The success of SGD in deep learning has been ascribed by prior works to the *implicit bias* induced by finite batch sizes (''SGD noise''). While prior works focused on *offline learning* (i.e., multiple-epoch training), we study the impact of SGD noise on *online* (i.e., single epoch) learning. Through an extensive empirical analysis of image and language data, we demonstrate that small batch sizes do *not* confer any implicit bias advantages in online learning. In contrast to offline learning, the benefits of SGD noise in online learning are strictly computational, facilitating more cost-effective gradient steps. This suggests that SGD in the online regime can be construed as taking noisy steps along the ''golden path'' of the noiseless *gradient descent* algorithm. We study this hypothesis and provide supporting evidence in loss and function space. Our findings challenge the prevailing understanding of SGD and offer novel insights into its role in online learning.

Large neural networks trained in the overparameterized regime are able to fit noise to zero train error. Recent work of Nakkiran and Bansal has empirically observed that such networks behave as “conditional samplers” from the noisy distribution. That is, they replicate the noise in the train data to unseen examples. We give a theoretical framework for studying this conditional sampling behavior in the context of learning theory. We relate the notion of such samplers to knowledge distillation, where a student network imitates the outputs of a teacher on unlabeled data. We show that samplers, while being bad classifiers, can be good teachers. Concretely, we prove that distillation from samplers is guaranteed to produce a student which approximates the Bayes optimal classifier. Finally, we show that some common learning algorithms (e.g., Nearest-Neighbours and Kernel Machines) can often generate samplers when applied in the overparameterized regime.

In this paper we study the problem of robust influence maximization in the independent cascade model under a hyperparametric assumption. In social networks users influence and are influenced by individuals with similar characteristics and as such they are associated with some features. A recent surging research direction in influence maximization focuses on the case where the edge probabilities on the graph are not arbitrary but are generated as a function of the features of the users and a global hyperparameter. We propose a model where the objective is to maximize the worst-case number of influenced users for any possible value of that hyperparameter. We provide theoretical results showing that proper robust solution in our model is NP-hard and an algorithm that achieves improper robust optimization. We make-use of sampling based techniques and of the renowned multiplicative weight updates algorithm. Additionally we validate our method empirically and prove that it outperforms the state-of-the-art robust influence maximization techniques.

We perform an experimental study of the dynamics of Stochastic Gradient Descent (SGD) in learning deep neural networks for several real and synthetic classification tasks. We show that in the initial epochs, almost all of the performance improvement of the classifier obtained by SGD can be explained by a linear classifier. More generally, we give evidence for the hypothesis that, as iterations progress, SGD learns functions of increasing complexity. This hypothesis can be helpful in explaining why SGD-learned classifiers tend to generalize well even in the over-parameterized regime. We also show that the linear classifier learned in the initial stages is ``retained'' throughout the execution even if training is continued to the point of zero training error, and complement this with a theoretical result in a simplified model. Key to our work is a new measure of how well one classifier explains the performance of another, based on conditional mutual information.