Generation through the lens of learning theory

We study generation through the lens of statistical learning theory. First, we abstract and formalize the results of Gold [1967], Angluin [1979, 1980], and Kleinberg and Mullainathan [2024] for language identification/generation in the limit in terms of a binary hypothesis class defined over an abstract instance space. Then, we formalize a different paradigm of generation studied by Kleinberg and Mullainathan [2024], which we call ``uniform generation," and provide a characterization of which hypothesis classes are uniformly generatable. As is standard in statistical learning theory, our characterization is in terms of the finiteness of a new combinatorial dimension we call the Closure dimension. By doing so, we are able to compare generatability with predictability (captured via PAC and online learnability) and show that these two properties of hypothesis classes are \emph{incompatible} - there are classes that are generatable but not predictable and vice versa.

Smoothed Online Classification can be Harder than Batch Classification

We study online classification under smoothed adversaries. In this setting, at each time point, the adversary draws an example from a distribution that has a bounded density with respect to a fixed base measure, which is known apriori to the learner. For binary classification and scalar-valued regression, previous works \citep{haghtalab2020smoothed, block2022smoothed} have shown that smoothed online learning is as easy as learning in the iid batch setting under PAC model. However, we show that smoothed online classification can be harder than the iid batch classification when the label space is unbounded. In particular, we construct a hypothesis class that is learnable in the iid batch setting under the PAC model but is not learnable under the smoothed online model. Finally, we identify a condition that ensures that the PAC learnability of a hypothesis class is sufficient for its smoothed online learnability.

Online Classification with Predictions

We study online classification when the learner has access to predictions about future examples. We design an online learner whose expected regret is never worse than the worst-case regret, gracefully improves with the quality of the predictions, and can be significantly better than the worst-case regret when the predictions of future examples are accurate. As a corollary, we show that if the learner is always guaranteed to observe data where future examples are easily predictable, then online learning can be as easy as transductive online learning. Our results complement recent work in online algorithms with predictions and smoothed online classification, which go beyond a worse-case analysis by using machine-learned predictions and distributional assumptions respectively.

The Complexity of Sequential Prediction in Dynamical Systems

We study the problem of learning to predict the next state of a dynamical system when the underlying evolution function is unknown. Unlike previous work, we place no parametric assumptions on the dynamical system, and study the problem from a learning theory perspective. We define new combinatorial measures and dimensions and show that they quantify the optimal mistake and regret bounds in the realizable and agnostic setting respectively.

Revisiting the Learnability of Apple Tasting

In online binary classification under \textit{apple tasting} feedback, the learner only observes the true label if it predicts "1". First studied by \cite{helmbold2000apple}, we revisit this classical partial-feedback setting and study online learnability from a combinatorial perspective. We show that the Littlestone dimension continues to prove a tight quantitative characterization of apple tasting in the agnostic setting, closing an open question posed by \cite{helmbold2000apple}. In addition, we give a new combinatorial parameter, called the Effective width, that tightly quantifies the minimax expected mistakes in the realizable setting. As a corollary, we use the Effective width to establish a \textit{trichotomy} of the minimax expected number of mistakes in the realizable setting. In particular, we show that in the realizable setting, the expected number of mistakes for any learner under apple tasting feedback can only be $\Theta(1), \Theta(\sqrt{T})$, or $\Theta(T)$.

Online Infinite-Dimensional Regression: Learning Linear Operators

We consider the problem of learning linear operators under squared loss between two infinite-dimensional Hilbert spaces in the online setting. We show that the class of linear operators with uniformly bounded $p$-Schatten norm is online learnable for any $p \in [1, \infty)$. On the other hand, we prove an impossibility result by showing that the class of uniformly bounded linear operators with respect to the operator norm is \textit{not} online learnable. Moreover, we show a separation between online uniform convergence and online learnability by identifying a class of bounded linear operators that is online learnable but uniform convergence does not hold. Finally, we prove that the impossibility result and the separation between uniform convergence and learnability also hold in the agnostic PAC setting.

Multiclass Online Learnability under Bandit Feedback

We study online multiclass classification under bandit feedback. We extend the results of (daniely2013price) by showing that the finiteness of the Bandit Littlestone dimension is necessary and sufficient for bandit online multiclass learnability even when the label space is unbounded. Our result complements the recent work by (hanneke2023multiclass) who show that the Littlestone dimension characterizes online multiclass learnability in the full-information setting when the label space is unbounded.

A Combinatorial Characterization of Online Learning Games with Bounded Losses

We study the online learnability of hypothesis classes with respect to arbitrary, but bounded, loss functions. We give a new scale-sensitive combinatorial dimension, named the sequential Minimax dimension, and show that it gives a tight quantitative characterization of online learnability. As applications, we give the first quantitative characterization of online learnability for two natural learning settings: vector-valued regression and multilabel classification.

Online Learning with Set-Valued Feedback

We study a variant of online multiclass classification where the learner predicts a single label but receives a \textit{set of labels} as feedback. In this model, the learner is penalized for not outputting a label contained in the revealed set. We show that unlike online multiclass learning with single-label feedback, deterministic and randomized online learnability are \textit{not equivalent} even in the realizable setting with set-valued feedback. Accordingly, we give two new combinatorial dimensions, named the Set Littlestone and Measure Shattering dimension, that tightly characterize deterministic and randomized online learnability respectively in the realizable setting. In addition, we show that the Measure Shattering dimension tightly characterizes online learnability in the agnostic setting. Finally, we show that practical learning settings like online multilabel ranking, online multilabel classification, and online interval learning are specific instances of our general framework.

On the Learnability of Multilabel Ranking

Multilabel ranking is a central task in machine learning with widespread applications to web search, news stories, recommender systems, etc. However, the most fundamental question of learnability in a multilabel ranking setting remains unanswered. In this paper, we characterize the learnability of multilabel ranking problems in both the batch and online settings for a large family of ranking losses. Along the way, we also give the first equivalence class of ranking losses based on learnability.