Understanding the training and generalization dynamics of deep neural networks as well as the actual accuracy of the network predictions when deployed in the wild are important open problems in machine learning. In this thesis, we study these two topics in the context of image classification. In the first part, we study the generalization properties of deep neural networks with respect to the regularization of the network training for standard image classification tasks. In the second part, we study the performance of conformal prediction based uncertainty estimation methods. Conformal prediction methods quantify the uncertainty of the predictions of a neural network in practical applications. We study the setup where the test distribution may induce a drop in the accuracy of the predictions due to distribution shift.

The training of deep neural networks is often regularized either implicitly, for example by early stopping the gradient descent, or explicitly, by adding an ℓ2-penalty to the loss function, in order to prevent overfitting to spurious patterns or noise. Even though these regularization methods are well established in the literature, recently it was uncovered that the test error of the network can exhibit novel phenomena such as yielding a double descent shape with respect to the regularization amount.

In the first part of this thesis, we develop a theoretical understanding of the double descent phenomenon with respect to model regularization. For this, we study regression tasks, in both the underparameterized and overparameterized regimes, for linear and non-linear models. We find that for linear regression, a double descent shaped risk is caused by a superposition of bias-variance tradeoffs corresponding to different parts of the data/model and can be mitigated by the proper scaling of the stepsizes or regularization strengths while improving the best-case performance. We next study a non-linear two-layer neural network and characterize the early-stopped gradient descent risk as a superposition of bias-variance tradeoffs and also show that double descent as a function of the L2-regularization coefficient occurs outside of the regime where the risk can be characterized using the existing tools in the literature. We empirically study deep networks trained on standard image classification datasets and show that our results well explain the dynamics of the network training.

In the second part of this thesis, we consider the effects of data distribution shift at test time for standard deep neural network classifiers. While recent uncertainty quantification methods like conformal prediction can generate provably valid confidence measures for any pre-trained black-box image classifier, these guarantees fail when there is a distribution shift. We propose a simple test-time recalibration method based on only unlabeled examples that provides excellent uncertainty estimates under natural distribution shifts. We show that our method provably succeeds on a theoretical toy distribution shift problem. Empirically, we show the success of our method for various natural distribution shifts of the popular ImageNet dataset.