arXiv subjects

Zhenan Fan

Publications and source records attributed to Zhenan Fan.

24 records · Page 2Linked to original sources

arXiv2022

Cardinality-constrained structured data-fitting problems

A memory-efficient framework is described for the cardinality-constrained structured data-fitting problem. Dual-based atom-identification rules are proposed that reveal the structure of the optimal primal solution from near-optimal dual solutions. These rules allow for a simple and computationally cheap algorithm for translating any feasible dual solution to a primal solution that satisfies the cardinality constraint. Rigorous guarantees are provided for obtaining a near-optimal primal solution given any dual-based method that generates dual iterates converging to an optimal dual solution. Numerical experiments on real-world datasets support confirm the analysis and demonstrate the efficiency of the proposed approach.

math.OC↗

arXiv2022

Polar Deconvolution of Mixed Signals

The signal demixing problem seeks to separate a superposition of multiple signals into its constituent components. This paper studies a two-stage approach that first decompresses and subsequently deconvolves the noisy and undersampled observations of the superposition using two convex programs. Probabilistic error bounds are given on the accuracy with which this process approximates the individual signals. The theory of polar convolution of convex sets and gauge functions plays a central role in the analysis and solution process. If the measurements are random and the noise is bounded, this approach stably recovers low-complexity and mutually incoherent signals, with high probability and with near-optimal sample complexity. We develop an efficient algorithm, based on level-set and conditional-gradient methods, that solves the convex optimization problems with sublinear iteration complexity and linear space requirements. Numerical experiments on both real and synthetic data confirm the theory and the efficiency of the approach.

cs.IR↗

arXiv2022

Improving Fairness for Data Valuation in Horizontal Federated Learning

Federated learning is an emerging decentralized machine learning scheme that allows multiple data owners to work collaboratively while ensuring data privacy. The success of federated learning depends largely on the participation of data owners. To sustain and encourage data owners' participation, it is crucial to fairly evaluate the quality of the data provided by the data owners and reward them correspondingly. Federated Shapley value, recently proposed by Wang et al. [Federated Learning, 2020], is a measure for data value under the framework of federated learning that satisfies many desired properties for data valuation. However, there are still factors of potential unfairness in the design of federated Shapley value because two data owners with the same local data may not receive the same evaluation. We propose a new measure called completed federated Shapley value to improve the fairness of federated Shapley value. The design depends on completing a matrix consisting of all the possible contributions by different subsets of the data owners. It is shown under mild conditions that this matrix is approximately low-rank by leveraging concepts and tools from optimization. Both theoretical analysis and empirical evaluation verify that the proposed measure does improve fairness in many circumstances.

cs.LG↗

arXiv2022

A dual approach for federated learning

We study the federated optimization problem from a dual perspective and propose a new algorithm termed federated dual coordinate descent (FedDCD), which is based on a type of coordinate descent method developed by Necora et al.[Journal of Optimization Theory and Applications, 2017]. Additionally, we enhance the FedDCD method with inexact gradient oracles and Nesterov's acceleration. We demonstrate theoretically that our proposed approach achieves better convergence rates than the state-of-the-art primal federated optimization algorithms under certain situations. Numerical experiments on real-world datasets support our analysis.

cs.LG↗

arXiv2019

Polar Alignment and Atomic Decomposition

Structured optimization uses a prescribed set of atoms to assemble a solution that fits a model to data. Polarity, which extends the familiar notion of orthogonality from linear sets to general convex sets, plays a special role in a simple and geometric form of convex duality. This duality correspondence yields a general notion of alignment that leads to an intuitive and complete description of how atoms participate in the final decomposition of the solution. The resulting geometric perspective leads to variations of existing algorithms effective for large-scale problems. We illustrate these ideas with many examples, including applications in matrix completion and morphological component analysis for the separation of mixtures of signals.

math.OC↗

arXiv2019

Bundle methods for dual atomic pursuit

The aim of structured optimization is to assemble a solution, using a given set of (possibly uncountably infinite) atoms, to fit a model to data. A two-stage algorithm based on gauge duality and bundle method is proposed. The first stage discovers the optimal atomic support for the primal problem by solving a sequence of approximations of the dual problem using a bundle-type method. The second stage recovers the approximate primal solution using the atoms discovered in the first stage. The overall approach leads to implementable and efficient algorithms for large problems.

math.OC↗