We propose a novel framework for analyzing the dynamics of distribution shift in real-world systems that captures the feedback loop between learning algorithms and the distributions on which they are deployed. Prior work largely models feedback-induced distribution shift as adversarial or via an overly simplistic distribution-shift structure. In contrast, we propose a coupled partial differential equation model that captures fine-grained changes in the distribution over time by accounting for complex dynamics that arise due to strategic responses to algorithmic decision-making, non-local endogenous population interactions, and other exogenous sources of distribution shift. We consider two common settings in machine learning: cooperative settings with information asymmetries, and competitive settings where a learner faces strategic users. For both of these settings, when the algorithm retrains via gradient descent, we prove asymptotic convergence of the retraining procedure to a steady-state, both in finite and in infinite dimensions, obtaining explicit rates in terms of the model parameters. To do so we derive new results on the convergence of coupled PDEs that extends what is known on multi-species systems. Empirically, we show that our approach captures well-documented forms of distribution shifts like polarization and disparate impacts that simpler models cannot capture.

Knowledge Graph Embeddings (KGEs) have shown promising performance on link prediction tasks by mapping the entities and relations from a knowledge graph into a geometric space.The capability of KGEs in preserving graph characteristics including structural aspects and semantics, highly depends on the design of their score function, as well as the inherited abilities from the underlying geometry.Many KGEs use the Euclidean geometry which renders them incapable of preserving complex structures and consequently causes wrong inferences by the models.To address this problem, we propose a neuro differential KGE that embeds nodes of a KG on the trajectories of Ordinary Differential Equations (ODEs).To this end, we represent each relation (edge) in a KG as a vector field on several manifolds.We specifically parameterize ODEs by a neural network to represent complex manifolds and complex vector fields on the manifolds.Therefore, the underlying embedding space is capable to assume the shape of various geometric forms to encode heterogeneous subgraphs.Experiments on synthetic and benchmark datasets using state-of-the-art KGE models justify the ODE trajectories as a means to enable structure preservation and consequently avoiding wrong inferences.

We analyze the spectral clustering procedure for identifying coarse structure in a data set $\mathbf{x}_1, \dots, \mathbf{x}_n$, and in particular study the geometry of graph Laplacian embeddings which form the basis for spectral clustering algorithms. More precisely, we assume that the data are sampled from a mixture model supported on a manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$, and pick a connectivity length-scale $\varepsilon>0$ to construct a kernelized graph Laplacian. We introduce a notion of a well-separated mixture model which only depends on the model itself, and prove that when the model is well separated, with high probability the embedded data set concentrates on cones that are centered around orthogonal vectors. Our results are meaningful in the regime where $\varepsilon = \varepsilon(n)$ is allowed to decay to zero at a slow enough rate as the number of data points grows. This rate depends on the intrinsic dimension of the manifold on which the data is supported.

Graph-based semi-supervised learning is the problem of propagating labels from a small number of labelled data points to a larger set of unlabelled data. This paper is concerned with the consistency of optimization-based techniques for such problems, in the limit where the labels have small noise and the underlying unlabelled data is well clustered. We study graph-based probit for binary classification, and a natural generalization of this method to multi-class classification using one-hot encoding. The resulting objective function to be optimized comprises the sum of a quadratic form defined through a rational function of the graph Laplacian, involving only the unlabelled data, and a fidelity term involving only the labelled data. The consistency analysis sheds light on the choice of the rational function defining the optimization.