| Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende Überarbeitung |
| p:ki:machinelearning4 [2024/04/07 10:33] – [1.3 Lernen der Gewichte] Ralf Kretzschmar | p:ki:machinelearning4 [2024/07/04 08:38] (aktuell) – [2. Das Gradientenverfahren von Hand durchrechnen] Ralf Kretzschmar |
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| 💡 Auch beim Ableiten wird von rechts nach links durch die Formel gegangen. | 💡 Auch beim Ableiten wird von rechts nach links durch die Formel von $J[k]$ gegangen. |
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| - Markiere diesen Auftrag als "Erledigt!". | - Markiere diesen Auftrag als "Erledigt!". |
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|
| |
| </WRAP> | </WRAP> |
| 👉 Hier lernst du, was es heisst, bei einem winzigen neuronalen Netz einen Lernschritt mit einem einzigen Sample durchzuführen und erlebst, wie dabei der Fehler minimiert wird. | 👉 Hier lernst du, was es heisst, bei einem winzigen neuronalen Netz einen Lernschritt mit einem einzigen Sample durchzuführen und erlebst, wie dabei der Fehler minimiert wird. |
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| ⚠️ In diesem Auftrag rechnest du einen Vorwärts-, dann einen Rückwärts- und anschliessend einen weiteren Vorwärtsschritt für das kleine FNN aus Abb.3 für ein einzigs Sample durch. Alle Formeln, welche du dafür brauchst, sind in diesem Auftrag enthalten. Ebenso gegeben sind die Eingangsgrössen $x_1$ und $x_2$ des Samples zusammen mit dem Desired Output $d_1$, alle Gewichte $v_{hi}[k]$ und $w_{ij}[k]$ zum Zeitpunkt $k$ und die Lernrate $\mu$. Alle weiteren Zahlen, welche du brauchst, wirst du Schritt für Schritt berechnen. | ⚠️ In diesem Auftrag rechnest du einen Vorwärts-, dann einen Rückwärts- und anschliessend einen weiteren Vorwärtsschritt für das kleine FNN aus Abb.3 für ein einziges Sample durch. Alle Formeln, welche du dafür brauchst, sind in diesem Auftrag enthalten. Ebenso gegeben sind die Eingangsgrössen $x_1$ und $x_2$ des Samples zusammen mit dem Desired Output $d_1$, alle Gewichte $v_{hi}[k]$ und $w_{ij}[k]$ zum Zeitpunkt $k$ und die Lernrate $\mu$. Alle weiteren Zahlen, welche du brauchst, wirst du Schritt für Schritt berechnen. |
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| - Berechne alle Zahlen und trage sie in die Textfelder ein. | - Berechne alle Zahlen ⚠️ auf drei Kommastellen genau und trage sie in die Textfelder ein. ⚠️ Rechne jeweils mit den auf drei Kommastellen gerundeten Zahlen weiter. |
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| \\ | \\ |
| | $a_1[k] = \displaystyle\sum_{h=0}^{n_i} v_{h1}[k] \cdot x_h = v_{01}[k] + v_{11}[k]\cdot x_1 + v_{21}[k]\cdot x_2 =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0ArKQOwgC+QA#3017266837e2a42a}} | | | $a_1[k] = \displaystyle\sum_{h=0}^{n_i} v_{h1}[k] \cdot x_h = v_{01}[k] + v_{11}[k]\cdot x_1 + v_{21}[k]\cdot x_2 =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0ArKQOwgC+QA#3017266837e2a42a}} | |
| | $\phi_1[k](.) = \displaystyle\frac{1}{1 + e^{-a_1[k]}} =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMAbACwgC+QA#8157b9f2738d5bad}} | | | $\phi_1[k](.) = \displaystyle\frac{1}{1 + e^{-a_1[k]}} =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMAbACwgC+QA#8157b9f2738d5bad}} | |
| | $y_1[k] = \displaystyle\sum_{i=0}^{n_h} w_{i1}[k] \cdot \phi_i[k](.) = w_{01}[k] + w_{11}[k]\cdot \phi_1[k](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMArPiAL5A#33570ee52b21cbaf}} | | | $y_1[k] = \displaystyle\sum_{i=0}^{n_h} w_{i1}[k] \cdot \phi_i[k](.) = w_{01}[k] + w_{11}[k]\cdot \phi_1[k](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMArANr4CMAuiAL5A#33570ee52b21cbaf}}| |
