Delay Complexity Dimension: Learning with Delayed and Censored Labels

We study realizable multiclass PAC learning with query feedback, where each training example provides a single binary response indicating whether its true label belongs to a chosen subset from a fixed query family. Our main contribution is the query dimension, a combinatorial parameter that captures both the structure of the hypothesis class and the ability of the available queries to distinguish labels. This parameter generalizes the bandit DS dimension, which it recovers under singleton queries. We establish a sample-complexity lower bound of $Ω(D/\varepsilon)$ and an upper bound of $\widetilde{O}(D^2/\varepsilon)$, suppressing factors polylogarithmic in the number of labels and the failure probability. When the available queries can split every candidate set into two parts of comparable size, we improve the upper bound to $\widetilde{O}(D/\varepsilon)$, yielding a characterization up to logarithmic factors.

We study the problem of multiclass PAC learning with bandit feedback in the realizable setting. In this framework, there is an unknown data distribution over an instance space $\mathcal{X}$ and a label space $\mathcal{Y}$, as in classical multiclass PAC learning, but the learner does not observe the labels of the i.i.d. training examples. Instead, in each round, it receives an unlabeled instance, predicts its label, and receives bandit feedback indicating only whether the prediction is correct. Despite this restriction, the goal remains the same as in classical PAC learning. We provide a general characterization of the optimal sample complexity of this problem, sharp for every concept class up to logarithmic factors. Our characterization is based on a new combinatorial dimension, termed the bandit $\mathrm{DS}$ dimension, defined via generalized combinatorial structures we call pseudo-boxes. These extend the pseudo-cubes underlying the $\mathrm{DS}$ dimension by allowing a different number of neighbors in each coordinate. In contrast to the $\mathrm{DS}$ dimension, which governs the full-information setting by counting the number of coordinates in the pseudo-cube, the bandit $\mathrm{DS}$ dimension aggregates the number of neighbors across coordinates, leading to a characterization in which the sample complexity scales with the total number of neighbors. We also propose a general learning algorithm achieving the upper bound, based on an algorithmic principle called ListCascade, which connects bandit learning to list learning and may be of independent interest.

How much does high confidence cost when a multiclass learner sees only whether its prediction is correct? We separate two behaviors in realizable bandit PAC learning. An anchored dimension gives a general lower bound that is attained, up to universal constants, by a family of product-block classes. In contrast, for $n\ge16$ divisible by four, an affine multiplexer with bandit DS dimension $2n-1$ has sample complexity $Θ((n\min\{n,\ln(1/δ)\}+\ln(1/δ))/\varepsilon)$ in the standard accuracy and confidence regime. Its lower bound applies to adaptive randomized learners with improper outputs. The obstruction is that evidence collected against one candidate group cannot be reused against the others. The better of two algorithms matches this cost. One finds the correct group through a positive response. The other exhausts the rank of successive groups. The resulting confidence profile rules out a universal additive-confidence bound linear in the bandit DS dimension. Moreover, classes with identical anchored dimension, bandit DS dimension, and pointwise label width can have polynomially different sample complexities. We also sharpen the confidence accounting in ListCascade and show how one positive response replaces the bandit DS dimension by the anchored dimension in a general upper bound.