Black-Box Resampling Techniques

• Black-box resampling is a suite of methods that leverage output queries from opaque models to reduce variance and quantify uncertainty.
• It includes techniques like overdispersed variational inference, kernelized Stein discrepancy-based weighting, and output randomization for adversarial defense.
• These approaches improve model interpretability, extraction, and optimization, operating efficiently under limited computational budgets.

Black-box resampling encompasses a suite of methodologies for leveraging samples, queries, or evaluations from a system whose internal mechanics and analytic forms are inaccessible. These techniques appear across statistical inference, optimization, model copying, adversarial defense, and counterfactual generation in various machine learning frameworks. Resampling acts on outputs of a "black-box"—including generative models, classifiers, simulators, or any system available only via input–output queries—for the purpose of variance reduction, optimal estimation, uncertainty quantification, synthesis, or interpretability, often under tight computational budgets and without auxiliary information such as gradients.

1. Variance Reduction in Black-Box Probabilistic Inference

Monte Carlo estimators in black-box variational inference (BBVI) suffer from high variance when naïvely sampling from the variational distribution $q(z;\lambda)$. Overdispersed black-box variational inference (OBBVI) proposes resampling from an overdispersed proposal $r(z;\lambda,\tau)$ within the same exponential family, constructing $$r(z;\lambda,\tau) = g(z,\tau) \exp[(\lambda^\top t(z) - A(\lambda))/\tau]$$ for $\tau \geq 1$ (Ruiz et al., 2016). Importance sampling weights $q(z;\lambda)/r(z;\lambda,\tau)$ correct the bias, and $\tau$ is adaptively tuned to minimize the empirical variance. This approach is strictly "black-box": it generalizes to any exponential family, requires no model-specific gradient derivation, and markedly reduces estimator variance (even exceeding BBVI using twice as many samples). In experiments on GNTS and Poisson DEF models, OBBVI delivers lower variance, faster ELBO convergence, and better predictive metrics. The computational overhead for importance weighting and proposal adaptation is negligible relative to these gains.

2. Black-Box Importance Sampling and Measure Correction

Traditional importance sampling relies on tractable evaluation of proposal densities. Black-box importance sampling (BBIS) circumvents this by calculating optimal empirical weights for arbitrary, often unknown, proposal mechanisms. BBIS formalizes the weighting via minimization of the kernelized Stein discrepancy (KSD):

$$\hat{w} = \underset{w: \sum w_i = 1,\, w_i \geq 0} {\arg\min}\, w^\top K_p w$$

where $K_p$ is the Steinized kernel matrix relative to the target $p(x)$ (Liu et al., 2016). The KSD is nonnegative and zero iff the weighted empirical measure matches $p(x)$ under mild regularity. BBIS only queries black-box outputs and uses test function bounds:

$$| \sum_i w_i h(x_i) - \mathbb{E}_p[h] | \leq C_h \sqrt{S(\{x_i, w_i\}, p)}$$

with $C_h$ dependent on the RKHS norm of $h$. This framework supports samples from implicit proposals, short MCMC runs, bootstraps, or policy off-distribution data. Empirically, BBIS reduces estimator MSE and delivers root-$n$ convergence rates (or faster with control variates) even in challenging multimodal and real-world tasks.

3. Resampling for Uncertainty Quantification in Expensive Black-Box Models

When only a limited number $K$ of expensive black-box evaluations are available, statistically optimal uncertainty quantification hinges on efficient resampling methodologies (He et al., 12 Aug 2024). CI construction proceeds in two stages: first, partitioning/model resampling to obtain $K$ estimates, second, forming a pivotal statistic (often Gaussian by CLT):

• Standard batching: Equal-sized non-overlapping partitions yield uncorrelated estimates; the classical $t$-interval formula applies.
• Uneven/Overlapping batching: Batches may overlap, and optimal CIs use affine combinations weighted by the inverse covariance $\Sigma$ of the batch estimates, formulated as:

$$CI_{GS}^{\Sigma}(Y_n) = \left( \frac{1^\top \Sigma^{-1} Y_n}{\lambda} \right) \pm \frac{t_{K-1,\,1-\alpha/2}}{\sqrt{\lambda(K-1)}} \sqrt{(Y_n - \frac{1^\top \Sigma^{-1} Y_n}{\lambda}1 )^\top \Sigma^{-1} (Y_n - \frac{1^\top \Sigma^{-1} Y_n}{\lambda}1 )}$$

with $\lambda = 1^\top \Sigma^{-1} 1$.

