PPI++: Power-Tuned High-Dimensional Inference

• PPI++ is a computational methodology that combines small labeled datasets with large-scale black-box predictions to construct valid confidence intervals for statistical parameters.
• It automatically tunes the reliance on predictions via an optimal scalar parameter, minimizing variance and ensuring efficiency improvements over classical and earlier PPI methods.
• The approach employs a single convex optimization step for point estimation and variance calculation, achieving both computational tractability and asymptotic accuracy.

PPI++ is a computational methodology for estimation and inference that leverages both a small labeled dataset and a much larger set of machine learning predictions to produce valid confidence sets for statistical parameters. It generalizes and improves upon the Prediction-Powered Inference (PPI) framework by automatically tuning its reliance on black-box predictors according to their empirical quality, thereby delivering improved statistical efficiency and computational tractability for high-dimensional inference problems (Angelopoulos et al., 2023).

1. Problem Setting and Core Notation

The PPI++ framework addresses the regime where $n$ labeled i.i.d. samples $\{(X_i, Y_i)\}_{i=1}^n$ from a joint distribution $\mathbb{P}$ are available but $n$ is small, and a large auxiliary set of i.i.d. inputs $\{\widetilde X_j\}_{j=1}^N$ ($N \gg n$) without labels is also observed. The key additional resource is access to a trained (possibly black-box) predictor $f:X \mapsto \hat Y$ that produces surrogates for unobserved responses.

The statistical objective is to estimate

$$\theta^* = \arg\min_{\theta \in \mathbb{R}^d} L(\theta)\qquad \text{where } L(\theta) \triangleq \mathbb{E}_{(X, Y) \sim \mathbb{P}}[\ell_\theta(X, Y)]$$

for a loss family $\ell_\theta$, and to construct valid $(1-\alpha)$ confidence sets for $\{(X_i, Y_i)\}_{i=1}^n$0 with higher efficiency than using the labeled data alone.

Key quantities:

• Population Loss: $\{(X_i, Y_i)\}_{i=1}^n$1
• Prediction-Powered Loss: $\{(X_i, Y_i)\}_{i=1}^n$2
• Empirical Label Loss: $\{(X_i, Y_i)\}_{i=1}^n$3
• Empirical Prediction Losses:
  • $\{(X_i, Y_i)\}_{i=1}^n$4
  • $\{(X_i, Y_i)\}_{i=1}^n$5

The central construct is the rectified loss

$\{(X_i, Y_i)\}_{i=1}^n$6

which is unbiased for $\{(X_i, Y_i)\}_{i=1}^n$7 for any $\{(X_i, Y_i)\}_{i=1}^n$8, controlling the contribution of prediction-powered versus label-only information.

2. The PPI++ Algorithm and Computational Pipeline

The PPI++ algorithm is designed for systematic plug-and-play deployment with standard convex optimization toolchains. Its steps are:

1. Power Tuning: Estimate the optimal weight $\{(X_i, Y_i)\}_{i=1}^n$9 that minimizes the trace of the limiting covariance,

$\mathbb{P}$0

where, under mild regularity,

$\mathbb{P}$1

$\mathbb{P}$2, and $\mathbb{P}$3. The empirical value $\mathbb{P}$4 is clipped to $\mathbb{P}$5.

2. Point Estimation: Compute

$\mathbb{P}$6

using Newton, quasi-Newton, gradient, or coordinate descent methods.

3. Variance Estimation: Estimate the Hessian and relevant variances at $\mathbb{P}$7,

$\mathbb{P}$8

4. Confidence Set Construction: The covariance estimate is

$\mathbb{P}$9

and the confidence interval for coordinate $n$0 is

$n$1

This pipeline requires only a single convex optimization, one pass over the data for gradient/Hessian accumulation, and a closed-form plug-in for power tuning.

The table below summarizes key steps and associated computational resources.

Step	Main Computation	Complexity
Power tuning	Plug-in variance/covariance estimation	Negligible (O(n+d))
Point estimation	Convex optimization in $n$2	$n$3 or $n$4 per iteration
Variance estimation	Single pass, gradient/Hessian accumulation	Linear in $n$5

3. Theoretical Properties and Statistical Guarantees

PPI++ achieves valid asymptotic coverage and strict efficiency improvements over both classical label-only inference and prior PPI methods. The following properties hold under regularity and smooth convex loss assumptions.

