Cassandra Marcussen

Anna Brandenberger, Cassandra Marcussen, Elchanan Mossel et al. · mit

We study processes of societal knowledge accumulation, where the validity of a new unit of knowledge depends both on the correctness of its derivation and on the validity of the units it depends on. A fundamental question in this setting is: If a constant fraction of the new derivations is wrong, can investing a constant fraction, bounded away from one, of effort ensure that a constant fraction of knowledge in society is valid? Ben-Eliezer, Mikulincer, Mossel, and Sudan (ITCS 2023) introduced a concrete probabilistic model to analyze such questions and showed an affirmative answer to this question. Their study, however, focuses on the simple case where each new unit depends on just one existing unit, and units attach according to a $\textit{preferential attachment rule}$. In this work, we consider much more general families of cumulative knowledge processes, where new units may attach according to varied attachment mechanisms and depend on multiple existing units. We also allow a (random) fraction of insertions of adversarial nodes. We give a robust affirmative answer to the above question by showing that for $\textit{all}$ of these models, as long as many of the units follow simple heuristics for checking a bounded number of units they depend on, all errors will be eventually eliminated. Our results indicate that preserving the quality of large interdependent collections of units of knowledge is feasible, as long as careful but not too costly checks are performed when new units are derived/deposited.

Renato Ferreira Pinto, Cassandra Marcussen, Elchanan Mossel et al.

We consider the problems of \emph{learning} and \emph{testing} real-valued convex functions over Gaussian space. Despite the extensive study of function convexity across mathematics, statistics, and computer science, its learnability and testability have largely been examined only in discrete or restricted settings -- typically with respect to the Hamming distance, which is ill-suited for real-valued functions. In contrast, we study these problems in high dimensions under the standard Gaussian measure, assuming sample access to the function and a mild smoothness condition, namely Lipschitzness. A smoothness assumption is natural and, in fact, necessary even in one dimension: without it, convexity cannot be inferred from finitely many samples. As our main results, we give: - Learning Convex Functions: An agnostic proper learning algorithm for Lipschitz convex functions that achieves error $\varepsilon$ using $n^{O(1/\varepsilon^2)}$ samples, together with a complementary lower bound of $n^{\mathrm{poly}(1/\varepsilon)}$ samples in the \emph{correlational statistical query (CSQ)} model. - Testing Convex Functions: A tolerant (two-sided) tester for convexity of Lipschitz functions with the same sample complexity (as a corollary of our learning result), and a one-sided tester (which never rejects convex functions) using $O(\sqrt{n}/\varepsilon)^n$ samples.

Xi Chen, Cassandra Marcussen

We give an algorithm for testing uniformity of distributions supported on hypergrids $[m_1] \times \cdots \times [m_n]$, which makes $\smash{\widetilde{O}(\text{poly}(m)\sqrt{n}/ε^2)}$ many queries to a subcube conditional sampling oracle with $m=\max_i m_i$. When $m$ is a constant, our algorithm is nearly optimal and strengthens the algorithm of [CCK+21] which has the same query complexity but works for hypercubes $\{\pm 1\}^n$ only. A key technical contribution behind the analysis of our algorithm is a proof of a robust version of Pisier's inequality for functions over hypergrids using Fourier analysis.