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في هذا المثال نوضح كيفية ملاءمة نماذج الانحدار باستخدام "الطبقات الاحتمالية" في TFP.
التبعيات والمتطلبات
يستورد
from pprint import pprint
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
import tensorflow.compat.v2 as tf
tf.enable_v2_behavior()
import tensorflow_probability as tfp
sns.reset_defaults()
#sns.set_style('whitegrid')
#sns.set_context('talk')
sns.set_context(context='talk',font_scale=0.7)
%matplotlib inline
tfd = tfp.distributions
اجعل الأمور سريعة!
قبل أن نتعمق ، دعنا نتأكد من أننا نستخدم وحدة معالجة الرسومات لهذا العرض التوضيحي.
للقيام بذلك ، حدد "وقت التشغيل" -> "تغيير نوع وقت التشغيل" -> "مسرع الأجهزة" -> "GPU".
سيتحقق المقتطف التالي من أن لدينا حق الوصول إلى وحدة معالجة الرسومات.
if tf.test.gpu_device_name() != '/device:GPU:0':
print('WARNING: GPU device not found.')
else:
print('SUCCESS: Found GPU: {}'.format(tf.test.gpu_device_name()))
WARNING: GPU device not found.
تحفيز
ألن يكون رائعًا إذا تمكنا من استخدام TFP لتحديد نموذج احتمالي ثم تقليل احتمال السجل السلبي ، على سبيل المثال ،
negloglik = lambda y, rv_y: -rv_y.log_prob(y)
حسنًا ، ليس هذا ممكنًا فحسب ، بل يوضح هذا الكولاب كيف! (في سياق مشاكل الانحدار الخطي.)
تجميع مجموعة البيانات.
w0 = 0.125
b0 = 5.
x_range = [-20, 60]
def load_dataset(n=150, n_tst=150):
np.random.seed(43)
def s(x):
g = (x - x_range[0]) / (x_range[1] - x_range[0])
return 3 * (0.25 + g**2.)
x = (x_range[1] - x_range[0]) * np.random.rand(n) + x_range[0]
eps = np.random.randn(n) * s(x)
y = (w0 * x * (1. + np.sin(x)) + b0) + eps
x = x[..., np.newaxis]
x_tst = np.linspace(*x_range, num=n_tst).astype(np.float32)
x_tst = x_tst[..., np.newaxis]
return y, x, x_tst
y, x, x_tst = load_dataset()
الحالة 1: لا شك
# Build model.
model = tf.keras.Sequential([
tf.keras.layers.Dense(1),
tfp.layers.DistributionLambda(lambda t: tfd.Normal(loc=t, scale=1)),
])
# Do inference.
model.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)
model.fit(x, y, epochs=1000, verbose=False);
# Profit.
[print(np.squeeze(w.numpy())) for w in model.weights];
yhat = model(x_tst)
assert isinstance(yhat, tfd.Distribution)
0.13032457 5.13029
الشكل 1: لا يوجد شك.
w = np.squeeze(model.layers[-2].kernel.numpy())
b = np.squeeze(model.layers[-2].bias.numpy())
plt.figure(figsize=[6, 1.5]) # inches
#plt.figure(figsize=[8, 5]) # inches
plt.plot(x, y, 'b.', label='observed');
plt.plot(x_tst, yhat.mean(),'r', label='mean', linewidth=4);
plt.ylim(-0.,17);
plt.yticks(np.linspace(0, 15, 4)[1:]);
plt.xticks(np.linspace(*x_range, num=9));
ax=plt.gca();
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position(('data', 0))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
#ax.spines['left'].set_smart_bounds(True)
#ax.spines['bottom'].set_smart_bounds(True)
plt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))
plt.savefig('/tmp/fig1.png', bbox_inches='tight', dpi=300)
الحالة 2: عدم اليقين البديل
# Build model.
model = tf.keras.Sequential([
tf.keras.layers.Dense(1 + 1),
tfp.layers.DistributionLambda(
lambda t: tfd.Normal(loc=t[..., :1],
scale=1e-3 + tf.math.softplus(0.05 * t[...,1:]))),
])
# Do inference.
model.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)
model.fit(x, y, epochs=1000, verbose=False);
