟩ − ⟨Y , X ˜ β(λk+1)⟩ ) /σ2 d −→ Exp(1) (n, p → ∞). 44 / 56

Outline 1 εύʔεਪఆͷϞσϧ 2 ͍Ζ͍Ζͳεύʔεਖ਼ଇԽ 3 εύʔεਪఆͷཧ࿦ n ≫ p ͷཧ࿦ n ≪ p ͷཧ࿦ 4 ߴ࣍ݩઢܗճؼͷݕఆ 5 εύʔεਪఆͷ࠷దԽख๏ 45 / 56

εύʔεਪఆʹ͓͚Δ࠷దԽͷ໰୊ҙࣝ R(β) = n ∑ i=1 ℓ(yi , x⊤ i β) f (β):ϩεؔ਺ + ψ(β) ਖ਼ଇԽ߲ = f (β) + ψ(β) ψ ͕ઑ͍ͬͯΔˠඍ෼ෆՄೳɽ ઑ͍ͬͯΔؔ਺ͷ࠷దԽ͸ ೉͍͠ɽ f ͸ͳΊΒ͔ͳ৔߹͕ଟ͍ɽ ψ ͷߏ଄Λར༻ ͢Ε͹ɼ͔͋ͨ΋ R ͕ͳΊΒ͔ Ͱ͋Δ͔ͷΑ͏ʹ ࠷దԽՄೳɽ 46 / 56

εύʔεਪఆʹ͓͚Δ࠷దԽͷ໰୊ҙࣝ R(β) = n ∑ i=1 ℓ(yi , x⊤ i β) f (β):ϩεؔ਺ + ψ(β) ਖ਼ଇԽ߲ = f (β) + ψ(β) ψ ͕ઑ͍ͬͯΔˠඍ෼ෆՄೳɽ ઑ͍ͬͯΔؔ਺ͷ࠷దԽ͸ ೉͍͠ɽ f ͸ͳΊΒ͔ͳ৔߹͕ଟ͍ɽ ψ ͷߏ଄Λར༻ ͢Ε͹ɼ͔͋ͨ΋ R ͕ͳΊΒ͔ Ͱ͋Δ͔ͷΑ͏ʹ ࠷దԽՄೳɽ యܕྫɿL1 ਖ਼ଇԽ ψ(β) = C ∑ p j=1 |βj | ˠ ࠲ඪ͝ͱʹ෼͔Ε͍ͯΔ. Ұ࣍ݩͷ࠷దԽ minb{(b − y)2 + C|b|} ͸ ؆୯. 46 / 56

࠲ඪ߱Լ๏ ࠲ඪ߱Լ๏ͷखॱ 1 ࠲ඪ j ∈ {1, . . . , p} Λ ԿΒ͔ͷํ๏ Ͱબ୒ɽ 2 j ൪໨ͷ࠲ඪ βj Λߋ৽ɽ ʢҎԼ͸ߋ৽ํ๏ͷྫʣ β(k+1) j ← argminβj R([β(k) 1 , . . . , βj , . . . , β(k) p ]). gj = ∂f (β(k)) ∂βj ͱͯ͠ɼ β(k+1) j ← argminβj ⟨gj , βj ⟩ + ψj (βj ) + ηk 2 ∥βj − β(k) j ∥. ࠲ඪΛҰͭͣͭͰ͸ͳ͘ෳ਺ݸબͿ͜ͱ΋ଟ͍: ϒϩοΫ࠲ඪ߱ Լ๏ɽ ࠲ඪ͸͋ΔϧʔϧʹैͬͯબΜͩΓɼϥϯμϜʹબΜͩΓ͢Δɽ 47 / 56

࠲ඪ߱Լ๏ͷऩଋϨʔτ ෼ղՄೳͳਖ਼ଇԽ߲ ψ(β) = ∑ p j=1 ψj (βj ) Λߟ͑Δɽ f ͸ඍ෼ՄೳͰޯ഑͕ L-Lipschitz ࿈ଓ (∇f (β) − ∇f (β′) ≤ L∥β − β′∥) ˠ ͜ͷͱ͖ɼf ͸ʮ׈Β͔ʯͰ͋Δͱݴ͏ɽ αΠΫϦοΫ (Saha and Tewari, 2013) R(β(k)) − R(ˆ β) ≤ L∥β(0) − ˆ β∥2 2k = O(1/k). ϥϯμϜબ୒ (Nesterov, 2012, Peter Richt´ arik, 2014) Ճ଎๏ͳ͠: O(1/k). Nesterov ͷՃ଎๏: O(1/k2) (Fercoq and Richt´ arik, 2013). f ͕ α-ڧತ: O(exp(−C(α/L)k)). f ͕ α-ڧತ+Ճ଎๏: O(exp(−C √ α/Lk)) (Lin et al., 2014). ࠲ඪͷબͼํ͸ϥϯμϜͰྑ͍ɽ 48 / 56

େن໛σʔλʹ͓͚Δ࠲ඪ߱Լ๏ Hydra: ฒྻ෼ࢄܭࢉΛ༻͍ͨ࠲ඪ߱Լ๏ (Richt´ arik and Tak´ aˇ c, 2013, Fercoq et al., 2014). େن໛ Lasso (p = 5 × 108, n = 109) ʹ͓͚Δ Hydra ͷܭࢉޮ཰ (Richt´ arik and Tak´ aˇ c, 2013)ɽ128 ϊʔυɼ4,096 ίΞɽ 49 / 56

