A PAC-Bayesian Perspective on the Interpolating Information Criterion

Deep learning is renowned for its theory-practice gap, whereby principled theory typically fails to provide much beneficial guidance for implementation in practice. This has been highlighted recently by the benign overfitting phenomenon: when neural networks become sufficiently large to interpolate the dataset perfectly, model performance appears to improve with increasing model size, in apparent contradiction with the well-known bias-variance tradeoff. While such phenomena have proven challenging to theoretically study for general models, the recently proposed Interpolating Information Criterion (IIC) provides a valuable theoretical framework to examine performance for overparameterized models. Using the IIC, a PAC-Bayes bound is obtained for a general class of models, characterizing factors which influence generalization performance in the interpolating regime. From the provided bound, we quantify how the test error for overparameterized models achieving effectively zero training error depends on the quality of the implicit regularization imposed by e.g. the combination of model, optimizer, and parameter-initialization scheme; the spectrum of the empirical neural tangent kernel; curvature of the loss landscape; and noise present in the data.

The Interpolating Information Criterion for Overparameterized Models

The problem of model selection is considered for the setting of interpolating estimators, where the number of model parameters exceeds the size of the dataset. Classical information criteria typically consider the large-data limit, penalizing model size. However, these criteria are not appropriate in modern settings where overparameterized models tend to perform well. For any overparameterized model, we show that there exists a dual underparameterized model that possesses the same marginal likelihood, thus establishing a form of Bayesian duality. This enables more classical methods to be used in the overparameterized setting, revealing the Interpolating Information Criterion, a measure of model quality that naturally incorporates the choice of prior into the model selection. Our new information criterion accounts for prior misspecification, geometric and spectral properties of the model, and is numerically consistent with known empirical and theoretical behavior in this regime.

Generalization Guarantees via Algorithm-dependent Rademacher Complexity

Algorithm- and data-dependent generalization bounds are required to explain the generalization behavior of modern machine learning algorithms. In this context, there exists information theoretic generalization bounds that involve (various forms of) mutual information, as well as bounds based on hypothesis set stability. We propose a conceptually related, but technically distinct complexity measure to control generalization error, which is the empirical Rademacher complexity of an algorithm- and data-dependent hypothesis class. Combining standard properties of Rademacher complexity with the convenient structure of this class, we are able to (i) obtain novel bounds based on the finite fractal dimension, which (a) extend previous fractal dimension-type bounds from continuous to finite hypothesis classes, and (b) avoid a mutual information term that was required in prior work; (ii) we greatly simplify the proof of a recent dimension-independent generalization bound for stochastic gradient descent; and (iii) we easily recover results for VC classes and compression schemes, similar to approaches based on conditional mutual information.

A Heavy-Tailed Algebra for Probabilistic Programming

Despite the successes of probabilistic models based on passing noise through neural networks, recent work has identified that such methods often fail to capture tail behavior accurately, unless the tails of the base distribution are appropriately calibrated. To overcome this deficiency, we propose a systematic approach for analyzing the tails of random variables, and we illustrate how this approach can be used during the static analysis (before drawing samples) pass of a probabilistic programming language compiler. To characterize how the tails change under various operations, we develop an algebra which acts on a three-parameter family of tail asymptotics and which is based on the generalized Gamma distribution. Our algebraic operations are closed under addition and multiplication; they are capable of distinguishing sub-Gaussians with differing scales; and they handle ratios sufficiently well to reproduce the tails of most important statistical distributions directly from their definitions. Our empirical results confirm that inference algorithms that leverage our heavy-tailed algebra attain superior performance across a number of density modeling and variational inference tasks.

When are ensembles really effective?

Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious. We study, both theoretically and empirically, the fundamental question of when ensembling yields significant performance improvements in classification tasks. Theoretically, we prove new results relating the \emph{ensemble improvement rate} (a measure of how much ensembling decreases the error rate versus a single model, on a relative scale) to the \emph{disagreement-error ratio}. We show that ensembling improves performance significantly whenever the disagreement rate is large relative to the average error rate; and that, conversely, one classifier is often enough whenever the disagreement rate is low relative to the average error rate. On the way to proving these results, we derive, under a mild condition called \emph{competence}, improved upper and lower bounds on the average test error rate of the majority vote classifier. To complement this theory, we study ensembling empirically in a variety of settings, verifying the predictions made by our theory, and identifying practical scenarios where ensembling does and does not result in large performance improvements. Perhaps most notably, we demonstrate a distinct difference in behavior between interpolating models (popular in current practice) and non-interpolating models (such as tree-based methods, where ensembling is popular), demonstrating that ensembling helps considerably more in the latter case than in the former.

Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes

The quality of many modern machine learning models improves as model complexity increases, an effect that has been quantified, for predictive performance, with the non-monotonic double descent learning curve. Here, we address the overarching question: is there an analogous theory of double descent for models which estimate uncertainty? We provide a partially affirmative and partially negative answer in the setting of Gaussian processes (GP). Under standard assumptions, we prove that higher model quality for optimally-tuned GPs (including uncertainty prediction) under marginal likelihood is realized for larger input dimensions, and therefore exhibits a monotone error curve. After showing that marginal likelihood does not naturally exhibit double descent in the input dimension, we highlight related forms of posterior predictive loss that do exhibit non-monotonicity. Finally, we verify empirically that our results hold for real data, beyond our considered assumptions, and we explore consequences involving synthetic covariates.

