In AI applications that are safety-critical, such as medical decision making and autonomous driving, or where the data is inherently noisy (for example, natural language understanding), it is important for a deep classifier to reliably quantify its uncertainty. The deep classifier should be able to be aware of its own limitations and when it should hand control over to the human experts. This tutorial shows how to improve a deep classifier's ability in quantifying uncertainty using a technique called Spectral-normalized Neural Gaussian Process (SNGP{.external}).
The core idea of SNGP is to improve a deep classifier's distance awareness by applying simple modifications to the network. A model's distance awareness is a measure of how its predictive probability reflects the distance between the test example and the training data. This is a desirable property that is common for gold-standard probabilistic models (for example, the Gaussian process{.external} with RBF kernels) but is lacking in models with deep neural networks. SNGP provides a simple way to inject this Gaussian-process behavior into a deep classifier while maintaining its predictive accuracy.
This tutorial implements a deep residual network (ResNet)-based SNGP model on scikit-learn’s two moons{.external} dataset, and compares its uncertainty surface with that of two other popular uncertainty approaches: Monte Carlo dropout{.external} and Deep ensemble{.external}.
This tutorial illustrates the SNGP model on a toy 2D dataset. For an example of applying SNGP to a real-world natural language understanding task using a BERT-base, check out the SNGP-BERT tutorial. For high-quality implementations of an SNGP model (and many other uncertainty methods) on a wide variety of benchmark datasets (such as CIFAR-100, ImageNet, Jigsaw toxicity detection, etc), refer to the Uncertainty Baselines{.external} benchmark.
About SNGP
SNGP is a simple approach to improve a deep classifier's uncertainty quality while maintaining a similar level of accuracy and latency. Given a deep residual network, SNGP makes two simple changes to the model:
It applies spectral normalization to the hidden residual layers.
It replaces the Dense output layer with a Gaussian process layer.
Compared to other uncertainty approaches (such as Monte Carlo dropout or Deep ensemble), SNGP has several advantages:
It works for a wide range of state-of-the-art residual-based architectures (for example, (Wide) ResNet, DenseNet, or BERT).
It is a single-model method—it does not rely on ensemble averaging). Therefore, SNGP has a similar level of latency as a single deterministic network, and can be scaled easily to large datasets like ImageNet{.external} and Jigsaw Toxic Comments classification{.external}.
It has strong out-of-domain detection performance due to the distance-awareness property.
The downsides of this method are:
The predictive uncertainty of SNGP is computed using the Laplace approximation{.external}. Therefore, theoretically, the posterior uncertainty of SNGP is different from that of an exact Gaussian process.
SNGP training needs a covariance reset step at the beginning of a new epoch. This can add a tiny amount of extra complexity to a training pipeline. This tutorial shows a simple way to implement this using Keras callbacks.
def make_training_data(sample_size=500):
"""Create two moon training dataset."""
train_examples, train_labels = sklearn.datasets.make_moons(
n_samples=2 * sample_size, noise=0.1)
# Adjust data position slightly.
train_examples[train_labels == 0] += [-0.1, 0.2]
train_examples[train_labels == 1] += [0.1, -0.2]
return train_examples, train_labels
Evaluate the model's predictive behavior over the entire 2D input space.
def make_testing_data(x_range=DEFAULT_X_RANGE, y_range=DEFAULT_Y_RANGE, n_grid=DEFAULT_N_GRID):
"""Create a mesh grid in 2D space."""
# testing data (mesh grid over data space)
x = np.linspace(x_range[0], x_range[1], n_grid)
y = np.linspace(y_range[0], y_range[1], n_grid)
xv, yv = np.meshgrid(x, y)
return np.stack([xv.flatten(), yv.flatten()], axis=-1)
To evaluate model uncertainty, add an out-of-domain (OOD) dataset that belongs to a third class. The model never observes these OOD examples during training.
