We present a new algorithm for automatically bounding the Taylor remainder series. In the special case of a scalar function $f: \mathbb{R} \to \mathbb{R}$, our algorithm takes as input a reference point $x_0$, trust region $[a, b]$, and integer $k \ge 1$, and returns an interval $I$ such that $f(x) - \sum_{i=0}^{k-1} \frac {1} {i!} f^{(i)}(x_0) (x - x_0)^i \in I (x - x_0)^k$ for all $x \in [a, b]$. As in automatic differentiation, the function $f$ is provided to the algorithm in symbolic form, and must be composed of known atomic functions. At a high level, our algorithm has two steps. First, for a variety of commonly-used elementary functions (e.g., $\exp$, $\log$), we use recently-developed theory to derive sharp polynomial upper and lower bounds on the Taylor remainder series. We then recursively combine the bounds for the elementary functions using an interval arithmetic variant of Taylor-mode automatic differentiation. Our algorithm can make efficient use of machine learning hardware accelerators, and we provide an open source implementation in JAX. We then turn our attention to applications. Most notably, in a companion paper we use our new machinery to create the first universal majorization-minimization optimization algorithms: algorithms that iteratively minimize an arbitrary loss using a majorizer that is derived automatically, rather than by hand. We also show that our automatically-derived bounds can be used for verified global optimization and numerical integration, and to prove sharper versions of Jensen's inequality.

Conventional federated learning algorithms train a single global model by leveraging all participating clients' data. However, due to heterogeneity in client generative distributions and predictive models, these approaches may not appropriately approximate the predictive process, converge to an optimal state, or generalize to new clients. We study personalization and generalization in stateless cross-device federated learning setups assuming heterogeneity in client data distributions and predictive models. We first propose a hierarchical generative model and formalize it using Bayesian Inference. We then approximate this process using Variational Inference to train our model efficiently. We call this algorithm Federated Variational Inference (FedVI). We use PAC-Bayes analysis to provide generalization bounds for FedVI. We evaluate our model on FEMNIST and CIFAR-100 image classification and show that FedVI beats the state-of-the-art on both tasks.

Ensembling has proven to be a powerful technique for boosting model performance, uncertainty estimation, and robustness in supervised learning. Advances in self-supervised learning (SSL) enable leveraging large unlabeled corpora for state-of-the-art few-shot and supervised learning performance. In this paper, we explore how ensemble methods can improve recent SSL techniques by developing a framework that permits data-dependent weighted cross-entropy losses. We refrain from ensembling the representation backbone; this choice yields an efficient ensemble method that incurs a small training cost and requires no architectural changes or computational overhead to downstream evaluation. The effectiveness of our method is demonstrated with two state-of-the-art SSL methods, DINO (Caron et al., 2021) and MSN (Assran et al., 2022). Our method outperforms both in multiple evaluation metrics on ImageNet-1K, particularly in the few-shot setting. We explore several weighting schemes and find that those which increase the diversity of ensemble heads lead to better downstream evaluation results. Thorough experiments yield improved prior art baselines which our method still surpasses; e.g., our overall improvement with MSN ViT-B/16 is 3.9 p.p. for 1-shot learning. These successes have encouraged increasingly advanced SSL techniques (e.g., Grill et al., 2020; Zbontar et al., 2021; He et al., 2022). Perhaps surprisingly however, a simple and otherwise common idea has received limited consideration: ensembling. Ensembling combines predictions from multiple trained models and has proven effective at improving model accuracy (Hansen & Salamon, 1990; Perrone & Cooper, 1992) and capturing predictive uncertainty in supervised learning (Lakshminarayanan et al., 2017; Ovadia et al., 2019). Ensembling in the SSL regime is nuanced, however; since the goal is to learn useful representations from unlabeled data, it is less obvious where and how to ensemble. We explore these questions in this work.

While the decision-theoretic optimality of the Bayesian formalism under correct model specification is well-known (Berger 2013), the Bayesian case becomes less clear under model misspecification (Grunwald 2017; Ramamoorthi 2015; Fushiki 2005). To formally understand the consequences of Bayesian misspecification, this work examines the relationship between posterior predictive risk and its sensitivity to correct model assumptions, i.e., choice of likelihood and prior. We present the multisample PAC$^m$-Bayes risk. This risk is justified by theoretical analysis based on PAC-Bayes as well as empirical study on a number of toy problems. The PAC$^m$-Bayes risk is appealing in that it entails direct minimization of the Monte-Carlo approximated posterior predictive risk yet recovers both the Bayesian formalism as well as the MLE in its limits. Our work is heavily influenced by Masegosa (2019); our contributions are to align training and generalization risks while offering a tighter bound which empirically performs at least as well and sometimes much better.