| | $J[k] = \displaystyle\frac{1}{2} (y_1[k]-d_1)^2 = $ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAjAHQAMArMSAL5A#572430543d67a8d1}} | | | $J[k] = \displaystyle\frac{1}{2} (y_1[k]-d_1)^2 = $ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAjAHQAMArMSAL5A#572430543d67a8d1}} | |
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| **Rückwärtsschritt zum Zeitpunkt [k] inklusive Gewicht-Update ** | **Rückwärtsschritt zum Zeitpunkt [k] inklusive Gewicht-Update ** |
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| | $(y_1[k]-d_1) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAIwB0ALAKwAMIAvkA#afc5eca7b986aab6}} | | | $(y_1[k]-d_1) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAIwB0ALKQJwgC+QA#afc5eca7b986aab6}}| |
| | $\phi'_1[k](.)= \phi_1[k](.) \cdot (1 - \phi_1[k](.)) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAjAHQAMIAvkA#8a6433a6600f1ecd}} | | | $\phi'_1[k](.)= \phi_1[k](.) \cdot (1 - \phi_1[k](.)) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQBMAzANoCMxAuiAL5A#8a6433a6600f1ecd}} | |
| | $\displaystyle\frac{\partial J[k]}{\partial w_{01}} = (y_1[k] - d_1) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAIwB0ALAKwAMIAvkA#3e1e02404ff2b201}} | | | $\displaystyle\frac{\partial J[k]}{\partial w_{01}} = (y_1[k] - d_1) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAIwB0ALKQJwgC+QA#3e1e02404ff2b201}} | |
| | $\displaystyle\frac{\partial J[k]}{\partial w_{11}} = (y_1[k] - d_1) \cdot \phi_1[k](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0ArAEwAcIAvkA#a5e510e211dd60dd}} | | | $\displaystyle\frac{\partial J[k]}{\partial w_{11}} = (y_1[k] - d_1) \cdot \phi_1[k](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0ArAEwDsIAvkA#a5e510e211dd60dd}} | |
| | $\displaystyle\frac{\partial J[k]}{\partial v_{01}} = (y_1[k] - d_1) \cdot w_{11}[k] \cdot \phi'_1[k](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQBMAHAJwgC+QA#8b42d6ec22e5a0e2}} | | | $\displaystyle\frac{\partial J[k]}{\partial v_{01}} = (y_1[k] - d_1) \cdot w_{11}[k] \cdot \phi'_1[k](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQBMAHANqkCcAuiAL5A#8b42d6ec22e5a0e2}} | |
| | $\displaystyle\frac{\partial J[k]}{\partial v_{11}} = (y_1[k] - d_1) \cdot w_{11}[k] \cdot \phi'_1[k](.) \cdot x_1[k] =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMALAOwgC+QA#f56b25c1d90bfd03}} | | | $\displaystyle\frac{\partial J[k]}{\partial v_{11}} = (y_1[k] - d_1) \cdot w_{11}[k] \cdot \phi'_1[k](.) \cdot x_1[k] =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMALANoBsA7ALogC+QA#f56b25c1d90bfd03}} | |
| | $\displaystyle\frac{\partial J[k]}{\partial v_{21}} = (y_1[k] - d_1) \cdot w_{11}[k] \cdot \phi'_1[k](.) \cdot x_2[k] =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0ATIaSAL5A#d160dc27095da0cb}} | | | $\displaystyle\frac{\partial J[k]}{\partial v_{21}} = (y_1[k] - d_1) \cdot w_{11}[k] \cdot \phi'_1[k](.) \cdot x_2[k] =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0ATIaSAL5A#d160dc27095da0cb}} | |
| | $w_{01}[k+1] = w_{01}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial w_{01}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQCcArACwgC+QA#eecf21da3b4d3638}} | | | $w_{01}[k+1] = w_{01}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial w_{01}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQCcArANoDMALALogC+QA#eecf21da3b4d3638}} | |
| | $w_{11}[k+1] = w_{11}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial w_{11}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0A7AKzkgC+QA#0f0c95084fd03ed1}} | | | $w_{11}[k+1] = w_{11}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial w_{11}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0A7AKwDa5AbALogC+QA#0f0c95084fd03ed1}} | |
