We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $\Omega(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity. This leads to an optimal refutation of the Alon-Saks-Seymour conjecture, as well as an optimal communication lower bound for the Clique versus Independent Set problem, improving the previous results of Balodis, Ben-David, G\"{o}\"{o}s, Jain and Kothari (FOCS 2021, SICOMP 2023) by several doubly logarithmic factors. As further applications of our construction to query complexity and learning theory, we exhibit: (a) a family of Boolean functions that has an optimal quartic separation between certificate complexity and approximate degree, and (b) a sample compression lower bound of $\Omega(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.
Optimal learners are tailored to exploit the i.i.d.\ data assumption underlying the classic PAC model. What if an i.i.d.\ training sample were corrupted with correctly labeled examples drawn from an otherwise unrelated, even adversarial source? This model of learning with monotone adversarial corruptions was recently introduced by Larsen et al. (2026), who demonstrated that all known optimal binary learners suffer increased error rates in this setting, from $O(d / n)$ in the PAC model to $\Omega (d \log(n / d) / n)$ under monotone corruption. Mehrotra (2026) proved this logarithmic factor to be necessary for binary classification, but left open the consequences of corruption for more general learning settings, such as multiclass classification and partial binary concept classes. As our primary result, we demonstrate that monotone adversaries are frighteningly more powerful in each of these settings. We exhibit a learnable multiclass problem, of DS dimension only 2, that becomes altogether unlearnable under a monotone adversary, and show an analogous result for partial binary concept classes. These results are achieved by an adaptive adversary permitted to view the original i.i.d.\ training set $S$ and to insert $b<\infty$ corrupted datapoints into $S$. In the multiclass example, the adversary need only insert a linear number $b = |S| = n$ of datapoints. We complement these impossibility results by proving that every class remains learnable when the number of adaptive additions is $o(n)$, which our previous multiclass lower bound proves to be tight. We further observe that the classic multiclass error rate of $O(d_{\mathrm{DS}} / n)$ remains achievable against adaptive adversaries restricted to a known constant budget $b = O(1)$, against semi-adaptive adversaries viewing only a $p$-fraction of $S$ for $p \in (0, 1)$, and against oblivious adversaries that cannot view $S$.
Julian Asilis, S. Dughmi, Chirag Pabbaraju· 0 citations
It is shown that the maximum hypergraph density of any multiclass hypothesis class is upper-bounded by its DS dimension, which proves a longstanding conjecture of Daniely and Shalev-Shwartz (2014) and determines the optimal dependence of the sample complexity on the DS dimension for multiclass as well as list learning.