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umz21/wyk/2011_Wielowarstwowe_sieci_neuronowe.ipynb

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Materiały na podstawie ubiegłorocznych
2021-03-02 08:32:40 +01:00
{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Uczenie maszynowe UMZ 2019/2020\n",
"### 26 maja 2020\n",
"# 11. Wielowarstwowe sieci neuronowe i algorytmy optymalizacji"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 11.1. Funkcje aktywacji"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"* Każda funkcja aktywacji ma swoje zalety i wady.\n",
"* Różne rodzaje funkcji aktywacji nadają się do różnych zastosowań."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/home/pawel/.local/lib/python2.7/site-packages/h5py/__init__.py:36: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.\n",
" from ._conv import register_converters as _register_converters\n",
"Using TensorFlow backend.\n"
]
}
],
"source": [
"%matplotlib inline\n",
"\n",
"import math\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import random\n",
"\n",
"import keras\n",
"from keras.datasets import mnist\n",
"from keras.models import Sequential\n",
"from keras.layers import Dense, Dropout, SimpleRNN, LSTM\n",
"from keras.optimizers import Adagrad, Adam, RMSprop, SGD\n",
"\n",
"from IPython.display import YouTubeVideo"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [],
"source": [
"def plot(fun):\n",
" x = np.arange(-3.0, 3.0, 0.01)\n",
" y = [fun(x_i) for x_i in x]\n",
" fig = plt.figure(figsize=(14, 7))\n",
" ax = fig.add_subplot(111)\n",
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
" ax.set_xlim(-3.0, 3.0)\n",
" ax.set_ylim(-1.5, 1.5)\n",
" ax.grid()\n",
" ax.plot(x, y)\n",
" plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Funkcja logistyczna"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$ g(x) = \\frac{1}{1 + e^{-x}} $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"* Przyjmuje wartości z przedziału $(0, 1)$."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### Funkcja logistyczna – wykres"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
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"text/plain": [
"<matplotlib.figure.Figure at 0x7fdda9490fd0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plot(lambda x: 1 / (1 + math.exp(-x)))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Tangens hiperboliczny"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$ g(x) = \\tanh x = \\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"* Przyjmuje wartości z przedziału $(-1, 1)$.\n",
"* Powstaje z funkcji logistycznej przez przeskalowanie i przesunięcie."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### Tangens hiperboliczny – wykres"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
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"text/plain": [
"<matplotlib.figure.Figure at 0x7fdda93e9590>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plot(lambda x: math.tanh(x))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### ReLU (_Rectifier Linear Unit_)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$ g(x) = \\max(0, x) $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### ReLU – zalety\n",
"* Mniej podatna na problem zanikającego gradientu (_vanishing gradient_) niż funkcje sigmoidalne, dzięki czemu SGD jest szybciej zbieżna.\n",
"* Prostsze obliczanie gradientu.\n",
"* Dzięki zerowaniu ujemnych wartości, wygasza neurony, „rozrzedzając” sieć (_sparsity_), co przyspiesza obliczenia."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"#### ReLU – wady\n",
"* Dla dużych wartości gradient może „eksplodować”.\n",
"* „Wygaszanie” neuronów."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### ReLU – wykres"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAA1cAAAG2CAYAAACTRXz+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nCoCVJawA6ElcAbCShBU
"text/plain": [
"<matplotlib.figure.Figure at 0x7fdda936c6d0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plot(lambda x: max(0, x))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Softplus"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$ g(x) = \\log(1 + e^{x}) $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"* Wygładzona wersja ReLU."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### Softplus – wykres"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAA1cAAAG2CAYAAACTRXz+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nG9bGOxpaO3XH0k26b9l
