Inspired by their powerful representation ability on graph-structured data, Graph Convolution Networks (GCNs) have been widely applied to recommender systems, and have shown superior performance. Despite their empirical success, there is a lack of theoretical explorations such as generalization properties. In this paper, we take a first step towards establishing a generalization guarantee for GCN-based recommendation models under inductive and transductive learning. We mainly investigate the roles of graph normalization and non-linear activation, providing some theoretical understanding, and construct extensive experiments to further verify these findings empirically. Furthermore, based on the proven generalization bound and the challenge of existing models in discrete data learning, we propose Item Mixture (IMix) to enhance recommendation. It models discrete spaces in a continuous manner by mixing the embeddings of positive-negative item pairs, and its effectiveness can be strictly guaranteed from empirical and theoretical aspects.

We are motivated by the problem of providing strong generalization guarantees in the context of meta-learning. Existing generalization bounds are either challenging to evaluate or provide vacuous guarantees in even relatively simple settings. We derive a probably approximately correct (PAC) bound for gradient-based metalearning using two different generalization frameworks in order to deal with the qualitatively different challenges of generalization at the "base" and "meta" levels. We employ bounds for uniformly stable algorithms at the base level and bounds from the PAC-Bayes framework at the meta level. The result of this approach is a novel PAC bound that is tighter when the base learner adapts quickly, which is precisely the goal of meta-learning. We show that our bound provides a tighter guarantee than other bounds on a toy non-convex problem on the unit sphere and a text-based classification example. We also present a practical regularization scheme motivated by the bound in settings where the bound is loose and demonstrate improved performance over baseline techniques.

We investigate a stochastic counterpart of majority votes over finite ensembles of classifiers, and study its generalization properties. While our approach holds for arbitrary distributions, we instantiate it with Dirichlet distributions: this allows for a closed-form and differentiable expression for the expected risk, which then turns the generalization bound into a tractable training objective. The resulting stochastic majority vote learning algorithm achieves state-of-the-art accuracy and benefits from (non-vacuous) tight generalization bounds, in a series of numerical experiments when compared to competing algorithms which also minimize PACBayes objectives - both with uninformed (data-independent) and informed (datadependent) priors.

We study learning to learn for regression problems through the lens of hyperparameter tuning. We propose the Langevin Gradient Descent Algorithm (LGD), which approximates the mean of the posterior distribution defined by the loss function and regularizer of a convex regression task. We prove the existence of an optimal hyperparameter configuration for which the LGD algorithm achieves the Bayes' optimal solution for squared loss. Subsequently, we study generalization guarantees on meta-learning optimal hyperparameters for the LGD algorithm from a given set of tasks in the data-driven setting. For a number of parameters $d$ and hyperparameter dimension $h$, we show a pseudo-dimension bound of $O(dh)$, upto logarithmic terms under mild assumptions on LGD. This matches the dimensional dependence of the bounds obtained in prior work for the elastic net, which only allows for $h=2$ hyperparameters, and extends their bounds to regression on convex loss. Finally, we show empirical evidence of the success of LGD and the meta-learning procedure for few-shot learning on linear regression using a few synthetically created datasets.

In this paper, we introduce and analyze Discriminative State-Space Models for forecasting non-stationary time series. We provide data-dependent generalization guarantees for learning these models based on the recently introduced notion of discrepancy. We provide an in-depth analysis of the complexity of such models. Finally, we also study the generalization guarantees for several structural risk minimization approaches to this problem and provide an efficient implementation for one of them which is based on a convex objective.

However, differential privacy's worst-case nature entails scaling such noise to the range of the queries even for

However, differential privacy's worst-case nature entails scaling such noise to the range of the queries even for