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機械学習勉強会09 2層フィードフォワードニューラルネット/MLStudy09

機械学習勉強会09 2層フィードフォワードニューラルネット/MLStudy09

機械学習勉強会09 2層フィードフォワードニューラルネット

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hachiilcane

March 03, 2022
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  1. χϡʔϩϯϞσϧ֮͑ͯ ·͢ʁ ࣠ࡧ γφϓε ిؾύϧε f = { 0 (w0

    + w1 x + w2 y ≤ 0) 1 (w0 + w1 x + w2 y > 0) x y ॏΈw1 ॏΈw2 f ग़ྗ ೖྗ ೖྗ 1 ͍ͭ΋1ͷ μϛʔೖྗ ॏΈw0 ೖྗۭؒΛઢͰ෼ ͚Δͱ͍͏ػೳ
  2. લճͷ෮शɿϩδεςΟο Ϋճؼͷ2ೖྗ3Ϋϥε෼ྨ x1 x2 a0 ग़ྗ ೖྗ ೖྗ x0 ͍ͭ΋1ͷ

    μϛʔೖྗ w00 ೖྗ૯࿨ a1 a2 w01 w02 w20 w21 w22 y0 = exp(a0 ) ∑K−1 k=0 exp(ak ) → P(t = 0|x) y1 = exp(a1 ) ∑K−1 k=0 exp(ak ) → P(t = 1|x) y2 = exp(a2 ) ∑K−1 k=0 exp(ak ) → P(t = 2|x) a0 = ∑2 i=0 w0i xi
  3. 2૚ϑΟʔυϑΥϫʔυχϡʔϥ ϧωοτͷ2ೖྗ3Ϋϥε෼ྨ͸ தؒ૚͕௥Ճ͞Ε͚ͨͩʂ 3૚ʹݟ͑Δ͚ͲɺॏΈύϥϝʔλ͕͋Δ૚͚ͩΛ਺͑ͯ2૚ͱ͍͏৔߹͕ଟ͍ x1 x2 a0 ग़ྗ૚ʢ2૚໨ʣ x0 όΠΞε߲

    w00 தؒ૚ʢ1૚໨ʣ w01 w02 v20 v21 v22 P(t = 0|x) y0 b1 z1 b2 z2 z0 a1 y1 a2 y2 ೖྗ૚ v00 v01 v02 = 1 = 1 όΠΞε߲ t0 t1 t2 P(t = 1|x) P(t = 2|x) D + 1 M + 1 K ؙͷࠨ͕ೖྗ૯࿨ɺӈ͕ͦΕ Λ׆ੑԽؔ਺ʹ௨ͨ͠஋ ؙͷࠨ͕ೖྗ૯࿨ɺӈ͕ͦΕΛ ιϑτϚοΫεؔ਺ʹ௨ͨ͠஋
  4. Ϟσϧͷ਺ࣜͷ֬ೝ h()͸ͱΓ͋͑ͣγάϞΠυؔ਺ͱ͢Δ தؒ૚ͷೖྗ૯࿨ɿbj = ∑D i=0 wji xi தؒ૚ͷग़ྗɿzj =

    h(bj ) ग़ྗ૚ͷೖྗ૯࿨ɿak = ∑M j=0 vkj zj ग़ྗ૚ͷग़ྗɿyk = exp(ak ) ∑K−1 l=0 exp(al ) = exp(ak ) u ೖྗ࣍ݩ:Dɺதؒ૚ͷχϡʔϩϯͷ਺:Mɺग़ྗ࣍ݩ:K b = wx b0 b1 b2 = w00 w01 w02 w10 w11 w12 w20 w21 w22 x0 x1 x2 ϕΫτϧදه͢ΔͳΒ͜Μͳ͔Μ͡ʢҰ෦͚ͩʣ
  5. ਤͱ਺ࣜΛηοτͰ֬ೝ x1 x2 a0 ग़ྗ૚ʢ2૚໨ʣ x0 όΠΞε߲ w00 தؒ૚ʢ1૚໨ʣ w01

    w02 v20 v21 v22 P(t = 0|x) y0 b1 z1 b2 z2 z0 a1 y1 a2 y2 ೖྗ૚ v00 v01 v02 = 1 = 1 όΠΞε߲ t0 t1 t2 P(t = 1|x) P(t = 2|x) D + 1 M + 1 K ೖྗ࣍ݩ:Dɺதؒ૚ͷχϡʔϩϯͷ਺:Mɺग़ྗ࣍ݩ:K bj = ∑D i=0 wji xi zj = h(bj ) ak = ∑M j=0 vkj zj yk = exp(ak ) ∑K−1 l=0 exp(al )
  6. ภඍ෼ͳΜ͔ͨ͘͠ͳ ͍ʂͱ͍͏৔߹͸ ਺஋ඍ෼๏Λ࢖͏ͱ͍͏ख΋͋Δɻ ඍ෼ͱ͸܏͖ͷ͜ͱͰɺ͋Δ஍఺w*ͷ܏͖ΛಘΔʹ͸ɺ͜ͷਤͰ͍͏hΛͲΜͲΜখ ͍͚ͯ͘͞͠͹ۙࣅతʹٻΊΒΕΔɻ ภඍ෼ͩͬͨΒɺண໨͢Δม਺Ҏ֎͸ݻఆʹ͢Δͱ͍͏ͷ͸਺஋ඍ෼๏Ͱ΋͓ͳ͡ɻ ͜ΕͰภඍ෼ಋؔ਺Λ௚઀ಋग़͠ͳͯ͘΋͋Δ஍఺ͷޯ഑ϕΫτϧΛ΋ͱΊΒΕΔɻ ඍ෼ͷఆٛɿ d dx

    f(x) = lim h→0 f(x + h) − f(x) h E(w) w w* w* + h w* − h 2h ͲΜͲΜhΛখ͍͚ͯ͘͞͠͹ɺ΍͕ͯw*Ͱ ͷ઀ઢʹۙ͘ɻͦΕ͢ͳΘͪ܏͖Ͱ͋Γɺͦ ͷॠؒͷมԽ཰Ͱ͋Γɺඍ෼Ͱ͋Δɻඍ෼ͷ ఆٛࣜͱ͍͍ͩͨಉ͜͡ͱΛݴ͍ͬͯΔɻ ∂E ∂w | w=w* ≃ E(w* + h) − E(w* − h) 2h
  7. ޡࠩٯ఻೻๏֓ཁ ೖྗΛೖΕͯग़ྗΛಘΔ ֤χϡʔϩϯͰͷޡࠩΛಘΔ ॏΈΛߋ৽͢Δ δ(2) k = yk − tk

    ←ୈ2૚ͷޡࠩ δ(1) j = h′ (bj ) K−1 ∑ k=0 vkj δ(2) k ←ୈ1૚ͷޡࠩ vkj := vkj − αδ(2) k zj /N wji := wji − αδ(1) j xi /N bj = ∑D i=0 wji xi zj = h(bj ) ak = ∑M j=0 vkj zj yk = exp(ak ) ∑K−1 l=0 exp(al ) x1 x2 a0 ग़ྗ૚ʢ2૚໨ʣ x0 w00 தؒ૚ʢ1૚໨ʣ w01 w02 v20 v21 v22 y0 b1 z1 b2 z2 z0 a1 y1 a2 y2 ೖྗ૚ v00 v01 v02 = 1 = 1 t0 t1 t2 D + 1 M + 1 K δ(1) j δ(2) k