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\usepackage{listings}
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\usepackage{amsmath,amssymb}

% \newfloat{lstfloat}{htbp}{lop}
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% \def\lstfloatautorefname{Listing}

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% %%
% %% Python definition (c) 1998 Michael Weber
% %% Additional definitions (2013) Alexis Dimitriadis
% %% modified by me (should not have empty lines)
% %%
% \lstdefinelanguage{iPython}{
%   morekeywords={access,and,break,class,continue,def,del,elif,else,except,exec,finally,for,from,global,if,import,
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%     %
%     % Built-ins
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%     %
%     sensitive=true,%
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%     morestring=[s]{'''}{'''},% used for documentation text (mulitiline strings)
%     morestring=[s]{"""}{"""},% added by Philipp Matthias Hahn
%     %
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%     morestring=[s]{u'}{'},% unicode strings
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% \begin{document}

% \begin{lstfloat}
% \begin{lstlisting}[language=iPython]
% import breeze.stats.distributions.Uniform
% import breeze.stats.distributions.Gaussian
% import scala.language.postfixOps

% object Activation {
%   def apply(x: Double): Double = math.max(0, x)

%   def d(x: Double): Double = if (x > 0) 1 else 0
% }

% class RSNN(val n: Int, val gamma: Double = 0.001) {
%     val g_unif = Uniform(-10, 10)
%     val g_gauss = Gaussian(0, 5)

%     val xis = g_unif.sample(n)
%     val vs = g_gauss.sample(n)
%     val bs = xis zip vs map {case(xi, v) => xi * v}

%     def computeL1(x: Double) = (bs zip vs) map {
%         case (b, v) => Activation(b + v * x) }
  
%     def computeL2(l1: Seq[Double], ws: Seq[Double]): Double =
%         (l1 zip ws) map { case (l, w) => w * l } sum
  
%     def output(ws: Seq[Double])(x: Double): Double =
%         computeL2(computeL1(x), ws)
  
%     def learn(data: Seq[(Double, Double)], ws: Seq[Double],
%         lamb: Double, gamma: Double): Seq[Double] = {
    
%         lazy val deltas = data.map {
%             case (x, y) =>
%                 val l1 = computeL1(x) // n
%                 val out = computeL2(l1, ws) // 1
%                 (l1 zip ws) map {case (l1, w) => (l1 * 2 * (out - y) +
%                     lam * 2 * w)  * gamma * -1}
%         }
  
%         deltas.foldRight(ws)(
%             (delta, ws) => ws zip (delta) map { case (w, d) => w + d })  
%     }

%     def train(data: Seq[(Double, Double)], iter: Int, lam: Double,
%         gamma: Double = gamma): (Seq[Double], Double => Double)= {
      
%         val ws = (1 to iter).foldRight((1 to n).map(
%             _ => 0.0) :Seq[Double])((i, w) => {
%             println(s"Training iteration $i")
%             println(w.sum/w.length)
%             learn(data, w, lam, gamma / 10)
%         })
%         (ws, output(ws))
%     }
% }
% \end{lstlisting}
% \caption{Scala code used to build and train the ridge penalized
%   randomized shallow neural network in .... The parameter \textit{lam}
% in the train function represents the $\lambda$ parameter in the error
% function. The parameters \textit{n} and \textit{gamma} set the number
% of hidden nodes and the stepsize for training.}
% \end{lstfloat}
% \clearpage

% \begin{lstlisting}[language=iPython]
% import tensorflow as tf
% import numpy as np
% from tensorflow.keras.callbacks import CSVLogger
% from tensorflow.keras.preprocessing.image import ImageDataGenerator

% mnist = tf.keras.datasets.mnist

% (x_train, y_train), (x_test, y_test) = mnist.load_data()
% x_train = x_train.reshape(x_train.shape[0], 28, 28, 1)
% x_train = x_train / 255.0
% x_test = x_test.reshape(x_test.shape[0], 28, 28, 1)
% x_test = x_test / 255.0

% y_train = tf.keras.utils.to_categorical(y_train)
% y_test = tf.keras.utils.to_categorical(y_test)

