PaperlayerView exact revision on arXiv

Source-linked AI summary

Contraction of R\'enyi Divergences for Discrete Channels

Adrien Vandenbroucque, Amedeo Roberto Esposito, Michael Gastpar

arXiv:2609.34570v1cs.ITmath.ITmath.PR

TL;DR

Contraction of Rényi divergences under discrete channels remains comparatively less understood. The paper analyzes their SDPI constants, showing that they are non-decreasing in α, have convex (α−1)-scaling for α≥1, and equal the χ²-SDPI constant for α∈[0,1].

  • Problem

    The contraction of Rényi divergences under stochastic channels remains comparatively less understood, despite its relevance across applications.

    Source: §I · p. 3

  • Method

    The paper characterizes Rényi-SDPI constants for discrete channels by analyzing their dependence on order and restricting the optimizing measures to small supports.

    Source: §I · pp. 2–3

  • Results

    The constants are non-decreasing in α, have convex (α−1)-scaling for α≥1, and equal the χ²-SDPI constant for α∈[0,1].

    Source: §I · p. 2

  • Takeaways & Limitations

    For pure-LDP mechanisms, the analysis yields sharp contraction guarantees.

    Source: §I · p. 2

Abstract

from arXiv · show

We investigate Strong Data-Processing Inequality (SDPI) constants for R\'enyi Divergences on finite spaces. We study their dependence on the R\'enyi order $\alpha$, proving that they are non-decreasing and that their scaling by $(\alpha-1)$ is convex for $\alpha\geq1$. We also identify several support restrictions on the probability measures involved in determining these constants. In particular, in the distribution-independent setting, measures supported on a common set of at most two points suffice to evaluate the R\'enyi-SDPI constant. For $\alpha\in[0,1]$, we further prove equality with the $\chi^2$-SDPI constant, while at order infinity we obtain a closed-form expression. In order to link contraction over product spaces to contraction along individual coordinates, we provide tensorisation bounds for arbitrary product channels analogous to those known for $\varphi$-Divergences. At finite orders, the R\'enyi-SDPI constants are bounded above and below through comparisons with the $\chi^2$-Divergence and Hellinger Divergences, with sharpness established in multiple cases. At order infinity, we instead relate these constants to the contraction of Total Variation Distance. Finally, our findings are applied to local differential privacy (LDP) and the analysis of Markov chains. This yields sharp contraction guarantees for pure-LDP mechanisms, and connects R\'enyi-LDP to R\'enyi-SDPI constants. For Markov chains, we derive finite-time convergence bounds and exhibit a family of chains for which R\'enyi-SDPIs improve on classical $\chi^2$-based bounds by arbitrarily large factors.

I. INTRODUCTION · A. Contributions and Organisation

The paper develops a systematic account of how Rényi-SDPI constants vary with order, which measures determine them, and how they behave on product spaces. It also compares these constants with other contraction measures and applies the results to privacy and Markov-chain convergence.

A. Contributions and Organisation

  • In the distribution-independent setting, measures supported on at most two points suffice; for α ≥ 2, ν can be a point mass and µ supported on that point and one other.These restrictions reduce the optimisation problems used to determine the constants.Source: §I · p. 2

I. INTRODUCTION

  • Motivated by Rényi-SDPI contraction being less understood than φ-Divergence contraction, the introduction frames the study around order dependence, determining measures, product spaces, and comparisons.It notes the breadth of Rényi-divergence applications as motivation for better understanding contraction under stochastic transformations.Source: §I · p. 3

