The goal is to learn a single classifier that also performs well on all the tasks.

We derive an online learning algorithm with improved regret guarantees for "easy"

The simplest nontrivial geometric concept is perhaps the halfspace.

However, learning theoretic guarantees have only been considered in very specific settings.

Prior known algorithms for both these settings have strong inherent limitations, making them impractical for real-world scenarios.

Recent works have shown that tokenisation is NP-complete. However, these works assume tokenisation is applied to inputs with unboundedly large alphabets -- an unrealistic assumption, given that in practice tokenisers operate over fixed-size alphabets, such as bytes or Unicode characters. We close this gap by analysing tokenisation over bounded $n$-ary alphabets, considering two natural variants: bottom-up tokenisation and direct tokenisation, where we must, respectively, select a sequence of merge operations or a vocabulary whose application optimally compresses a dataset. First, we note that proving hardness results for an $n$-ary alphabet proves the same results for alphabets of any larger size. We then prove that even with binary alphabets, both variants are not only NP-complete, but admit no polynomial-time approximation scheme (unless P=NP). We further show that direct tokenisation remains NP-complete even when applied to unary alphabets. While unary alphabets may not be practically useful, this result establishes that the computational intractability of tokenisation is not an artifact of large alphabets or complex constructions, but a fundamental barrier. Overall, our results explain why practical algorithms such as BPE and UnigramLM are heuristic, and points toward approximation algorithms being an important path going forward for tokenisation research.

We examine the relationship between learnability and robust (or agnostic) learn-ability for the problem of distribution learning.

The fundamental theorem of statistical learning states that binary PAC learning is governed by a single parameter -- the Vapnik-Chervonenkis (VC) dimension -- which determines both learnability and sample complexity. Extending this to multiclass classification has long been challenging, since Natarajan's work in the late 80s proposing the Natarajan dimension (Nat) as a natural analogue of VC. Daniely and Shalev-Shwartz (2014) introduced the DS dimension, later shown by Brukhim et al. (2022) to characterize multiclass learnability. Brukhim et al. also showed that Nat and DS can diverge arbitrarily, suggesting that multiclass learning is governed by DS rather than Nat. We show that agnostic multiclass PAC sample complexity is in fact governed by two distinct dimensions. Specifically, we prove nearly tight agnostic sample complexity bounds that, up to log factors, take the form $\frac{DS^{1.5}}ε + \frac{Nat}{ε^2}$ where $ε$ is the excess risk. This bound is tight up to a $\sqrt{DS}$ factor in the first term, nearly matching known $Nat/ε^2$ and $DS/ε$ lower bounds. The first term reflects the DS-controlled regime, while the second shows that the Natarajan dimension still dictates asymptotic behavior for small $ε$. Thus, unlike binary or online classification -- where a single dimension (VC or Littlestone) controls both phenomena -- multiclass learning inherently involves two structural parameters. Our technical approach departs from traditional agnostic learning methods based on uniform convergence or reductions to realizable cases. A key ingredient is a novel online procedure based on a self-adaptive multiplicative-weights algorithm performing a label-space reduction, which may be of independent interest.

Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension. We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.