• Cheap/Weighted bootstrap: Resample estimates using exchangeable weights, combine as above, and adjust variability.
• Batched jackknife: Leave-one-batch-out estimators.

All such approaches are proven to be asymptotically uniformly most accurate unbiased (UMA) within the class of homogeneous two-sided intervals; thus, under computational constraints, they yield statistically shortest CIs given the information structure.

4. Resampling for Model Extraction and Knowledge Distillation

In adversarial scenarios, resampling methodologies enable the replication of black-box models when internal operations and raw training data are undisclosed. The Black-Box Ripper framework utilizes an evolutionary optimization to "resample" in the synthetic latent space (Barbalau et al., 2020). Given only API access to output probabilities, it trains a generator (e.g., GAN, VAE) on a proxy dataset and iteratively perturbs the generator’s latent codes to produce samples that—when fed into the black-box model—yield high-confidence predictions for a target class. This evolutionary resampling proceeds until the teacher model outputs a distribution close to a target class one-hot vector. The student network is then trained via a cross-entropy loss using the teacher's soft predictions. Empirical comparisons to glass-box and knockoff methods show Black-Box Ripper achieves competitive or superior accuracy. The method’s constraint is query-efficiency; future work aims to minimize API calls and counter adversarial extraction strategies.

5. Black-Box Resampling in Adversarial Robustness

Output randomization acts as a black-box resampling technique to defend against query-based adversarial attacks. Instead of perturbing inputs or internal layers, the defense adds noise $\epsilon \sim \mathcal{N}(0, \sigma^2 I)$ directly to model outputs $p$:

$d(p) = p + \epsilon$

This randomization effectively corrupts finite difference gradient estimates used in attacks like ZOO, sharply suppressing attack success rates (empirically, from nonzero to $0\%$ at modest $\sigma^2$) while maintaining classification accuracy within prescribed bounds (Park et al., 2021). The method allows for precise control of misclassification probability by mathematically relating the required noise level to confidence gaps via the inverse Gaussian CDF. Output randomization can be trained (for white-box defense), is computationally lightweight, and generalizes to uncertainty quantification contexts where deliberate stochasticity at the output level can be interpreted as resampling for robustness.

6. Counterfactual Resampling for Interpretability

Black-box resampling also underpins counterfactual explanation generation in classification models. There are two techniques (Delaunay et al., 23 Apr 2024):

• Transparent methods: Perturb the sparse word matrix directly, $z = X + \epsilon$, with $\epsilon \in \{-1, 0, 1\}$ and clipping to binary values. These resampling steps correspond to concrete add/remove/replace operations.
• Opaque methods: Map text to a latent space, perform additive noise perturbation, invert back to text $z = g^{-1}(g(x) + \epsilon)$.

Empirical evidence on NLP tasks (fake news, sentiment, spam) indicates transparent resampling delivers more minimal, plausible, and computationally efficient counterfactuals. Opaque (latent) approaches introduce complexity without notable performance gain or interpretive value.

7. Generative Resampling for Black-Box Optimization

Resampling in offline black-box optimization is advanced by inverse generative methods like Denoising Diffusion Optimization Models (DDOM) (Krishnamoorthy et al., 2023). Here, diffusion models learn $p(x|y)$: the (one-to-many) mapping from function values $y$ to input candidates $x$. DDOM incorporates reweighting during training to emphasize higher-achieving samples and uses classifier-free guidance in the conditional score to generalize beyond dataset maxima. Sampling proceeds by reverse diffusion guided toward high function values. DDOM empirically achieves leading normalized scores on Design-Bench tasks and demonstrates flexibility in adapting resampling focus via loss weights and guidance parameters.

Sharpness-aware black-box optimization (SABO) further extends the framework by reparameterizing the objective via a Gaussian search distribution, then iteratively resampling to compute gradients at worst-case points within a KL-constrained neighborhood and updating the distribution (Ye et al., 16 Oct 2024). SABO empirically outperforms conventional evolution strategies, enhancing generalization in both synthetic and prompt tuning tasks, with convergence and generalization theoretically characterized.

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Black-box resampling is thus a foundational concept uniting variance reduction, estimator optimality, interpretable explanation, robust defense, generative candidate synthesis, and distillation in settings where only input–output access is possible. It leverages stochastic reweighting, latent perturbation, output randomization, generative modeling, and adaptive querying, each tailored for its respective application but sharing the principle that judicious sampling and reweighting from black-box outputs grants measurable control over estimation quality, robustness, and approximation accuracy.