A. Contributions and Organisation

  • Rényi-SDPI constants are non-decreasing in α, and their scaling by α−1 is convex for α ≥ 1.These properties are derived from a functional representation of the constants.Source: §I · p. 2, A. Contributions and Organisation · p. 3
  • For α ∈ [0, 1], the distribution-independent Rényi-SDPI constant equals the χ2-SDPI constant, and the paper characterises when strict contraction fails.The contributions section identifies this equality and characterisation as part of the distribution-independent analysis.Source: §I · p. 2, A. Contributions and Organisation · p. 3
  • The paper establishes tensorisation bounds for Rényi-SDPI constants on product spaces.The bounds conclude the section’s analysis of how contraction behaves in product spaces.Source: §I · p. 2, A. Contributions and Organisation · p. 3
  • At finite orders, upper bounds complement the universal lower bound provided by the χ2-SDPI constant, with further comparisons obtained via Hellinger divergences; at order infinity, upper and lower bounds are derived in terms of Total Variation Distance.Source: A. Contributions and Organisation · p. 3
  • For pure-LDP mechanisms, the paper bounds the Rényi-SDPI constant and proves the bound is attained by a randomised response mechanism.It also relates privacy guarantees expressed using Rényi divergences to contraction at the same order.Source: §I · p. 2, A. Contributions and Organisation · pp. 3–4
  • For reversible Markov chains, finite-order Rényi divergences contract asymptotically at the χ2 rate; the paper also gives finite-time bounds and a family improving on the classical χ2-SDPI bound.For irreducible, aperiodic chains, the stated limit is lim t→∞ ηα(π, K^t)^(1/t) = ηχ2(π, K) for α ∈ (0, ∞).Source: §I · p. 2, A. Contributions and Organisation · p. 4

II. BACKGROUND AND DEFINITIONS … A. Functional Representation and Dependence on the Order

For finite-alphabet channels, the section defines SDPI constants as contraction coefficients and describes how Rényi-SDPI constants depend on the order. It establishes monotonicity for distribution-dependent constants at orders at least 1, and continuity and convexity of the order-scaled constant for a fixed reference measure, with these order properties extending to distribution-independent constants subject to an essential discontinuity caveat at order one.

A. Markov Kernels

  • A Markov kernel transforms input measures from the left and output functions from the right, and has a dual kernel defined relative to an input measure.The kernel maps µ to µK, while its action on functions maps f to Kf.Source: A. Markov Kernels · pp. 4–5

B. Divergences and (Strong) Data-Processing Inequalities

  • SDPI constants tighten the data-processing inequality with a contraction coefficient, and the order-zero convention gives ηα(K) = η1−α(K) for α ∈ [0, 1].The order-zero definition uses reverse KL, preserving the stated symmetry at the endpoints.Source: B. Divergences and (Strong)… · p. 6

A. Functional Representation and Dependence on the Order

B. Properties of Extremal Distributions · C. Simplified Representations and Conditions for Trivial Contraction

This section examines the distributions and representations relevant to Rényi-SDPI constants, characterizing extremizers and support restrictions. It also gives conditions for the constant to equal 1 across the stated ranges of Rényi order.

B. Properties of Extremal Distributions

  • For α ∈[2, ∞], if ηα(µ, K) > ηχ2(µ, K), every extremal distribution has support strictly contained in supp(µ).The result strengthens the support restriction known for α = 2 to every extremizer in this order range.Source: B. Properties of Extremal Distributions · p. 10
  • At α = ∞, extremizers are restrictions of µ to non-trivial measurable sets; when 0 < η∞(µ, K) < 1, every extremizer has this form.The restriction representation applies to the distributions achieving η∞(µ, K).Source: B. Properties of Extremal Distributions · p. 10
  • For every order α ∈[0, ∞], distribution-independent Rényi-SDPI optimization reduces to binary-supported pairs; for α ∈[2, ∞], one measure need only be a point mass.These support restrictions apply to the optimization defining ηα(K).Source: B. Properties of Extremal Distributions · p. 12

C. Simplified Representations and Conditions for Trivial Contraction

  • For α = ∞, the distribution-independent SDPI constant has a Dirac-measure characterization, obtained using pairs ν = δx and µ = (1 − t)δx + tδx′.Unlike the total-variation characterization, this uses the same Dirac measure for both distributions in the limiting argument.Source: C. Simplified Representations and… · pp. 12–13
  • For every Markov kernel K and α ∈[0, 1], ηα(K) = ηχ2(K); this equality also holds for Chernoff information.The order restriction matters: for a Z-channel with crossover probability λ ∈(0, 1), ηχ2(K) = λ < 1 while ηα(K) = 1 for all α ∈(1, ∞].Source: C. Simplified Representations and… · p. 14
  • The section establishes necessary-and-sufficient structural conditions on K for ηα(K) = 1, with the conditions depending on the range of α.The result strengthens an earlier sufficient condition based on channel conditionals with disjoint supports.Source: C. Simplified Representations and… · p. 15
  • For α∈[0,1], ηα(K)=1 if and only if there exist x,x′∈X satisfying the stated support condition.The supplied passage truncates before specifying the condition.Source: C. Simplified Representations and… · p. 15
  • For α∈(1,∞], ηα(K)=1 if and only if there exist x,x′∈X satisfying the stated support condition.The supplied passage truncates before specifying the condition.Source: C. Simplified Representations and… · p. 15

