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情報幾何の応用と最近の機械学習の動向

Oshita Noriaki
November 03, 2018

 情報幾何の応用と最近の機械学習の動向

勉強会での発表した資料です。

Oshita Noriaki

November 03, 2018
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  1. ۂ཰ɾᎇ཰ʢΕ͍Γͭʣ ͦͷͱ͖Ճ଎౓ϕΫτϧΛߟ͑ͯΈΔͱ Ͱ͋Δ͔Βɼɹɹ͕ɹɹɹʹ௚ަ͍ͯ͠Δɽɹɹͷ௕͞Λ ͱॻ͖ɼۂઢ1 T ͷۂ཰ͱݺͿɽ͢ͳΘͪ ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɾ e′ 1 (s)

    0 = d (e1 (s) ⋅ e1 (s)) ds = 2e′ i (s) ⋅ e1 (s) e′ 1 (s) e1 (s) e′ 1 (s) k(s) κ(s) = e′ 1 (s) ⋅ e′ 1 (s) = x′′(s)2 + y′′(s)2 + z′′(s)2
  2. ۂ཰ɾᎇ཰ʢΕ͍Γͭʣ w ᎇ཰ e′ 1 e′ 2 e′ 3 =

    0 k 0 −κ 0 τ 0 −τ 0 e1 e2 e3 ಋग़͸লུ͠·͢ɽ
  3. ૒ରΞϑΝΠϯ઀ଓ ΞϑΝΠϯ઀ଓ∇Λ࣋ͭ3JFNBOOଟ༷ମ(M, g)ʹ͓͍ͯ Xg (Y, Z) = g(∇X Y, Z)

    + g(Y, ∇* X Z) (X, Y, Z) ∈ (M) Ͱఆٛ͞ΕΔΞϑΝΠϯ઀ଓ∇*Λܭྔgʹؔ͢Δ∇ͷ૒ରΞϑΝΠϯ઀ଓͱ͍͏ɽ
  4. ৘ใزԿͷԠ༻ w ࣗવޯ഑๏ w αϙʔτϕΫτϧϚγϯ w #PPTUJOH w ओ੒෼෼ੳ w

    ͳͲɼ͋Γͱ͋ΒΏΔ΋ͷʹԠ༻͞Ε͍ͯΔ w ৄ͘͠͸৘ใزԿֶͷ৽ల։ ⋮
  5. Χʔωϧ๏ͷॏཁͳఆཧ L Y Z ू߹Њ্ͷਖ਼ఆ஋Χʔωϧ Њ্ͷؔ਺͔ΒͳΔώϧϕϧτۭؒͰ࣍ͷࡾͭΛຬͨ͢ ΋ͷ͕Ұҙʹଘࡏ͢Δɻ   ͸೚ҙʹݻఆ

      ༗ݶ࿨ͷܗͷݩ͸ͷதͰ᜚ີ 
  ࠶ੜੑ ⟨f, k( ⋅ , x)⟩ℋ = f(x) (∀x ∈ , ∀f ∈ ℋ) ʢ.PPSF"SPOT[KOͷఆཧʣ Hk k(· ,x) ∈ Hk x ∈ Ω f = n ∑ i=1 ci k(· ,xi ) Hk
  6. ਖ਼ఆ஋Χʔωϧ ɽɽɹਖ਼ఆ஋Χʔωϧͷఆٛͱجຊతੑ࣭ ·ͣɼ࣮਺஋ͷਖ਼ఆ஋Χʔωϧ͔Βఆٛ͢Δɽ Λू߹ͱ͢Δͱ͖ɼ࣍ͷ৚݅Λຬͨ͢Χʔωϧ ɹɹɹɹɹɹɹɹΛਖ਼ఆ஋Χʔωϧ QPTJUJWF EFpOFLFSOFM ͱ͍͏ɽ w ରশੑɿ೚ҙͷɹɹɹɹɹɹʹର͠ɹɹ

    w ਖ਼஋ੑɿ೚ҙͷ ʹର͠ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɽ k : × → ℝ (্ͷ) x, y ∈ k(x, y) = k(y, x) n ∈ ℕ, x1 , …, xn ∈ , c1 , …, cn ∈ ℝ n ∑ i,j=1 ci cj k (xi , xj) ≥ 0
  7. ਖ਼ఆ஋ɾ൒ਖ਼ఆ஋ zTMz > 0 zTMz ≥ 0 ͕ඞͣͳΓͨͭͱ͖ਖ਼ఆ஋ͱ͍͏ɽ ͕੒Γཱͭͱ͖൒ਖ਼ఆ஋ͱ͍͏ɽ z

