Researchers have resolved a question posed by Nikolov and Ullman about how accurately one can answer many statistical queries while satisfying pure differential privacy (that is, privacy with no extra failure probability). They show that for a database of size n, universe size T, k bounded queries, and privacy parameter ε>0, there is a pure ε‑differentially private mechanism whose expected worst‑coordinate error is on the order of O(min{1, sqrt( log(2T)·log(2k) / (ε n) )}). In plain terms: the worst single answer (in expectation) can shrink roughly like 1/√n in the high‑dimensional regimes where known lower bounds apply. Here “worst‑coordinate error” means the largest absolute error among the released answers; T is the number of possible data values and k is the number of queries.

The construction starts from a known private algorithm called private multiplicative weights (PMW), but not the usual numerical answers. Instead the method records a “selection‑only” transcript that notes which signed queries the PMW procedure picked at each step. That transcript distribution is accurate but does not by itself give the pointwise likelihood guarantees needed for pure ε‑privacy. To fix this, the authors replace the transcript’s probability mass function by a distance‑penalized likelihood envelope. Concretely, for each possible transcript ω they form e_p_x(ω) = max_y e^{-(ε/2) d_H(x,y)} p_y(ω), where d_H(x,y) is the Hamming distance (the number of differing rows between databases x and y) and p_y is the PMW transcript law for database y. They then normalize these envelope weights and sample from the resulting distribution. Penalizing by e^{-(ε/2) d_H} makes the mechanism satisfy a pointwise likelihood ratio bound that gives pure ε‑differential privacy.

The technical heart of the proof shows that this envelope does not wreck accuracy. The envelope can add probability mass from other databases, so the authors must bound both the extra mass and the extra error it brings. They use a likelihood‑level Maurey argument to show that, inside a Hamming ball around the true database, the maximum transcript likelihood can be upper bounded by a small family of auxiliary PMW laws. For nearby data they control these maxima via finite‑order Rényi moment bounds; for far data they use a direct mixture bound. Crucially, they group Hamming radii into blocks of width about 1/ε so that summing over distances does not introduce an extra 1/ε factor in the final error rate.