"text/plain": [
"<matplotlib.figure.Figure at 0x7fdde8452e10>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plot(lambda x: math.log(1 + math.exp(x)))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Problem zanikającego gradientu (_vanishing gradient problem_)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"* Sigmoidalne funkcje aktywacji ograniczają wartości na wyjściach neuronów do niewielkich przedziałów ($(-1, 1)$, $(0, 1)$ itp.).\n",
"* Jeżeli sieć ma wiele warstw, to podczas propagacji wstecznej mnożymy przez siebie wiele małych wartości → obliczony gradient jest mały.\n",
"* Im więcej warstw, tym silniejszy efekt zanikania."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"#### Sposoby na zanikający gradient\n",
"\n",
"* Modyfikacja algorytmu optymalizacji (_RProp_, _RMSProp_)\n",
"* Użycie innej funckji aktywacji (ReLU, softplus)\n",
"* Dodanie warstw _dropout_\n",
"* Nowe architektury (LSTM itp.)\n",
"* Więcej danych, zwiększenie mocy obliczeniowej"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 11.2. Wielowarstwowe sieci neuronowe w&nbsp;praktyce"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Przykład: MNIST\n",
"\n",
"_Modified National Institute of Standards and Technology database_"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"* Zbiór cyfr zapisanych pismem odręcznym\n",
"* 60 000 przykładów uczących, 10 000 przykładów testowych\n",
"* Rozdzielczość każdego przykładu: 28 × 28 = 784 piksele"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"# źródło: https://github.com/keras-team/keras/examples/minst_mlp.py\n",
"\n",
"import keras\n",
"from keras.datasets import mnist\n",
"\n",
"# załaduj dane i podziel je na zbiory uczący i testowy\n",
"(x_train, y_train), (x_test, y_test) = mnist.load_data()"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [],
"source": [
"def draw_examples(examples, captions=None):\n",
" plt.figure(figsize=(16, 4))\n",
" m = len(examples)\n",
" for i, example in enumerate(examples):\n",
" plt.subplot(100 + m * 10 + i + 1)\n",
" plt.imshow(example, cmap=plt.get_cmap('gray'))\n",
" plt.show()\n",
" if captions is not None:\n",
" print(6 * ' ' + (10 * ' ').join(str(captions[i]) for i in range(m)))"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
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"text/plain": [
"<matplotlib.figure.Figure at 0x7fdda922aad0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" 5 0 4 1 9 2 1\n"
]
}
],
"source": [
"draw_examples(x_train[:7], captions=y_train)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"60000 przykładów uczących\n",
"10000 przykładów testowych\n"
]
}
],
"source": [
"num_classes = 10\n",
"\n",
"x_train = x_train.reshape(60000, 784) # 784 = 28 * 28\n",
"x_test = x_test.reshape(10000, 784)\n",
"x_train = x_train.astype('float32')\n",
"x_test = x_test.astype('float32')\n",
"x_train /= 255\n",
"x_test /= 255\n",
"print('{} przykładów uczących'.format(x_train.shape[0]))\n",
"print('{} przykładów testowych'.format(x_test.shape[0]))\n",
"\n",
"# przekonwertuj wektory klas na binarne macierze klas\n",
"y_train = keras.utils.to_categorical(y_train, num_classes)\n",
"y_test = keras.utils.to_categorical(y_test, num_classes)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"scrolled": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"_________________________________________________________________\n",
"Layer (type) Output Shape Param # \n",
"=================================================================\n",
"dense_1 (Dense) (None, 512) 401920 \n",
"_________________________________________________________________\n",
"dropout_1 (Dropout) (None, 512) 0 \n",
"_________________________________________________________________\n",
"dense_2 (Dense) (None, 512) 262656 \n",
"_________________________________________________________________\n",
"dropout_2 (Dropout) (None, 512) 0 \n",
"_________________________________________________________________\n",
"dense_3 (Dense) (None, 10) 5130 \n",
"=================================================================\n",
"Total params: 669,706\n",
"Trainable params: 669,706\n",
"Non-trainable params: 0\n",
"_________________________________________________________________\n"
]
}
],
"source": [
