吴恩达机器学习-C2W2-Relu(线性整流函数)

引入

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.gridspec import GridSpec
plt.style.use('./deeplearning.mplstyle')
import tensorflow as tf
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense, LeakyReLU
from tensorflow.keras.activations import linear, relu, sigmoid
%matplotlib widget
from matplotlib.widgets import Slider
from lab_utils_common import dlc
from autils import plt_act_trio
from lab_utils_relu import *
import warnings
warnings.simplefilter(action='ignore', category=UserWarning)

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2 - ReLU激活

本周，引入了一种新的激活方式——整流线性单元(ReLU)。
a = max(0,z) \quad\quad\text {# ReLU function}

plt_act_trio()
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运行结果：

右边的例子展示了ReLU的一个应用。在这个例子中，派生的“意识”特征不是二元的，而是具有连续范围的值。s型最适合开/关或二进制的情况。ReLU提供了一个连续的线性关系。此外，它有一个输出为零的“关闭”范围。“关闭”功能使ReLU成为非线性激活。为什么需要这样做?让我们在下面检查一下。

为什么是非线性激活?

所示的函数由线性块(分段线性)组成。斜率在直线段是一致的，然后在过渡点突然变化。在过渡点处，添加一个新的线性函数，当添加到现有函数时，将产生新的斜率。新函数是在过渡点添加的，但在此之前对输出没有贡献。非线性激活函数负责在过渡点之前或之后禁用输入。下面的练习提供了一个更具体的例子。

这个练习将在一个回归问题中使用下面的网络，你必须对一个分段线性目标建模:
网络在第一层有3个单元。每人负责一块靶子。单元0是预先编程和固定映射到第一段。您将修改第1单元和第2单元中的权重和偏差，以对第2和第3部分进行建模。输出单元也是固定的，简单地对第一层的输出求和。
使用下面的滑块，修改权重和偏差以匹配目标。提示:从w1和b1开始，让w2和b2为零，直到匹配第二段。点击比滑动更快。如果你有问题，不要担心，下面的文字将详细描述这一点

_ = plt_relu_ex()
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这个练习的目标是了解ReLU的非线性行为是如何提供所需的能力来关闭函数，直到需要它们为止。让我们在这个例子中看看这是如何工作的。
右边的图包含了第一层单元的输出。
从顶部开始，单位0负责标记为1的第一个段。线性函数$z$和ReLU $a$后面的函数都显示出来了。可以看到，ReLU在区间[0,1]之后切断了函数。这是关键，因为它可以防止干扰以下部分。
第一单元负责第二部分。在这里，ReLU保持这个单位安静，直到x等于1。因为第一个单位没有作用，所以第一个单位$w_1^{[1]}$的斜率，就是目标直线的斜率。
第二单元负责第三部分。ReLU再次将输出归零，直到x达到正确的值。必须设置单位$w_2^{[1]}$的斜率，使单位1和2的和具有所需的斜率。再次调整偏置以保持输出为负，直到x达到2。
ReLU激活的“关闭”或“禁用”功能使模型能够将线性段拼接在一起，以模拟复杂的非线性函数。

=====新材料结束=

新活动

本周引入了一种新的激活方式——整流线性单元(ReLU)。

def plt_act_trio():
    X = np.linspace(-5,5,100)
    fig,ax = plt.subplots(1,3, figsize=(6,2))
    widgvis(fig)
    ax[0].plot(X,tf.keras.activations.linear(X))
    ax[0].axvline(0, lw=0.3, c="black")
    ax[0].axhline(0, lw=0.3, c="black")
    ax[0].set_title("linear")
    ax[1].plot(X,tf.keras.activations.sigmoid(X))
    ax[1].axvline(0, lw=0.3, c="black")
    ax[1].axhline(0, lw=0.3, c="black")
    ax[1].set_title("sigmoid")
    ax[2].plot(X,tf.keras.activations.relu(X))
    ax[2].axhline(0, lw=0.3, c="black")
    ax[2].axvline(0, lw=0.3, c="black")
    ax[2].set_title("relu")
    fig.suptitle("Common Activation Functions", fontsize=14)
    fig.tight_layout(pad=0.2)
    plt.show()
plt_act_trio()
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运行结果：