# Profit.
[print(np.squeeze(w.numpy())) for w in model.weights];
yhat = model(x_tst)
assert isinstance(yhat, tfd.Distribution)
[0.14738432 0.1815331 ] [4.4812164 1.2219843]
الشكل 2: عدم اليقين البديل
plt.figure(figsize=[6, 1.5]) # inches
plt.plot(x, y, 'b.', label='observed');
m = yhat.mean()
s = yhat.stddev()
plt.plot(x_tst, m, 'r', linewidth=4, label='mean');
plt.plot(x_tst, m + 2 * s, 'g', linewidth=2, label=r'mean + 2 stddev');
plt.plot(x_tst, m - 2 * s, 'g', linewidth=2, label=r'mean - 2 stddev');
plt.ylim(-0.,17);
plt.yticks(np.linspace(0, 15, 4)[1:]);
plt.xticks(np.linspace(*x_range, num=9));
ax=plt.gca();
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position(('data', 0))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
#ax.spines['left'].set_smart_bounds(True)
#ax.spines['bottom'].set_smart_bounds(True)
plt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))
plt.savefig('/tmp/fig2.png', bbox_inches='tight', dpi=300)
الحالة 3: عدم اليقين المعرفي
# Specify the surrogate posterior over `keras.layers.Dense` `kernel` and `bias`.
def posterior_mean_field(kernel_size, bias_size=0, dtype=None):
n = kernel_size + bias_size
c = np.log(np.expm1(1.))
return tf.keras.Sequential([
tfp.layers.VariableLayer(2 * n, dtype=dtype),
tfp.layers.DistributionLambda(lambda t: tfd.Independent(
tfd.Normal(loc=t[..., :n],
scale=1e-5 + tf.nn.softplus(c + t[..., n:])),
reinterpreted_batch_ndims=1)),
])
# Specify the prior over `keras.layers.Dense` `kernel` and `bias`.
def prior_trainable(kernel_size, bias_size=0, dtype=None):
n = kernel_size + bias_size
return tf.keras.Sequential([
tfp.layers.VariableLayer(n, dtype=dtype),
tfp.layers.DistributionLambda(lambda t: tfd.Independent(
tfd.Normal(loc=t, scale=1),
reinterpreted_batch_ndims=1)),
])
# Build model.
model = tf.keras.Sequential([
tfp.layers.DenseVariational(1, posterior_mean_field, prior_trainable, kl_weight=1/x.shape[0]),
tfp.layers.DistributionLambda(lambda t: tfd.Normal(loc=t, scale=1)),
])
# Do inference.
model.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)
model.fit(x, y, epochs=1000, verbose=False);
# Profit.
[print(np.squeeze(w.numpy())) for w in model.weights];
yhat = model(x_tst)
assert isinstance(yhat, tfd.Distribution)
[ 0.1387333 5.125723 -4.112224 -2.2171402] [0.12476114 5.147452 ]
الشكل 3: عدم اليقين المعرفي
plt.figure(figsize=[6, 1.5]) # inches
plt.clf();
plt.plot(x, y, 'b.', label='observed');
yhats = [model(x_tst) for _ in range(100)]
avgm = np.zeros_like(x_tst[..., 0])
for i, yhat in enumerate(yhats):
m = np.squeeze(yhat.mean())
s = np.squeeze(yhat.stddev())
if i < 25:
plt.plot(x_tst, m, 'r', label='ensemble means' if i == 0 else None, linewidth=0.5)
avgm += m
plt.plot(x_tst, avgm/len(yhats), 'r', label='overall mean', linewidth=4)
plt.ylim(-0.,17);
plt.yticks(np.linspace(0, 15, 4)[1:]);
plt.xticks(np.linspace(*x_range, num=9));
ax=plt.gca();
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position(('data', 0))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
#ax.spines['left'].set_smart_bounds(True)
#ax.spines['bottom'].set_smart_bounds(True)
plt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))
plt.savefig('/tmp/fig3.png', bbox_inches='tight', dpi=300)
الحالة 4: عدم اليقين الوراثي والمعرفي
# Build model.
model = tf.keras.Sequential([
tfp.layers.DenseVariational(1 + 1, posterior_mean_field, prior_trainable, kl_weight=1/x.shape[0]),
tfp.layers.DistributionLambda(
lambda t: tfd.Normal(loc=t[..., :1],
scale=1e-3 + tf.math.softplus(0.01 * t[...,1:]))),
])
# Do inference.
model.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=negloglik)
model.fit(x, y, epochs=1000, verbose=False);
# Profit.
[print(np.squeeze(w.numpy())) for w in model.weights];
yhat = model(x_tst)
assert isinstance(yhat, tfd.Distribution)
[ 0.12753433 2.7504077 5.160624 3.8251898 -3.4283297 -0.8961645 -2.2378397 0.1496858 ] [0.14511648 2.7104297 5.1248145 3.7724588 ]
الشكل 4: كلا من عدم اليقين الوراثي والمعرفي
plt.figure(figsize=[6, 1.5]) # inches
plt.plot(x, y, 'b.', label='observed');
yhats = [model(x_tst) for _ in range(100)]
avgm = np.zeros_like(x_tst[..., 0])
for i, yhat in enumerate(yhats):
m = np.squeeze(yhat.mean())
s = np.squeeze(yhat.stddev())
if i < 15:
plt.plot(x_tst, m, 'r', label='ensemble means' if i == 0 else None, linewidth=1.)
