Proximal Accuracy Rule (PAR)

• PAR is a unifying principle that ensures each iterative or collaborative update meets a problem-specific accuracy metric, as seen in proximal-gradient and best proximity point methods.
• It guarantees convergence in optimization and sampling by enforcing tight residual controls and descent conditions, even in heavy-tailed and nonconvex settings.
• In collaborative decision fusion, PAR maximizes performance by combining judgments with minimal accuracy differences, thereby optimizing the overall system outcome.

The Proximal Accuracy Rule (PAR) is a unifying principle that underlies a spectrum of results in optimization, sampling, information fusion, and statistical estimation. It stipulates that algorithmic or collaborative accuracy is maximized—whether through iterative numerical methods, probabilistic inference, or ensemble decision strategies—when each update step or combination is “proximal” to optimality, either in an explicit proximity metric or in baseline capabilities. PAR appears in diverse forms: as a rule for inexact iterative solvers to enforce sufficient accuracy at each step, as a fusion benefit predictor in human–machine teams, and as a separator of attainable accuracy in stochastic sampling schemes.

1. Theoretical Foundations and General Formulations

At its core, PAR formalizes the demand that each iterative or collaborative step is sufficiently accurate relative to a metric that is problem-dependent. In nonlinear programming and best proximity point theory, this is reflected in the existence and uniqueness of best proximity points with iterative sequences designed to achieve minimal separation, e.g., $d(g x_{n+1}, T x_n) = d(A,B)$ in best proximity point theorems under completeness rather than compactness assumptions (Fernández-León, 2012).

In proximal optimization, the rule is operationalized as a requirement that each proximal (or projected/proximal-gradient) step achieves a prespecified residual, often linked to the subdifferential or an $M$-stationarity measure, e.g.,

$$r_k = \left\| \frac{1}{\gamma_k}(x^{k-1} - x^k) - (\nabla f(x^{k-1}) - \nabla f(x^k)) \right\|$$

with algorithmic termination when $r_k \leq \varepsilon$ (Marchi, 2022).

For inexact Riemannian proximal-gradient methods, the rule is formalized as a requirement that the computed direction $n_k$ satisfies $\|n_k - n_k^\star\| \leq q(E_k, \|n_k\|)$ and a descent condition in the local surrogate function, ensuring both global and local convergence with rates modulated by the inexactness schedule (Huang et al., 2021).

2. Proximal Accuracy in Optimization Algorithms

PAR is integral to proximal distance algorithms, in which an MM (majorization-minimization) surrogate is constructed at each iteration, converting a constrained problem into a penalized, unconstrained one with a smoothed distance penalty:

$F_\rho(x) = f(x) + \rho\, \varphi_S(x),$

where $$\varphi_S(x) \approx \sqrt{(\operatorname{dist}(x,S))^2 + \varepsilon}$$. The penalized term is majorized at each step by projection, so that the update is

$$x_{n+1} = \operatorname{prox}_{(1/w_n) f}(P_S(x_n)),$$

where $w_n$ is adaptively computed. Achieving proximal accuracy requires that this proximal subproblem be solved to sufficient tightness, ensuring monotonic decrease of the penalty function and overall convergence (Lange et al., 2015).

Proximal gradient methods for composite objectives use the residuals of their fixed-point iteration as a measure of accuracy, and recent nonmonotone variants have been shown to efficiently achieve global $O(1/\sqrt{k})$ convergence in achieving the rule, even without convexity or monotonicity requirements (Marchi, 2022).

Empirically, higher problem complexity (e.g., singular regression design) can be handled by explicit construction of “ridgeless-type” proximal estimators which apply proximal operators to initial estimators, yielding $\sqrt{n}$-consistency and asymptotic normality—properties shown to adhere closely to the limiting distributions predicted by the accuracy criteria set out by PAR (Quaini et al., 2022).

3. Proximal Accuracy in Sampling and Inference

A novel and striking instance of PAR governs the attainable accuracy of proximal samplers used for heavy-tailed distributions. Here, PAR demarcates a separation between Gaussian (Brownian) and stable (Lévy) oracles:

• With a Brownian (Gaussian) oracle, the iteration complexity to reach $\varepsilon$-accuracy (in $\chi^2$-divergence) scales polynomially in $1/\varepsilon$:

$N \geq \tilde{\mathcal{O}}(\varepsilon^{-2/\nu}),$

where $\nu$ denotes tail index parameters for the generalized Cauchy class.