右边的示例显示了ReLu的一个应用。在本例中，“感知”特性不是二进制的，而是具有从0到更大值的范围。s型最适合开/关或二进制的情况。ReLu提供了一个线性关系和一个输出为零的“关闭”范围。“关闭”功能使ReLu成为非线性激活。为什么需要这样做?让我们用下面的例子来检查一下。

X = np.linspace(0,2*np.pi, 100)
y = np.cos(X)+1
y[50:100]=0
fig,ax = plt.subplots(1,1, figsize=(2,2))
widgvis(fig)
ax.plot(X,y)
plt.show()
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运行结果：

w10 = np.array([[-1]])
b10 = np.array([2.6])
d10 = Dense(1, activation = "linear", input_shape = (1,), weights=[w10,b10])
z10 = d10(X.reshape(-1,1))
a10 = relu(z10)
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def plt_act1(y,z,a):
    fig,ax = plt.subplots(1,3, figsize=(6,2.5))
    widgvis(fig)
    ax[0].plot(X,y,label="target")
    ax[0].axvline(0, lw=0.3, c="black")
    ax[0].axhline(0, lw=0.3, c="black")
    ax[0].set_title("y - target")
    ax[1].plot(X,y, label="target")
    ax[1].plot(X,z, c=dlc["dldarkred"],label="z")
    ax[1].axvline(0, lw=0.3, c="black")
    ax[1].axhline(0, lw=0.3, c="black")
    ax[1].set_title("z = wX+b")
    ax[1].legend(loc="upper center")
    ax[2].plot(X,y, label="target")
    ax[2].plot(X,a, c=dlc["dldarkred"],label="ReLu(z)")
    ax[2].axhline(0, lw=0.3, c="black")
    ax[2].axvline(0, lw=0.3, c="black")
    ax[2].set_title("with relu")
    ax[2].legend()
    fig.suptitle("Role of Activation", fontsize=14)
    fig.tight_layout(pad=0.2)
    return(ax)


def plt_add_notation(ax):
    ax[1].annotate(text = "matches\n here", xy =(1.5,1.0), 
                   xytext = (0.1,-1.5), fontsize=10,
                  arrowprops=dict(facecolor=dlc["dlpurple"],width=2, headwidth=8))
    ax[1].annotate(text = "but not\n here", xy =(5,-2.5), 
                   xytext = (1,-3), fontsize=10,
                  arrowprops=dict(facecolor=dlc["dlpurple"],width=2, headwidth=8))
    ax[2].annotate(text = "ReLu\n 'off'", xy =(2.6,0), 
                   xytext = (0.1,0.1), fontsize=10,
                  arrowprops=dict(facecolor=dlc["dlpurple"],width=2, headwidth=8))
    
ax = plt_act1(y,z10,a10)
plt_add_notation(ax)
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X = np.linspace(0,2*np.pi, 100)
y = np.cos(X)+1
y[0:49]=0
fig,ax = plt.subplots(1,1, figsize=(2,2))
widgvis(fig)
ax.plot(X,y)
plt.show()
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w11 = np.array([[1]])
b11 = np.array([-3.7])
d11 = Dense(1, activation = "linear", input_shape = (1,), weights=[w11,b11])
z11 = d11(X.reshape(-1,1))
a11 = relu(z11)
plt_act1(y,z11,a11)
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X = np.linspace(0,2*np.pi, 100)
y = np.cos(X)+1
X=X.reshape(-1,1)
yhat = relu(d10(X)) + relu(d11(X))

fig,ax = plt.subplots(1,2, figsize=(4,2))
widgvis(fig)
ax[0].plot(X,y)
ax[1].plot(X,y)
ax[1].plot(X,yhat)
plt.show()
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X=X.reshape(-1,1)
yhat = relu(d10(X)) + relu(d11(X))
fig,ax = plt.subplots(1,1, figsize=(2,2))
widgvis(fig)
ax.plot(X,y)
ax.plot(X,yhat)
plt.show()
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yhat = model.predict(X.reshape(-1,1))
fig,ax = plt.subplots(1,1, figsize=(2,2))
widgvis(fig)
ax.plot(X,y)
ax.plot(X,yhat)
plt.show()
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model = Sequential(
    [ 
        d10,
        tf.layers.Activation(activations.relu),
        d11,
        tf.layers.Activation(activations.relu),
        Dense(1, activation = 'linear')
    ]
)
model.compile(
    loss=tf.keras.losses.MeanSquaredError(),
    optimizer=tf.keras.optimizers.Adam(0.1),
)

