We study agnostic online learning from continuous-time data streams, a setting that naturally arises in applications such as environmental monitoring, personalized recommendation, and high-frequency trading. Unlike classical discrete-time models, learners in this setting must interact with a continually evolving data stream while making queries and updating models only at sparse, strategically selected times. We develop a general theoretical framework for learning from both *oblivious* and *adaptive* data streams, which may be noisy and non-stationary. For oblivious streams, we present a black-box reduction to classical online learning that yields a regret bound of $T \cdot R(S)/S$ for any class with discrete-time regret $R(S)$, where $T$ is the time horizon and $S$ is the *query budget*. For adaptive streams, which can evolve in response to learner actions, we design a dynamic query strategy in conjunction with a novel importance weighting scheme that enables unbiased loss estimation. In particular, for hypothesis class $\mathcal{H}$ with a finite Littlestone dimension, we establish a tight regret bound of $\tilde{\Theta}(T \cdot \sqrt{\mathsf{Ldim}(\mathcal{H})/S})$ that holds in both settings. Our results provide the first *quantitative* characterization of agnostic learning in continuous-time online environments with limited interaction.

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 between 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.

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 *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 *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.

Models capturing parameterized random walks on graphs have been widely adopted in wildlife conservation to study species dispersal as a function of landscape features. Learning the probabilistic model empowers ecologists to understand animal responses to conservation strategies. By exploiting the connection between random walks and simple electric networks, we show that learning a random walk model can be reduced to finding the optimal graph Laplacian for a circuit. We propose a moment matching strategy that correlates the model’s hitting and commuting times with those observed empirically. To find the best Laplacian, we propose a neural network capable of back-propagating gradients through the matrix inverse in an end-to-end fashion. We developed a scalable method called CGInv which back-propagates the gradients through a neural network encoding each layer as a conjugate gradient iteration. To demonstrate its effectiveness, we apply our computational framework to applications in landscape connectivity modeling. Our experiments successfully demonstrate that our framework effectively and efficiently recovers the ground-truth configurations.