"model = Sequential()\n",
"model.add(Dense(512, activation='relu', input_shape=(784,)))\n",
"model.add(Dropout(0.2))\n",
"model.add(Dense(512, activation='relu'))\n",
"model.add(Dropout(0.2))\n",
"model.add(Dense(num_classes, activation='softmax'))\n",
"model.summary()"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"((60000, 784), (60000, 10))\n"
]
}
],
"source": [
"print(x_train.shape, y_train.shape)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Train on 60000 samples, validate on 10000 samples\n",
"Epoch 1/5\n",
"60000/60000 [==============================] - 9s 153us/step - loss: 0.2489 - acc: 0.9224 - val_loss: 0.1005 - val_acc: 0.9706\n",
"Epoch 2/5\n",
"60000/60000 [==============================] - 9s 151us/step - loss: 0.1042 - acc: 0.9683 - val_loss: 0.0861 - val_acc: 0.9740\n",
"Epoch 3/5\n",
"60000/60000 [==============================] - 9s 153us/step - loss: 0.0742 - acc: 0.9782 - val_loss: 0.0733 - val_acc: 0.9796\n",
"Epoch 4/5\n",
"60000/60000 [==============================] - 9s 154us/step - loss: 0.0603 - acc: 0.9824 - val_loss: 0.0713 - val_acc: 0.9800\n",
"Epoch 5/5\n",
"60000/60000 [==============================] - 9s 157us/step - loss: 0.0512 - acc: 0.9848 - val_loss: 0.0749 - val_acc: 0.9795\n"
]
},
{
"data": {
"text/plain": [
"<keras.callbacks.History at 0x7fdda4f97110>"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model.compile(loss='categorical_crossentropy', optimizer=RMSprop(), metrics=['accuracy'])\n",
"\n",
"model.fit(x_train, y_train, batch_size=128, epochs=5, verbose=1,\n",
" validation_data=(x_test, y_test))"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Test loss: 0.074858742202\n",
"Test accuracy: 0.9795\n"
]
}
],
"source": [
"score = model.evaluate(x_test, y_test, verbose=0)\n",
"\n",
"print('Test loss: {}'.format(score[0]))\n",
"print('Test accuracy: {}'.format(score[1]))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Warstwa _dropout_ to metoda regularyzacji, służy zapobieganiu nadmiernemu dopasowaniu sieci. Polega na tym, że część węzłów sieci jest usuwana w sposób losowy."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"_________________________________________________________________\n",
"Layer (type) Output Shape Param # \n",
"=================================================================\n",
"dense_4 (Dense) (None, 512) 401920 \n",
"_________________________________________________________________\n",
"dense_5 (Dense) (None, 512) 262656 \n",
"_________________________________________________________________\n",
"dense_6 (Dense) (None, 10) 5130 \n",
"=================================================================\n",
"Total params: 669,706\n",
"Trainable params: 669,706\n",
"Non-trainable params: 0\n",
"_________________________________________________________________\n",
"Train on 60000 samples, validate on 10000 samples\n",
"Epoch 1/5\n",
"60000/60000 [==============================] - 8s 139us/step - loss: 0.2237 - acc: 0.9303 - val_loss: 0.0998 - val_acc: 0.9676\n",
"Epoch 2/5\n",
"60000/60000 [==============================] - 8s 136us/step - loss: 0.0818 - acc: 0.9748 - val_loss: 0.0788 - val_acc: 0.9770\n",
"Epoch 3/5\n",
"60000/60000 [==============================] - 8s 136us/step - loss: 0.0538 - acc: 0.9831 - val_loss: 0.1074 - val_acc: 0.9695\n",
"Epoch 4/5\n",
"60000/60000 [==============================] - 10s 161us/step - loss: 0.0397 - acc: 0.9879 - val_loss: 0.0871 - val_acc: 0.9763\n",
"Epoch 5/5\n",
"60000/60000 [==============================] - 12s 195us/step - loss: 0.0299 - acc: 0.9910 - val_loss: 0.0753 - val_acc: 0.9812\n"
]
},
{
"data": {
"text/plain": [
"<keras.callbacks.History at 0x7fdda3dcad50>"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Bez warstw Dropout\n",
"\n",
"num_classes = 10\n",
"\n",
"(x_train, y_train), (x_test, y_test) = mnist.load_data()\n",
"\n",
"x_train = x_train.reshape(60000, 784) # 784 = 28 * 28\n",
"x_test = x_test.reshape(10000, 784)\n",
"x_train = x_train.astype('float32')\n",
"x_test = x_test.astype('float32')\n",
"x_train /= 255\n",
"x_test /= 255\n",
"\n",
"y_train = keras.utils.to_categorical(y_train, num_classes)\n",
"y_test = keras.utils.to_categorical(y_test, num_classes)\n",
"\n",
"model_no_dropout = Sequential()\n",
"model_no_dropout.add(Dense(512, activation='relu', input_shape=(784,)))\n",
"model_no_dropout.add(Dense(512, activation='relu'))\n",