Here, the blue and orange represent the positive and negative classes, and the red represents the OOD data. A model that quantifies the uncertainty well is expected to be confident when close to training data (i.e., \(p(x_{test})\) close to 0 or 1), and be uncertain when far away from the training data regions (i.e., \(p(x_{test})\) close to 0.5).
The deterministic model
Define model
Start from the (baseline) deterministic model: a multi-layer residual network (ResNet) with dropout regularization.
classDeepResNet(tf.keras.Model):"""Defines a multi-layer residual network."""def__init__(self,num_classes,num_layers=3,num_hidden=128,dropout_rate=0.1,**classifier_kwargs):super().__init__()#Definesclassmetadata.self.num_hidden=num_hiddenself.num_layers=num_layersself.dropout_rate=dropout_rateself.classifier_kwargs=classifier_kwargs#Definesthehiddenlayers.self.input_layer=tf.keras.layers.Dense(self.num_hidden,trainable=False)self.dense_layers=[self.make_dense_layer() for _ in range(num_layers)]#Definestheoutputlayer.self.classifier=self.make_output_layer(num_classes)defcall(self,inputs):#Projectsthe2dinputdatatohighdimension.hidden=self.input_layer(inputs)#ComputestheResNethiddenrepresentations.foriinrange(self.num_layers):resid=self.dense_layers[i](hidden)resid=tf.keras.layers.Dropout(self.dropout_rate)(resid)hidden+=residreturnself.classifier(hidden)defmake_dense_layer(self):"""Uses the Dense layer as the hidden layer."""returntf.keras.layers.Dense(self.num_hidden,activation="relu")defmake_output_layer(self,num_classes):"""Uses the Dense layer as the output layer."""returntf.keras.layers.Dense(num_classes,**self.classifier_kwargs)
This tutorial uses a six-layer ResNet with 128 hidden units.
defplot_uncertainty_surface(test_uncertainty,ax,cmap=None):"""Visualizes the 2D uncertainty surface. For simplicity, assume these objects already exist in the memory: test_examples: Array of test examples, shape (num_test, 2). train_labels: Array of train labels, shape (num_train, ). train_examples: Array of train examples, shape (num_train, 2). Arguments: test_uncertainty: Array of uncertainty scores, shape (num_test,). ax: A matplotlib Axes object that specifies a matplotlib figure. cmap: A matplotlib colormap object specifying the palette of the predictive surface. Returns: pcm: A matplotlib PathCollection object that contains the palette information of the uncertainty plot. """#Normalizeuncertaintyforbettervisualization.test_uncertainty=test_uncertainty/np.max(test_uncertainty)#Setviewlimits.ax.set_ylim(DEFAULT_Y_RANGE)ax.set_xlim(DEFAULT_X_RANGE)#Plotnormalizeduncertaintysurface.pcm=ax.imshow(np.reshape(test_uncertainty,[DEFAULT_N_GRID,DEFAULT_N_GRID]),cmap=cmap,origin="lower",extent=DEFAULT_X_RANGE+DEFAULT_Y_RANGE,vmin=DEFAULT_NORM.vmin,vmax=DEFAULT_NORM.vmax,interpolation='bicubic',aspect='auto')#Plottrainingdata.ax.scatter(train_examples[:,0],train_examples[:,1],c=train_labels,cmap=DEFAULT_CMAP,alpha=0.5)ax.scatter(ood_examples[:,0],ood_examples[:,1],c="red",alpha=0.1)returnpcm
Now visualize the predictions of the deterministic model. First plot the class probability:
\[p(x) = softmax(logit(x))\]
resnet_logits = resnet_model(test_examples)
resnet_probs = tf.nn.softmax(resnet_logits, axis=-1)[:, 0] # Take the probability for class 0.
In this plot, the yellow and purple are the predictive probabilities for the two classes. The deterministic model did a good job in classifying the two known classes—blue and orange—with a nonlinear decision boundary. However, it is not distance-aware, and classified the never-observed red out-of-domain (OOD) examples confidently as the orange class.
