TFP 概率层：回归

在 TensorFlow.org 上查看 在 Google Colab 中运行 在 GitHub 上查看源代码 下载笔记本

在本例中，我们将展示如何使用 TFP 的“概率层”拟合回归模型。

依赖项和先决条件

导入

加速！

在深入研究之前，让我们确保我们正在为此演示使用 GPU。

为此，请选择“运行时”->“更改运行时类型”->“硬件加速器”->“GPU”。

以下代码段将验证我们是否可以访问 GPU。

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.

动机

如果我们可以使用 TFP 指定概率模型，然后简单地最小化负对数似然，那不是很好吗，即

negloglik = lambda y, rv_y: -rv_y.log_prob(y)

好吧，这不仅是可能的，而且此 Colab 展示了如何做到！（在线性回归问题的背景下。）

合成数据集。

案例 1：无不确定性

# 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_keras.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

图 1：无不确定性。

png

案例 2：偶然不确定性

# 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_keras.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]

图 2：偶然不确定性

png

案例 3：认知不确定性

# 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_keras.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  ]

图 3：认知不确定性

png

案例 4：偶然和认知不确定性

# 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_keras.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 ]

图 4：偶然和认知不确定性

png

案例 5：函数不确定性

自定义 PSD 内核

# 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_keras.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)

图 5：函数不确定性

png