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We consider a line-search method for continuous optimization under a stochastic setting where the function values and gradients are available only through inexact probabilistic zeroth and first-order oracles. These oracles capture multiple stan- dard settings including expected loss minimization and zeroth-order optimization. Moreover, our framework is very general and allows the function and gradient estimates to be biased. The proposed algorithm is simple to describe, easy to im- plement, and uses these oracles in a similar way as the standard deterministic line search uses exact function and gradient values. Under fairly general conditions on the oracles, we derive a high probability tail bound on the iteration complexity of the algorithm when applied to non-convex smooth functions. These results are stronger than those for other existing stochastic line search methods and apply in more general settings. 

Tractably modelling distributions over manifolds has long been an important goal in the natural sciences. Recent work has focused on developing general machine learning models to learn such distributions. However, for many applications these distributions must respect manifold symmetries—a trait which most previous models disregard. In this paper, we lay the theoretical foundations for learning symmetry-invariant distributions on arbitrary manifolds via equivariant manifold flows. We demonstrate the utility of our approach by learning quantum field theory-motivated invariant SU(n) densities and by correcting meteor impact dataset bias. 

Hyperbolic space is particularly useful for embedding data with hierarchical structure; however, representing hyperbolic space with ordinary floating-point numbers greatly affects the performance due to its \emph{ineluctable} numerical errors. Simply increasing the precision of floats fails to solve the problem and incurs a high computation cost for simulating greater-than-double-precision floats on hardware such as GPUs, which does not support them. In this paper, we propose a simple, feasible-on-GPUs, and easy-to-understand solution for numerically accurate learning on hyperbolic space. We do this with a new approach to represent hyperbolic space using multi-component floating-point (MCF) in the Poincar{\'e} upper-half space model. Theoretically and experimentally we show our model has small numerical error, and on embedding tasks across various datasets, models represented by multi-component floating-points gain more capacity and run significantly faster on GPUs than prior work. 

Adversarial examples are a widely studied phenomenon in machine learning models. While most of the attention has been focused on neural networks, other practical models also suffer from this issue. In this work, we propose an algorithm for evaluating the adversarial robustness of k-nearest neighbor classification, i.e., finding a minimum-norm adversarial example. Diverging from previous proposals, we propose the first geometric approach by performing a search that expands outwards from a given input point. On a high level, the search radius expands to the nearby higher-order Voronoi cells until we find a cell that classifies differently from the input point. To scale the algorithm to a large k, we introduce approximation steps that find perturbation with smaller norm, compared to the baselines, in a variety of datasets. Furthermore, we analyze the structural properties of a dataset where our approach outperforms the competition. 

We develop nested variational inference (NVI), a family of methods that learn proposals for nested importance samplers by minimizing an forward or reverse KL divergence at each level of nesting. NVI is applicable to many commonly-used importance sampling strategies and provides a mechanism for learning intermediate densities, which can serve as heuristics to guide the sampler. Our experiments apply NVI to (a) sample from a multimodal distribution using a learned annealing path (b) learn heuristics that approximate the likelihood of future observations in a hidden Markov model and (c) to perform amortized inference in hierarchical deep generative models. We observe that optimizing nested objectives leads to improved sample quality in terms of log average weight and effective sample size. 

Bootstrapping has been a primary tool for ensemble and uncertainty quantification in machine learning and statistics. However, due to its nature of multiple training and resampling, bootstrapping deep neural networks is computationally burdensome; hence it has difficulties in practical application to the uncertainty estimation and related tasks. To overcome this computational bottleneck, we propose a novel approach called Neural Bootstrapper (NeuBoots), which learns to generate bootstrapped neural networks through single model training. NeuBoots injects the bootstrap weights into the high-level feature layers of the backbone network and outputs the bootstrapped predictions of the target, without additional parameters and the repetitive computations from scratch. We apply NeuBoots to various machine learning tasks related to uncertainty quantification, including prediction calibrations in image classification and semantic segmentation, active learning, and detection of out-of-distribution samples. Our empirical results show that NeuBoots outperforms other bagging based methods under a much lower computational cost without losing the validity of bootstrapping.