Sampling-Free Bounds via Rényi Divergence

• The paper introduces sampling-free analytic bounds that leverage Rényi divergence to provide sharp, non-asymptotic guarantees across diverse fields including communication complexity and privacy.
• It employs both classical and quantum formulations, using analytic and optimization-based methods to bypass empirical sampling for deterministic risk control.
• Key implications include robust risk quantification and improved privacy accounting, with applications ranging from hypothesis testing to quantum cryptography.

Sampling-free bounds based on Rényi divergence provide explicit, non-empirical guarantees for a wide spectrum of information-theoretic, statistical, privacy, and learning problems. These bounds rely on analytic or optimization-based control of Rényi divergences—classically or quantumly—rather than requiring empirical sampling or Monte Carlo estimation. They yield sharp, non-asymptotic results in communication complexity, differential privacy, statistical hypothesis testing, sensitivity analysis, generalization error, quantum information theory, and large deviations, and are crucial whenever deterministic risk control or robust worst-case quantification is needed.

1. Rényi Divergence: Definitions and Key Properties

For probability measures $P,Q$ on a measurable space $(\Omega, \mathcal F)$, and $\alpha \in (0,1)\cup(1, \infty)$, the Rényi divergence of order $\alpha$ is

$$D_\alpha(P \| Q) = \frac{1}{\alpha - 1} \log \int \left( \frac{dP}{dQ} \right)^\alpha dQ,$$

whenever $P \ll Q$. Rényi divergence recovers various important quantities:

• $D_1(P\|Q)$ is the Kullback-Leibler divergence,
• $D_0(P\|Q) = -\log Q(\operatorname{supp} P)$,
• $D_\infty(P\|Q) = \log \sup_x \frac{P(x)}{Q(x)}$ for finite alphabets.

In the quantum setting, Petz and sandwiched Rényi divergences further generalize this to density operators, with the sandwiched version $\widetilde D_\alpha(\rho\|\sigma)$ defined for $\alpha \in [1/2,1)\cup(1,\infty)$ and satisfying a data processing inequality for $\alpha \ge 1/2$ (Warsi et al., 16 May 2025, Bluhm et al., 2023).

2. Communication Complexity: Sampling-Free Bounds in Exact Sampling

Hill, Alajaji, and Linder establish tight, sampling-free bounds for the communication complexity of exact sampling with shared randomness, characterizing both lower and upper bounds on exponential cost (Campbell cost) and Rényi entropy in terms of Rényi divergences of various orders (Hill et al., 13 Jun 2025). For any $\alpha \in (0,1)$ and $t = (1-\alpha)/\alpha$, the following holds:

• Lower bound:

$$L(t) \geq \max\{ D_{1/\alpha}(P\|Q) + O(1), D_{2-\alpha}(P\|Q) + O(1) \}$$

• Upper bound via Poisson functional representation (PFR):

$$L(t) \leq (1+\epsilon)D_{(1+\epsilon(1-\alpha))/\alpha}(P\|Q) + c_1(\alpha, \epsilon)$$

where $L(t)$ is the exponential cost, and the gap is within $O(10)$ bits across a wide range of $\alpha$. As $\alpha \rightarrow 1$, these bounds recover classical KL-based results.

The method is inherently sampling-free: it relies only on analytic computation of Rényi divergences or upper bounds thereof. For distributions with tractable densities, no empirical estimation is necessary.

3. Privacy, Differential Privacy, and Information Leakage

Sampling-free Rényi bounds critically underpin modern privacy accounting under (Rényi) differential privacy (RDP). For matrix mechanisms under random allocation (balls-into-bins model), tight, deterministic RDP bounds are computable via dynamic programming for banded strategy matrices, scaling polynomially in the bandwidth (i.e., $O(bp\,\alpha^{2p})$ time) and holding for arbitrary batchings. Conditional composition lemmas further allow strict privacy accounting in small-$\epsilon$ (high privacy) regimes. These approaches dominate Monte Carlo-based estimators, as the latter incur polynomial or inverse dependence on $\delta$ (privacy parameter) (Schuchardt et al., 29 Jan 2026).

