Uniform Convergence Beyond Glivenko-Cantelli

We characterize conditions under which collections of distributions on $\{0,1\}^\mathbb{N}$ admit uniform estimation of their mean. Prior work from Vapnik and Chervonenkis (1971) has focused on uniform convergence using the empirical mean estimator, leading to the principle known as $P-$ Glivenko-Cantelli. We extend this framework by moving beyond the empirical mean estimator and introducing Uniform Mean Estimability, also called $UME-$ learnability, which captures when a collection permits uniform mean estimation by any arbitrary estimator. We work on the space created by the mean vectors of the collection of distributions. For each distribution, the mean vector records the expected value in each coordinate. We show that separability of the mean vectors is a sufficient condition for $UME-$ learnability. However, we show that separability of the mean vectors is not necessary for $UME-$ learnability by constructing a collection of distributions whose mean vectors are non-separable yet $UME-$ learnable using techniques fundamentally different from those used in our separability-based analysis. Finally, we establish that countable unions of $UME-$ learnable collections are also $UME-$ learnable, solving a conjecture posed in Cohen et al. (2025).

Learning from Snapshots of Discrete and Continuous Data Streams

Imagine a smart camera trap selectively clicking pictures to understand animal movement patterns within a particular habitat. These "snapshots", or pieces of data captured from a data stream at adaptively chosen times, provide a glimpse of different animal movements unfolding through time. Learning a continuous-time process through snapshots, such as smart camera traps, is a central theme governing a wide array of online learning situations. In this paper, we adopt a learning-theoretic perspective in understanding the fundamental nature of learning different classes of functions from both discrete data streams and continuous data streams. In our first framework, the \textit{update-and-deploy} setting, a learning algorithm discretely queries from a process to update a predictor designed to make predictions given as input the data stream. We construct a uniform sampling algorithm that can learn with bounded error any concept class with finite Littlestone dimension. Our second framework, known as the \emph{blind-prediction} setting, consists of a learning algorithm generating predictions independently of observing the process, only engaging with the process when it chooses to make queries. Interestingly, we show a stark contrast in learnability where non-trivial concept classes are unlearnable. However, we show that adaptive learning algorithms are necessary to learn sets of time-dependent and data-dependent functions, called pattern classes, in either framework. Finally, we develop a theory of pattern classes under discrete data streams for the blind-prediction setting.

The Dimension of Self-Directed Learning

Understanding the self-directed learning complexity has been an important problem that has captured the attention of the online learning theory community since the early 1990s. Within this framework, the learner is allowed to adaptively choose its next data point in making predictions unlike the setting in adversarial online learning. In this paper, we study the self-directed learning complexity in both the binary and multi-class settings, and we develop a dimension, namely $SDdim$, that exactly characterizes the self-directed learning mistake-bound for any concept class. The intuition behind $SDdim$ can be understood as a two-player game called the "labelling game". Armed with this two-player game, we calculate $SDdim$ on a whole host of examples with notable results on axis-aligned rectangles, VC dimension $1$ classes, and linear separators. We demonstrate several learnability gaps with a central focus on self-directed learning and offline sequence learning models that include either the best or worst ordering. Finally, we extend our analysis to the self-directed binary agnostic setting where we derive upper and lower bounds.