Computational and Statistical Tradeoffs via Convex Relaxation (1211.1073v2)

Abstract: In modern data analysis, one is frequently faced with statistical inference problems involving massive datasets. Processing such large datasets is usually viewed as a substantial computational challenge. However, if data are a statistician's main resource then access to more data should be viewed as an asset rather than as a burden. In this paper we describe a computational framework based on convex relaxation to reduce the computational complexity of an inference procedure when one has access to increasingly larger datasets. Convex relaxation techniques have been widely used in theoretical computer science as they give tractable approximation algorithms to many computationally intractable tasks. We demonstrate the efficacy of this methodology in statistical estimation in providing concrete time-data tradeoffs in a class of denoising problems. Thus, convex relaxation offers a principled approach to exploit the statistical gains from larger datasets to reduce the runtime of inference algorithms.

• The paper introduces time-data complexity classes and shows how convex relaxation allows trading data volume for reduced computation in inference tasks.
• Convex relaxation provides tractable approximate solutions, allowing algorithms to leverage data for computational efficiency in inference tasks.
• The framework significantly impacts high-dimensional statistics under constraints by integrating algorithmic efficiency into statistical design.

Computational and Statistical Tradeoffs via Convex Relaxation

The paper entitled "Computational and Statistical Tradeoffs via Convex Relaxation" by Venkat Chandrasekaran and Michael I. Jordan presents a theoretical framework addressing the role of convex relaxation in reducing computational complexity associated with large-scale statistical inference problems. This work explores how massive datasets, often considered burdensome in contemporary data analysis contexts, can instead be leveraged as assets to optimize the tradeoffs between computation and statistical accuracy.

Overview

The central theme of the paper is the interpretation of statistical datasets as resources akin to time and space in computational tasks, thereby proposing a blended approach from computer science and statistics to tackle massive data problems. The authors define a "time-data complexity class" $TD(t(p),n(p),\epsilon(p))$, which formalizes the parameters of computational runtime, sample size, and statistical accuracy. This novel construct allows for analysis of the tradeoffs between runtime and sample complexity given a fixed level of accuracy.

Convex relaxation is identified as an effective method for designing approximation algorithms for inferential tasks, where it serves not only to provide tractable solutions to otherwise computationally intractable problems but also to take advantage of additional data for efficient computation. The core methodology involves convex optimization, specifically using outer approximations to simplify algorithmic complexity while maintaining inferential performance.

Key Contributions

1. Time-Data Complexity Classes: The paper introduces the concept of time-data complexity classes to characterize tradeoffs in computational and statistical settings. This formulation provides a structured way to analyze the computational cost savings achieved via larger datasets without assuming the classical separation of computational and statistical objectives.
2. Convex Relaxation Techniques: The authors articulate a procedure where convex relaxation reduces computational expenses by allowing more data to be processed through weaker, computationally efficient relaxations that deliver acceptable estimation quality.
3. Practical Examples and Computations: Several examples illustrate explicit time-data tradeoffs employing convex relaxation in high-dimensional inference problems, such as denoising in sequence models. The examples also address applications like collaborative filtering, sparse PCA, and network identification, confirming that substantial computational speedups can be realized even with constant factor increases in dataset size.

Implications and Future Directions

The implications of this research are significant for the field of high-dimensional statistics, particularly under resource constraints. Practically, the ability to trade additional data for computational efficiency impacts domains involving real-time data processing and large-scale machine learning systems. Theoretically, this shifts perspective towards integrating algorithmic efficiency within statistical frameworks rather than treating them as separate design considerations.

The findings prompt further research into:

• Streaming Data Environments: Extending the framework to settings where data arrives in streams and decisions must be made under tight runtime constraints.
• Alternative Algorithm Weakening Mechanisms: Exploration of other approaches, such as dimensionality reduction or data quantization, which could serve similar roles in balancing computational loads.
• Universal Principles in Convex Approximation: Studying the broader applicability of these time-data tradeoff principles across various forms of convex relaxation and approximation problem classes.

Overall, the paper presents a sophisticated analysis of computational strategies in statistical inference, providing a solid foundation for future developments in the fusion of computational efficiency and statistical rigor.