• Asymptotic Normality: For consistent estimation of $n$6 and $n$7,

$n$8

where $n$9 is defined as above.

• Optimal Weighting: There exists a closed-form $\{\widetilde X_j\}_{j=1}^N$0 that minimizes the total variance, leading to confidence intervals never wider than classical intervals and strictly tighter whenever $\{\widetilde X_j\}_{j=1}^N$1.
• GLM Convexity: For generalized linear models (GLMs), $\{\widetilde X_j\}_{j=1}^N$2 is convex for $\{\widetilde X_j\}_{j=1}^N$3; unique point estimation and valid confidence intervals result.
• Coverage: The constructed intervals achieve asymptotic nominal level $\{\widetilde X_j\}_{j=1}^N$4.
• Test-inversion Equivalence: The confidence sets via convex optimization are asymptotically equivalent to the test-inversion regions of the original intractable PPI procedure.

4. Computational Advantages and Practical Implementation

PPI++ is computationally tractable in arbitrary dimensions, unlike the original PPI, which, for $\{\widetilde X_j\}_{j=1}^N$5, required an infeasible grid search or inversion procedure for each $\{\widetilde X_j\}_{j=1}^N$6 candidate. All components—point estimation, plug-in variance/covariance estimation, and power tuning—are compatible with standard convex optimization libraries and GLM solvers.

• Point Estimation: Performed by a single convex optimization.
• Variance and Hessian Estimation: Accumulation can be streamlined within any iterative convex solver.
• Power Tuning: Simple plug-in updates for optimal $\{\widetilde X_j\}_{j=1}^N$7 based on empirical covariance traces.

The approach is extensible to any loss of interest and does not require bespoke code for each problem instance; it is suitable as a modular addition to existing statistical pipelines.

5. Comparative Analysis: Classical Inference, PPI, and PPI++

PPI++ interpolates between classical inference (using only labeled data) and PPI (fully relying on the predictor) by optimally weighting the imputed information:

• Classical Label-Only Inference: Corresponds to $\{\widetilde X_j\}_{j=1}^N$8, variance is only a function of labeled data.
• PPI ($\{\widetilde X_j\}_{j=1}^N$9): When the black-box predictor $N \gg n$0 is highly accurate, substantial variance reduction is possible, to the order $N \gg n$1. However, PPI can inflate variance if $N \gg n$2 is poor.
• PPI++ (Power-Tuned): Selects $N \gg n$3 to minimize total variance; it is never worse than either classical or PPI and, empirically, often strictly better. PPI++ is computationally efficient and yields tighter confidence intervals across regimes.

PPI++ thus unifies the two approaches, always exploiting any signal in $N \gg n$4 while retaining validity in the worst case.

6. Empirical Performance and Illustrations

Empirical studies demonstrate the adaptability and efficiency gains of PPI++ in both synthetic and real-world scenarios:

• Mean Estimation without Covariates: For $N \gg n$5 and $N \gg n$6, as the input noise $N \gg n$7 increases, PPI++ smoothly recovers PPI for low noise and classical inference for high noise, always maintaining nominal coverage. In intermediate regimes, PPI++ delivers strictly narrower intervals.
• Linear and Logistic Regression ($N \gg n$8): Behaves analogously with similar gains.
• Real Data:
  • Amazon Deforestation (Binary Outcome): PPI++ surpasses both classical and PPI baselines for all sample sizes.
  • SDSS Galaxies (Spiral/Not): When predictions are excellent, PPI and PPI++ coincide, both substantially outperforming classical intervals.
  • AlphaFold (Odds-Ratio Estimation): Up to 25% narrowing of confidence intervals.
  • Census Income (OLS, Logistic): PPI++ matches PPI when $N \gg n$9 is high quality, both dramatically superior to classical baseline.

These findings underscore PPI++'s strict efficiency gains in leveraging predictive models for valid uncertainty quantification.

7. Summary and Significance

PPI++ achieves estimation and inference that are always at least as efficient as label-only procedures and often outperform them wherever black-box predictions contain signal. The method employs a convex optimization framework with a control-variates-style loss rectification, augmented by automatic, closed-form tuning of a scalar "power" parameter. Statistical guarantees ensure asymptotic validity and optimal interval width, while the practical computational requirements are minimal, facilitating its integration into standard statistical and machine learning workflows (Angelopoulos et al., 2023).