D. Tensorisation · IV. BOUNDS ON THE CONTRACTION COEFFICIENT

The tensorisation results extend Rényi-divergence bounds to products of arbitrary Markov kernels and derive bounds for both distribution-dependent and distribution-independent Rényi-SDPI constants. The following section shifts from exact computation of general contraction coefficients to upper and lower bounds using divergences with understood contraction properties.

D. Tensorisation

  • The product-reference result applies for α∈(1, ∞], while the distribution-independent result covers α∈(0, 1) ∪ (1, ∞], with ν≪γ required when α>1.The product-reference assertion assumes ν≪µ1⊗···⊗µn.Source: D. Tensorisation · p. 16
  • The tensorisation extension of covers arbitrary product Markov kernels and a distribution-independent version with correlated reference measures.The latter version also holds for a larger range of Rényi orders than the product-reference result.Source: D. Tensorisation · p. 16
  • Theorem 10 yields tensorisation bounds of the type in Eq. (18) for both distribution-dependent and distribution-independent Rényi-SDPI constants.Source: D. Tensorisation · p. 16
  • The distribution-dependent tensorisation results omit α∈(0, 1), and Example 8 provides a counterexample in that range.Source: D. Tensorisation · p. 17
  • Rényi divergences do not generally admit tensorisation identities of the form in Eq. (17), as an example at α=∞ illustrates.Source: D. Tensorisation · p. 17

IV. BOUNDS ON THE CONTRACTION COEFFICIENT

  • Exact computation of a general Markov kernel’s Rényi-SDPI constant at arbitrary order is difficult, motivating a focus on bounds.Unlike χ²-contraction, which essentially reduces to an eigenvalue problem, evaluating Rényi-SDPI constants is demanding.Source: §IV · p. 18
  • Systematic upper and lower bounds for ηα(µ, K) and ηα(K) use divergences with understood contraction properties to bound Rényi-SDPI constants.The stated aim is to capture channel contractivity across α, including where closed-form expressions are unattainable.Source: §IV · p. 18, N. Proof of Theorem 10 · p. 53

A. Linear Bounds between Rényi-SDPI Constants and χ2-SDPI Constants · B. Bounds via One-To-One Mapping with Hellinger Divergences

The sections bound Rényi-SDPI constants using χ²- and Hellinger-SDPI constants, identifying when these comparisons are informative and when Hellinger-based bounds are tight. They also specify the orders for which a total-variation upper bound holds.

A. Linear Bounds between Rényi-SDPI Constants and χ2-SDPI Constants

  • The χ²-SDPI constant is a lower bound for SDPI constants of twice-differentiable φ-divergences with φ′′(1) > 0, both with and without a fixed input distribution.Source: A. Linear Bounds between Rényi-SDPI… · p. 18
  • The χ² comparison yields bounds that sandwich Rényi-SDPI constants, but their dependence on support minima and α can make them vacuous.The bounds are typically informative only for small relevant supports and may become vacuous near α = 0 or sufficiently above 2.Source: A. Linear Bounds between Rényi-SDPI… · pp. 18–19
  • The χ²-based Rényi bounds vary by order regime: the section states results for α > 2 and for α ∈ [1, 2].The supplied passages do not include the bounds’ formulas.Source: A. Linear Bounds between Rényi-SDPI… · p. 19

B. Bounds via One-To-One Mapping with Hellinger Divergences

  • The one-to-one mapping between Rényi and Hellinger divergences gives Hellinger-SDPI-based bounds, including new lower bounds and an upper bound for α ∈ (0, 1).For α > 1, the Hellinger-based lower bound can improve on the χ²-based lower bound because ηχ2(K) ≤ ηHα(K).Source: B. Bounds via One-To-One Mapping with… · pp. 19–20
  • The total-variation upper bound ηα(K) ≤ ηTV(K) holds for α ∈ [0, 1], but for every α > 1 some kernel has ηα(K) > ηTV(K).The χ² lower bound ηα(K) ≥ ηχ2(K) holds for α > 0.Source: B. Bounds via One-To-One Mapping with… · p. 21
  • Hellinger-based distribution-dependent upper bounds are tight for every α > 1, as shown using a binary erasure channel.The paper also shows that at α = 2 the bound recovers the exact Rényi-SDPI constant of the binary symmetric channel.Source: B. Bounds via One-To-One Mapping with… · pp. 22–23
  • At α = 2, the distribution-dependent bound recovers the exact binary symmetric channel Rényi-SDPI constant, matching the value computed in [24, Corollary 6].Source: B. Bounds via One-To-One Mapping with… · p. 23

