Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.

Recent advances in deep learning theory

Understanding the generalization behaviour of deep neural networks is a topic of recent interest that has driven the production of many studies, notably the development and evaluation of generalization "explainability" measures that quantify model generalization ability. Generalization measures have also proven useful in the development of powerful layer-wise model tuning and optimization algorithms, though these algorithms require specific kinds of generalization measures which can probe individual layers. The purpose of this paper is to explore the neglected subtopic of probeable generalization measures; to establish firm ground for further investigations, and to inspire and guide the development of novel model tuning and optimization algorithms. We evaluate and compare measures, demonstrating effectiveness and robustness across model variations, dataset complexities, training hyperparameters, and training stages. We also introduce a new dataset of trained models and performance metrics, GenProb, for testing generalization measures, model tuning algorithms and optimization algorithms.

In Search of Probeable Generalization Measures

Conformal prediction has emerged as a rigorous means of providing deep learning models with reliable uncertainty estimates and safety guarantees. Yet, its performance is known to degrade under distribution shift and long-tailed class distributions, which are often present in real world applications. Here, we characterize the performance of several post-hoc and training-based conformal prediction methods under these settings, providing the first empirical evaluation on large-scale datasets and models. We show that across numerous conformal methods and neural network families, performance greatly degrades under distribution shifts violating safety guarantees. Similarly, we show that in long-tailed settings the guarantees are frequently violated on many classes. Understanding the limitations of these methods is necessary for deployment in real world and safety-critical applications.

Empirically Validating Conformal Prediction on Modern Vision Architectures Under Distribution Shift and Long-tailed Data

As a powerful Bayesian non-parameterized algorithm, the Gaussian process (GP) has performed a signiﬁcant role in Bayesian optimization and signal processing. GPs have also advanced online decision-making systems because their posterior distribution has a closed-form solution. However, its training and inference process requires all historic data to be stored and the GP model to be trained from scratch. For those reasons, several online GP algorithms, such as O-SGPR and O-SVGP, have been speciﬁcally designed for streaming settings. In this paper, we present a new theoretical framework for online GPs based on the online probably approximately correct (PAC) Bayes theory. The framework offers both a guarantee of generalized performance and good accuracy. Instead of minimizing the marginal likelihood, our algorithm optimizes both the empirical risk function and a regularization item, which is in proportion to the divergence between the prior distribution and posterior distribution of parameters. In addition to its theoretical appeal, the algorithm performs well empirically on several regression datasets. Compared to other online GP algorithms, ours yields a generalization guarantee and very competitive accuracy.

Streaming PAC-Bayes Gaussian process regression with a performance guarantee for online decision making

Several uncertainty estimation methods have been recently proposed for machine translation evaluation. While these methods can provide a useful indication of when not to trust model predictions, we show in this paper that the majority of them tend to underestimate model uncertainty, and as a result they often produce misleading confidence intervals that do not cover the ground truth. We propose as an alternative the use of conformal prediction, a distribution-free method to obtain confidence intervals with a theoretically established guarantee on coverage. First, we demonstrate that split conformal prediction can ``correct'' the confidence intervals of previous methods to yield a desired coverage level. Then, we highlight biases in estimated confidence intervals, both in terms of the translation language pairs and the quality of translations. We apply conditional conformal prediction techniques to obtain calibration subsets for each data subgroup, leading to equalized coverage.

Conformalizing Machine Translation Evaluation

Identifying optimal values for a high-dimensional set of hyperparameters is a problem that has received growing attention given its importance to large-scale machine learning applications such as neural architecture search. Recently developed optimization methods can be used to select thousands or even millions of hyperparameters. Such methods often yield overfit models, however, leading to poor performance on unseen data. We argue that this overfitting results from using the standard hyperparameter optimization objective function. Here we present an alternative objective that is equivalent to a Probably Approximately Correct-Bayes (PAC-Bayes) bound on the expected out-of-sample error. We then devise an efficient gradient-based algorithm to minimize this objective; the proposed method has asymptotic space and time complexity equal to or better than other gradient-based hyperparameter optimization methods. We show that this new method significantly reduces out-of-sample error when applied to hyperparameter optimization problems known to be prone to overfitting.

Efficient hyperparameter optimization by way of PAC-Bayes bound minimization.

We consider the problem of constructing distribution-free prediction sets with finite-sample conditional guarantees. Prior work has shown that it is impossible to provide exact conditional coverage universally in finite samples. Thus, most popular methods only provide marginal coverage over the covariates. This paper bridges this gap by defining a spectrum of problems that interpolate between marginal and conditional validity. We motivate these problems by reformulating conditional coverage as coverage over a class of covariate shifts. When the target class of shifts is finite dimensional, we show how to simultaneously obtain exact finite sample coverage over all possible shifts. For example, given a collection of protected subgroups, our algorithm outputs intervals with exact coverage over each group. For more flexible, infinite dimensional classes where exact coverage is impossible, we provide a simple procedure for quantifying the gap between the coverage of our algorithm and the target level. Moreover, by tuning a single hyperparameter, we allow the practitioner to control the size of this gap across shifts of interest. Our methods can be easily incorporated into existing split conformal inference pipelines, and thus can be used to quantify the uncertainty of modern black-box algorithms without distributional assumptions.

Conformal Prediction With Conditional Guarantees

Before deploying a black-box model in high-stakes problems, it is important to evaluate the model's performance on sensitive subpopulations. For example, in a recidivism prediction task, we may wish to identify demographic groups for which our prediction model has unacceptably high false positive rates or certify that no such groups exist. In this paper, we frame this task, often referred to as"fairness auditing,"in terms of multiple hypothesis testing. We show how the bootstrap can be used to simultaneously bound performance disparities over a collection of groups with statistical guarantees. Our methods can be used to flag subpopulations affected by model underperformance, and certify subpopulations for which the model performs adequately. Crucially, our audit is model-agnostic and applicable to nearly any performance metric or group fairness criterion. Our methods also accommodate extremely rich -- even infinite -- collections of subpopulations. Further, we generalize beyond subpopulations by showing how to assess performance over certain distribution shifts. We test the proposed methods on benchmark datasets in predictive inference and algorithmic fairness and find that our audits can provide interpretable and trustworthy guarantees.

John J. Cherian

Papers

Efficient hyperparameter optimization by way of PAC-Bayes bound minimization.

Conformal Prediction With Conditional Guarantees

Statistical Inference for Fairness Auditing