画像解析（MNIST）

Έ΍͜ ΋͘΋͘ձ ./*45 ϓϨθϯλʔɹTomohiro Iida

ΞδΣϯμ ࣗݾ঺հˍຊ೔ͷΰʔϧઆ໌ʢ෼ʣ ࢀՃऀͦΕͧΕͷࣗݾ঺հʢ෼ʣ ./*45ϋϯζΦϯʢ෼ʣ

ຊ೔ͷΰʔϧ ࣗ෼ͷखॻ͖਺ࣈΛ ൑ผ͢ΔϞσϧΛ࡞Δ

ࢯ໊ʢϋϯυϧωʔϜՄʣ झຯ ʮ͜ΜͳJQIPOF͸ݏͩʂʯΛڭ͍͑ͯͩ͘͞ ࢀՃऀࣗݾ঺հ

ࣗݾ঺հˍຊ೔ͷΰʔϧઆ໌ʢ෼ʣ ࢀՃऀͦΕͧΕͷࣗݾ঺հʢ෼ʣ ./*45ϋϯζΦϯʢ෼ʣ

./*45ͱ͸ ./*45ͱ͸ɺը૾σʔληοτͷ͜ͱɻ  ʢखॻ͖਺ࣈը૾ ຕͱɺςετը૾ ຕΛूΊͨ΋ͷʣ  ҎԼ͕ϖΞʹͳ͍ͬͯΔɻ खॻ͖ͷʙͷ਺ࣈ͕ॻ͔Εͨը૾σʔλͱɺ ͦͷը૾ʹॻ͔Ε࣮ͨࡍͷ਺ࣈσʔλ

ը૾ೝࣝͱ͸ IUUQTTVQFSTNBTICSPTGBOEPNDPNKBXJLJ&""&#&"' ը૾ೝࣝͱ͸ ը૾σʔλ͔Βɺਓؒʹͱͬͯҙຯͷ͋Δ৘ใΛ நग़͢ΔλεΫͷ͜ͱΛ͍͏ɻ ୯ͳΔ਺ࣈʹ͗͢ͳ͍

./*45ͱ͸ IUUQTXFCMBCPPTDBTJFSSBOFUQZUIPOLFSBTNOJTUTBNQMFIUNM σʔλͷ༷ࢠ

./*45ͱ͸ IUUQTXFCMBCPPTDBTJFSSBOFUQZUIPOLFSBTNOJTUTBNQMFIUNM ग़ྗ͸͜Μͳײ͡ ਺ࣈ ֬཰ 0 0.014 1 0.001 2
0.013 3 0.719 4 0.034 5 0.016 6 0.023 7 0.065 8 0.086 9 0.029

Ͱ͸ɺ΍ͬͯΈ·͠ΐ͏ ࣮૷

4FRVFOUJBMϞσϧ IUUQNBSVQFLFDPN*,%"%7@%-@/P@,FSBTIUNM %FOTF

3FMVؔ਺ ೖྗ͕ҎԼͳΒɺҎ্ͳΒͦͷ··ग़ྗ͢Δؔ਺ͷ͜ͱɻ ҎԼͷ਺ࣜͰදͤΔɻ I Y \Y Y Y ^
IUUQTUFDICMPHHNPBQKQ&"'&"#&"%"&#'&&"&#$&&"&#$&#&#&#&"&#%#'&'&$&#&'#"&$"$&"&""

%SPQPVU σʔλ͕มʹͳͬͯ͠·͏͜ͱΛ๷͙ɻ IUUQTRJJUBDPNNJOFJUFNTECBCCFGD

3.4QSPQ lޯ഑߱Լ๏zΛ ࠷దԽ͢ΔΞϧΰϦζϜͷͻͱͭ దԠతֶश཰ͷํࣜͷ࣮ݱ

ޯ഑߱Լ๏ ޯ഑߱Լ๏ͷߟ͑ํ˰্͔ΒϘʔϧΛམͱ͠Ұ൪௿͍ͱ͜Ζ΁୧Γண͘·Ͱ଴ͭ IUUQXXXQMBODWUJUFDIBDKQGVLVEBMBCCBDCEEBCDCBGDDGDQEG

ޯ഑߱Լ๏ ࣌ࠁͰ ʹ͋ͬͨϘʔϧΛޯ഑ͷٯํ޲΁ಈ͔͢ࡍͷϧʔϧ (+1) = () + ∆() , ∆()
= − (4.9) lͱॱ࣍܁Γฦ͢͜ͱͰɼޡࠩؔ਺άϥϑͷఈ΁߱Γ͍ͯ͘ ˎ̍εςοϓͷҠಈڑ཭∆ ͸lֶश཰zͰܾఆ͞ΕΔ IUUQXXXQMBODWUJUFDIBDKQGVLVEBMBCCBDCEEBCDCBGDDGDQEG