C. Bounds at Order α = ∞ · V. APPLICATIONS · A. Connection to Local Differential Privacy

At order infinity, Rényi-SDPI bounds connect to Total Variation contraction, which also upper-bounds contraction at every Rényi order. The applications relate Rényi local differential privacy to Rényi-SDPI constants, with an equivalence for pure LDP at order infinity and sharp finite-order bounds attained by binary randomized response.

C. Bounds at Order α = ∞

  • For finite α, a small perturbation yields O(t^2) Rényi divergence proportional to χ², but at α = ∞ it grows as O(|t|), so the χ² approximation fails.The Hellinger link also becomes uninformative at infinity because the limiting Hellinger divergence degenerates and its SDPI constant is zero by convention.Source: C. Bounds at Order α = ∞ · p. 24
  • The distribution-dependent TV coefficient has a closed form, and η∞(µ, K) ≥ ηTV(µ, K), improving a prior lower bound by removing its min_x µ(x) factor.The formula is ηTV(µ, K) = max over x ∈ X and x′ ∈ supp(µ) of ||K(·|x) − K(·|x′)||TV.Source: C. Bounds at Order α = ∞ · pp. 24–25
  • ηφ(K) ≤ ηTV(K) ≤ η∞(K), and ηα(K) ≤ η∞(K) for every α ∈ [0, ∞], making η∞ a common upper bound for φ-divergence and Rényi contraction coefficients.This extends the traditional upper-bound role of ηTV(K).Source: C. Bounds at Order α = ∞ · p. 25
  • The condition ηTV(µ, K) ≤ η∞(µ, K) depends on support: it holds exactly when η∞(µ, K) ≥ Dinter, with Dintra ≥ Dinter a sufficient condition.Here Dintra and Dinter are the maximal TV distances between channel outputs for input pairs both inside supp(µ) and across supp(µ) and its complement, respectively.Source: C. Bounds at Order α = ∞ · pp. 25–26

A. Connection to Local Differential Privacy

  • For α ∈ [2, ∞], (α, ε)-RLDP implies a bound on the corresponding Rényi-SDPI constant, and at infinity ε-LDP is equivalent to η∞(K) ≤ 1 − e^−ε.At finite α, the reverse implication fails: ηα(K) < 1 − e^−ε can hold even when K is not (α, ε)-Rényi LDP.Source: A. Connection to Local Differential… · p. 27
  • The paper identifies sharp Rényi-divergence contraction bounds for ε-LDP mechanisms at every order, attained by binary randomized response, and the bound Υα,ε is non-decreasing in α.Binary randomized response is the BSC mechanism described as ε-LDP in the passage.Source: A. Connection to Local Differential… · p. 28

B. Convergence of Markov Chains

Rényi-SDPI coefficients yield finite-time convergence bounds for Markov chains, and at every finite positive order their contraction coefficients decay at the same exponential rate as the χ²-SDPI coefficient. In a constructed family, Rényi-SDPI bounds can outperform χ²-based bounds by arbitrarily large factors.

