Size sensitive packing number for Hamming cube and its consequences

We prove a size-sensitive version of Haussler’s Packing lemma~\cite{Haussler92spherepacking} for set-systems with bounded primal shatter dimension, which have an additional {\em size-sensitive property}. This answers a question asked by Ezra~\cite{Ezra-sizesendisc-soda-14}. We also partially address another point raised by Ezra regarding overcounting of sets in her chaining procedure. As a consequence of these improvements, we get an improvement on the size-sensitive discrepancy bounds for set systems with the above property. Improved bounds on the discrepancy for these special set systems also imply an improvement in the sizes of {\em relative $(\varepsilon, \delta)$-approximations} and $(\nu, \alpha)$-samples.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of set systems with bounded primal shatter dimension by establishing a size‑sensitive analogue of Haussler’s classic packing lemma. Haussler’s original result states that for any set system (X,S) with primal shatter dimension d, any maximal δ‑separated family P satisfies |P| = O((n/δ)^d). While powerful, this bound does not exploit information about the sizes of the individual sets. Ezra introduced a refined notion of size‑sensitive shattering dimensions d₁ and d₂ (with d₁+d₂ = d) and proved a packing bound that contained an extra polylogarithmic factor O(j^{d₂}), conjecturing that this factor could be eliminated.

The authors revisit Haussler’s probabilistic proof and modify it to accommodate the size‑sensitive setting. Instead of requiring a random sample to behave like a uniform (ε,δ/n)‑approximation, they partition the packing family according to set size, denoting by P_l the subfamily of sets of size l. By a careful analysis of the expected intersection sizes between a random sample and the sets in each size class, they obtain the following uniform bound for every l:

 M(l) = |P_l| ≤ c*·(n/δ)^{d₁}·(l/δ)^{d₂},

where c* is an absolute constant independent of n, l, and δ. This eliminates the unwanted logarithmic factor and matches the lower bound up to constant factors.

Armed with this sharper packing lemma, the paper revisits two major applications:

1. Size‑sensitive discrepancy – The authors combine the new packing bound with Ezra’s size‑sensitive chaining construction and the classic Beck‑Spencer partial‑coloring method (as made constructive by Lovett and Meka). The resulting discrepancy bounds improve Ezra’s Theorem 5. For d₁ > 1 they obtain

 disc(S) = O( |S|^{d₂/(2d)}·n^{(d₁−1)/(2d)}·(1+2·log(1+log min{n,|S|})) ),

and for d₁ = 1

 disc(S) = O( |S|^{d₂/(2d)}·log n·(1+2·log(1+log min{n,|S|})) ).

Thus the extra factor j^{d₂} disappears, and the logarithmic dependence is reduced to a double‑log term. Moreover, for low‑degree set systems (each element belongs to at most t sets) the authors improve the classic O(√t log n) bound to

 O( t^{1/2−1/(2d)}·√{log log t}·log n ),

which is constructive via the Lovett‑Meka algorithm.

2. Relative (ε,δ)-approximations and (ν,α)-samples – By exploiting the known relationship between discrepancy and sampling, the improved discrepancy bounds translate into smaller relative approximations. When d₁ > 1 the size of a relative (ε,δ)-approximation becomes

 O( (1/(εδ))·ε^{d/(d₁)}·δ^{2d/(d₁)}·log log n ),

and for d₁ = 1 it becomes

 O( (log n)/(εδ)·ε^{δ/(d+1)}·δ^{2d/(d+1)} ).

These improve upon the best previously known bounds, which contained larger logarithmic factors.

The paper also provides a detailed exposition of the underlying tools: the definition of δ‑separated packings, the Hamming‑cube interpretation of set systems, the chaining decomposition (including Ezra’s size‑sensitive refinement), and the constructive partial‑coloring lemma of Lovett and Meka. The authors discuss the intuition behind why a naïve extension of Haussler’s proof fails for size‑sensitive parameters and how their modified sampling argument overcomes this obstacle.

In conclusion, the work delivers a clean, size‑sensitive packing theorem that removes extraneous logarithmic terms, leading to tighter discrepancy bounds, more efficient relative approximations, and constructive algorithms for low‑degree set systems. The results close the gap left by Ezra’s conjecture and open avenues for extending size‑sensitive packing to other metric spaces and streaming models.