Stable Algorithms Lower Bounds for Estimation

2026-03-23

AI Summary

This work investigates the limitations of stable algorithms in statistical estimation, focusing on whether efficient algorithms must necessarily fail under conditions of minimum mean squared error (MMSE) instability. By introducing MMSE instability as a universal criterion for the first time and integrating low-degree polynomial methods, algorithmic stability theory, and classical algorithms—such as Dijkstra’s algorithm, Gaussian elimination, and lattice basis reduction—from concrete tasks in coding, graph theory, and lattice-based cryptography, the study systematically reveals a fundamental gap between the capabilities of stable algorithms and polynomial-time algorithms. The core contribution establishes that, for three classes of MMSE-unstable problems, all low-degree polynomial algorithms are stable, thereby achieving a strict separation from known efficient algorithms, providing new upper bounds for low-degree MMSE, and forging a novel connection between statistical physics phase transitions and computational complexity.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this work, we show that for all statistical estimation problems, a natural MMSE instability (discontinuity) condition implies the failure of stable algorithms, serving as a version of OGP for estimation tasks. Using this criterion, we establish separations between stable and polynomial-time algorithms for the following MMSE-unstable tasks (i) Planted Shortest Path, where Dijkstra's algorithm succeeds, (ii) random Parity Codes, where Gaussian elimination succeeds, and (iii) Gaussian Subset Sum, where lattice-based methods succeed. For all three, we further show that all low-degree polynomials are stable, yielding separations against low-degree methods and a new method to bound the low-degree MMSE. In particular, our technique highlights that MMSE instability is a common feature for Shortest Path and the noiseless Parity Codes and Gaussian subset sum. Last, we highlight that our work places rigorous algorithmic footing on the long-standing physics belief that first-order phase transitions--which in this setting translates to MMSE-instability impose fundamental limits on classes of efficient algorithms.

Problem

• MMSE instability
• stable algorithms
• statistical estimation
• computational hardness
• phase transitions

Innovation

• MMSE instability
• stable algorithms
• low-degree polynomials
• statistical estimation
• phase transitions