Approximate message passing for nonconvex sparse regularization with stability and asymptotic analysis

We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show that the stability condition corresponds to the de Almeida--Thouless condition in spin glass literature. Through asymptotic analysis, we show the correspondence between the density evolution of SCAD-AMP and the replica symmetric solution. Numerical experiments confirm that for a sufficiently large system size, SCAD-AMP achieves the optimal performance predicted by the replica method. Through replica analysis, a phase transition between replica symmetric (RS) and replica symmetry breaking (RSB) region is found in the parameter space of SCAD. The appearance of the RS region for a nonconvex penalty is a significant advantage that indicates the region of smooth landscape of the optimization problem. Furthermore, we analytically show that the statistical representation performance of the SCAD penalty is better than that of L1-based methods, and the minimum representation error under RS assumption is obtained at the edge of the RS/RSB phase. The correspondence between the convergence of the existing coordinate descent algorithm and RS/RSB transition is also indicated

Statistical Mechanics of High-Dimensional Inference

To model modern large-scale datasets, we need efficient algorithms to infer a set of $P$ unknown model parameters from $N$ noisy measurements. What are fundamental limits on the accuracy of parameter inference, given finite signal-to-noise ratios, limited measurements, prior information, and computational tractability requirements? How can we combine prior information with measurements to achieve these limits? Classical statistics gives incisive answers to these questions as the measurement density $\alpha = \frac{N}{P}\rightarrow \infty$. However, these classical results are not relevant to modern high-dimensional inference problems, which instead occur at finite $\alpha$. We formulate and analyze high-dimensional inference as a problem in the statistical physics of quenched disorder. Our analysis uncovers fundamental limits on the accuracy of inference in high dimensions, and reveals that widely cherished inference algorithms like maximum likelihood (ML) and maximum-a posteriori (MAP) inference cannot achieve these limits. We further find optimal, computationally tractable algorithms that can achieve these limits. Intriguingly, in high dimensions, these optimal algorithms become computationally simpler than MAP and ML, while still outperforming them. For example, such optimal algorithms can lead to as much as a 20% reduction in the amount of data to achieve the same performance relative to MAP. Moreover, our analysis reveals simple relations between optimal high dimensional inference and low dimensional scalar Bayesian inference, insights into the nature of generalization and predictive power in high dimensions, information theoretic limits on compressed sensing, phase transitions in quadratic inference, and connections to central mathematical objects in convex optimization theory and random matrix theory.Comment: See http://ganguli-gang.stanford.edu/pdf/HighDimInf.Supp.pdf for supplementary materia

Replica Creation Algorithm for Data Grids

Data grid system is a data management infrastructure that facilitates reliable access and sharing of large amount of data, storage resources, and data transfer services that can be scaled across distributed locations. This thesis presents a new replication algorithm that improves data access performance in data grids by distributing relevant data copies around the grid. The new Data Replica Creation Algorithm (DRCM) improves performance of data grid systems by reducing job execution time and making the best use of data grid resources (network bandwidth and storage space). Current algorithms focus on number of accesses in deciding which file to replicate and where to place them, which ignores resources’ capabilities. DRCM differs by considering both user and resource perspectives; strategically placing replicas at locations that provide the lowest transfer cost. The proposed algorithm uses three strategies: Replica Creation and Deletion Strategy (RCDS), Replica Placement Strategy (RPS), and Replica Replacement Strategy (RRS). DRCM was evaluated using network simulation (OptorSim) based on selected performance metrics (mean job execution time, efficient network usage, average storage usage, and computing element usage), scenarios, and topologies. Results revealed better job execution time with lower resource consumption than existing approaches. This research contributes replication strategies embodied in one algorithm that enhances data grid performance, capable of making a decision on creating or deleting more than one file during same decision. Furthermore, dependency-level-between-files criterion was utilized and integrated with the exponential growth/decay model to give an accurate file evaluation