Perhaps surprisingly, recent studies have shown probabilistic model likelihoods have poor specificity for out-of-distribution (OOD) detection and often assign higher likelihoods to OOD data than in-distribution data. To ameliorate this issue we propose DoSE, the density of states estimator. Drawing on the statistical physics notion of ``density of states,'' the DoSE decision rule avoids direct comparison of model probabilities, and instead utilizes the ``probability of the model probability,'' or indeed the frequency of any reasonable statistic. The frequency is calculated using nonparametric density estimators (e.g., KDE and one-class SVM) which measure the typicality of various model statistics given the training data and from which we can flag test points with low typicality as anomalous. Unlike many other methods, DoSE requires neither labeled data nor OOD examples. DoSE is modular and can be trivially applied to any existing, trained model. We demonstrate DoSE's state-of-the-art performance against other unsupervised OOD detectors on previously established ``hard'' benchmarks.

Modern machine learning methods including deep learning have achieved great success in predictive accuracy for supervised learning tasks, but may still fall short in giving useful estimates of their predictive {\em uncertainty}. Quantifying uncertainty is especially critical in real-world settings, which often involve input distributions that are shifted from the training distribution due to a variety of factors including sample bias and non-stationarity. In such settings, well calibrated uncertainty estimates convey information about when a model's output should (or should not) be trusted. Many probabilistic deep learning methods, including Bayesian-and non-Bayesian methods, have been proposed in the literature for quantifying predictive uncertainty, but to our knowledge there has not previously been a rigorous large-scale empirical comparison of these methods under dataset shift. We present a large-scale benchmark of existing state-of-the-art methods on classification problems and investigate the effect of dataset shift on accuracy and calibration. We find that traditional post-hoc calibration does indeed fall short, as do several other previous methods. However, some methods that marginalize over models give surprisingly strong results across a broad spectrum of tasks.

Discriminative neural networks offer little or no performance guarantees when deployed on data not generated by the same process as the training distribution. On such out-of-distribution (OOD) inputs, the prediction may not only be erroneous, but confidently so, limiting the safe deployment of classifiers in real-world applications. One such challenging application is bacteria identification based on genomic sequences, which holds the promise of early detection of diseases, but requires a model that can output low confidence predictions on OOD genomic sequences from new bacteria that were not present in the training data. We introduce a genomics dataset for OOD detection that allows other researchers to benchmark progress on this important problem. We investigate deep generative model based approaches for OOD detection and observe that the likelihood score is heavily affected by population level background statistics. We propose a likelihood ratio method for deep generative models which effectively corrects for these confounding background statistics. We benchmark the OOD detection performance of the proposed method against existing approaches on the genomics dataset and show that our method achieves state-of-the-art performance. We demonstrate the generality of the proposed method by showing that it significantly improves OOD detection when applied to deep generative models of images.

Hamiltonian Monte Carlo is a powerful algorithm for sampling from difficult-to-normalize posterior distributions. However, when the geometry of the posterior is unfavorable, it may take many expensive evaluations of the target distribution and its gradient to converge and mix. We propose neural transport (NeuTra) HMC, a technique for learning to correct this sort of unfavorable geometry using inverse autoregressive flows (IAF), a powerful neural variational inference technique. The IAF is trained to minimize the KL divergence from an isotropic Gaussian to the warped posterior, and then HMC sampling is performed in the warped space. We evaluate NeuTra HMC on a variety of synthetic and real problems, and find that it significantly outperforms vanilla HMC both in time to reach the stationary distribution and asymptotic effective-sample-size rates.

We present a simple case study, demonstrating that Variational Information Bottleneck (VIB) can improve a network's classification calibration as well as its ability to detect out-of-distribution data. Without explicitly being designed to do so, VIB gives two natural metrics for handling and quantifying uncertainty.

Recent work in unsupervised representation learning has focused on learning deep directed latent-variable models. Fitting these models by maximizing the marginal likelihood or evidence is typically intractable, thus a common approximation is to maximize the evidence lower bound (ELBO) instead. However, maximum likelihood training (whether exact or approximate) does not necessarily result in a good latent representation, as we demonstrate both theoretically and empirically. In particular, we derive variational lower and upper bounds on the mutual information between the input and the latent variable, and use these bounds to derive a rate-distortion curve that characterizes the tradeoff between compression and reconstruction accuracy. Using this framework, we demonstrate that there is a family of models with identical ELBO, but different quantitative and qualitative characteristics. Our framework also suggests a simple new method to ensure that latent variable models with powerful stochastic decoders do not ignore their latent code.