The task of tuning regularization coefficients in regularized regression models with provable guarantees across problem instances still poses a significant challenge in the literature. This paper investigates the sample complexity of tuning regularization parameters in linear and logistic regressions under $\ell_1$ and $\ell_2$-constraints in the data-driven setting. For the linear regression problem, by more carefully exploiting the structure of the dual function class, we provide a new upper bound for the pseudo-dimension of the validation loss function class, which significantly improves the best-known results on the problem. Remarkably, we also instantiate the first matching lower bound, proving our results are tight. For tuning the regularization parameters of logistic regression, we introduce a new approach to studying the learning guarantee via an approximation of the validation loss function class. We examine the pseudo-dimension of the approximation class and construct a uniform error bound between the validation loss function class and its approximation, which allows us to instantiate the first learning guarantee for the problem of tuning logistic regression regularization coefficients.

We present a transductive learning algorithm that takes as input training examples from a distribution P and arbitrary (unlabeled) test examples, possibly chosen by an adversary. This is unlike prior work that assumes that test examples are small perturbations of P. Our algorithm outputs a selective classifier, which abstains from predicting on some examples. By considering selective transductive learning, we give the first nontrivial guarantees for learning classes of bounded VC dimension with arbitrary train and test distributions--no prior guarantees were known even for simple classes of functions such as intervals on the line. In particular, for any function in a class C of bounded VC dimension, we guarantee a low test error rate and a low rejection rate with respect to P. Our algorithm is efficient given an Empirical Risk Minimizer (ERM) for C. Our guarantees hold even for test examples chosen by an unbounded white-box adversary. We also give guarantees for generalization, agnostic, and unsupervised settings.

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 present a general theoretical analysis of structured prediction with a series of new results.

Last, we provide the detailed proofs for Lemma 1 and Theorem 1. A.1 Background on Rademacher complexity and the contraction inequality Definition 1

V arious evaluation measures have been developed for multi-label classification, including Hamming Loss (HL), Subset Accuracy (SA) and Ranking Loss (RL).

This is obtained by combining the regret bound in Eq.(2) with

The task of tuning regularization coefficients in regularized regression models with provable guarantees across problem instances still poses a significant challenge in the literature. This paper investigates the sample complexity of tuning regularization parameters in linear and logistic regressions under \ell_1 and \ell_2 -constraints in the data-driven setting. For the linear regression problem, by more carefully exploiting the structure of the dual function class, we provide a new upper bound for the pseudo-dimension of the validation loss function class, which significantly improves the best-known results on the problem. Remarkably, we also instantiate the first matching lower bound, proving our results are tight. For tuning the regularization parameters of logistic regression, we introduce a new approach to studying the learning guarantee via an approximation of the validation loss function class.

Stochastic compositional optimization (SCO) problem constitutes a class of optimization problems characterized by the objective function with a compositional form, including the tasks with known derivatives, such as AUC maximization, and the derivative-free tasks exemplified by black-box vertical federated learning (VFL). From the learning theory perspective, the learning guarantees of SCO algorithms with known derivatives have been studied in the literature. However, the potential impacts of the derivative-free setting on the learning guarantees of SCO remains unclear and merits further investigation. This paper aims to reveal the impacts by developing a theoretical analysis for two derivative-free algorithms, black-box SCGD and SCSC. Specifically, we first provide the sharper generalization upper bounds of convex SCGD and SCSC based on a new stability analysis framework more effective than prior work under some milder conditions, which is further developed to the non-convex case using the almost co-coercivity property of smooth function.