B. Convergence of Markov Chains

  • For irreducible, aperiodic, reversible chains, every finite positive-order Rényi-SDPI coefficient decays at the same exponential rate as ηχ2(π,K)^t.The Rényi-SDPI coefficient directly controls convergence speed, and the result gives its exact asymptotic rate.Source: B. Convergence of Markov Chains · p. 29
  • Iterating the one-step SDPI gives Dα(νK^t∥π) ≤ ηα(π,K)^t Dα(ν∥π), avoiding direct analysis of K^t and potentially improving χ²-based convergence bounds.Direct evaluation of ηα(π,K^t) is generally difficult.Source: B. Convergence of Markov Chains · p. 29
  • In the community-chain family, ηχ2(π,K) < η2(π,K) < 1 makes the classical bound eventually sharper, although the nonlinear Rényi-SDPI bound can remain tighter for a substantial time.The comparison trades off the classical bound’s smaller contraction coefficient against the nonlinear bound’s logarithmic dependence on initial divergence.Source: B. Convergence of Markov Chains · p. 30
  • 329 steps versus 459: from the smallest community, the Rényi-SDPI bound reaches χ²(νK^t∥π) ≤ 1 sooner than the χ²-SDPI bound.For the five-community example with p = 0.99, Fig. 5 compares both bounds with the exact divergence.Source: B. Convergence of Markov Chains · pp. 30–31, Figure 5 · p. 31
  • The first transition erases within-community location; scaling community sizes preserves post-step exact divergence while increasing initial χ², allowing an arbitrarily large gap between the bounds.The construction also leaves η2(π,K) unchanged under this scaling.Source: B. Convergence of Markov Chains · p. 31

APPENDIX A TECHNICAL RESULTS ON MONOTONICITY … C. Proof of Theorem 1

The appendix develops monotonicity tools and variational arguments for the Rényi-SDPI results, culminating in an order comparison that also covers the endpoint α = 1.

APPENDIX A TECHNICAL RESULTS ON MONOTONICITY

  • A convexity or concavity criterion for auxiliary functions yields monotonicity of function ratios, with strict curvature giving strict monotonicity.The appendix motivates this criterion as a way to handle recurring ratio-monotonicity arguments.Source: Appendix A · p. 34
  • For g(t) = log(ae^kt + b), ab ≥ 0 and a + b ≤ 1 imply non-decreasing behavior of the associated function, strict when ab > 0.The proof uses the sign of g′′(t) to establish convexity and applies the ratio-monotonicity criterion.Source: Appendix A · p. 34

A. Proof of Proposition 1

  • The proof of Proposition 1 represents admissible measures ν ≪ µ through their densities, identifying them with Dµ and excluding the constant density 1 for ν ≠ µ.It uses this correspondence and the variational representation to obtain Eq. (6), with equality at f = 1.Source: A. Proof of Proposition 1 · p. 35

B. A First-Order Optimality Lemma

  • At a maximizer, the nonnegative first-order slack vanishes wherever the optimizing perturbation is positive.The proof obtains this from a zero inner product between the perturbation and the slack, which is nonnegative on supp(µ).Source: B. A First-Order Optimality Lemma · p. 36

C. Proof of Theorem 1

  • The three-order comparison is derived by applying weighted AM–GM to the first-order condition, with exponent choices that return the quantities to order α.The proof identifies the exponent relations used to convert the product and conjugate norms back to order α.Source: C. Proof of Theorem 1 · pp. 37–38
  • The proof establishes that the Rényi-SDPI constants are non-decreasing in the order, including α = 1 by taking the limit from above.It rules out intervals where the order comparison fails, then uses convergence of Rényi divergence to KL divergence to include the endpoint.Source: C. Proof of Theorem 1 · pp. 39–40

D. Proof of Theorem 2 … G. Proof of Theorem 3

These sections establish convexity of the scaled Rényi-SDPI constants and continuity results, then show that when the order-α constant strictly exceeds its χ² counterpart, it is attained and lies below one.

D. Proof of Theorem 2

  • For fixed µ and K, (α−1)ηα(µ,K) is convex on (1,∞).The proof establishes that the function lies below the chord joining its values at arbitrary α and β.Source: D. Proof of Theorem 2 · p. 42

E. Proof of Corollary 1

  • For fixed µ and K, ηα(µ,K) is continuous for α in [1,∞], including at both endpoints.Continuity at one and infinity is established separately from continuity on (1,∞).Source: E. Proof of Corollary 1 · pp. 42–43

F. Proof of Corollary 2

  • For distribution-independent ηα(K), monotonicity and convexity of (α−1)ηα(K) follow from pointwise suprema; ηα(K) is continuous on (1,∞) and at infinity.The scaled function is a pointwise supremum of convex functions, and continuity at infinity follows from Corollary 1.Source: F. Proof of Corollary 2 · p. 43