ޯ഑߱Լ๏ ֶश཰ͷऔΓํ͸τϥΠΞϯυΤϥʔʹجͮ͘ w ֶश཰͕େ͖͍  ˰ࠁΈ͕ߥ͗ͯ͢ޡࠩؔ਺ͷܗঢ়Λଊ͑ΒΕͣɼऩଋ͠ͳ͍  w ֶश཰͕খ͍͞  ˰ֶश͕Ұ޲ʹਐߦ͠ͳ͍ IUUQXXXQMBODWUJUFDIBDKQGVLVEBMBCCBDCEEBCDCBGDDGDQEG

$//ͱ͸ ը૾σʔλʹదͨ͠ܗͷ  χϡʔϥϧωοτϫʔΫ ը૾Λͦͷ··ͷܗʹ̎࣍ݩͰ  ѻ͏͜ͱ͕Ͱ͖Δ $POWPMVUJPOBM/FVSBM/FUXPSL

$//ͷશମ૾ $//͸ɺ৞ΈࠐΈ΍ϓʔϦϯάʹΑͬͯಘΒΕͨಛ௃Ϛοϓͷ֤ϐΫηϧσʔλΛɺ  χϡʔϥϧωοτϫʔΫͷೖྗ஋ͱͯ͠ར༻͢Δɻ IUUQTXXXBUNBSLJUDPKQBJUBSUJDMFTOFXTIUNM

$// ৞ΈࠐΈ IUUQTULFOHPHJUIVCJPCMPHVOEFSTUBOEJOHDPOWPMVUJPOBMOFVSBMOFUXPSLTGPSOMQ
Χʔωϧ  ʢϑΟϧλʔʣ

$// ϓʔϦϯά IUUQTULFOHPHJUIVCJPCMPHVOEFSTUBOEJOHDPOWPMVUJPOBMOFVSBMOFUXPSLTGPSOMQ ը૾αΠζΛܾΊΒΕͨϧʔϧʹैͬͯখ͘͢͞Δ NBYϓʔϦϯά

$// શ݁߹૚ ग़ྗͷܗΛ̍࣍ݩʹมܗ͢Δ ௨ৗͷχϡʔϥϧωοτϫʔΫͱಉ͡  ʮӅΕ૚ʯʮग़ྗ૚ʯʹ઀ଓ͠ɺ

͓͠·͍ pO

画像解析（MNIST）

画像解析（MNIST）

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Έ΍͜ ΋͘΋͘ձ ./*45 ϓϨθϯλʔɹTomohiro Iida

ΞδΣϯμ ࣗݾ঺հˍຊ೔ͷΰʔϧઆ໌ʢ෼ʣ ࢀՃऀͦΕͧΕͷࣗݾ঺հʢ෼ʣ ./*45ϋϯζΦϯʢ෼ʣ

ΞδΣϯμ ࣗݾ঺հˍຊ೔ͷΰʔϧઆ໌ʢ෼ʣ ࢀՃऀͦΕͧΕͷࣗݾ঺հʢ෼ʣ ./*45ϋϯζΦϯʢ෼ʣ

ܦྺ

ຊ೔ͷΰʔϧ ࣗ෼ͷखॻ͖਺ࣈΛ ൑ผ͢ΔϞσϧΛ࡞Δ

ࢯ໊ʢϋϯυϧωʔϜՄʣ झຯ ʮ͜ΜͳJQIPOF͸ݏͩʂʯΛڭ͍͑ͯͩ͘͞ ࢀՃऀࣗݾ঺հ

ࣗݾ঺հˍຊ೔ͷΰʔϧઆ໌ʢ෼ʣ ࢀՃऀͦΕͧΕͷࣗݾ঺հʢ෼ʣ ./*45ϋϯζΦϯʢ෼ʣ

./45ͱ͸ ./45ͱ͸ɺը૾σʔληοτͷ͜ͱɻ  ʢखॻ͖਺ࣈը૾ ຕͱɺςετը૾ ຕΛूΊͨ΋ͷʣ  ҎԼ͕ϖΞʹͳ͍ͬͯΔɻ खॻ͖ͷʙͷ਺ࣈ͕ॻ͔Εͨը૾σʔλͱɺ ͦͷը૾ʹॻ͔Ε࣮ͨࡍͷ਺ࣈσʔλ

