Neural Information Processing Systems http://nips.cc/

For example, in image object recognition and language models building[4,5,6,7,8,9], the number of classes scales as the number of possible objects or the dictionary size respectively.

Partial concepts are functions that can be undefined in certain parts of the input space.

Under the framework of Probably Approximately Correct (PAC) learning, we first show that the graph dimension and the Natarajan dimension, which characterize the standard multiclass learnability, are no longer applicable in robust learning problem. We then generalize these notions to the robust learning setting, denoted as the adversarial graph dimension (AG-dimension) and the adversarial Natarajan dimension (AN-dimension). Upper and lower bounds of the sample complexity of robust multiclass learning are rigorously derived based on the AG-dimension and AN-dimension, respectively. Moreover, we calculate the AG-dimension and AN-dimension of the class of linear multiclass predictors, and show that the graph (Natarajan) dimension is of the same order as the AG(AN)-dimension. Finally, we prove that the AG-dimension and AN-dimension are not equivalent.

The theoretical analysis motivates us to devise two effective multi-class kernel learning algorithms with statistical guarantees.

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.

Classification is one of the most common tasks in machine learning. Within classification, there is normally a split between binary classification (only two possible outputs) and multiclass classification (more than two possible outputs). The theoretical analysis of these settings shares the same split. Under the P AC-learning model, binary classification learnability under the 0-1 loss is known to be characterized by the VC-dimension [V apnik and Chervonenkis, 1974, Shalev-Shwartz and Ben-David, 2014]. For multiclass classification, there has also been a further split between finite and infinite label cases.