ۙ઀ޯ഑๏ܕख๏ f (β) ઢܗۙࣅ +ψ(β) gk ∈ ∂f (β(k)), ¯ gk = 1 k ∑ k τ=1 gτ . ۙ઀ޯ഑๏: β(k+1) = arg min β∈Rp { g⊤ k β + ψ(β) + ηk 2 ∥β − β(k)∥2 } . ਖ਼ଇԽ૒ରฏۉ๏ (Xiao, 2009, Nesterov, 2009): β(k+1) = arg min β∈Rp { ¯ g⊤ k β + ψ(β) + ηk 2 ∥β∥2 } . ݤͱͳΔܭࢉ͸ۙ઀ࣸ૾: prox(q|ψ) := arg min x { ψ(x) + 1 2 ∥x − q∥2 } . L1 ਖ਼ଇԽͳΒ؆୯ʹܭࢉͰ͖Δ (Soft-thresholding ؔ਺)ɽ 50 / 56

ۙ઀ࣸ૾ͷྫ: L1 ਖ਼ଇԽ prox(q|C∥ · ∥1) = arg min x { C∥x∥1 + 1 2 ∥x − q∥2 } = (sign(qj ) max(|qj | − C, 0))j . → Soft-thresholding ؔ਺. ղੳղ! 51 / 56

ۙ઀ޯ഑๏ܕख๏ͷऩଋϨʔτ f ͷੑ࣭ ׈Β͔ ඇ׈Β͔ ڧತ exp(− √ α/Lk) 1 k ඇڧತ 1 k2 1 √ k 1 ׈Β͔ͳ৔߹ɼNesterov ͷՃ଎๏Λ࢖ͬͨ࣌ͷऩଋϨʔτΛࣔͯ͠ ͍Δ (Nesterov, 2007, Zhang et al., 2010)ɽ 2 Ճ଎๏Λ࢖Θͳ͚Ε͹ɼͦΕͧΕ exp(−(α/L)k), 1 k ʹͳΔɽ 3 ্ͷΦʔμʔ͸ޯ഑৘ใͷΈΛ༻͍Δํ๏ (First order method) ͷ தͰ࠷దɽ 52 / 56

֦ுϥάϥϯδΞϯܕख๏ min β f (β) + ψ(β) ⇔ min x,y f (x) + ψ(y) s.t. x = y. ࠷దԽͷ೉͠͞Λ ෼཭ɽ ֦ுϥάϥϯδΞϯ: L(x, y, λ) = f (x) + ψ(y) + λ⊤(y − x) + ρ 2 ∥y − x∥2. ৐਺๏ (Hestenes, 1969, Powell, 1969, Rockafellar, 1976) 1 (x(k+1), y(k+1)) = argminx,y L(x, y, λ(k)). 2 λ(k+1) = λ(k) − ρ(y(k+1) − x(k+1)). x, y Ͱͷಉ࣌࠷దԽ͸΍΍໘౗ ˠ ަޓํ޲৐਺๏ɽ 53 / 56

ަޓํ޲৐਺๏ min x,y f (x) + ψ(y) s.t. x = y. L(x, y, λ) = f (x) + ψ(y) + λ⊤(y − x) + ρ 2 ∥y − x∥2. ަޓํ޲৐਺๏ (Gabay and Mercier, 1976) x(k+1) = arg min x f (x) − λ(k)⊤x + ρ 2 ∥y(k) − x∥2 y(k+1) = arg min y ψ(y) + λ(k)⊤y + ρ 2 ∥y − x(k+1)∥2(= prox(x(k+1) − λ(k)/ρ|ψ/ρ)) λ(k+1) = λ(k) − ρ(x(k+1) − y(k+1)) x, y ͷಉ࣌࠷దԽ͸ަޓʹ࠷దԽ͢Δ͜ͱͰճආɽ y ͷߋ৽͸ۙ઀ࣸ૾ɽL1 ਖ਼ଇԽͷΑ͏ͳ৔߹͸؆୯ɽ ߏ଄తਖ਼ଇԽ΁ͷ֦ு΋༰қɽ ࠷దղʹऩଋ͢Δอূ͋Γɽ Ұൠతʹ͸ O(1/k) (He and Yuan, 2012), ڧತͳΒ͹ઢܗऩଋ (Deng and Yin, 2012, Hong and Luo, 2012)ɽ 54 / 56

֬཰త࠷దԽ αϯϓϧ਺ͷଟ͍େن໛σʔλ Ͱ༗༻ɽ Ұճͷߋ৽ʹ͢΂ͯͷαϯϓϧΛಡΈࠐ·ͳ͍Ͱ΋େৎ෉. ■ ΦϯϥΠϯܕ FOBOS (Duchi and Singer, 2009) RDA (Xiao, 2009) ■ όονܕ SVRG (Stochastic Variance Reduced Gradient) (Johnson and Zhang, 2013) SDCA (Stochastic Dual Coordinate Ascent) (Shalev-Shwartz and Zhang, 2013) SAG (Stochastic Averaging Gradient) (Le Roux et al., 2013) ֬཰తަޓํ޲৐਺๏: Suzuki (2013), Ouyang et al. (2013), Suzuki (2014). 55 / 56

·ͱΊ ༷ʑͳεύʔεϞσϦϯά L1 ਖ਼ଇԽ άϧʔϓਖ਼ଇԽ τϨʔεϊϧϜਖ਼ଇԽ Lasso ͷ઴ۙతৼΔ෣͍ ઴ۙ෼෍ Adaptive Lasso ͷΦϥΫϧϓϩύςΟ ੍ݶݻ༗஋৚݅ˠ ∥ˆ β − β∗∥2 = Op (d log(p)/n) ݕఆ όΠΞεআڈ๏ɼڞ෼ࢄݕఆ౷ܭྔ ࠷దԽख๏ ࠲ඪ߱Լ๏ ۙ઀ޯ഑๏ (ަޓํ޲) ৐਺๏ 56 / 56

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