Sami Davies

Sami Davies, Benjamin Moseley, Heather Newman

The $\ell_p$-norm objectives for correlation clustering present a fundamental trade-off between minimizing total disagreements (the $\ell_1$-norm) and ensuring fairness to individual nodes (the $\ell_\infty$-norm). Surprisingly, in the offline setting it is possible to simultaneously approximate all $\ell_p$-norms with a single clustering. Can this powerful guarantee be achieved in an online setting? This paper provides the first affirmative answer. We present a single algorithm for the online-with-a-sample (AOS) model that, given a small constant fraction of the input as a sample, produces one clustering that is simultaneously $O(\log^4 n)$-competitive for all $\ell_p$-norms with high probability, $O(\log n)$-competitive for the $\ell_\infty$-norm with high probability, and $O(1)$-competitive for the $\ell_1$-norm in expectation. This work successfully translates the offline "all-norms" guarantee to the online world. Our setting is motivated by a new hardness result that demonstrates a fundamental separation between these objectives in the standard random-order (RO) online model. Namely, while the $\ell_1$-norm is trivially $O(1)$-approximable in the RO model, we prove that any algorithm in the RO model for the fairness-promoting $\ell_\infty$-norm must have a competitive ratio of at least $Ω(n^{1/3})$. This highlights the necessity of a different beyond-worst-case model. We complement our algorithm with lower bounds, showing our competitive ratios for the $\ell_1$- and $\ell_\infty$- norms are nearly tight in the AOS model.

Sami Davies, Arya Mazumdar, Soumyabrata Pal et al.

Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically difficult to obtain tight lower bounds for mixtures. Exploiting a connection between total variation distance and the characteristic function of the mixture, we provide fairly tight functional approximations. This enables us to derive new lower bounds on the total variation distance between pairs of two-component Gaussian mixtures that have a shared covariance matrix.

Sami Davies, Miklos Z. Racz, Cyrus Rashtchian et al.

In the usual trace reconstruction problem, the goal is to exactly reconstruct an unknown string of length $n$ after it passes through a deletion channel many times independently, producing a set of traces (i.e., random subsequences of the string). We consider the relaxed problem of approximate reconstruction. Here, the goal is to output a string that is close to the original one in edit distance while using much fewer traces than is needed for exact reconstruction. We present several algorithms that can approximately reconstruct strings that belong to certain classes, where the estimate is within $n/\mathrm{polylog}(n)$ edit distance, and where we only use $\mathrm{polylog}(n)$ traces (or sometimes just a single trace). These classes contain strings that require a linear number of traces for exact reconstruction and which are quite different from a typical random string. From a technical point of view, our algorithms approximately reconstruct consecutive substrings of the unknown string by aligning dense regions of traces and using a run of a suitable length to approximate each region. To complement our algorithms, we present a general black-box lower bound for approximate reconstruction, building on a lower bound for distinguishing between two candidate input strings in the worst case. In particular, this shows that approximating to within $n^{1/3 - δ}$ edit distance requires $n^{1 + 3δ/2}/\mathrm{polylog}(n)$ traces for $0< δ< 1/3$ in the worst case.