PAC-Learning for Strategic Classification

The study of strategic or adversarial manipulation of testing data to fool a classifier has attracted much recent attention. Most previous works have focused on two extreme situations where any testing data point either is completely adversarial or always equally prefers the positive label. In this paper, we generalize both of these through a unified framework for strategic classification, and introduce the notion of strategic VC-dimension (SVC) to capture the PAC-learnability in our general strategic setup. SVC provably generalizes the recent concept of adversarial VC-dimension (AVC) introduced by Cullina et al. arXiv:1806.01471. We instantiate our framework for the fundamental strategic linear classification problem. We fully characterize: (1) the statistical learnability of linear classifiers by pinning down its SVC; (2) its computational tractability by pinning down the complexity of the empirical risk minimization problem. Interestingly, the SVC of linear classifiers is always upper bounded by its standard VC-dimension. This characterization also strictly generalizes the AVC bound for linear classifiers in arXiv:1806.01471.

Small-Support Uncertainty Principles on $mathbb{Z}/p$ over Finite Fields

We establish an uncertainty principle for functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ with constant support (where $p mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ for which $|text{supp}; {f}| = S$ must satisfy $|text{supp}; hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredis theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.