• In contrast, a stable oracle exploiting jump processes matched to the target's tail behavior enables iteration complexity only logarithmic in $1/\varepsilon$:

$$N = \mathcal{O}\left(\log\frac{1}{\varepsilon}\right).$$

This forms a fundamental barrier for Gaussian-based samplers and provides a rigorous quantitative foundation for choosing stable oracles when exactness is required in heavy-tailed settings (He et al., 27 May 2024).

4. Proximal Accuracy in Collaborative and Human–Machine Decision Fusion

PAR, as extended to decision fusion contexts, governs the improvement achievable by combining independent decision systems (e.g., a human and a face recognition algorithm). Here, the rule states:

• The fusion benefit, defined as

$$\mathrm{Fusion\ Benefit} = \mathrm{AUC}_{fused} - \mathrm{AUC}_{best},$$

is maximized when the absolute difference in baseline accuracies is minimal. Empirical findings show a strong negative correlation (e.g., $r \approx -0.74$ to $-0.90$) between fusion benefit and $|\Delta \mathrm{AUC}|$.

• There exists a “critical fusion zone” such that even when one decision maker is inferior, if its accuracy is not distant from the superior partner, fusing their judgments leads to performance gains—quantified, for instance, by thresholds $\lambda \approx 0.10$–$0.20$ AUC below machine accuracy in face identification assessments (Phillips et al., 2 Oct 2025).

This rule provides a principled, quantitative criterion for adaptive or “intelligent” fusion: only combine judgments when $$|\mathrm{AUC}_{human} - \mathrm{AUC}_{machine}| \leq \lambda$$, otherwise rely solely on the higher-performing system, empirically maximizing overall system accuracy.

5. Algorithmic Applications and Numerical Evidence

Algorithm design under PAR leverages accuracy-driven stopping criteria and adaptive stepsizes. For example:

• In Riemannian optimization, accuracy schedules for inexact proximal subproblem solves trade off per-iteration complexity and global/local convergence rate, with stricter rules enforced locally when finer guarantees are required (Huang et al., 2021).
• List-based enhancements to proximal decoding for LDPC codes exploit the principle that, after oscillatory saturation, residual errors are sparse and located in bits exhibiting minimal oscillation in the gradient. By using an empirical measure $\Delta_i^{(h)}$ for error detection and subsequently launching an “ML-in-the-list” local search, up to 1 dB performance improvement is achieved—an instantiation of PAR capitalizing on near-perfect convergence and targeted error correction (Tsouchlos et al., 11 Sep 2024).

Empirical results across different domains (matrix completion, sparse PCA, face recognition, heavy-tailed sampling) confirm that adherence to PAR—in proximal updates, termination tests, and fusion policies—delivers convergence and accuracy consistent with theoretical guarantees.

6. Contextual Extensions and Significance

The generality of PAR renders it robust across diverse settings, including:

• Relaxation of compactness to completeness in fixed point and best proximity analyses, significantly broadening the class of applicable spaces in which iterative schemes achieve optimal accuracy (Fernández-León, 2012).
• Meta-optimization frameworks, such as Amortized Proximal Optimization, where meta-learned update rules are adapted through online gradients to closely mimic idealized proximal steps, thus ensuring low computational overhead and numerical stability with high proximal accuracy (Bae et al., 2022).

In descriptive modeling, as in pairwise association rules for label ranking, the decomposition of complex ranking structures into local, accurate pairwise relations embodies the principle that accurate local (proximal) inference enables robust descriptive modeling in large and noisy spaces (Sá et al., 2019).

7. Implications, Limitations, and Future Trajectories

PAR serves as both a design guideline and an analytic tool for certifying progress and controlling error in optimization and inference algorithms. Its instantiations—including explicit step residuals, error thresholds, and performance-based fusion rules—are adapted to the geometry and information structure of the problem domain.

A direct implication is that the choice of proximal operator, oracle, or fusion policy should be determined by the accuracy landscape (e.g., tail properties for samplers, baseline ability differences in group decisions). Adherence to PAR often enables computational economies (e.g., logarithmic vs. polynomial complexity in sampling) and identifies when further effort in approximation or collaboration is unlikely to yield benefits.

Limitations arise when the presupposed conditions (e.g., properties of subgradients, existence of projections, or reliable measures of baseline performance) are not met or not accessible, constraining the direct applicability of PAR. When the structure is available, however, performance guarantees and empirical efficiency follow predictably.

In summary, the Proximal Accuracy Rule encapsulates a unifying principle that accuracy—whether in iterative numerical methods, stochastic simulation, statistical estimation, or collaborative decision making—is maximized via updates or combinations that are as “proximal” as possible to the local structure or peer capability. This rule, instantiated throughout multiple domains, provides a rigorous foundation for the design and analysis of accurate and efficient algorithms and systems.