Many important problems can be formulated as reasoning in knowledge graphs. Representation learning has proved extremely effective for transductive reasoning, in which one needs to make new predictions for already observed entities. This is true for both attributed graphs(where each entity has an initial feature vector) and non-attributed graphs (where the only initial information derives from known relations with other entities). For out-of-sample reasoning, where one needs to make predictions for entities that were unseen at training time, much prior work considers attributed graph. However, this problem is surprisingly under-explored for non-attributed graphs. In this paper, we study the out-of-sample representation learning problem for non-attributed knowledge graphs, create benchmark datasets for this task, develop several models and baselines, and provide empirical analyses and comparisons of the proposed models and baselines.

The gradient descent algorithm is an approach to find the minimum point or optimal solution for a given dataset. It follows the steepest descent approach. That is it moves in the negative gradient direction to find the local or global minima, starting out from a random point. We use gradient descent to reach the lowest point of the cost function. In Machine Learning, it is used to update the coefficients of our model.

And since we now have probabilities we can calculate the Cross Entropy as we have reviewed earlier.

Gaussian processes are an effective model class for learning unknown functions, particularly in settings where accurately representing predictive uncertainty is of key importance. Motivated by applications in the physical sciences, the widely-used Mat\'ern class of Gaussian processes has recently been generalized to model functions whose domains are Riemannian manifolds, by re-expressing said processes as solutions of stochastic partial differential equations. In this work, we propose techniques for computing the kernels of these processes on compact Riemannian manifolds via spectral theory of the Laplace-Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods. We also extend the generalization from the Mat\'ern to the widely-used squared exponential Gaussian process. By allowing Riemannian Mat\'ern Gaussian processes to be trained using well-understood techniques, our work enables their use in mini-batch, online, and non-conjugate settings, and makes them more accessible to machine learning practitioners.

One of the principal scientific challenges in deep learning is explaining generalization, i.e., why the particular way the community now trains networks to achieve small training error also leads to small error on held-out data from the same population. It is widely appreciated that some worst-case theories -- such as those based on the VC dimension of the class of predictors induced by modern neural network architectures -- are unable to explain empirical performance. A large volume of work aims to close this gap, primarily by developing bounds on generalization error, optimization error, and excess risk. When evaluated empirically, however, most of these bounds are numerically vacuous. Focusing on generalization bounds, this work addresses the question of how to evaluate such bounds empirically. Jiang et al. (2020) recently described a large-scale empirical study aimed at uncovering potential causal relationships between bounds/measures and generalization. Building on their study, we highlight where their proposed methods can obscure failures and successes of generalization measures in explaining generalization. We argue that generalization measures should instead be evaluated within the framework of distributional robustness.

Fine-tuned pre-trained language models can suffer from severe miscalibration for both in-distribution and out-of-distribution (OOD) data due to over-parameterization. To mitigate this issue, we propose a regularized fine-tuning method. Our method introduces two types of regularization for better calibration: (1) On-manifold regularization, which generates pseudo on-manifold samples through interpolation within the data manifold. Augmented training with these pseudo samples imposes a smoothness regularization to improve in-distribution calibration. (2) Off-manifold regularization, which encourages the model to output uniform distributions for pseudo off-manifold samples to address the over-confidence issue for OOD data. Our experiments demonstrate that the proposed method outperforms existing calibration methods for text classification in terms of expectation calibration error, misclassification detection, and OOD detection on six datasets. Our code can be found at https://github.com/Lingkai-Kong/Calibrated-BERT-Fine-Tuning.

Prevailing methods for graphs require abundant label and edge information for learning. When data for a new task are scarce, meta-learning can learn from prior experiences and form much-needed inductive biases for fast adaption to new tasks. Here, we introduce G-Meta, a novel meta-learning algorithm for graphs. G-Meta uses local subgraphs to transfer subgraph-specific information and learn transferable knowledge faster via meta gradients. G-Meta learns how to quickly adapt to a new task using only a handful of nodes or edges in the new task and does so by learning from data points in other graphs or related, albeit disjoint label sets. G-Meta is theoretically justified as we show that the evidence for a prediction can be found in the local subgraph surrounding the target node or edge. Experiments on seven datasets and nine baseline methods show that G-Meta outperforms existing methods by up to 16.3%. Unlike previous methods, G-Meta successfully learns in challenging, few-shot learning settings that require generalization to completely new graphs and never-before-seen labels. Finally, G-Meta scales to large graphs, which we demonstrate on a new Tree-of-Life dataset comprising of 1,840 graphs, a two-orders of magnitude increase in the number of graphs used in prior work.

We study a natural extension of classical empirical risk minimization, where the hypothesis space is a random subspace of a given space. In particular, we consider possibly data dependent subspaces spanned by a random subset of the data, recovering as a special case Nystr\"om approaches for kernel methods. Considering random subspaces naturally leads to computational savings, but the question is whether the corresponding learning accuracy is degraded. These statistical-computational tradeoffs have been recently explored for the least squares loss and self-concordant loss functions, such as the logistic loss. Here, we work to extend these results to convex Lipschitz loss functions, that might not be smooth, such as the hinge loss used in support vector machines. This extension requires developing new proofs, that use different technical tools. Our main results show the existence of different settings, depending on how hard the learning problem is, for which computational efficiency can be improved with no loss in performance. Theoretical results are illustrated with simple numerical experiments.

Tensor linear regression is an important and useful tool for analyzing tensor data. To deal with high dimensionality, CANDECOMP/PARAFAC (CP) low-rank constraints are often imposed on the coefficient tensor parameter in the (penalized) $M$-estimation. However, we show that the corresponding optimization may not be attainable, and when this happens, the estimator is not well-defined. This is closely related to a phenomenon, called CP degeneracy, in low-rank tensor approximation problems. In this article, we provide useful results of CP degeneracy in tensor regression problems. In addition, we provide a general penalized strategy as a solution to overcome CP degeneracy. The asymptotic properties of the resulting estimation are also studied. Numerical experiments are conducted to illustrate our findings.

Directional data consist of observations distributed on a (hyper)sphere, and appear in many applied fields, such as astronomy, ecology, and environmental science. This paper studies both statistical and computational problems of kernel smoothing for directional data. We generalize the classical mean shift algorithm to directional data, which allows us to identify local modes of the directional kernel density estimator (KDE). The statistical convergence rates of the directional KDE and its derivatives are derived, and the problem of mode estimation is examined. We also prove the ascending property of our directional mean shift algorithm and investigate a general problem of gradient ascent on the unit hypersphere. To demonstrate the applicability of our proposed algorithm, we evaluate it as a mode clustering method on both simulated and real-world datasets.