"model_no_dropout.add(Dense(num_classes, activation='softmax'))\n",
"model_no_dropout.summary()\n",
"\n",
"model_no_dropout.compile(loss='categorical_crossentropy',\n",
" optimizer=RMSprop(),\n",
" metrics=['accuracy'])\n",
"\n",
"model_no_dropout.fit(x_train, y_train,\n",
" batch_size=128,\n",
" epochs=5,\n",
" verbose=1,\n",
" validation_data=(x_test, y_test))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Test loss (no dropout): 0.0753162465898\n",
"Test accuracy (no dropout): 0.9812\n"
]
}
],
"source": [
"# Bez warstw Dropout\n",
"\n",
"score = model_no_dropout.evaluate(x_test, y_test, verbose=0)\n",
"\n",
"print('Test loss (no dropout): {}'.format(score[0]))\n",
"print('Test accuracy (no dropout): {}'.format(score[1]))"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"_________________________________________________________________\n",
"Layer (type) Output Shape Param # \n",
"=================================================================\n",
"dense_7 (Dense) (None, 2500) 1962500 \n",
"_________________________________________________________________\n",
"dense_8 (Dense) (None, 2000) 5002000 \n",
"_________________________________________________________________\n",
"dense_9 (Dense) (None, 1500) 3001500 \n",
"_________________________________________________________________\n",
"dense_10 (Dense) (None, 1000) 1501000 \n",
"_________________________________________________________________\n",
"dense_11 (Dense) (None, 500) 500500 \n",
"_________________________________________________________________\n",
"dense_12 (Dense) (None, 10) 5010 \n",
"=================================================================\n",
"Total params: 11,972,510\n",
"Trainable params: 11,972,510\n",
"Non-trainable params: 0\n",
"_________________________________________________________________\n",
"Train on 60000 samples, validate on 10000 samples\n",
"Epoch 1/10\n",
"60000/60000 [==============================] - 145s 2ms/step - loss: 1.4242 - acc: 0.5348 - val_loss: 0.4426 - val_acc: 0.8638\n",
"Epoch 2/10\n",
"60000/60000 [==============================] - 140s 2ms/step - loss: 0.3245 - acc: 0.9074 - val_loss: 0.2231 - val_acc: 0.9360\n",
"Epoch 3/10\n",
"60000/60000 [==============================] - 137s 2ms/step - loss: 0.1993 - acc: 0.9420 - val_loss: 0.1694 - val_acc: 0.9485\n",
"Epoch 4/10\n",
"60000/60000 [==============================] - 136s 2ms/step - loss: 0.1471 - acc: 0.9571 - val_loss: 0.1986 - val_acc: 0.9381\n",
"Epoch 5/10\n",
"60000/60000 [==============================] - 132s 2ms/step - loss: 0.1189 - acc: 0.9650 - val_loss: 0.1208 - val_acc: 0.9658\n",
"Epoch 6/10\n",
"60000/60000 [==============================] - 131s 2ms/step - loss: 0.0983 - acc: 0.9711 - val_loss: 0.1260 - val_acc: 0.9637\n",
"Epoch 7/10\n",
"60000/60000 [==============================] - 129s 2ms/step - loss: 0.0818 - acc: 0.9753 - val_loss: 0.0984 - val_acc: 0.9727\n",
"Epoch 8/10\n",
"60000/60000 [==============================] - 129s 2ms/step - loss: 0.0710 - acc: 0.9784 - val_loss: 0.1406 - val_acc: 0.9597\n",
"Epoch 9/10\n",
"60000/60000 [==============================] - 129s 2ms/step - loss: 0.0611 - acc: 0.9811 - val_loss: 0.0987 - val_acc: 0.9727\n",
"Epoch 10/10\n",
"60000/60000 [==============================] - 136s 2ms/step - loss: 0.0533 - acc: 0.9837 - val_loss: 0.1070 - val_acc: 0.9718\n"
]
},
{
"data": {
"text/plain": [
"<keras.callbacks.History at 0x7fdd95c86610>"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Więcej warstw, inna funkcja aktywacji\n",
"\n",
"num_classes = 10\n",
"\n",
"(x_train, y_train), (x_test, y_test) = mnist.load_data()\n",
"\n",
"x_train = x_train.reshape(60000, 784) # 784 = 28 * 28\n",
"x_test = x_test.reshape(10000, 784)\n",
"x_train = x_train.astype('float32')\n",
"x_test = x_test.astype('float32')\n",
"x_train /= 255\n",
"x_test /= 255\n",
"\n",
"y_train = keras.utils.to_categorical(y_train, num_classes)\n",
"y_test = keras.utils.to_categorical(y_test, num_classes)\n",
"\n",
"model3 = Sequential()\n",
"model3.add(Dense(2500, activation='tanh', input_shape=(784,)))\n",
"model3.add(Dense(2000, activation='tanh'))\n",
"model3.add(Dense(1500, activation='tanh'))\n",
"model3.add(Dense(1000, activation='tanh'))\n",
"model3.add(Dense(500, activation='tanh'))\n",
"model3.add(Dense(num_classes, activation='softmax'))\n",