Fat-Tailed Variational Inference with Anisotropic Tail Adaptive Flows

While fat-tailed densities commonly arise as posterior and marginal distributions in robust models and scale mixtures, they present challenges when Gaussian-based variational inference fails to capture tail decay accurately. We first improve previous theory on tails of Lipschitz flows by quantifying how the tails affect the rate of tail decay and by expanding the theory to non-Lipschitz polynomial flows. Then, we develop an alternative theory for multivariate tail parameters which is sensitive to tail-anisotropy. In doing so, we unveil a fundamental problem which plagues many existing flow-based methods: they can only model tail-isotropic distributions (i.e., distributions having the same tail parameter in every direction). To mitigate this and enable modeling of tail-anisotropic targets, we propose anisotropic tail-adaptive flows (ATAF). Experimental results on both synthetic and real-world targets confirm that ATAF is competitive with prior work while also exhibiting appropriate tail-anisotropy.

Evaluating natural language processing models with generalization metrics that do not need access to any training or testing data

The search for effective and robust generalization metrics has been the focus of recent theoretical and empirical work. In this paper, we discuss the performance of natural language processing (NLP) models, and we evaluate various existing and novel generalization metrics. Compared to prior studies, we (i) focus on NLP instead of computer vision (CV), (ii) focus on generalization metrics that predict test error instead of the generalization gap, (iii) focus on generalization metrics that do not need the access to data, and (iv) focus on the heavy-tail (HT) phenomenon that has received comparatively less attention in the study of deep neural networks (NNs). We extend recent HT-based work which focuses on power law (PL) distributions, and we study exponential (EXP) and exponentially truncated power law (E-TPL) fitting to the empirical spectral densities (ESDs) of weight matrices. Our detailed empirical studies show that (i) \emph{shape metrics}, or the metrics obtained from fitting the shape of the ESDs, perform uniformly better at predicting generalization performance than \emph{scale metrics} commonly studied in the literature, as measured by the \emph{average} rank correlations with the generalization performance for all of our experiments; (ii) among forty generalization metrics studied in our paper, the \RANDDISTANCE metric, a new shape metric invented in this paper that measures the distance between empirical eigenvalues of weight matrices and those of randomly initialized weight matrices, achieves the highest worst-case rank correlation with generalization performance under a variety of training settings; and (iii) among the three HT distributions considered in our paper, the E-TPL fitting of ESDs performs the most robustly.

Generalization Properties of Stochastic Optimizers via Trajectory Analysis

Despite the ubiquitous use of stochastic optimization algorithms in machine learning, the precise impact of these algorithms on generalization performance in realistic non-convex settings is still poorly understood. In this paper, we provide an encompassing theoretical framework for investigating the generalization properties of stochastic optimizers, which is based on their dynamics. We first prove a generalization bound attributable to the optimizer dynamics in terms of the celebrated Fernique-Talagrand functional applied to the trajectory of the optimizer. This data- and algorithm-dependent bound is shown to be the sharpest possible in the absence of further assumptions. We then specialize this result by exploiting the Markovian structure of stochastic optimizers, deriving generalization bounds in terms of the (data-dependent) transition kernels associated with the optimization algorithms. In line with recent work that has revealed connections between generalization and heavy-tailed behavior in stochastic optimization, we link the generalization error to the local tail behavior of the transition kernels. We illustrate that the local power-law exponent of the kernel acts as an effective dimension, which decreases as the transitions become "less Gaussian". We support our theory with empirical results from a variety of neural networks, and we show that both the Fernique-Talagrand functional and the local power-law exponent are predictive of generalization performance.

Taxonomizing local versus global structure in neural network loss landscapes

Viewing neural network models in terms of their loss landscapes has a long history in the statistical mechanics approach to learning, and in recent years it has received attention within machine learning proper. Among other things, local metrics (such as the smoothness of the loss landscape) have been shown to correlate with global properties of the model (such as good generalization). Here, we perform a detailed empirical analysis of the loss landscape structure of thousands of neural network models, systematically varying learning tasks, model architectures, and/or quantity/quality of data. By considering a range of metrics that attempt to capture different aspects of the loss landscape, we demonstrate that the best test accuracy is obtained when: the loss landscape is globally well-connected; ensembles of trained models are more similar to each other; and models converge to locally smooth regions. We also show that globally poorly-connected landscapes can arise when models are small or when they are trained to lower quality data; and that, if the loss landscape is globally poorly-connected, then training to zero loss can actually lead to worse test accuracy. Based on these results, we develop a simple one-dimensional model with load-like and temperature-like parameters, we introduce the notion of an \emph{effective loss landscape} depending on these parameters, and we interpret our results in terms of a \emph{rugged convexity} of the loss landscape. When viewed through this lens, our detailed empirical results shed light on phases of learning (and consequent double descent behavior), fundamental versus incidental determinants of good generalization, the role of load-like and temperature-like parameters in the learning process, different influences on the loss landscape from model and data, and the relationships between local and global metrics, all topics of recent interest.