In many applications, especially in the sciences, data and tasks have known invariances. Encoding such invariances directly into a machine learning model can improve learning outcomes, while it also poses challenges on efficient model design. In the first part of the talk, we will focus on the invariances relevant to eigenvectors and eigenspaces being inputs to a neural network. Such inputs are important, for instance, for graph representation learning, point clouds and graphics. We will discuss targeted architectures that express the relevant invariances or equivariances - sign flips and changes of basis - and their theoretical and empirical benefits in different applications. Second, we will take a broader, theoretical perspective. Empirically, it is known that encoding invariances into the machine learning model can reduce sample complexity. What can we say theoretically? We will look at example results for various settings and models.

This talk is based on joint work with Derek Lim, Joshua Robinson, Behrooz Tahmasebi, Thien Le, Hannah Lawrence, Bobak Kiani, Lingxiao Zhao, Tess Smidt, Suvrit Sra, Haggai Maron and Melanie Weber.

Conformal prediction equips machine learning models with a reasonable notion of uncertainty quantification without making strong distributional assumptions. It wraps around any black-box prediction model and converts point predictions into set predictions that have a predefined marginal coverage guarantee. However, conformal prediction only works if we fix the underlying machine learning model in advance. A relatively unaddressed issue in conformal prediction is that of model selection and/or aggregation: for a given problem, which of the plethora of prediction methods (random forests, neural nets, regularized linear models, etc.) should we conformalize? This talk presents a new approach towards conformal model aggregation in online settings that is based on combining the prediction sets from several algorithms by voting, where weights on the models are adapted over time based on past performance.