"model3.summary()\n",
"\n",
"model3.compile(loss='categorical_crossentropy',\n",
" optimizer=RMSprop(),\n",
" metrics=['accuracy'])\n",
"\n",
"model3.fit(x_train, y_train,\n",
" batch_size=128,\n",
" epochs=10,\n",
" verbose=1,\n",
" validation_data=(x_test, y_test))"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Test loss: 0.107020105763\n",
"Test accuracy: 0.9718\n"
]
}
],
"source": [
"# Więcej warstw, inna funkcja aktywacji\n",
"\n",
"score = model3.evaluate(x_test, y_test, verbose=0)\n",
"\n",
"print('Test loss: {}'.format(score[0]))\n",
"print('Test accuracy: {}'.format(score[1]))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Przykład: 4-pikselowy aparat fotograficzny"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
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"text/html": [
"\n",
" <iframe\n",
" width=\"800\"\n",
" height=\"600\"\n",
" src=\"https://www.youtube.com/embed/ILsA4nyG7I0\"\n",
" frameborder=\"0\"\n",
" allowfullscreen\n",
" ></iframe>\n",
" "
],
"text/plain": [
"<IPython.lib.display.YouTubeVideo at 0x7f4d3ff03110>"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"YouTubeVideo('ILsA4nyG7I0', width=800, height=600)"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"def generate_example(description):\n",
" variant = random.choice([1, -1])\n",
" if description == 's': # solid\n",
" return (np.array([[ 1.0, 1.0], [ 1.0, 1.0]]) if variant == 1 else\n",
" np.array([[-1.0, -1.0], [-1.0, -1.0]]))\n",
" elif description == 'v': # vertical\n",
" return (np.array([[ 1.0, -1.0], [ 1.0, -1.0]]) if variant == 1 else\n",
" np.array([[-1.0, 1.0], [-1.0, 1.0]]))\n",
" elif description == 'd': # diagonal\n",
" return (np.array([[ 1.0, -1.0], [-1.0, 1.0]]) if variant == 1 else\n",
" np.array([[-1.0, 1.0], [ 1.0, -1.0]]))\n",
" elif description == 'h': # horizontal\n",
" return (np.array([[ 1.0, 1.0], [-1.0, -1.0]]) if variant == 1 else\n",
" np.array([[-1.0, -1.0], [ 1.0, 1.0]]))\n",
" else:\n",
" return np.array([[random.uniform(-1, 1), random.uniform(-1, 1)],\n",
" [random.uniform(-1, 1), random.uniform(-1, 1)]])"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"num_classes = 4\n",
"\n",
"trainset_size = 4000\n",
"testset_size = 1000\n",
"\n",
"y4_train = np.array([random.choice(['s', 'v', 'd', 'h']) for i in range(trainset_size)])\n",
"x4_train = np.array([generate_example(desc) for desc in y4_train])\n",
"\n",
"y4_test = np.array([random.choice(['s', 'v', 'd', 'h']) for i in range(testset_size)])\n",
"x4_test = np.array([generate_example(desc) for desc in y4_test])"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAA60AAACQCAYAAADjqY0xAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n+EkdMK/DIz3PE0n+NMl
"text/plain": [
"<matplotlib.figure.Figure at 0x7f4d3ffc2ed0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
" s s d s h s v\n"
]
}
],
"source": [
"draw_examples(x4_train[:7], captions=y4_train)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"x4_train = x4_train.reshape(trainset_size, 4)\n",
"x4_test = x4_test.reshape(testset_size, 4)\n",
"x4_train = x4_train.astype('float32')\n",
"x4_test = x4_test.astype('float32')\n",
"\n",
"y4_train = np.array([{'s': 0, 'v': 1, 'd': 2, 'h': 3}[desc] for desc in y4_train])\n",
"y4_test = np.array([{'s': 0, 'v': 1, 'd': 2, 'h': 3}[desc] for desc in y4_test])\n",
"\n",
"y4_train = keras.utils.to_categorical(y4_train, num_classes)\n",
"y4_test = keras.utils.to_categorical(y4_test, num_classes)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"_________________________________________________________________\n",
"Layer (type) Output Shape Param # \n",
"=================================================================\n",
"dense_16 (Dense) (None, 4) 20 \n",
"_________________________________________________________________\n",
"dense_17 (Dense) (None, 4) 20 \n",
"_________________________________________________________________\n",
"dense_18 (Dense) (None, 8) 40 \n",
"_________________________________________________________________\n",
"dense_19 (Dense) (None, 4) 36 \n",
"=================================================================\n",
"Total params: 116\n",
"Trainable params: 116\n",
"Non-trainable params: 0\n",
"_________________________________________________________________\n"
]
}
],
"source": [
"model4 = Sequential()\n",
"model4.add(Dense(4, activation='tanh', input_shape=(4,)))\n",