model.fit(
    X,y,
    epochs=1000
)
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model = Sequential(
    [ 
        Dense(1,activation="relu", name = 'l1'),
        Dense(1,activation="linear", name = 'l2')
    ]
)
model.compile(
    loss=tf.keras.losses.MeanSquaredError(),
    optimizer=tf.keras.optimizers.Adam(0.01),
)

model.fit(
    X,y,
    epochs=10
)
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yhat = model.predict(X)
yhat[0:5]
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fig,ax = plt.subplots(1,1, figsize=(4,4))
ax.plot(X,y)
ax.plot(X,yhat, c=dlc["dldarkred"])
plt.show()
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l1 = model.get_layer('l1')
l2 = model.get_layer('l2')
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l1.get_weights()
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l2.get_weights()
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l1 = model.get_layer('l1')
l2 = model.get_layer('l2')
l1.get_weights()
l2.get_weights()

w1 = np.array([[-1]])
b1 = np.array([1])
l1.set_weights([w1,b1])

w2 = np.array([[1]])
b2 = np.array([0])
l2.set_weights([w2,b2])
model.fit(
    X,y,
    epochs=100
)
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l2.set_weights([w2,b2])
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yhat = model.predict(X)
fig,ax = plt.subplots(1,1, figsize=(4,4))
ax.plot(X,y)
ax.plot(X,yhat, c=dlc["dldarkred"])
plt.show()
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2-D

import matplotlib.pyplot as plt
import numpy as np
import matplotlib as mpl
import warnings
from matplotlib import cm
from matplotlib.patches import FancyArrowPatch
from matplotlib.colors import ListedColormap, LinearSegmentedColormap
import matplotlib.colors as colors
from lab_utils_common import dlc
dkcolors = plt.cm.Paired((1,3,7,9,5,11))
ltcolors = plt.cm.Paired((0,2,6,8,4,10))
dkcolors_map = mpl.colors.ListedColormap(dkcolors)
ltcolors_map = mpl.colors.ListedColormap(ltcolors)


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def plt_mc_data(ax, X, y, classes,  class_labels=None, map=plt.cm.Paired, 
                legend=False, size=50, m='o', equal_xy = False):
    """ Plot multiclass data. Note, if equal_xy is True, setting ylim on the plot may not work """
    for i in range(classes):
        idx = np.where(y == i)
        col = len(idx[0])*[i]
        label = class_labels[i] if class_labels else "c{}".format(i)
        ax.scatter(X[idx, 0], X[idx, 1],  marker=m,
                    c=col, vmin=0, vmax=map.N, cmap=map,
                    s=size, label=label)
    if legend: ax.legend()
    if equal_xy: ax.axis("equal")

def plt_mc(X_train,y_train,classes):
    css = np.unique(y_train)
    fig,ax = plt.subplots(1,1,figsize=(3,3))
    fig.canvas.toolbar_visible = False
    fig.canvas.header_visible = False
    fig.canvas.footer_visible = False
    plt_mc_data(ax, X_train,y_train,classes, map=dkcolors_map, legend=True, size=10, equal_xy = False)
    ax.set_title("Multiclass Data")
    ax.set_xlabel("x0")
    ax.set_ylabel("x1")
    return(ax)
    
def plot_cat_decision_boundary_mc(ax, X, predict , class_labels=None, legend=False, vector=True):

    # create a mesh to points to plot
    x_min, x_max = X[:, 0].min(), X[:, 0].max()
    y_min, y_max = X[:, 1].min(), X[:, 1].max()
    h = max(x_max-x_min, y_max-y_min)/200
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
                         np.arange(y_min, y_max, h))
    points = np.c_[xx.ravel(), yy.ravel()]
    #print("points", points.shape)
    #print("xx.shape", xx.shape)

    #make predictions for each point in mesh
    if vector:
        Z = predict(points)
    else:
        Z = np.zeros((len(points),))
        for i in range(len(points)):
            Z[i] = predict(points[i].reshape(1,2))
    Z = Z.reshape(xx.shape)