% model = tf.keras.models.Sequential()
% model.add(tf.keras.layers.Conv2D(24,kernel_size=5,padding='same',activation='relu',input_shape=(28,28,1)))
% model.add(tf.keras.layers.MaxPool2D())
% model.add(tf.keras.layers.Conv2D(64,kernel_size=5,padding='same',activation='relu'))
% model.add(tf.keras.layers.MaxPool2D(padding='same'))
% model.add(tf.keras.layers.Flatten())
% model.add(tf.keras.layers.Dense(256, activation='relu'))
% model.add(tf.keras.layers.Dropout(0.2))
% model.add(tf.keras.layers.Dense(10, activation='softmax'))
% model.compile(optimizer='adam', loss="categorical_crossentropy",
%     metrics=["accuracy"])

% datagen = ImageDataGenerator(
%     rotation_range = 30,
%     zoom_range = 0.15,
%     width_shift_range=2,
%     height_shift_range=2,
%     shear_range = 1)

% csv_logger = CSVLogger(<Target File>)
    
% history = model.fit(datagen.flow(x_train, y_train, batch_size=50),
% validation_data=(x_test, y_test),
%     epochs=125, callbacks=[csv_logger],
%     steps_per_epoch = x_train.shape[0]//50) 

% \end{lstlisting}
% \clearpage
% \begin{lstlisting}[language=iPython]
% import tensorflow as tf
% import numpy as np
% from tensorflow.keras.callbacks import CSVLogger
% from tensorflow.keras.preprocessing.image import ImageDataGenerator
% mnist = tf.keras.datasets.fashion_mnist

% (x_train, y_train), (x_test, y_test) = mnist.load_data()
% x_train = x_train.reshape(x_train.shape[0], 28, 28, 1)
% x_test = x_test.reshape(x_test.shape[0], 28, 28, 1)
% x_train, x_test = x_train / 255.0, x_test / 255.0

% y_train = tf.keras.utils.to_categorical(y_train)
% y_test = tf.keras.utils.to_categorical(y_test)

% model = tf.keras.Sequential()
% model.add(tf.keras.layers.Conv2D(filters = 32, kernel_size = (3, 3), activation='relu',
%     input_shape = (28, 28, 1), padding='same'))
% model.add(tf.keras.layers.Conv2D(filters = 32, kernel_size = (2, 2), activation='relu', padding = 'same'))
% model.add(tf.keras.layers.MaxPool2D(strides=(2,2)))
% model.add(tf.keras.layers.Conv2D(filters = 64, kernel_size = (3, 3), activation='relu', padding='same'))
% model.add(tf.keras.layers.Conv2D(filters = 64, kernel_size = (3, 3), activation='relu', padding='same'))
% model.add(tf.keras.layers.MaxPool2D(strides=(2,2)))
% model.add(tf.keras.layers.Flatten())
% model.add(tf.keras.layers.Dense(256, activation='relu'))
% model.add(tf.keras.layers.Dropout(0.2))
% model.add(tf.keras.layers.Dense(10, activation='softmax'))

% model.compile(optimizer=tf.keras.optimizers.Adam(lr = 1e-3), loss="categorical_crossentropy", metrics=["accuracy"])

% datagen = ImageDataGenerator(
%     rotation_range = 15,
%     zoom_range = 0.1,
%     width_shift_range=2,
%     height_shift_range=2,
%     shear_range = 0.5,
%     fill_mode = 'constant',
%     cval = 0)
        
%     csv_logger = CSVLogger(<Target File>)
    
%     history = model.fit(datagen.flow(x_train, y_train, batch_size=30),
%         steps_per_epoch=2000,
%         validation_data=(x_test, y_test),
%         epochs=125, callbacks=[csv_logger],
%         shuffle=True)

% \end{lstlisting}
% \begin{lstlisting}[language=iPython]
% def get_random_sample(a, b, number_of_samples=10):
%     x = []
%     y = []
%     for category_number in range(0,10):
%     # get all samples of a category
%         train_data_category = a[b==category_number]
%         # pick a number of random samples from the category
%         train_data_category = train_data_category[np.random.randint(
%             train_data_category.shape[0], size=number_of_samples), :]
%         x.extend(train_data_category)
%         y.append([category_number]*number_of_samples)

%         return (np.asarray(x).reshape(-1, 28, 28, 1),
%             np.asarray(y).reshape(10*number_of_samples,1))
% \end{lstlisting}
\begin{document}

\begin{align}
  \makebox[2cm][c]{$\overset{\text{Lem. A.6}}{\underset{\delta \text{
  small enough}}{=}} $} 
\end{align}

\end{document} 

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