    ≡ ඇθϩྻϕΫτϧ M ≡ OºOͷ࣮਺ରশߦྻ .͕ਖ਼ఆ஋ͱ͸ .͕ਖ਼ఆ஋ͱ͸ zTMz = [z1 , z2 , ⋯, zn] m11 m12 ⋯ m1n ⋮ ⋱ mn1 m21 ⋯ mnn z1 z2 ⋮ zn
  8. "-JOFBS5JNF,FSOFM(PPEOFTTPG'JU5FTU w/*14ͷϕετϖʔύʔʂʂ w ෼෍ͷ෼͔Βͳ͍ͱ͜ΖΛਪఆ͢Δɽ w ྫ͑͹͜Ε͸ඪ४ਖ਼ن෼෍ͱ͍ͬͯྑ͍ͷ͔Ͳ͏͔Λ൑ ఆ͢Δ w ",FSOFMJ[FE4UFJO%JTDSFQBODZGPS(PPEOFTTPGpU 5FTUTʢ2JBOH-JV

    +BTPO%-FF .JDIBFM*+PSEBOʣ ͱ͍͏ख๏Λ༻͍ͯʢޙ೔2JJUBʹ֓ཁΛॻ͘༧ఆ ͭͷ ֬཰෼෍͕ࣅ͍ͯΔ͔Λઢܗ࣌ؒͰଌఆ͢Δɽ IUUQTBSYJWPSHBCT
  9. A Linear-Time Kernel Goodness-of-Fit Test Wittawat Jitkrittum1 Wenkai Xu1 Zolt´