    #contour plot highlights boundaries between values - classes in this case
    ax.contour(xx, yy, Z, linewidths=1) 
    #ax.axis('tight')

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X = np.random.rand(300, 2)
y = np.sqrt( X[:,0]**2 + X[:,1]**2 ) < 0.6
#y = np.logical_and( X[:,0] < 0.5, X[:,1] < 0.5 ).astype(int)
y.shape
plt_mc(X,y,2,)
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model = Sequential(
    [ 
        Dense(2,activation="relu",    name = 'l1'),
        Dense(1,activation="sigmoid", name = 'l2')
    ]
)

model.compile(
    loss=tf.keras.losses.MeanSquaredError(),
    optimizer=tf.keras.optimizers.Adam(0.01),
)

model.fit(
    X,y,
    epochs=150
)
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ax = plt_mc(X,y,2,)
predict = lambda  x: (model.predict(x) > 0.5).astype(int)
plot_cat_decision_boundary_mc(ax, X, predict, legend = True, vector=True)
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l1 = model.get_layer("l1")
W1,b1 = l1.get_weights()
l2 = model.get_layer("l2")
W2,b2 = l2.get_weights()
print(W1,b1)
print(W2,b2)
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x0 = np.array([0.4,0.60])
np.dot( np.dot(x0,W1) + b1, W2) + b2
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import time
import warnings

import numpy as np
import matplotlib.pyplot as plt

from sklearn import cluster, datasets, mixture
from sklearn.neighbors import kneighbors_graph
from sklearn.preprocessing import StandardScaler
from itertools import cycle, islice

np.random.seed(0)

# ============
# Generate datasets. We choose the size big enough to see the scalability
# of the algorithms, but not too big to avoid too long running times
# ============
n_samples = 500
noisy_circles = datasets.make_circles(n_samples=n_samples, factor=0.5, noise=0.05)
noisy_moons = datasets.make_moons(n_samples=n_samples, noise=0.05)
blobs = datasets.make_blobs(n_samples=n_samples, random_state=8)
no_structure = np.random.rand(n_samples, 2), None

# Anisotropicly distributed data
random_state = 170
X, y = datasets.make_blobs(n_samples=n_samples, random_state=random_state)
transformation = [[0.6, -0.6], [-0.4, 0.8]]
X_aniso = np.dot(X, transformation)
aniso = (X_aniso, y)

# blobs with varied variances
varied = datasets.make_blobs(
    n_samples=n_samples, cluster_std=[1.0, 2.5, 0.5], random_state=random_state
)

# ============
# Set up cluster parameters
# ============
plt.figure(figsize=(9 * 2 + 3, 13))
plt.subplots_adjust(
    left=0.02, right=0.98, bottom=0.001, top=0.95, wspace=0.05, hspace=0.01
)

plot_num = 1

default_base = {
    "quantile": 0.3,
    "eps": 0.3,
    "damping": 0.9,
    "preference": -200,
    "n_neighbors": 3,
    "n_clusters": 3,
    "min_samples": 7,
    "xi": 0.05,
    "min_cluster_size": 0.1,
}

datasets = [
    (
        noisy_circles,
        {
            "damping": 0.77,
            "preference": -240,
            "quantile": 0.2,
            "n_clusters": 2,
            "min_samples": 7,
            "xi": 0.08,
        },
    ),
    (
        noisy_moons,
        {
            "damping": 0.75,
            "preference": -220,
            "n_clusters": 2,
            "min_samples": 7,
            "xi": 0.1,
        },
    ),
    (
        varied,
        {
            "eps": 0.18,
            "n_neighbors": 2,
            "min_samples": 7,
            "xi": 0.01,
            "min_cluster_size": 0.2,
        },
    ),
    (
        aniso,
        {
            "eps": 0.15,
            "n_neighbors": 2,
            "min_samples": 7,
            "xi": 0.1,
            "min_cluster_size": 0.2,
        },
    ),
    (blobs, {"min_samples": 7, "xi": 0.1, "min_cluster_size": 0.2}),
    (no_structure, {}),
]

datasets = [
    (no_structure, {}),
]

for i_dataset, (dataset, algo_params) in enumerate(datasets):
    # update parameters with dataset-specific values
    params = default_base.copy()
    params.update(algo_params)

