Chen et al. [2022] proves that in such a binary-classification problem, an MoE layer converges to an Proof: We will prove this by contradiction. Thus, we show that vanilla MoE does not guarantee convergence with mixture of datasets.

In fact, the bound can be improved by considering a data-dependent measure, e.g., Rademacher

In particular, we consider sequential prediction with an adversary that injects adversarial (or out-of-distribution) examples in a stream of i.i.d.

The sample complexity of multi-group learning is shown to improve in the group-realizable setting over the agnostic setting, even when the family of groups is infinite so long as it has finite VC dimension. The improved sample complexity is obtained by empirical risk minimization over the class of group-realizable concepts, which itself could have infinite VC dimension. Implementing this approach is also shown to be computationally intractable, and an alternative approach is suggested based on improper learning.

As machine learning models become increasingly embedded in societal infrastructure, auditing them for bias is of growing importance. However, in real-world deployments, auditing is complicated by the fact that model owners may adaptively update their models in response to changing environments, such as financial markets. These updates can alter the underlying model class while preserving certain properties of interest, raising fundamental questions about what can be reliably audited under such shifts. In this work, we study group fairness auditing under arbitrary updates. We consider general shifts that modify the pre-audit model class while maintaining invariance of the audited property. Our goals are two-fold: (i) to characterize the information complexity of allowable updates, by identifying which strategic changes preserve the property under audit; and (ii) to efficiently estimate auditing properties, such as group fairness, using a minimal number of labeled samples. We propose a generic framework for PAC auditing based on an Empirical Property Optimization (EPO) oracle. For statistical parity, we establish distribution-free auditing bounds characterized by the SP dimension, a novel combinatorial measure that captures the complexity of admissible strategic updates. Finally, we demonstrate that our framework naturally extends to other auditing objectives, including prediction error and robust risk.

Learning with identical train and test distributions has been extensively investigated both practically and theoretically. Much remains to be understood, however, in statistical learning under distribution shifts. This paper focuses on a distribution shift setting where train and test distributions can be related by classes of (data) transformation maps. We initiate a theoretical study for this framework, investigating learning scenarios where the target class of transformations is either known or unknown. We establish learning rules and algorithmic reductions to Empirical Risk Minimization (ERM), accompanied with learning guarantees. We obtain upper bounds on the sample complexity in terms of the VC dimension of the class composing predictors with transformations, which we show in many cases is not much larger than the VC dimension of the class of predictors. We highlight that the learning rules we derive offer a game-theoretic viewpoint on distribution shift: a learner searching for predictors and an adversary searching for transformation maps to respectively minimize and maximize the worst-case loss.

A private learner is trained on a sample of labeled points and generates a hypothesis that can be used for predicting the labels of newly sampled points while protecting the privacy of the training set [Kasiviswannathan et al., FOCS 2008]. Past research uncovered that private learners may need to exhibit significantly higher sample complexity than non-private learners as is the case of learning of one-dimensional threshold functions [Bun et al., FOCS 2015, Alon et al., STOC 2019].We explore prediction as an alternative to learning. A predictor answers a stream of classification queries instead of outputting a hypothesis. Earlier work has considered a private prediction model with a single classification query [Dwork and Feldman, COLT 2018]. We observe that when answering a stream of queries, a predictor must modify the hypothesis it uses over time, and in a manner that cannot rely solely on the training set.We introduce {\em private everlasting prediction} taking into account the privacy of both the training set {\em and} the (adaptively chosen) queries made to the predictor. We then present a generic construction of private everlasting predictors in the PAC model.The sample complexity of the initial training sample in our construction is quadratic (up to polylog factors) in the VC dimension of the concept class. Our construction allows prediction for all concept classes with finite VC dimension, and in particular threshold functions over infinite domains, for which (traditional) private learning is known to be impossible.

Algorithmic machine teaching studies the interaction between a teacher and a learner where the teacher selects labeled examples aiming at teaching a target hypothesis. In a quest to lower teaching complexity and to achieve more natural teacher-learner interactions, several teaching models and complexity measures have been proposed for both the batch settings (e.g., worst-case, recursive, preference-based, and non-clashing models) as well as the sequential settings (e.g., local preference-based model). To better understand the connections between these different batch and sequential models, we develop a novel framework which captures the teaching process via preference functions $\Sigma$. In our framework, each function $\sigma \in \Sigma$ induces a teacher-learner pair with teaching complexity as $\TD(\sigma)$. We show that the above-mentioned teaching models are equivalent to specific types/families of preference functions in our framework. This equivalence, in turn, allows us to study the differences between two important teaching models, namely $\sigma$ functions inducing the strongest batch (i.e., non-clashing) model and $\sigma$ functions inducing a weak sequential (i.e., local preference-based) model. Finally, we identify preference functions inducing a novel family of sequential models with teaching complexity linear in the VC dimension of the hypothesis class: this is in contrast to the best known complexity result for the batch models which is quadratic in the VC dimension.

In this work, we investigate the expressiveness of the conditional mutual information (CMI) framework of Steinke and Zakynthinou (2020) and the prospect of using it to provide a unified framework for proving generalization bounds in the realizable setting. We first demonstrate that one can use this framework to express non-trivial (but sub-optimal) bounds for any learning algorithm that outputs hypotheses from a class of bounded VC dimension. We then explore two directions of strengthening this bound: (i) Can the CMI framework express optimal bounds for VC classes?