In learning theory and adaptive data analysis, Sibson's mutual information of order $\alpha$ (a Rényi information quantity) provides non-asymptotic, sampling-free generalization and event probability bounds: $$P(E) \le \exp\left( (\alpha-1) D_\alpha(P_{XY}\|P_X P_Y) \right) \cdot \left[ P_X P_Y(E) \right]^{(\alpha-1)/\alpha}$$ with $E$ an event and $P_{XY}$ the joint law (Esposito et al., 2019). This includes maximal leakage as $\alpha \to \infty$. These bounds interpolate between concentration-inequality-based results and worst-case, leakage-based regimes, and do not require empirical evaluation of information quantities.

4. Hypothesis Testing, Statistical Estimation, and Robustness

For binary hypothesis testing, reverse Rényi divergence yields sharp, explicit, sampling-free finite-sample (non-asymptotic) bounds on Type II error, exhibiting genuine phase transitions (strong converses) that are tighter than KL-based weak converses and scale linearly for product measures (Bruno et al., 14 Jan 2026).

In robust large deviations and rare-event sensitivity analysis, information-theoretic dualities allow bounding not only probabilities but also their derivatives (sensitivities) with respect to model parameters, via risk-sensitive or exponential family expectations and Rényi divergence constraints. These bounds do not require simulating rare events or the target model, but only analytic control of Rényi divergences and moment-generating functions under a reference law (Dupuis et al., 2018, Atar et al., 2020). Optimization over controls reduces to tractable convex programs.

5. Quantum Rényi Divergence: Learning and Cryptography

Quantum extensions of Rényi divergence enable tight, sampling-free generalization error bounds in quantum learning and device-independent cryptography. When loss observables and state manipulations are tractable, variational representations for the Petz and sandwiched Rényi divergences yield expected and high-probability generalization bounds for quantum learners: $$\overline{\mathrm{gen}} \leq \inf_{\alpha \in (0,1)} \left\{ \mathbb{E}_{w,s} \sqrt{\frac{2\mu^2}{\alpha}\overline D_{\alpha}(\cdot \| \cdot)} + \cdots \right\}$$ where $\overline D_\alpha$ is a generalized (modified) sandwiched divergence that exhibits improved numerical and analytical tightness over the Petz Rényi divergence (Warsi et al., 16 May 2025).

For device-independent cryptography, sampling-free, variational bounds on quantum Rényi divergences directly certify operational quantities such as smooth min-entropy and advantage-distillation thresholds. These SBP-friendly and SDP-friendly bounds never require tomography or state-sampling, and outperform previous entropy accumulation theorems in noise tolerance and key rate (Hahn et al., 2024).

6. Continuity and Stability: Sandwiched Rényi Divergences

Uniform continuity of sandwiched Rényi divergence and related entropic quantities (conditional entropy, mutual information) admits explicit, sampling-free analytic bounds. Three technical approaches—axiomatic (additivity), operator-space (interpolation norms), and hybrid—yield optimal continuity guarantees at different Rényi orders and system sizes (Bluhm et al., 2023). For quantum Markov chains and quantum information processing stability, these results provide dimension-independent continuity rates that can be computed deterministically without empirical methods.

7. Estimation and Concentration Properties

Mirrored-kernel plug-in estimators for Rényi divergence on smooth density classes satisfy exponential concentration inequalities, with explicit non-asymptotic (variance-type and bias) rates. Under regularity and boundedness conditions, bias decays at optimal minimax rates in sample size, and variance enjoys exponential tails without recourse to empirical sampling for control (Singh et al., 2016). In finite alphabets, Rényi divergence can be bounded above in terms of total variation distance and alphabet cardinalities, yielding powerful, deterministic bounds for discrete settings (Sason et al., 2015).

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In summary, sampling-free Rényi divergence bounds provide a foundational toolset for analytic risk control, privacy analysis, generalization error, statistical inference, and quantum information, offering deterministic guarantees that circumvent the limitations of empirical or simulation-based estimation. These approaches are characterized by analytic optimization, explicit continuity and tightness, and wide applicability across classical and quantum probabilistic models.