Interpretable Deep Gaussian Process
We propose interpretable deep Gaussian Processes (GPs) that combine the expressiveness of deep Neural Networks (NNs) with quantified uncertainty of deep GPs. Our approach is based on approximating deep GP as a GP, which allows explicit, analytic forms for compositions of a wide variety of kernels. Consequently, our approach admits interpretation as both NNs with specified activation functions and as a variational approximation to deep GPs. We provide general recipes for deriving the effective kernels for deep GPs of two, three, or infinitely many layers, composed of homogeneous or heterogeneous kernels. Results illustrate the expressiveness of our effective kernels through samples from the prior and inference on simulated data and demonstrate advantages of interpretability by analysis of analytic forms, drawing relations and equivalences across kernels, and a priori identification of non-pathological regimes of hyperparameter space. …

Probabilistic Face Embedding (PFE)
Embedding methods have achieved success in face recognition by comparing facial features in a latent semantic space. However, in a fully unconstrained face setting, the features learned by the embedding model could be ambiguous or may not even be present in the input face, leading to noisy representations. We propose Probabilistic Face Embeddings (PFEs), which represent each face image as a Gaussian distribution in the latent space. The mean of the distribution estimates the most likely feature values while the variance shows the uncertainty in the feature values. Probabilistic solutions can then be naturally derived for matching and fusing PFEs using the uncertainty information. Empirical evaluation on different baseline models, training datasets and benchmarks show that the proposed method can improve the face recognition performance of deterministic embeddings by converting them into PFEs. The uncertainties estimated by PFEs also serve as good indicators of the potential matching accuracy, which are important for a risk-controlled recognition system. …

Generative Parameter Sampler (GPS)
Uncertainty quantification has been a core of the statistical machine learning, but its computational bottleneck has been a serious challenge for both Bayesians and frequentists. We propose a model-based framework in quantifying uncertainty, called predictive-matching Generative Parameter Sampler (GPS). This procedure considers an Uncertainty Quantification (UQ) distribution on the targeted parameter, which is defined as the minimizer of a distance between the empirical distribution and the resulting predictive distribution. This framework adopts a hierarchical modeling perspective such that each observation is modeled by an individual parameter. This individual parameterization permits the resulting inference to be computationally scalable and robust to outliers. Our approach is illustrated for linear models, Poisson processes, and deep neural networks for classification. The results show that the GPS is successful in providing uncertainty quantification as well as additional flexibility beyond what is allowed by classical statistical procedures under the postulated statistical models. …

Early Stopping
In machine learning, early stopping is a form of regularization used to avoid overfitting when training a learner with an iterative method, such as gradient descent. Such methods update the learner so as to make it better fit the training data with each iteration. Up to a point, this improves the learner’s performance on data outside of the training set. Past that point, however, improving the learner’s fit to the training data comes at the expense of increased generalization error. Early stopping rules provide guidance as to how many iterations can be run before the learner begins to over-fit. Early stopping rules have been employed in many different machine learning methods, with varying amounts of theoretical foundation. …