| | $v_{01}[k+1] = v_{01}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial v_{01}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQCMA7KSAL5A#cae0163ecf055c43}} | | | $v_{01}[k+1] = v_{01}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial v_{01}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQCMA7ANqkAcAuiAL5A#cae0163ecf055c43}} | |
| | $v_{11}[k+1] = v_{11}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial v_{11}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0AzAJwBMIAvkA#58d65877583d207f}} | | | $v_{11}[k+1] = v_{11}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial v_{11}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0AzAJwBMIAvkA#58d65877583d207f}} | |
| | $v_{21}[k+1] = v_{21}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial v_{21}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQBsAjABwgC+QA#bbff551455d4ad14}} | | | $v_{21}[k+1] = v_{21}[k] - \mu \cdot \displaystyle\frac{\partial J[k]}{\partial v_{21}}=$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQBsAjABwgC+QA#bbff551455d4ad14}} | |
| **Vorwärtsschritt zum Zeitpunkt [k] inklusive Berechnung des Fehlerwerts ** | **Vorwärtsschritt zum Zeitpunkt [k] inklusive Berechnung des Fehlerwerts ** |
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| | $a_1[k+1] = \displaystyle\sum_{h=0}^{n_i} v_{h1}[k+1] \cdot x_h = v_{01}[k+1] + v_{11}[k+1]\cdot x_1 + v_{21}[k+1]\cdot x_2 =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0A7AEykgC+QA#a1e3255cf55bfbfd}} | | | $a_1[k+1] = \displaystyle\sum_{h=0}^{n_i} v_{h1}[k+1] \cdot x_h = v_{01}[k+1] + v_{11}[k+1]\cdot x_1 + v_{21}[k+1]\cdot x_2 =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iAtAAwB0A7AEwDaArAGwC6IAvkA#a1e3255cf55bfbfd}} | |
| | $\phi_1[k+1](.) = \displaystyle\frac{1}{1 + e^{-a_1[k+1]}} =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMATAGwgC+QA#34d89511cfbb4c40}} | | | $\phi_1[k+1](.) = \displaystyle\frac{1}{1 + e^{-a_1[k+1]}} =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDMATANoCsAbALogC+QA#34d89511cfbb4c40}} | |
| | $y_1[k+1] = \displaystyle\sum_{i=0}^{n_h} w_{i1}[k+1] \cdot \phi_i[k+1](.) = w_{01}[k+1] + w_{11}[k+1]\cdot \phi_1[k+1](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDs+AHCAL5A#8fba5b9000846eb1}} | | | $y_1[k+1] = \displaystyle\sum_{i=0}^{n_h} w_{i1}[k+1] \cdot \phi_i[k+1](.) = w_{01}[k+1] + w_{11}[k+1]\cdot \phi_1[k+1](.) =$ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQDs+A2sQBwC6IAvkA#8fba5b9000846eb1}} | |
| | $J[k+1] = \displaystyle\frac{1}{2} (y_1[k+1]-d_1)^2 = $ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQCsAnAOwgC+QA#188774b08b6a540e}} | | | $J[k+1] = \displaystyle\frac{1}{2} (y_1[k+1]-d_1)^2 = $ | {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQCsAnANoBsA7ALogC+QA#188774b08b6a540e}} | |
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| **Vergleich der Fehlerwerte $J[k+1]$ und $J[k]$** | **Vergleich der Fehlerwerte $J[k+1]$ und $J[k]$** |
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| Der Fehler hat durch das Training um so viel abgenommen (d.h. trage das Resultat von $J[k] - J[k+1]$ ein): {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQAsArAMwgC+QA#9dff039c86a546be}} | Der Fehler hat durch das Training um so viel abgenommen (d.h. trage das Resultat von $J[k] - J[k+1]$ ein): {{gem/match?0=N4IgLgpgHmIFwhAGhAJwgc2gB3iADAHQAsArANoDMxAuiAL5A#9dff039c86a546be}} |
| </WRAP> | </WRAP> |
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