TFP 概率層：迴歸

在 TensorFlow.org 上檢視 在 Google Colab 中執行 在 GitHub 上檢視原始碼 下載筆記本

在此範例中，我們示範如何使用 TFP 的「機率層」來擬合迴歸模型。

依附元件與先決條件

匯入

加速運算！

在我們深入探討之前，請先確認我們在此示範中使用了 GPU。

若要執行此操作，請選取「執行階段」->「變更執行階段類型」->「硬體加速器」->「GPU」。

以下程式碼片段將驗證我們是否可以存取 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.

動機

如果我們可以使用 TFP 指定機率模型，然後簡單地最小化負對數概似，那不是很好嗎？例如：

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

不僅有可能，而且這個 Colab 還示範了如何做到！ (以線性迴歸問題為例。)

合成資料集。

案例 1：沒有不確定性

# 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

圖 1：沒有不確定性。

png

案例 2：隨機不確定性

# 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]

圖 2：隨機不確定性

png

案例 3：認知不確定性

# 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  ]

圖 3：認知不確定性

png

案例 4：隨機與認知不確定性

# 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 ]

圖 4：隨機與認知不確定性

png

案例 5：函數不確定性

自訂 PSD 核心

# 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)

圖 5：函數不確定性

png