Addressing Budget Allocation and Revenue Allocation in Data Market Environments Using an Adaptive Sampling Algorithm

High-quality machine learning models are dependent on access to high-quality training data. When the data are not already available, it is tedious and costly to obtain them. Data markets help with identifying valuable training data: model consumers pay to train a model, the market uses that budget to identify data and train the model (the budget allocation problem), and finally the market compensates data providers according to their data contribution (revenue allocation problem). For example, a bank could pay the data market to access data from other financial institutions to train a fraud detection model. Compensating data contributors requires understanding data's contribution to the model; recent efforts to solve this revenue allocation problem based on the Shapley value are inefficient to lead to practical data markets. In this paper, we introduce a new algorithm to solve budget allocation and revenue allocation problems simultaneously in linear time. The new algorithm employs an adaptive sampling process that selects data from those providers who are contributing the most to the model. Better data means that the algorithm accesses those providers more often, and more frequent accesses corresponds to higher compensation. Furthermore, the algorithm can be deployed in both centralized and federated scenarios, boosting its applicability. We provide theoretical guarantees for the algorithm that show the budget is used efficiently and the properties of revenue allocation are similar to Shapley's. Finally, we conduct an empirical evaluation to show the performance of the algorithm in practical scenarios and when compared to other baselines. Overall, we believe that the new algorithm paves the way for the implementation of practical data markets.

Latent Multimodal Functional Graphical Model Estimation

Joint multimodal functional data acquisition, where functional data from multiple modes are measured simultaneously from the same subject, has emerged as an exciting modern approach enabled by recent engineering breakthroughs in the neurological and biological sciences. One prominent motivation to acquire such data is to enable new discoveries of the underlying connectivity by combining multimodal signals. Despite the scientific interest, there remains a gap in principled statistical methods for estimating the graph underlying multimodal functional data. To this end, we propose a new integrative framework that models the data generation process and identifies operators mapping from the observation space to the latent space. We then develop an estimator that simultaneously estimates the transformation operators and the latent graph. This estimator is based on the partial correlation operator, which we rigorously extend from the multivariate to the functional setting. Our procedure is provably efficient, with the estimator converging to a stationary point with quantifiable statistical error. Furthermore, we show recovery of the latent graph under mild conditions. Our work is applied to analyze simultaneously acquired multimodal brain imaging data where the graph indicates functional connectivity of the brain. We present simulation and empirical results that support the benefits of joint estimation.

L-SVRG and L-Katyusha with Adaptive Sampling

Stochastic gradient-based optimization methods, such as L-SVRG and its accelerated variant L-Katyusha [12], are widely used to train machine learning models. Theoretical and empirical performance of L-SVRG and L-Katyusha can be improved by sampling the observations from a non-uniform distribution [17]. However, to design a desired sampling distribution, Qian et al.[17] rely on prior knowledge of smoothness constants that can be computationally intractable to obtain in practice when the dimension of the model parameter is high. We propose an adaptive sampling strategy for L-SVRG and L-Katyusha that learns the sampling distribution with little computational overhead, while allowing it to change with iterates, and at the same time does not require any prior knowledge on the problem parameters. We prove convergence guarantees for L-SVRG and L-Katyusha for convex objectives when the sampling distribution changes with iterates. These results show that even without prior information, the proposed adaptive sampling strategy matches, and in some cases even surpasses, the performance of the sampling scheme in Qian et al.[17]. Extensive simulations support our theory and the practical utility of the proposed sampling scheme on real data.

Adaptive Client Sampling in Federated Learning via Online Learning with Bandit Feedback

In federated learning (FL) problems, client sampling plays a key role in the convergence speed of training algorithm. However, while being an important problem in FL, client sampling is lack of study. In this paper, we propose an online learning with bandit feedback framework to understand the client sampling problem in FL. By adapting an Online Stochastic Mirror Descent algorithm to minimize the variance of gradient estimation, we propose a new adaptive client sampling algorithm. Besides, we use online ensemble method and doubling trick to automatically choose the tuning parameters in the algorithm. Theoretically, we show dynamic regret bound with comparator as the theoretically optimal sampling sequence; we also include the total variation of this sequence in our upper bound, which is a natural measure of the intrinsic difficulty of the problem. To the best of our knowledge, these theoretical contributions are novel to existing literature. Moreover, by implementing both synthetic and real data experiments, we show empirical evidence of the advantages of our proposed algorithms over widely-used uniform sampling and also other online learning based sampling strategies in previous studies. We also examine its robustness to the choice of tuning parameters. Finally, we discuss its possible extension to sampling without replacement and personalized FL objective. While the original goal is to solve client sampling problem, this work has more general applications on stochastic gradient descent and stochastic coordinate descent methods.