In this plot, the yellow indicates high uncertainty, and the purple indicates low uncertainty. A deterministic ResNet's uncertainty depends only on the test examples' distance from the decision boundary. This leads the model to be over-confident when out of the training domain. The next section shows how SNGP behaves differently on this dataset.
The SNGP model
Define SNGP model
Let's now implement the SNGP model. Both the SNGP components, SpectralNormalization and RandomFeatureGaussianProcess, are available at the tensorflow_model's built-in layers.
Let's inspect these two components in more detail. (You can also jump to the full SNGP model section to learn how SNGP is implemented.)
SpectralNormalization wrapper
SpectralNormalization{.external} is a Keras layer wrapper. It can be applied to an existing Dense layer like this:
Spectral normalization regularizes the hidden weight \(W\) by gradually guiding its spectral norm (that is, the largest eigenvalue of \(W\)) toward the target value norm_multiplier).
The Gaussian Process (GP) layer
RandomFeatureGaussianProcess{.external} implements a random-feature based approximation{.external} to a Gaussian process model that is end-to-end trainable with a deep neural network. Under the hood, the Gaussian process layer implements a two-layer network:
Here, \(x\) is the input, and \(W\) and \(b\) are frozen weights initialized randomly from Gaussian and Uniform distributions, respectively. (Therefore, \(\Phi(x)\) are called "random features".) \(\beta\) is the learnable kernel weight similar to that of a Dense layer.
num_inducing: The dimension \(M\) of the hidden weight \(W\). Default to 1024.
normalize_input: Whether to apply layer normalization to the input \(x\).
scale_random_features: Whether to apply the scale \(\sqrt{2/M}\) to the hidden output.
gp_cov_momentum controls how the model covariance is computed. If set to a positive value (for example, 0.999), the covariance matrix is computed using the momentum-based moving average update (similar to batch normalization). If set to -1, the covariance matrix is updated without momentum.
Given a batch input with shape (batch_size, input_dim), the GP layer returns a logits tensor (shape (batch_size, num_classes)) for prediction, and also covmat tensor (shape (batch_size, batch_size)) which is the posterior covariance matrix of the batch logits.
Theoretically, it is possible to extend the algorithm to compute different variance values for different classes (as introduced in the original SNGP paper{.external}). However, this is difficult to scale to problems with large output spaces (such as classification with ImageNet or language modeling).
The full SNGP model
Given the base class DeepResNet, the SNGP model can be implemented easily by modifying the residual network's hidden and output layers. For compatibility with Keras model.fit() API, also modify the model's call() method so it only outputs logits during training.
classDeepResNetSNGP(DeepResNet):def__init__(self,spec_norm_bound=0.9,**kwargs):self.spec_norm_bound=spec_norm_boundsuper().__init__(**kwargs)defmake_dense_layer(self):"""Applies spectral normalization to the hidden layer."""dense_layer=super().make_dense_layer()returnnlp_layers.SpectralNormalization(dense_layer,norm_multiplier=self.spec_norm_bound)defmake_output_layer(self,num_classes):"""Uses Gaussian process as the output layer."""returnnlp_layers.RandomFeatureGaussianProcess(num_classes,gp_cov_momentum=-1,**self.classifier_kwargs)defcall(self,inputs,training=False,return_covmat=False):# Gets logits and a covariance matrix from the GP layer.logits,covmat=super().call(inputs)# Returns only logits during training.ifnottrainingandreturn_covmat:returnlogits,covmatreturnlogits
Use the same architecture as the deterministic model.
resnet_config
sngp_model = DeepResNetSNGP(**resnet_config)
sngp_model.build((None, 2))
sngp_model.summary()
Implement a Keras callback to reset the covariance matrix at the beginning of a new epoch.
classResetCovarianceCallback(tf.keras.callbacks.Callback):defon_epoch_begin(self,epoch,logs=None):"""Resets covariance matrix at the beginning of the epoch."""ifepoch > 0:self.model.classifier.reset_covariance_matrix()
Add this callback to the DeepResNetSNGP model class.