G. Proof of Theorem 3

  • When ηα(µ,K) > ηχ2(µ,K), an extremal distribution exists and attains ηα(µ,K).Compactness and continuity on the domain excluding the constant density yield attainment.Source: G. Proof of Theorem 3 · p. 44
  • If ηα(µ,K) > ηχ2(µ,K), then ηχ2(µ,K) < ηα(µ,K) < 1.The proof first separates the equality case from the strict case.Source: G. Proof of Theorem 3 · pp. 43–44
  • For an arbitrary extremal distribution ν⋆, the proof derives Eq. (10), with equality on supp(ν⋆).The argument applies Lemma 4 to the density associated with ν⋆.Source: G. Proof of Theorem 3 · p. 44

H. Proof of Theorem 4 · I. Proof of Theorem 5 · J. Proof of Theorem 6

These proofs establish support restrictions for extremizers and reduce the distribution-independent R’enyi-SDPI optimization to small supports. They also characterize the order-infinity constant through conditional input measures on events.

H. Proof of Theorem 4

  • The proof exhibits an extremal distribution ν⋆ and concludes that its support is strictly contained in supp(µ), i.e., supp(ν⋆) ⊊ supp(µ), establishing the support restriction for this exhibited distribution.Source: H. Proof of Theorem 4 · p. 45

I. Proof of Theorem 5

  • The event-based contraction parameter equals η∞(µ, K), using conditional input measures for the lower bound and an extension from events to bounded functions for the upper bound.The extension uses the layer-cake representation and Hölder’s inequality.Source: §I · pp. 46–47
  • When 0 < η∞(µ, K) < 1, an extremal input has density proportional to an event indicator and is therefore µ conditioned on a nontrivial event.Equality in Hölder forces the extremal density to take only the values zero and its essential maximum.Source: §I · p. 47

J. Proof of Theorem 6

  • For finite orders, the distribution-independent R’enyi-SDPI supremum can be restricted to pairs supported on at most two common points.The proof treats orders 0 and 1 via their equality with the forward and reverse KL constants, and handles intermediate and larger finite orders separately.Source: J. Proof of Theorem 6 · pp. 47–48
  • At order infinity, the distribution-independent optimization also reduces to pairs supported on at most two common points.This case follows by comparison with finite orders and the order-infinity relation established in the proof.Source: J. Proof of Theorem 6 · pp. 47–48
  • For α in [2, ∞], a two-point reference measure reduces the extremal input to a Dirac measure at one of its support points.The support restriction for finite α and the conditional-event restriction at infinity both force this form.Source: J. Proof of Theorem 6 · pp. 48–49

K. Proof of Theorem 7 … N. Proof of Theorem 10

These proofs derive the order-infinity contraction calculation, establish equality with the χ²-SDPI constant for orders in [0,1], characterize when support patterns force contraction one, and prove tensorisation bounds for product channels.

K. Proof of Theorem 7

  • For Theorem 7, the order-infinity calculation reduces to inputs ν=δx and μt=(1−t)δx+tδx′, with its supremum approached as t→0.Optimizing over arbitrary x,x′ then yields the SDPI constant.Source: K. Proof of Theorem 7 · p. 49

L. Proof of Theorem 8

  • For every α∈[0,1], ηα(K)=ηχ²(K).For 0<α<1, the proof sandwiches ηα(K) between ηχ²(K) and the equal Hellinger-SDPI constant; the endpoint cases reduce to KL-SDPI. [4, Corollary III.1]Source: L. Proof of Theorem 8 · p. 49

M. Proof of Theorem 9

  • For 0<α<1, ηα(K)=1 exactly when some two channel rows have disjoint supports.If all row supports intersect pairwise, the total-variation contraction is below one, which yields ηα(K)<1.Source: M. Proof of Theorem 9 · p. 50

N. Proof of Theorem 10

  • For α>1, including α=∞, the product-channel argument uses Rényi-SDPI bounds; for 0<α<1, it uses the Hellinger divergence and a φ-divergence inequality.The Hellinger divergence is generated by an operator-convex function in this range [4, Corollary III.1], and the proof invokes [6, Theorem 7].Source: N. Proof of Theorem 10 · pp. 52–53
  • Theorem 10’s product-channel bound is proved by induction over coordinates, applying component kernels in stages and then taking the auxiliary space to be a singleton.The induction combines terms according to whether their coordinate subsets contain the coordinate processed at that step.Source: N. Proof of Theorem 10 · pp. 52–53

O. Proof of Corollary 3 … C. Comparing SDPI Constants of Functionally Related Divergences

The proofs establish product-channel SDPI bounds and derive comparisons between SDPI constants for divergences related by a function, with additional bounds depending on Rényi order and support assumptions.