    X, y = dataset

    # normalize dataset for easier parameter selection
    X = StandardScaler().fit_transform(X)

    # estimate bandwidth for mean shift
    bandwidth = cluster.estimate_bandwidth(X, quantile=params["quantile"])

    # connectivity matrix for structured Ward
    connectivity = kneighbors_graph(
        X, n_neighbors=params["n_neighbors"], include_self=False
    )
    # make connectivity symmetric
    connectivity = 0.5 * (connectivity + connectivity.T)

    # ============
    # Create cluster objects
    # ============
    ms = cluster.MeanShift(bandwidth=bandwidth, bin_seeding=True)
    two_means = cluster.MiniBatchKMeans(n_clusters=params["n_clusters"])
    ward = cluster.AgglomerativeClustering(
        n_clusters=params["n_clusters"], linkage="ward", connectivity=connectivity
    )
    spectral = cluster.SpectralClustering(
        n_clusters=params["n_clusters"],
        eigen_solver="arpack",
        affinity="nearest_neighbors",
    )
    dbscan = cluster.DBSCAN(eps=params["eps"])
    optics = cluster.OPTICS(
        min_samples=params["min_samples"],
        xi=params["xi"],
        min_cluster_size=params["min_cluster_size"],
    )
    affinity_propagation = cluster.AffinityPropagation(
        damping=params["damping"], preference=params["preference"], random_state=0
    )
    average_linkage = cluster.AgglomerativeClustering(
        linkage="average",
        affinity="cityblock",
        n_clusters=params["n_clusters"],
        connectivity=connectivity,
    )
    birch = cluster.Birch(n_clusters=params["n_clusters"])
    gmm = mixture.GaussianMixture(
        n_components=params["n_clusters"], covariance_type="full"
    )

    clustering_algorithms = (
        ("MiniBatch\nKMeans", two_means),
        ("Affinity\nPropagation", affinity_propagation),
        ("MeanShift", ms),
        ("Spectral\nClustering", spectral),
        ("Ward", ward),
        ("Agglomerative\nClustering", average_linkage),
        ("DBSCAN", dbscan),
        ("OPTICS", optics),
        ("BIRCH", birch),
        ("Gaussian\nMixture", gmm),
    )

    for name, algorithm in clustering_algorithms:
        t0 = time.time()

        # catch warnings related to kneighbors_graph
        with warnings.catch_warnings():
            warnings.filterwarnings(
                "ignore",
                message="the number of connected components of the "
                + "connectivity matrix is [0-9]{1,2}"
                + " > 1. Completing it to avoid stopping the tree early.",
                category=UserWarning,
            )
            warnings.filterwarnings(
                "ignore",
                message="Graph is not fully connected, spectral embedding"
                + " may not work as expected.",
                category=UserWarning,
            )
            print(X.shape,algorithm)
            algorithm.fit(X)

        t1 = time.time()
        if hasattr(algorithm, "labels_"):
            y_pred = algorithm.labels_.astype(int)
        else:
            y_pred = algorithm.predict(X)

        plt.subplot(len(datasets), len(clustering_algorithms), plot_num)
        if i_dataset == 0:
            plt.title(name, size=18)

        colors = np.array(
            list(
                islice(
                    cycle(
                        [
                            "#377eb8",
                            "#ff7f00",
                            "#4daf4a",
                            "#f781bf",
                            "#a65628",
                            "#984ea3",
                            "#999999",
                            "#e41a1c",
                            "#dede00",
                        ]
                    ),
                    int(max(y_pred) + 1),
                )
            )
        )
        # add black color for outliers (if any)
        colors = np.append(colors, ["#000000"])
        plt.scatter(X[:, 0], X[:, 1], s=10, color=colors[y_pred])

        plt.xlim(-2.5, 2.5)
        plt.ylim(-2.5, 2.5)
        plt.xticks(())
        plt.yticks(())
        plt.text(
            0.99,
            0.01,
            ("%.2fs" % (t1 - t0)).lstrip("0"),
            transform=plt.gca().transAxes,
            size=15,
            horizontalalignment="right",
        )
        plot_num += 1

plt.show()
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