"model4.add(Dense(4, activation='tanh'))\n",
"model4.add(Dense(8, activation='relu'))\n",
"model4.add(Dense(num_classes, activation='softmax'))\n",
"model4.summary()"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"model4.layers[0].set_weights(\n",
" [np.array([[ 1.0, 0.0, 1.0, 0.0],\n",
" [ 0.0, 1.0, 0.0, 1.0],\n",
" [ 1.0, 0.0, -1.0, 0.0],\n",
" [ 0.0, 1.0, 0.0, -1.0]],\n",
" dtype=np.float32), np.array([0., 0., 0., 0.], dtype=np.float32)])\n",
"model4.layers[1].set_weights(\n",
" [np.array([[ 1.0, -1.0, 0.0, 0.0],\n",
" [ 1.0, 1.0, 0.0, 0.0],\n",
" [ 0.0, 0.0, 1.0, -1.0],\n",
" [ 0.0, 0.0, -1.0, -1.0]],\n",
" dtype=np.float32), np.array([0., 0., 0., 0.], dtype=np.float32)])\n",
"model4.layers[2].set_weights(\n",
" [np.array([[ 1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0],\n",
" [ 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 0.0, 0.0],\n",
" [ 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0],\n",
" [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0]],\n",
" dtype=np.float32), np.array([0., 0., 0., 0., 0., 0., 0., 0.], dtype=np.float32)])"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"model4.layers[3].set_weights(\n",
" [np.array([[ 1.0, 0.0, 0.0, 0.0],\n",
" [ 1.0, 0.0, 0.0, 0.0],\n",
" [ 0.0, 1.0, 0.0, 0.0],\n",
" [ 0.0, 1.0, 0.0, 0.0],\n",
" [ 0.0, 0.0, 1.0, 0.0],\n",
" [ 0.0, 0.0, 1.0, 0.0],\n",
" [ 0.0, 0.0, 0.0, 1.0],\n",
" [ 0.0, 0.0, 0.0, 1.0]],\n",
" dtype=np.float32), np.array([0., 0., 0., 0.], dtype=np.float32)])\n",
"\n",
"model4.compile(loss='categorical_crossentropy',\n",
" optimizer=Adagrad(),\n",
" metrics=['accuracy'])"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[array([[ 1., 0., 1., 0.],\n",
" [ 0., 1., 0., 1.],\n",
" [ 1., 0., -1., 0.],\n",
" [ 0., 1., 0., -1.]], dtype=float32), array([ 0., 0., 0., 0.], dtype=float32)]\n",
"[array([[ 1., -1., 0., 0.],\n",
" [ 1., 1., 0., 0.],\n",
" [ 0., 0., 1., -1.],\n",
" [ 0., 0., -1., -1.]], dtype=float32), array([ 0., 0., 0., 0.], dtype=float32)]\n",
"[array([[ 1., -1., 0., 0., 0., 0., 0., 0.],\n",
" [ 0., 0., 1., -1., 0., 0., 0., 0.],\n",
" [ 0., 0., 0., 0., 1., -1., 0., 0.],\n",
" [ 0., 0., 0., 0., 0., 0., 1., -1.]], dtype=float32), array([ 0., 0., 0., 0., 0., 0., 0., 0.], dtype=float32)]\n",
"[array([[ 1., 0., 0., 0.],\n",
" [ 1., 0., 0., 0.],\n",
" [ 0., 1., 0., 0.],\n",
" [ 0., 1., 0., 0.],\n",
" [ 0., 0., 1., 0.],\n",
" [ 0., 0., 1., 0.],\n",
" [ 0., 0., 0., 1.],\n",
" [ 0., 0., 0., 1.]], dtype=float32), array([ 0., 0., 0., 0.], dtype=float32)]\n"
]
}
],
"source": [
"for layer in model4.layers:\n",
" print(layer.get_weights())"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 0.17831734, 0.17831734, 0.17831734, 0.46504799]], dtype=float32)"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model4.predict([np.array([[1.0, 1.0], [-1.0, -1.0]]).reshape(1, 4)])"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Test loss: 0.765614629269\n",
"Test accuracy: 1.0\n"
]
}
],
"source": [
"score = model4.evaluate(x4_test, y4_test, verbose=0)\n",
"\n",
"print('Test loss: {}'.format(score[0]))\n",
"print('Test accuracy: {}'.format(score[1]))"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"_________________________________________________________________\n",
"Layer (type) Output Shape Param # \n",
"=================================================================\n",
"dense_20 (Dense) (None, 4) 20 \n",
"_________________________________________________________________\n",
"dense_21 (Dense) (None, 4) 20 \n",
"_________________________________________________________________\n",
"dense_22 (Dense) (None, 8) 40 \n",
"_________________________________________________________________\n",
"dense_23 (Dense) (None, 4) 36 \n",
"=================================================================\n",
"Total params: 116\n",
"Trainable params: 116\n",
"Non-trainable params: 0\n",
"_________________________________________________________________\n"
]
}
],
"source": [
"model5 = Sequential()\n",
"model5.add(Dense(4, activation='tanh', input_shape=(4,)))\n",
"model5.add(Dense(4, activation='tanh'))\n",
"model5.add(Dense(8, activation='relu'))\n",
"model5.add(Dense(num_classes, activation='softmax'))\n",
"model5.compile(loss='categorical_crossentropy',\n",
" optimizer=RMSprop(),\n",
" metrics=['accuracy'])\n",
