Extreme Points of the $(0,δ)$-LDP Polytope with Small Input Size and Arbitrary Output Sizes

Abstract: The structure of locally differentially private (LDP) mechanisms can be understood through the geometry of the corresponding privacy polytope. While the extreme points of the ( (ε,0))-LDP polytope are well characterized (Kairouz \emph{et al.}, 2014; Holohan \emph{et al.}, 2017; Pensia \emph{et al.}, 2017), comparatively little is known for the ((ε,δ))-LDP polytope with (δ>0). Recent work (Elangovan and Jog, 2024) has shown that even in the special case (ε=0), the ( (0,δ) )-LDP privacy polytope exhibits fundamentally different behaviour. In this work, we provide complete characterizations of the extreme points for the low-input-alphabet regime (k=2) and (k=3) and with arbitrary output alphabet size (m ). We also identify new extreme mechanisms for larger input alphabet sizes $k$, of the star configuration type, as introduced by Elangovan and Jog (2024).

• The paper completely characterizes extreme points for input sizes k=2 and k=3 with arbitrary output sizes, identifying all allowable support patterns and confirming that non-singleton extremes for k=2 require k+1 outputs.
• A structural lemma shows that any extreme configuration containing a singleton row has a specific δ-separated form, while every row in any extreme mechanism must have another row at total variation distance exactly δ.
• The paper introduces new star-type extreme mechanisms for general k whose existence depends on number-theoretic conditions such as greatest-common-divisor relationships, while leaving classification for k≥4 open.

Motivation and problem setting

Local differential privacy (LDP) constrains each randomized response mechanism $M:\mathcal{X}\to\mathcal{Y}$ so that, for all inputs $x,x'$ and measurable $S\subseteq\mathcal{Y}$,

$$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$

A mechanism is a row-stochastic matrix $Q\in[0,1]^{k\times m}$, and the set $\mathcal{P}_\delta$ of matrices satisfying the $(0,\delta)$-LDP constraint — equivalently, pairwise total variation (TV) distance at most $\delta$ between rows — is a convex polytope. Because divergence-based objectives in LDP hypothesis testing and estimation are convex in their arguments, optimal mechanisms can be sought among extreme points of this polytope. For pure DP ($\delta=0$), extreme points are well characterized and output alphabet sizes larger than the input size $k$ are unnecessary; for $x,x'$0, prior work by Elangovan and Jog showed that even at $x,x'$1 the geometry changes qualitatively: extreme points with $x,x'$2 exist once $x,x'$3, and no general upper bound on $x,x'$4 is known, making a full characterization for arbitrary $x,x'$5 appear intractable.

This paper narrows that gap by completely characterizing extreme points of $x,x'$6 for input sizes $x,x'$7 and $x,x'$8 with arbitrary output size $x,x'$9, proving a structural lemma valid for all $S\subseteq\mathcal{Y}$0, and exhibiting new star-type extreme configurations for general $S\subseteq\mathcal{Y}$1.

Tools from the perturbation framework

The analysis rests on two results imported from Elangovan and Jog. The first is a characterization of perturbation matrices $S\subseteq\mathcal{Y}$2 such that both $S\subseteq\mathcal{Y}$3 and $S\subseteq\mathcal{Y}$4 remain in $S\subseteq\mathcal{Y}$5: row sums of $S\subseteq\mathcal{Y}$6 must vanish, entries must vanish where $S\subseteq\mathcal{Y}$7, entries must agree across columns where tight rows coincide, and a signed-sum condition must hold on columns where tight rows differ. A matrix is extreme precisely when the only feasible perturbation is $S\subseteq\mathcal{Y}$8. The second is a localization lemma: every extreme configuration has a unique coordinate $S\subseteq\mathcal{Y}$9 such that all entries outside column $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$0 are at most $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$1. Both tools convert extremality into finite systems of linear constraints, which the paper resolves case by case.

Extreme configurations containing a singleton row

The paper's most broadly applicable result holds for all $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$2: if an extreme configuration contains a singleton row (a row with a single 1), then, up to row and column permutations, it must be the matrix whose first row is $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$3 and whose remaining rows are $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$4 with the $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$5 placed in a distinct column per row. The proof proceeds by showing that any row with support on two or more non-localized columns admits a feasible perturbation, and that any column beyond the first supported by multiple rows likewise does. Consequently, every pair of points in such a configuration is tight (at TV distance exactly $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$6).

This lemma immediately yields a supporting result: in any extreme configuration, every row has some other row at TV distance exactly $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$7 from it — otherwise a mass-shifting perturbation within the row's support would be feasible.

Complete characterization for $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$8

For $$\Pr[M(x)\in S]\le e^{\epsilon}\Pr[M(x')\in S]+\delta.$$9, the only extreme configurations, up to permutation, are:

1. the singleton row configuration $Q\in[0,1]^{k\times m}$0;
2. the two-row configuration with a singleton row, $Q\in[0,1]^{k\times m}$1;
3. the two-row configuration without a singleton row, $Q\in[0,1]^{k\times m}$2.

