We study computational aspects of algorithmic replicability, a notion of stability introduced by Impagliazzo, Lei, Pitassi, and Sorrell [2022]. Motivated by a recent line of work that established strong statistical connections between replicability and other notions of learnability such as online learning, private learning, and SQ learning, we aim to understand better the computational connections between replicability and these learning paradigms. Our first result shows that there is a concept class that is efficiently replicably PAC learnable, but, under standard cryptographic assumptions, no efficient online learner exists for this class. Subsequently, we design an efficient replicable learner for PAC learning parities when the marginal distribution is far from uniform, making progress on a question posed by Impagliazzo et al. [2022]. To obtain this result, we design a replicable lifting framework inspired by Blanc, Lange, Malik, and Tan [2023] that transforms in a black-box manner efficient replicable PAC learners under the uniform marginal distribution over the Boolean hypercube to replicable PAC learners under any marginal distribution, with sample and time complexity that depends on a certain measure of the complexity of the distribution. Finally, we show that any pure DP learner can be transformed to a replicable one in time polynomial in the accuracy, confidence parameters and exponential in the representation dimension of the underlying hypothesis class.

Most work in the area of learning theory has focused on designing effective Probably Approximately Correct (PAC) learners. Recently, other models of learning such as transductive error have seen more scrutiny. We move toward showing that these problems are equivalent by reducing agnostic learning with a PAC guarantee to agnostic learning with a transductive guarantee by adding a small number of samples to the dataset. We first rederive the result of Aden-Ali et al. arXiv:2304.09167 reducing PAC learning to transductive learning in the realizable setting using simpler techniques and at more generality as background for our main positive result. Our agnostic transductive to PAC conversion technique extends the aforementioned argument to the agnostic case, showing that an agnostic transductive learner can be efficiently converted to an agnostic PAC learner. Finally, we characterize the performance of the agnostic one inclusion graph algorithm of Asilis et al. arXiv:2309.13692 for binary classification, and show that plugging it into our reduction leads to an agnostic PAC learner that is essentially optimal. Our results imply that transductive and PAC learning are essentially equivalent for supervised learning with pseudometric losses in the realizable setting, and for binary classification in the agnostic setting. We conjecture this is true more generally for the agnostic setting.

Multilabel ranking is a central task in machine learning. However, the most fundamental question of learnability in a multilabel ranking setting with relevance-score feedback remains unanswered. In this work, we characterize the learnability of multilabel ranking problems in both batch and online settings for a large family of ranking losses. Along the way, we give two equivalence classes of ranking losses based on learnability that capture most, if not all, losses used in practice.

We introduce definitions of computable PAC learning for binary classification over computable metric spaces. We provide sufficient conditions for learners that are empirical risk minimizers (ERM) to be computable, and bound the strong Weihrauch degree of an ERM learner under more general conditions. We also give a presentation of a hypothesis class that does not admit any proper computable PAC learner with computable sample function, despite the underlying class being PAC learnable.

We provide sample complexity upper bounds for agnostically learning multivariate Gaussians under the constraint of approximate differential privacy. These are the first finite sample upper bounds for general Gaussians which do not impose restrictions on the parameters of the distribution. Our bounds are near-optimal in the case when the covariance is known to be the identity, and conjectured to be near-optimal in the general case. From a technical standpoint, we provide analytic tools for arguing the existence of global "locally small" covers from local covers of the space. These are exploited using modifications of recent techniques for differentially private hypothesis selection. Our techniques may prove useful for privately learning other distribution classes which do not possess a finite cover.

We design differentially private learning algorithms that are agnostic to the learning model assuming access to a limited amount of unlabeled public data. First, we provide a new differentially private algorithm for answering a sequence of m online classification queries (given by a sequence of m unlabeled public feature vectors) based on a private training set. Our algorithm follows the paradigm of subsample-and-aggregate, in which any generic non-private learner is trained on disjoint subsets of the private training set, and then for each classification query, the votes of the resulting classifiers ensemble are aggregated in a differentially private fashion. Our private aggregation is based on a novel combination of the distance-to-instability framework [26], and the sparse-vector technique [15, 18]. We show that our algorithm makes a conservative use of the privacy budget. In particular, if the underlying non-private learner yields a classification error of at most α (0, 1), then our construction answers more queries, by at least a factor of 1/α in some cases, than what is implied by a straightforward application of the advanced composition theorem for differential privacy. Next, we apply the knowledge transfer technique to construct a private learner that outputs a classifier, which can be used to answer an unlimited number of queries. In the PAC model, we analyze our construction and prove upper bounds on the sample complexity for both the realizable and the non-realizable cases. Similar to non-private sample complexity, our bounds are completely characterized by the VC dimension of the concept class.

We design differentially private learning algorithms that are agnostic to the learning modelassuming access to a limited amount of unlabeled public data. First, we provide a new differentially private algorithm for answering a sequence of m online classification queries (given by a sequence of m unlabeled public feature vectors) based on a private training set. Our algorithm follows the paradigm of subsample-and-aggregate, in which any generic non-private learner is trained on disjoint subsets of the private training set, and then for each classification query, the votes of the resulting classifiers ensemble are aggregated in a differentially private fashion. Our private aggregation is based on a novel combination of the distance-to-instability framework [26], and the sparse-vector technique [15, 18]. We show that our algorithm makes a conservative use of the privacy budget. In particular, if the underlying non-private learner yields a classification error of at most α (0, 1), then our construction answers more queries, by at least a factor of1/α in some cases, than what is implied by a straightforward application of the advanced composition theorem for differential privacy. Next, we apply the knowledge transfer technique to construct a private learner that outputs a classifier, which can be used to answer an unlimited number of queries. In the PAC model, we analyze our construction and prove upper bounds on the sample complexity for both the realizable and the non-realizable cases. Similar to non-private sample complexity, our bounds are completely characterized by the VC dimension of the concept class.