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We present a generalization of Nesterov's accelerated gradient descent algorithm. Our algorithm (AGNES) provably achieves acceleration for smooth convex and strongly convex minimization tasks with noisy gradient estimates if the noise intensity is proportional to the magnitude of the gradient at every point. Nesterov's method converges at an accelerated rate if the constant of proportionality is below 1, while AGNES accommodates any signal-to-noise ratio. The noise model is motivated by applications in overparametrized machine learning. AGNES requires only two parameters in convex and three in strongly convex minimization tasks, improving on existing methods. We further provide clear geometric interpretations and heuristics for the choice of parameters. 

We consider the problem of determining the manifold $$n$$-widths of Sobolev and Besov spaces with error measured in the $$L_p$$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $$q$$ satisfies $$q\leq p$$ or $$1 \leq p \leq 2$$. We close this gap and obtain sharp lower bounds for all $$1 \leq p,q \leq \infty$$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $$\ell^M_q$$-balls in the $$\ell_p$$-norm when $$p\leq q$$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases. 

Despite significant progress in vaccine research, the level of protection provided by vaccination can vary significantly across individuals. As a result, understanding immunologic variation across individuals in response to vaccination is important for developing next-generation efficacious vaccines. Accurate outcome prediction and identification of predictive biomarkers would represent a significant step towards this goal. Moreover, in early phase vaccine clinical trials, small datasets are prevalent, raising the need and challenge of building a robust and explainable prediction model that can reveal heterogeneity in small datasets. We propose a new model named Generative Mixture of Logistic Regression (GeM-LR), which combines characteristics of both a generative and a discriminative model. In addition, we propose a set of model selection strategies to enhance the robustness and interpretability of the model. GeM-LR extends a linear classifier to a non-linear classifier without losing interpretability and empowers the notion of predictive clustering for characterizing data heterogeneity in connection with the outcome variable. We demonstrate the strengths and utility of GeM-LR by applying it to data from several studies. GeM-LR achieves better prediction results than other popular methods while providing interpretations at different levels. 

Canonicalization provides an architecture-agnostic method for enforcing equivariance, with generalizations such as frame-averaging recently gaining prominence as a lightweight and flexible alternative to equivariant architectures. Recent works have found an empirical benefit to using probabilistic frames instead, which learn weighted distributions over group elements. In this work, we provide strong theoretical justification for this phenomenon: for commonly-used groups, there is no efficiently computable choice of frame that preserves continuity of the function being averaged. In other words, unweighted frame-averaging can turn a smooth, non-symmetric function into a discontinuous, symmetric function. To address this fundamental robustness problem, we formally define and construct weighted frames, which provably preserve continuity, and demonstrate their utility by constructing efficient and continuous weighted frames for the actions of SO(d), O(d), and Sn on point clouds. 

In this paper we propose a method for the optimal allocation of observations between an intrinsically explainable glass box model and a black box model. An optimal allocation being defined as one which, for any given explainability level (i.e. the proportion of observations for which the explainable model is the prediction function), maximizes the performance of the ensemble on the underlying task, and maximizes performance of the explainable model on the observations allocated to it, subject to the maximal ensemble performance condition. The proposed method is shown to produce such explainability optimal allocations on a benchmark suite of tabular datasets across a variety of explainable and black box model types. These learned allocations are found to consistently maintain ensemble performance at very high explainability levels (explaining 74% of observations on average), and in some cases even outperform both the component explainable and black box models while improving explainability.