The proof shows that two rows may share support in at most one column (otherwise a common perturbation preserves feasibility), that localization plus tightness forces the shared entry to equal $Q\in[0,1]^{k\times m}$3, and that each row can carry at most one non-localized nonzero entry, which must equal $Q\in[0,1]^{k\times m}$4. Notably, unlike the pure-DP case, these extremes require exactly $Q\in[0,1]^{k\times m}$5 columns in the non-singleton case — the output alphabet cannot be compressed to $Q\in[0,1]^{k\times m}$6 without leaving the polytope's extreme set, confirming the qualitative departure from the $Q\in[0,1]^{k\times m}$7 theory.

Complete characterization for $Q\in[0,1]^{k\times m}$8

For three inputs, the paper first proves that every extreme configuration contains a tight point, resolving affirmatively a conjecture of Elangovan and Jog in the $Q\in[0,1]^{k\times m}$9 case. The argument uses the fact that each row has a nonempty tightness set $\mathcal{P}_\delta$0; since tightness is symmetric and $\mathcal{P}_\delta$1 is nonempty among only three rows, one of rows 1 or 2 must be tight against both others.

The main theorem then states that the only possible three-row extreme configurations are:

1. the singleton-row family of the general lemma above;
2. the "all-singleton-column" configuration with rows $\mathcal{P}_\delta$2, $\mathcal{P}_\delta$3, $\mathcal{P}_\delta$4;
3. two families without singleton rows, one mixing a row of the form $\mathcal{P}_\delta$5 with two $\mathcal{P}_\delta$6-type rows, and one built entirely from rows of the form $\mathcal{P}_\delta$7 with pairwise-disjoint $\mathcal{P}_\delta$8-positions.

The proof reduces any candidate to a canonical support structure (three rows sharing the localized column, with at most one additional shared-support column per row pair), then applies a counting argument: the perturbation system has ten free variables subject to three row-sum constraints and at most three distance constraints, so extremality requires at least four structural constraints (e.g., $\mathcal{P}_\delta$9, $(0,\delta)$0, or paired zero conditions). Case analysis over which singleton columns $(0,\delta)$1 are positive, aided by an explicit decomposition machinery using basis rows $(0,\delta)$2 and their two-$(0,\delta)$3 variants $(0,\delta)$4, eliminates all remaining candidates. A recurring technique is showing that whenever a parameter lies strictly between $(0,\delta)$5 and $(0,\delta)$6, a perturbation moving mass between its support positions keeps all pairwise TV distances at or below $(0,\delta)$7, so parameters must be pinned to $(0,\delta)$8.

New star configurations for general $(0,\delta)$9

Star configurations, introduced by Elangovan and Jog, are parameterized by a positive tuple $\delta$0 summing to 2, with center point $\delta$1 and satellite points indexed by binary vectors $\delta$2 satisfying $\delta$3. Extremality reduces to whether the linear system $\delta$4 over all subsets $\delta$5 with $\delta$6 has only the trivial solution.

The paper contributes two new extreme families:

• Three-group reciprocal tuples: tuples consisting of $\delta$7 copies of $\delta$8, $\delta$9 copies of $\delta=0$0, and $\delta=0$1 copies of $\delta=0$2, with $\delta=0$3 and $\delta=0$4, $\delta=0$5. The proof chains swapping arguments across index groups: shared gcds force equality of perturbation variables within groups, and the full-group sum constraint then zeroes each group out.
• Mixed tuples: tuples with one entry $\delta=0$6, $\delta=0$7 copies of $\delta=0$8, and $\delta=0$9 copies of $k$0, extreme when $k$1, with strict inequality required unless $k$2. As a corollary, two-group tuples of $k$3 copies of $k$4 and $k$5 copies of $k$6 with $k$7 are extreme; setting $k$8 recovers the earlier theorem of Elangovan and Jog as a special case.

These results enlarge the known catalogue of extreme mechanisms with $k$9 and clarify how number-theoretic properties of the tuple (gcd structure) govern extremality.

Limitations and open questions

The complete characterizations are confined to $x,x'$00; for $x,x'$01 the paper offers only additional examples rather than a classification, and the tight-point conjecture of Elangovan and Jog is verified here only for $x,x'$02. The counting argument underlying the $x,x'$03 proof relies on the specific support reduction of the perturbation framework and does not obviously extend, since the number of free perturbation variables grows with $x,x'$04 while the constraint budget grows only with $x,x'$05. Whether a uniform bound on the output alphabet size of extreme points exists — even for fixed small $x,x'$06 — remains open, as does the tight-point conjecture in general.

Conclusion

This paper delivers the first complete descriptions of extreme points of the $x,x'$07-LDP polytope for input alphabets of size two and three with unbounded output alphabets, establishes a structural lemma for singleton-row extremes valid at all input and output sizes, confirms the tight-point conjecture for $x,x'$08, and expands the family of known star-type extreme mechanisms for larger inputs. Together these results sharpen the geometric picture of approximate LDP while delineating precisely where current techniques cease to suffice.