"model5.summary()"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"scrolled": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Train on 4000 samples, validate on 1000 samples\n",
"Epoch 1/8\n",
"4000/4000 [==============================] - 0s - loss: 1.1352 - acc: 0.5507 - val_loss: 1.0160 - val_acc: 0.7330\n",
"Epoch 2/8\n",
"4000/4000 [==============================] - 0s - loss: 0.8918 - acc: 0.8722 - val_loss: 0.8094 - val_acc: 0.8580\n",
"Epoch 3/8\n",
"4000/4000 [==============================] - 0s - loss: 0.6966 - acc: 0.8810 - val_loss: 0.6283 - val_acc: 0.8580\n",
"Epoch 4/8\n",
"4000/4000 [==============================] - 0s - loss: 0.5284 - acc: 0.8810 - val_loss: 0.4697 - val_acc: 0.8580\n",
"Epoch 5/8\n",
"4000/4000 [==============================] - 0s - loss: 0.3797 - acc: 0.9022 - val_loss: 0.3312 - val_acc: 1.0000\n",
"Epoch 6/8\n",
"4000/4000 [==============================] - 0s - loss: 0.2555 - acc: 1.0000 - val_loss: 0.2166 - val_acc: 1.0000\n",
"Epoch 7/8\n",
"4000/4000 [==============================] - 0s - loss: 0.1612 - acc: 1.0000 - val_loss: 0.1318 - val_acc: 1.0000\n",
"Epoch 8/8\n",
"4000/4000 [==============================] - 0s - loss: 0.0939 - acc: 1.0000 - val_loss: 0.0732 - val_acc: 1.0000\n"
]
},
{
"data": {
"text/plain": [
"<keras.callbacks.History at 0x7f4d34067510>"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model5.fit(x4_train, y4_train, epochs=8, validation_data=(x4_test, y4_test))"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 0.00708295, 0.00192736, 0.02899081, 0.96199888]], dtype=float32)"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model5.predict([np.array([[1.0, 1.0], [-1.0, -1.0]]).reshape(1, 4)])"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Test loss: 0.0731911802292\n",
"Test accuracy: 1.0\n"
]
}
],
"source": [
"score = model5.evaluate(x4_test, y4_test, verbose=0)\n",
"\n",
"print('Test loss: {}'.format(score[0]))\n",
"print('Test accuracy: {}'.format(score[1]))"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [],
"source": [
"import contextlib\n",
"\n",
"@contextlib.contextmanager\n",
"def printoptions(*args, **kwargs):\n",
" original = np.get_printoptions()\n",
" np.set_printoptions(*args, **kwargs)\n",
" try:\n",
" yield\n",
" finally: \n",
" np.set_printoptions(**original)"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[array([[-0.2, -0.5, 0.8, 1. ],\n",
" [-0.9, 0.1, -0.8, 0.2],\n",
" [-0.2, 0.4, 0.1, -0.4],\n",
" [-0.8, 0.8, 1. , 0.3]], dtype=float32), array([ 0. , -0. , 0.1, -0.1], dtype=float32)]\n",
"[array([[-0.4, 0.9, -1.3, 1.7],\n",
" [-0.4, -0.7, 0.3, -0.3],\n",
" [ 0.8, -0.9, -1.1, -0.2],\n",
" [ 1.3, 0.5, 0.4, -0.2]], dtype=float32), array([-0. , -0. , 0.2, 0. ], dtype=float32)]\n",
"[array([[-1.6, 0.3, 0.3, -0.3, -1.1, 1.2, 0.7, -1. ],\n",
" [ 0.4, 1.3, -0.9, 0.8, -0.4, -0.7, -1.2, -1. ],\n",
" [ 0.6, 1. , 0.9, -1. , -1.1, -0.2, -0.4, -0.3],\n",
" [ 1.1, 0.1, -0.9, 1.3, -0.3, -0.2, 0.2, -0.4]], dtype=float32), array([-0. , 0.2, -0.1, 0. , -0.1, -0. , -0.1, 0.1], dtype=float32)]\n",
"[array([[ 0.6, -1.5, 1.3, -1.4],\n",
" [-0.4, -1.6, -0.3, 1.2],\n",
" [ 1.2, 1.1, -0.3, -1.5],\n",
" [ 0.6, 1.4, -1.5, -1.2],\n",
" [ 0.2, -1.3, -0.9, 0.8],\n",
" [ 0.6, -1.5, 0.8, -1. ],\n",
" [ 0.4, -1.3, 0.4, 0.3],\n",
" [-1.3, 0.5, -0.9, 0.8]], dtype=float32), array([-0.8, 0.7, 0.4, 0.1], dtype=float32)]\n"
]
}
],
"source": [
"with printoptions(precision=1, suppress=True):\n",
" for layer in model5.layers:\n",
" print(layer.get_weights())"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 12.3. Odmiany metody gradientu prostego\n",
"\n",
"* Batch gradient descent\n",
"* Stochastic gradient descent\n",
"* Mini-batch gradient descent"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### _Batch gradient descent_\n",
"\n",
"* Klasyczna wersja metody gradientu prostego\n",
"* Obliczamy gradient funkcji kosztu względem całego zbioru treningowego:\n",
" $$ \\theta := \\theta - \\alpha \\cdot \\nabla_\\theta J(\\theta) $$\n",
"* Dlatego może działać bardzo powoli\n",
"* Nie można dodawać nowych przykładów na bieżąco w trakcie trenowania modelu (_online learning_)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### _Stochastic gradient descent_ (SGD)\n",
"\n",
"* Aktualizacja parametrów dla każdego przykładu:\n",
" $$ \\theta := \\theta - \\alpha \\cdot \\nabla_\\theta \\, J \\! \\left( \\theta, x^{(i)}, y^{(i)} \\right) $$\n",