plt.plot(x_tst, m + 2 * s, 'g', linewidth=0.5, label='ensemble means + 2 ensemble stdev' if i == 0 else None);
plt.plot(x_tst, m - 2 * s, 'g', linewidth=0.5, label='ensemble means - 2 ensemble stdev' if i == 0 else None);
avgm += m
plt.plot(x_tst, avgm/len(yhats), 'r', label='overall mean', linewidth=4)
plt.ylim(-0.,17);
plt.yticks(np.linspace(0, 15, 4)[1:]);
plt.xticks(np.linspace(*x_range, num=9));
ax=plt.gca();
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position(('data', 0))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
#ax.spines['left'].set_smart_bounds(True)
#ax.spines['bottom'].set_smart_bounds(True)
plt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))
plt.savefig('/tmp/fig4.png', bbox_inches='tight', dpi=300)
الحالة 5: عدم اليقين الوظيفي
نواة PSD مخصصة
class RBFKernelFn(tf.keras.layers.Layer):
def __init__(self, **kwargs):
super(RBFKernelFn, self).__init__(**kwargs)
dtype = kwargs.get('dtype', None)
self._amplitude = self.add_variable(
initializer=tf.constant_initializer(0),
dtype=dtype,
name='amplitude')
self._length_scale = self.add_variable(
initializer=tf.constant_initializer(0),
dtype=dtype,
name='length_scale')
def call(self, x):
# Never called -- this is just a layer so it can hold variables
# in a way Keras understands.
return x
@property
def kernel(self):
return tfp.math.psd_kernels.ExponentiatedQuadratic(
amplitude=tf.nn.softplus(0.1 * self._amplitude),
length_scale=tf.nn.softplus(5. * self._length_scale)
)
# For numeric stability, set the default floating-point dtype to float64
tf.keras.backend.set_floatx('float64')
# Build model.
num_inducing_points = 40
model = tf.keras.Sequential([
tf.keras.layers.InputLayer(input_shape=[1]),
tf.keras.layers.Dense(1, kernel_initializer='ones', use_bias=False),
tfp.layers.VariationalGaussianProcess(
num_inducing_points=num_inducing_points,
kernel_provider=RBFKernelFn(),
event_shape=[1],
inducing_index_points_initializer=tf.constant_initializer(
np.linspace(*x_range, num=num_inducing_points,
dtype=x.dtype)[..., np.newaxis]),
unconstrained_observation_noise_variance_initializer=(
tf.constant_initializer(np.array(0.54).astype(x.dtype))),
),
])
# Do inference.
batch_size = 32
loss = lambda y, rv_y: rv_y.variational_loss(
y, kl_weight=np.array(batch_size, x.dtype) / x.shape[0])
model.compile(optimizer=tf.optimizers.Adam(learning_rate=0.01), loss=loss)
model.fit(x, y, batch_size=batch_size, epochs=1000, verbose=False)
# Profit.
yhat = model(x_tst)
assert isinstance(yhat, tfd.Distribution)
الشكل 5: عدم اليقين الوظيفي
y, x, _ = load_dataset()
plt.figure(figsize=[6, 1.5]) # inches
plt.plot(x, y, 'b.', label='observed');
num_samples = 7
for i in range(num_samples):
sample_ = yhat.sample().numpy()
plt.plot(x_tst,
sample_[..., 0].T,
'r',
linewidth=0.9,
label='ensemble means' if i == 0 else None);
plt.ylim(-0.,17);
plt.yticks(np.linspace(0, 15, 4)[1:]);
plt.xticks(np.linspace(*x_range, num=9));
ax=plt.gca();
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['left'].set_position(('data', 0))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
#ax.spines['left'].set_smart_bounds(True)
#ax.spines['bottom'].set_smart_bounds(True)
plt.legend(loc='center left', fancybox=True, framealpha=0., bbox_to_anchor=(1.05, 0.5))
plt.savefig('/tmp/fig5.png', bbox_inches='tight', dpi=300)