High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach

Undirected graphical models have been widely used to model the conditional independence structure of high-dimensional random vector data for years. In many modern applications such as EEG and fMRI data, the observations are multivariate random functions rather than scalars. To model the conditional independence of this type of data, functional graphical models are proposed and have attracted an increasing attention in recent years. In this paper, we propose a neighborhood selection approach to estimate Gaussian functional graphical models. We first estimate the neighborhood of all nodes via function-on-function regression, and then we can recover the whole graph structure based on the neighborhood information. By estimating conditional structure directly, we can circumvent the need of a well-defined precision operator which generally does not exist. Besides, we can better explore the effect of the choice of function basis for dimension reduction. We give a criterion for choosing the best function basis and motivate two practically useful choices, which we justified by both theory and experiments and show that they are better than expanding each function onto its own FPCA basis as in previous literature. In addition, the neighborhood selection approach is computationally more efficient than fglasso as it is more easy to do parallel computing. The statistical consistency of our proposed methods in high-dimensional setting are supported by both theory and experiment.

Personalized Federated Learning: A Unified Framework and Universal Optimization Techniques

We study the optimization aspects of personalized Federated Learning (FL). We develop a universal optimization theory applicable to all convex personalized FL models in the literature. In particular, we propose a general personalized objective capable of recovering essentially any existing personalized FL objective as a special case. We design several optimization techniques to minimize the general objective, namely a tailored variant of Local SGD and variants of accelerated coordinate descent/accelerated SVRCD. We demonstrate the practicality and/or optimality of our methods both in terms of communication and local computation. Lastly, we argue about the implications of our general optimization theory when applied to solve specific personalized FL objectives.

FuDGE: Functional Differential Graph Estimation with fully and discretely observed curves

We consider the problem of estimating the difference between two functional undirected graphical models with shared structures. In many applications, data are naturally regarded as high-dimensional random function vectors rather than multivariate scalars. For example, electroencephalography (EEG) data are more appropriately treated as functions of time. In these problems, not only can the number of functions measured per sample be large, but each function is itself an infinite dimensional object, making estimation of model parameters challenging. In practice, curves are usually discretely observed, which makes graph structure recovery even more challenging. We formally characterize when two functional graphical models are comparable and propose a method that directly estimates the functional differential graph, which we term FuDGE. FuDGE avoids separate estimation of each graph, which allows for estimation in problems where individual graphs are dense, but their difference is sparse. We show that FuDGE consistently estimates the functional differential graph in a high-dimensional setting for both discretely observed and fully observed function paths. We illustrate finite sample properties of our method through simulation studies. In order to demonstrate the benefits of our method, we propose Joint Functional Graphical Lasso as a competitor, which is a generalization of the Joint Graphical Lasso. Finally, we apply our method to EEG data to uncover differences in functional brain connectivity between alcoholics and control subjects.

Direct Estimation of Differential Functional Graphical Models

We consider the problem of estimating the difference between two functional undirected graphical models with shared structures. In many applications, data are naturally regarded as high-dimensional random function vectors rather than multivariate scalars. For example, electroencephalography (EEG) data are more appropriately treated as functions of time. In these problems, not only can the number of functions measured per sample be large, but each function is itself an infinite dimensional object, making estimation of model parameters challenging. We develop a method that directly estimates the difference of graphs, avoiding separate estimation of each graph, and show it is consistent in certain high-dimensional settings. We illustrate finite sample properties of our method through simulation studies. Finally, we apply our method to EEG data to uncover differences in functional brain connectivity between alcoholics and control subjects.