"* Dużo szybszy niż _batch gradient descent_\n",
"* Można dodawać nowe przykłady na bieżąco w trakcie trenowania (_online learning_)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### _Stochastic gradient descent_ (SGD)\n",
"\n",
"* Częsta aktualizacja parametrów z dużą wariancją:\n",
"\n",
"<img src=\"http://ruder.io/content/images/2016/09/sgd_fluctuation.png\" style=\"margin: auto;\" width=\"50%\" />\n",
"\n",
"* Z jednej strony dzięki temu nie utyka w złych minimach lokalnych, ale z drugiej strony może „wyskoczyć” z dobrego minimum"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### _Mini-batch gradient descent_\n",
"\n",
"* Kompromis między _batch gradient descent_ i SGD\n",
" $$ \\theta := \\theta - \\alpha \\cdot \\nabla_\\theta \\, J \\left( \\theta, x^{(i : i+n)}, y^{(i : i_n)} \\right) $$\n",
"* Stabilniejsza zbieżność dzięki redukcji wariancji aktualizacji parametrów\n",
"* Szybszy niż klasyczny _batch gradient descent_\n",
"* Typowa wielkość batcha: między 50 a 256 przykładów"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Wady klasycznej metody gradientu prostego, czyli dlaczego potrzebujemy optymalizacji\n",
"\n",
"* Trudno dobrać właściwą szybkość uczenia (_learning rate_)\n",
"* Jedna ustalona wartość stałej uczenia się dla wszystkich parametrów\n",
"* Funkcja kosztu dla sieci neuronowych nie jest wypukła, więc uczenie może utknąć w złym minimum lokalnym lub punkcie siodłowym"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 12.4. Algorytmy optymalizacji metody gradientu\n",
"\n",
"* Momentum\n",
"* Nesterov Accelerated Gradient\n",
"* Adagrad\n",
"* Adadelta\n",
"* RMSprop\n",
"* Adam\n",
"* Nadam\n",
"* AMSGrad"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Momentum\n",
"\n",
"* SGD źle radzi sobie w „wąwozach” funkcji kosztu\n",
"* Momentum rozwiązuje ten problem przez dodanie współczynnika $\\gamma$, który można trakować jako „pęd” spadającej piłki:\n",
" $$ v_t := \\gamma \\, v_{t-1} + \\alpha \\, \\nabla_\\theta J(\\theta) $$\n",
" $$ \\theta := \\theta - v_t $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Przyspiesony gradient Nesterova (_Nesterov Accelerated Gradient_, NAG)\n",
"\n",
"* Momentum czasami powoduje niekontrolowane rozpędzanie się piłki, przez co staje się „mniej sterowna”\n",
"* Nesterov do piłki posiadającej pęd dodaje „hamulec”, który spowalnia piłkę przed wzniesieniem:\n",
" $$ v_t := \\gamma \\, v_{t-1} + \\alpha \\, \\nabla_\\theta J(\\theta - \\gamma \\, v_{t-1}) $$\n",
" $$ \\theta := \\theta - v_t $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Adagrad\n",
"\n",
"* “<b>Ada</b>ptive <b>grad</b>ient”\n",
"* Adagrad dostosowuje współczynnik uczenia (_learning rate_) do parametrów: zmniejsza go dla cech występujących częściej, a zwiększa dla występujących rzadziej\n",
"* Świetny do trenowania na rzadkich (_sparse_) zbiorach danych\n",
"* Wada: współczynnik uczenia może czasami gwałtownie maleć"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Adadelta i RMSprop\n",
"* Warianty algorytmu Adagrad, które radzą sobie z problemem gwałtownych zmian współczynnika uczenia"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Adam\n",
"\n",
"* “<b>Ada</b>ptive <b>m</b>oment estimation”\n",
"* Łączy zalety algorytmów RMSprop i Momentum\n",
"* Można go porównać do piłki mającej ciężar i opór\n",
"* Obecnie jeden z najpopularniejszych algorytmów optymalizacji"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Nadam\n",
"* “<b>N</b>esterov-accelerated <b>ada</b>ptive <b>m</b>oment estimation”\n",
"* Łączy zalety algorytmów Adam i Nesterov Accelerated Gradient"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### AMSGrad\n",
"* Wariant algorytmu Adam lepiej dostosowany do zadań takich jak rozpoznawanie obiektów czy tłumaczenie maszynowe"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"<img src=\"contours_evaluation_optimizers.gif\" style=\"margin: auto;\" width=\"80%\" />"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"<img src=\"saddle_point_evaluation_optimizers.gif\" style=\"margin: auto;\" width=\"80%\" />"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"celltoolbar": "Slideshow",
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.3"
},
"livereveal": {
"start_slideshow_at": "selected",
"theme": "amu"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
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