O. Proof of Corollary 3

  • For product-channel upper bounds, the proof applies Theorem 10 to marginals and uses the divergence decomposition, including the zero-divergence empty marginal.Source: O. Proof of Corollary 3 · pp. 54–55
  • For product channels, ηα(µ,K) is at least the largest coordinate-wise ηα(µi,Ki), and Theorem 10 supplies an upper bound by marginalization.The same reasoning gives a bound for the distribution-independent constant.Source: O. Proof of Corollary 3 · p. 55

A. Proof of Lemma 1

  • For α≥2, the proof strengthens the bound using integral Taylor expansion and the monotonicity of Cα.The resulting bound is stronger than Eq. (71).Source: A. Proof of Lemma 1 · pp. 55–56

B. Proof of Proposition 2

C. Comparing SDPI Constants of Functionally Related Divergences

D. Proof of Proposition 4 … H. Proof of Theorem 12

These proofs establish endpoint conditions for equality between the order-infinity and total-variation contraction coefficients, and derive Rényi contraction bounds through Hellinger and total-variation comparisons. They also show a Z-channel example where the Rényi coefficient exceeds the total-variation coefficient.

D. Proof of Proposition 4

  • ηα(K) = 1 > ηTV(K) for the constructed Z-channel, whose total-variation coefficient is λ.The proof evaluates Rényi divergences between δ1 and με = [1 − ε, ε].Source: D. Proof of Proposition 4 · p. 58

E. Proof of Proposition 5

  • When Mα(μ,K) = 0, the distribution-dependent coefficient ηα(μ,K) is 0; the distribution-independent proof likewise treats Mα(K) = 0 as giving ηα(K) = 0.Source: E. Proof of Proposition 5 · pp. 59–60
  • The distribution-dependent bound is refined using Hα(νK∥μK) ≤ Mα(μ,K), and the refinement is never worse than the initial bound.The maximizing argument uses t⋆ = Mα(μ,K)/ηHα(μ,K), with t⋆ ≤ Cμ.Source: E. Proof of Proposition 5 · p. 59

F. Proof of Proposition 6

  • η∞(μ,K) ≤ ηTV(μ,K)/min_y∈supp(μK) μK(y), by bounding the order-infinity divergence through total variation.The proof uses the condition ν ≪ μ and ν ≠ μ for finite, positive D∞(ν∥μ).Source: F. Proof of Proposition 6 · p. 60

G. Proof of Lemma 2

  • The total-variation contraction proof decomposes ν − μ into positive and negative parts and uses a variational representation, including the zero-contraction case.Source: G. Proof of Lemma 2 · pp. 60–61

H. Proof of Theorem 12

APPENDIX D … E. Computations for Example 14

The appendix proves contraction bounds and convergence estimates, and computes chi-square contraction for a broader class of Markov chains that includes Example 14.

A. Proof of Theorem 13

  • For α≥2, ηα(K) reduces to point-mass versus two-point inputs and is at most 1−e^-ε, strictly when ε>0; ε-LDP equals (∞,ε)-Rényi LDP.Source: A. Proof of Theorem 13 · p. 63

B. Proof of Theorem 14

  • For α>1, convexity and two-point support reduction yield a comparison with a binary symmetric channel at parameter 1/(1+e^ε); limits extend the inequalities to orders one and infinity.Source: B. Proof of Theorem 14 · p. 64
  • At α=0, the proof uses reverse KL, applies the order-one relation with the measures interchanged, and then repeats the remaining steps.Source: B. Proof of Theorem 14 · p. 64

C. Proof of Corollary 4

  • For every α∈(0,∞), a finite constant cα,π independent of t bounds the relevant contraction rate, using χ2-SDPI decay under reversibility and stationarity.Source: C. Proof of Corollary 4 · p. 64

D. Proof of Proposition 7

  • Iterating the Rényi-SDPI gives Dα(νK^t∥π)≤ηα(π,K)^tDα(ν∥π), which yields Eq. (37) through the Rényi–Hellinger relationship.Source: D. Proof of Proposition 7 · p. 65

E. Computations for Example 14

Related papers

Loading 2609.34570v1…