Learning with Monotone Adversarial Corruptions

A monotone adversary observes an i.i.d. labeled sample and appends a finite number of further examples of its choice, every one of them labeled correctly by the target hypothesis. The learner sees a uniform shuffle of the combined sample and is scored on the original distribution. Every example is correctly labeled, but the insertions depend on the clean sample, so the combined sample is not exchangeable. Larsen, Pabbaraju, and Shetty, who introduced this model, showed that empirical risk minimization attains expected error $O((d/n)\log(n/d))$ for classes of VC dimension $d$, and that every known optimal learner can be pushed away from the $Θ(d/n)$ rate, optimal for PAC learning. They asked whether the extra logarithm is an artifact of those particular algorithms or an inherent consequence of the lack of exchangeability. We show that this additional cost is inherent beyond VC dimension one. In the worst case over classes of VC dimension $d$ and over known finite insertion budgets, the minimax expected error is $Θ(1/n)$ at $d=1$ and $Θ((d/n)\log(n/d))$ for $d\geq 2$. The same rates hold with Littlestone dimension $d_{\mathrm L}$ in place of $d$, so the clean online-to-batch rate $O(d_{\mathrm L}/n)$ is unattainable as well. Thus, somewhat counterintuitively, adding correctly labeled examples can make learning harder by a logarithmic factor, even for classes that admit finite mistake bounds in online learning. The dimension-one upper bound is achieved by a simple improper learner whose analysis adapts the leave-one-out argument underlying the one-inclusion graph. All of our lower bounds are elementary and come from a single construction: an explicit class and prior on which two target hypothesis, which differ a point of nonnegligible mass, produce the same sample.

Standard Importance Sampling (IS) collapses under label corruption because high-norm examples, prioritized for variance reduction, are often adversarial outliers. We formalize this misalignment using an $\varepsilon$-contamination model and propose Disagreement-Regularized Importance Sampling (DR-IS), a sub-sampling method based on loss rank-disagreement across independent proxy ensemble. We prove finite-sample concentration bounds showing that the empirical rank disagreement of bulk corrupted examples is bounded above, and that of boundary-clean examples bounded below, both at rate $O(\sqrt{\log(N/δ)/K})$ with probability $1-δ$; when the structural expectation gap $Δ'$ between the two groups is positive and the boundary-clean set is at least as large as the selected subset, these bounds certify strict separation and control the contamination rate of the selected subset. Empirically, DR-IS remains robust under targeted high-norm attacks that break magnitude-based methods such as the Error $L_2$-norm (EL2N) on benchmark datasets. DR-IS complements training-dynamics approaches like Area Under the Margin ranking (AUM), offering improved robustness in the loss-aligned regime alongside explicit finite-sample concentration certificates and a contamination bound limiting noise leakage from the statistical tail of corrupted points.

Learning from imperfect data is a central theme in machine learning, connecting practical questions of robustness to fundamental questions of learnability. Here we examine attribute noise: learning from corrupted inputs while keeping the labels intact, a setting that has received considerably less analytical attention than its label-noise counterpart. We consider two types of corruption models: additive noise and replacement noise. Through experiments with multi-layer perceptrons (MLPs) on corrupted classification datasets, we find that neural networks remain robust, maintaining well-above-chance accuracy even when inputs are >90% corrupted -- far beyond human recognition. To understand this robustness, we analyze infinite-width networks in the heavy-corruption regime using a mean-field-inspired approach and derive a leading-order decision rule for the classification outcome: the network implements a prototype rule, the nearest-class-mean, assigning each test point to the class whose training-set average it most closely resembles. This leading-order decision rule is universal across a broad range of MLP architectures, holding for any depth, as well as a wide class of activation functions and noise distributions. The same centroid mechanism closely matches finite-width network behavior in our experiments and provides an interpretable and analytically tractable account of why learning can succeed even when individual training examples carry almost no signal.