    an Szab´ o2 Kenji Fukumizu3 Arthur Gretton1 1Gatsby Unit, University College London 2CMAP, ´ Ecole Polytechnique 3The Institute of Statistical Mathematics A Linear-Time Kernel Goodness-of-Fit Test Wittawat Jitkrittum1 Wenkai Xu1 Zolt´ an Szab´ o2 Kenji Fukumizu3 Arthur Gretton1 1Gatsby Unit, University College London 2CMAP, ´ Ecole Polytechnique 3The Institute of Statistical Mathematics Summary •Given: {x i}n i=1 ⇠ q (unknown), and a density p. •Goal: Test H0 : p = q vs H1 : p 6= q quickly. •New multivariate goodness-of-fit test (FSSD): 1.Nonparametric: arbitrary, unnormalized p. x 2 Rd. 2.Linear-time: O(n) runtime complexity. Fast. 3.Interpretable: tell where p does not fit the data. Previous: Kernel Stein Discrepancy (KSD) •Let x(x, v) := 1 p(x)rx[k(x, v)p(x)] 2 Rd. Stein witness function: g(v) = E x⇠q [x(x, v)] where g = (g1 , . . . , gd ) and each gi 2 F, an RKHS associated with kernel k. - 4 - 2 2 4 - 0.2 0.2 0.4 p(x) q(x) g(x) Known: Under some conditions, kgkFd = 0 () p = q. [Chwialkowski et al., 2016, Liu et al., 2016] Statistic: KSD2 = kgk2 Fd = double sums z }| { E x⇠q E y⇠q hp (x, y) ⇡ 2 n(n 1) P i<j hp (x i , x j ). where hp (x, y) := [rx log p(x)] k(x, y) [ry log p(y)] + rxryk(x, y) + [ry log p(y)] rxk(x, y) + [rx log p(x)] ryk(x, y). Characteristics of KSD: 3 Nonparametric. Applicable to a wide range of p. 3 Do not need the normalizer of p. 7 Runtime: O(n 2). Computationally expensive. Linear-Time KSD (LKS) Test: [Liu et al., 2016] kgk2 Fd ⇡ 2 n P n/2 i=1 hp (x2i 1 , x2i ). 3 Runtime: O(n). 7 High variance. Low test power. The Finite Set Stein Discrepancy (FSSD) Idea: Evaluate witness g at J locations {v1 , . . . , v J}. Fast. FSSD2 = 1 dJ J X j=1 kg(v j )k 2 2 . Proposition (FSSD is a discrepancy measure). Main conditions: 1.(Nice kernel) Kernel k is C0 -universal, and real analytic (Taylor series at any point converges) e.g., Gaussian kernel. 2.(Vanishing boundary) lim kxk!1 p(x)g(x) = 0. 3.(Avoid “blind spots” ) Locations {v1 , . . . , v J} are drawn from a distribution h which has a density. Then, for any J 1, h-a.s. FSSD2 = 0 () p = q. Characteristics of FSSD: 3 Nonparametric. 3 Do not need the normalizer of p. 3 Runtime: O(n). 3 Higher test power than LKS. Model Criticism with FSSD Proposal: Optimize locations {v1 , . . . , v J} and kernel bandwidth by arg max score = FSSD2/s H1 (runtime: O(n)). Proposition: This procedure maximizes the true positive rate = P(detect di↵erence | p 6= q). score: 0.034 score: 0.44 Interpretable Features for Model Criticism 12K robbery events in Chicago in 2016 Model p = 10-component Gaussian mixture F = v⇤ = where model does not fit well. Maximization objective FSSD2/s H1 . Optimized v⇤ is highly interpretable. Bahadur Slope and Bahadur E ciency •Bahadur slope u rate of p-value ! 0 of statistic Tn under H1 . High = good. •Bahadur e ciency = ratio slope(1) slope(2) of slopes of two tests. > 1 means test(1) better. •Results: Slopes of FSSD and LKS tests when p = N(0, 1) and q = N(µ q , 1). 0 50 100 n 0.0 0.5 1.0 p-value T(1) n T(2) n Proposition. Let s2 k , k2 be kernel bandwidths of FSSD and LKS. Fix s2 k = 1. Then, 8 µ q 6= 0, 9v 2 R, 8 k2 > 0, the Bahadur e ciency slope(FSSD)(µ q , v, s2 k ) slope(LKS)(µ q , k2) > 2. FSSD is statistically more e cient than LKS. Experiment: Restricted Boltzmann Machine •40 binary hidden units. d = 50 visible units. Significance level a = 0.05. · · · · · · Model p Perturb one weight to get q. 2000 4000 Sample size n 0.00 0.25 0.50 0.75 Rejection rate 1000 2000 3000 4000 Sample size n 0 100 200 300 Time (s) 2000 4000 Sample size n 0.0 0.5 1.0 Rejection rate FSSD-opt FSSD-rand KSD LKS MMD-opt ME-opt Better •FSSD-opt, (FSSD-rand) = Proposed tests. J = 5 optimized, (random) locations. •MMD-opt [Gretton et al., 2012] = State-of-the-art two-sample test (quadratic-time). •ME-opt [Jitkrittum et al., 2016] = Linear-time two-sample test with optimized locations. •Key: FSSD (O(n)), KSD (O(n 2)) have comparable power. FSSD is much faster. WJ, WX, and AG thank the Gatsby Charitable Foundation for the financial support. ZSz was financially supported by the Data Science Initiative. KF has been supported by KAKENHI Innovative Areas 25120012. Contact: [email protected] Code: github.com/wittawatj/kernel-gof
  10. Ψ΢εաఔͷجૅ Ͱ͸͇͕࣍ݩͷ৔߹͸Ͳ͏ͳΔͰ͠ΐ͏ ౴͑͸  EJN   EJN  

     EJN y = WTϕ(x1 , x2 ) 212 = 441 213 = 9261 ⋮ 2110 = 16,679,880,978,201
  11. ࢀߟจݙ w ৘ใزԿֶͷ৽ల։ɿ؁རढ़Ұ w ৘ใزԿֶͷجૅɿ౻ݪজ෉ w ۂઢͱۂ໘ͷඍ෼زԿɿখྛতࣣ w Χʔωϧ๏ೖ໳ɿ෱ਫ݈࣍ w

    (BVTTJBO1SPDFTTGPS.BDIJOF-FBSOJOH ʢIUUQXXXHBVTTJBOQSPDFTTPSHHQNMʣɹ  w Ψ΢εաఔͷجૅͱڭࢣͳֶ͠श IUUQXXXJTNBDKQdEBJDIJMFDUVSFT)(BVTTJBO1SPDFTTHQMFDUVSFEBJDIJQEG  w ʰΨ΢εաఔͱػցֶशʱαϙʔτϖʔδ IUUQDIBTFOPSHdEBJUJNHQCPPL