classDeepResNetSNGPWithCovReset(DeepResNetSNGP):deffit(self,*args,**kwargs):"""Adds ResetCovarianceCallback to model callbacks."""kwargs["callbacks"]=list(kwargs.get("callbacks",[]))kwargs["callbacks"].append(ResetCovarianceCallback())returnsuper().fit(*args,**kwargs)
Now compute the posterior predictive probability. The classic method for computing the predictive probability of a probabilistic model is to use Monte Carlo sampling, i.e.,
where \(M\) is the sample size, and \(logit_m(x)\) are random samples from the SNGP posterior \(MultivariateNormal\)(sngp_logits,sngp_covmat). However, this approach can be slow for latency-sensitive applications such as autonomous driving or real-time bidding. Instead, you can approximate \(E(p(x))\) using the mean-field method{.external}:
Visualize the class probability (left) and the predictive uncertainty (right) of the SNGP model.
plot_predictions(sngp_probs, model_name="SNGP")
Remember that in the class probability plot (left), the yellow and purple are class probabilities. When close to the training data domain, SNGP correctly classifies the examples with high confidence (i.e., assigning near 0 or 1 probability). When far away from the training data, SNGP gradually becomes less confident, and its predictive probability becomes close to 0.5 while the (normalized) model uncertainty rises to 1.
Compare this to the uncertainty surface of the deterministic model:
As mentioned earlier, a deterministic model is not distance-aware. Its uncertainty is defined by the distance of the test example from the decision boundary. This leads the model to produce overconfident predictions for the out-of-domain examples (red).
by averaging over multiple Dropout-enabled forward passes \(\{logit_m(x)\}_{m=1}^M\).
def mc_dropout_sampling(test_examples):
# Enable dropout during inference.
return resnet_model(test_examples, training=True)
# Monte Carlo dropout inference.
dropout_logit_samples = [mc_dropout_sampling(test_examples) for _ in range(num_ensemble)]
dropout_prob_samples = [tf.nn.softmax(dropout_logits, axis=-1)[:, 0] for dropout_logits in dropout_logit_samples]
dropout_probs = tf.reduce_mean(dropout_prob_samples, axis=0)
Deep ensemble is a state-of-the-art (but expensive) method for deep learning uncertainty. To train a Deep ensemble, first train \(M\) ensemble members.
# Deep ensemble training
resnet_ensemble = []
for _ in range(num_ensemble):
resnet_model = DeepResNet(**resnet_config)
resnet_model.compile(optimizer=optimizer, loss=loss, metrics=metrics)
resnet_model.fit(train_examples, train_labels, verbose=0, **fit_config)
resnet_ensemble.append(resnet_model)
Collect logits and compute the mean predictive probability \(E(p(x)) = \frac{1}{M}\sum_{m=1}^M softmax(logit_m(x))\).
# Deep ensemble inference
ensemble_logit_samples = [model(test_examples) for model in resnet_ensemble]
ensemble_prob_samples = [tf.nn.softmax(logits, axis=-1)[:, 0] for logits in ensemble_logit_samples]
ensemble_probs = tf.reduce_mean(ensemble_prob_samples, axis=0)
Both the Monte Carlo Dropout and Deep ensemble methods improve the model's uncertainty ability by making the decision boundary less certain. However, they both inherit the deterministic deep network's limitation in lacking distance awareness.
Summary
In this tutorial, you have:
Implemented the SNGP model on a deep classifier to improve its distance awareness.
Trained the SNGP model end-to-end using Keras Model.fit API.
Visualized the uncertainty behavior of SNGP.
Compared the uncertainty behavior between SNGP, Monte Carlo dropout and deep ensemble models.
Resources and further reading
Check out the SNGP-BERT tutorial for an example of applying SNGP on a BERT model for uncertainty-aware natural language understanding.
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