ը૾ೝࣝͱ͸ IUUQTTVQFSTNBTICSPTGBOEPNDPNKBXJLJ&""&#&"' ը૾ೝࣝͱ͸ ը૾σʔλ͔Βɺਓؒʹͱͬͯҙຯͷ͋Δ৘ใΛ நग़͢ΔλεΫͷ͜ͱΛ͍͏ɻ ୯ͳΔ਺ࣈʹ͗͢ͳ͍

./*45ͱ͸ IUUQTXFCMBCPPTDBTJFSSBOFUQZUIPOLFSBTNOJTUTBNQMFIUNM σʔλͷ༷ࢠ

./*45ͱ͸ IUUQTXFCMBCPPTDBTJFSSBOFUQZUIPOLFSBTNOJTUTBNQMFIUNM ग़ྗ͸͜Μͳײ͡ ਺ࣈ ֬཰ 0 0.014 1 0.001 2

Ͱ͸ɺ΍ͬͯΈ·͠ΐ͏ ࣮૷

4FRVFOUJBMϞσϧ IUUQNBSVQFLFDPN*,%"%7@%-@/P@,FSBTIUNM %FOTF

3FMVؔ਺ ೖྗ͕ҎԼͳΒɺҎ্ͳΒͦͷ··ग़ྗ͢Δؔ਺ͷ͜ͱɻ ҎԼͷ਺ࣜͰදͤΔɻ I Y \Y Y Y ^

%SPQPVU σʔλ͕มʹͳͬͯ͠·͏͜ͱΛ๷͙ɻ IUUQTRJJUBDPNNJOFJUFNTECBCCFGD

3.4QSPQ lޯ഑߱Լ๏zΛ ࠷దԽ͢ΔΞϧΰϦζϜͷͻͱͭ దԠతֶश཰ͷํࣜͷ࣮ݱ

ޯ഑߱Լ๏ ޯ഑߱Լ๏ͷߟ͑ํ˰্͔ΒϘʔϧΛམͱ͠Ұ൪௿͍ͱ͜Ζ΁୧Γண͘·Ͱ଴ͭ IUUQXXXQMBODWUJUFDIBDKQGVLVEBMBCCBDCEEBCDCBGDDGDQEG

ޯ഑߱Լ๏ ࣌ࠁͰ ʹ͋ͬͨϘʔϧΛޯ഑ͷٯํ޲΁ಈ͔͢ࡍͷϧʔϧ (+1) = () + ∆() , ∆()

ޯ഑߱Լ๏ ֶश཰ͷऔΓํ͸τϥΠΞϯυΤϥʔʹجͮ͘ w ֶश཰͕େ͖͍  ˰ࠁΈ͕ߥ͗ͯ͢ޡࠩؔ਺ͷܗঢ়Λଊ͑ΒΕͣɼऩଋ͠ͳ͍  w ֶश཰͕খ͍͞  ˰ֶश͕Ұ޲ʹਐߦ͠ͳ͍ IUUQXXXQMBODWUJUFDIBDKQGVLVEBMBCCBDCEEBCDCBGDDGDQEG

$//ͱ͸ ը૾σʔλʹదͨ͠ܗͷ  χϡʔϥϧωοτϫʔΫ ը૾Λͦͷ··ͷܗʹ̎࣍ݩͰ  ѻ͏͜ͱ͕Ͱ͖Δ $POWPMVUJPOBM/FVSBM/FUXPSL

$//ͷશମ૾ $//͸ɺ৞ΈࠐΈ΍ϓʔϦϯάʹΑͬͯಘΒΕͨಛ௃Ϛοϓͷ֤ϐΫηϧσʔλΛɺ  χϡʔϥϧωοτϫʔΫͷೖྗ஋ͱͯ͠ར༻͢Δɻ IUUQTXXXBUNBSLJUDPKQBJUBSUJDMFTOFXTIUNM

$// ৞ΈࠐΈ IUUQTULFOHPHJUIVCJPCMPHVOEFSTUBOEJOHDPOWPMVUJPOBMOFVSBMOFUXPSLTGPSOMQ

$// ϓʔϦϯά IUUQTULFOHPHJUIVCJPCMPHVOEFSTUBOEJOHDPOWPMVUJPOBMOFVSBMOFUXPSLTGPSOMQ ը૾αΠζΛܾΊΒΕͨϧʔϧʹैͬͯখ͘͢͞Δ NBYϓʔϦϯά

$// શ݁߹૚ ग़ྗͷܗΛ̍࣍ݩʹมܗ͢Δ ௨ৗͷχϡʔϥϧωοτϫʔΫͱಉ͡  ʮӅΕ૚ʯʮग़ྗ૚ʯʹ઀ଓ͠ɺ

͓͠·͍ pO