Trustworthy and Scalable Nonconvex Statistical Estimation for Sample-Starved Multi-Modal Data Models
Abstract
This research program aims to develop a novel suite of modern information processing algorithmsfor sample-starved nonconvex low-complexity models. Particular emphasis is placed on the design of scalable, robust, and model-agnostic nonconvex optimization algorithms under multi-modal and heterogeneous data models. This plays a critical role in enabling efficient extraction of critical information and actionable intelligence from complex and messy data sources, which is of imminent need to real-time decision making and is at the core of the ArmyÕs missionÑparticularly for those applications that have to deal with time urgency, incomplete knowledge of the environment, and adversarial attacks. To achieve this goal, one needs to address a number of fundamental challenges, e.g. how to efficiently deal with the high degree of nonconvexity due to multi-modal data and/or nonlinear sensing mechanisms, how to ensure robustness against potential model mismatch, and how to enable model-agnostic bias reduction and inference under data heterogeneity? The main technical approach proposed in this project lies in the integrated consideration of statistics and optimization. New insights and novel foundational techniques for high-dimensional statistics, mathematical optimization, and statistical learning will be developed to meet the research objectives. Specifically, the proposed research program aims at the following major thrusts. Thrust 1: randomly initialized nonconvex methods for multi-modal mixture models. Dueto the presence of multi-modal data, the estimation tasks often involve solving highly nonconvex optimization problems. The theory of nonconvex optimization, however, is far from mature. In order to guarantee fast convergence, prior theory often recommends careful initialization, which either relies too much on the statistical models or is often found unnecessary in practice. This thrust explores model-agnostic initialization of nonconvex methods under proper statistical models, which uncovers the effectiveness of nonconvex optimization for multi-modal information fusion. Thrust 2: uncertainty quantification for nonconvex statistical estimation. Due to the stringent robustness requirements in defense applications, one is often asked to produce an uncertainty rangeor a confidence band of the obtained estimate. This is, however, highly challenging when it comes to an estimate returned by a nonconvex optimization algorithm. The aim of this thrust is to take a substantial step towards addressing this issue by developing a powerful distributional theory for nonconvex optimization, which in turn enables trustworthy and optimal uncertainty assessment. Thrust 3: model-agnostic inference under data heterogeneity Ñ a data asymmetrization approach. In many low-rank estimation problems, conventional spectral methods often suffer from a significant bias issue, thus hampering their statistical accuracy. While proper bias correction has been suggested, such results typically rely heavily on the homogeneous noise assumption, which is very sensitive to model mismatch and is not adaptive to noise heteroskedasticity. This thrust proposes to tackle this bias issue via an ÒasymmetrizedÓ spectral method, which allows to optimally reduce the estimation bias without requiring any prior knowledge about the noise statistics.Thrust 4: bridging convex and nonconvex optimization in uncertainty quantification. Convexrelaxation has achieved remarkable efficacy when solving several nonconvex estimation problems. However, the current theoretical support of convex relaxation is often far from optimal in the noisy settings. This raises serious concerns regarding whether we can trust convex relaxation in highly noisy environments. This thrust addresses this issue by exploiting an untold gift of nonconvex optimization Ñ an intimate connection between convex and nonconvex optimization. This in turn allows for robust and model-agnostic uncertainty quantification for conv
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 09, 2020
- Source ID
- W911NF2010097
Entities
People
- Yuxin Chen
Organizations
- Army Contracting Command
- Princeton University
- United States Army