We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Convex Analysisの二,三の進展について

Characteristics of and Important Lessons From the Coronavirus Disease 2019 (COVID-19) Outbreak in China

In federated optimization, heterogeneity in the clients' local datasets and computation speeds results in large variations in the number of local updates performed by each client in each communication round. Naive weighted aggregation of such models causes objective inconsistency, that is, the global model converges to a stationary point of a mismatched objective function which can be arbitrarily different from the true objective. This paper provides a general framework to analyze the convergence of federated heterogeneous optimization algorithms. It subsumes previously proposed methods such as FedAvg and FedProx and provides the first principled understanding of the solution bias and the convergence slowdown due to objective inconsistency. Using insights from this analysis, we propose FedNova, a normalized averaging method that eliminates objective inconsistency while preserving fast error convergence.

/pdf/tackling-the-objective-inconsistency-problem-in-54vle77xqz.pdf

Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization

The general principles of Bayesian data analysis imply that models for survey responses should be constructed conditional on all variables that affect the probability of inclusion and nonresponse, which are also the variables used in survey weighting and clustering. However, such models can quickly become very complicated, with potentially thousands of poststratification cells. It is then a challenge to develop general families of multilevel probability models that yield reasonable Bayesian inferences. We discuss in the context of several ongoing public health and social surveys. This work is currently open-ended, and we conclude with thoughts on how research could proceed to solve these problems.

Struggles with survey weighting and regression modeling

Motivated by federated learning, we consider the hub-and-spoke model of distributed optimization in which a central authority coordinates the computation of a solution among many agents while limiting communication. We first study some past procedures for federated optimization, and show that their fixed points need not correspond to stationary points of the original optimization problem, even in simple convex settings with deterministic updates. In order to remedy these issues, we introduce FedSplit, a class of algorithms based on operator splitting procedures for solving distributed convex minimization with additive structure. We prove that these procedures have the correct fixed points, corresponding to optima of the original optimization problem, and we characterize their convergence rates under different settings. Our theory shows that these methods are provably robust to inexact computation of intermediate local quantities. We complement our theory with some simple experiments that demonstrate the benefits of our methods in practice.

FedSplit: An algorithmic framework for fast federated optimization

The relative case fatality rates (CFRs) between groups and countries are key measures of relative risk that guide policy decisions regarding scarce medical resource allocation during the ongoing COVID-19 pandemic. In the middle of an active outbreak when surveillance data is the primary source of information, estimating these quantities involves compensating for competing biases in time series of deaths, cases, and recoveries. These include time- and severity- dependent reporting of cases as well as time lags in observed patient outcomes. In the context of COVID-19 CFR estimation, we survey such biases and their potential significance. Further, we analyze theoretically the effect of certain biases, like preferential reporting of fatal cases, on naive estimators of CFR. We provide a partially corrected estimator of these naive estimates that accounts for time lag and imperfect reporting of deaths and recoveries. We show that collection of randomized data by testing the contacts of infectious individuals regardless of the presence of symptoms would mitigate bias by limiting the covariance between diagnosis and death. Our analysis is supplemented by theoretical and numerical results and a simple and fast open-source codebase .

/pdf/on-identifying-and-mitigating-bias-in-the-estimation-of-the-7ny1tinf6x.pdf

On Identifying and Mitigating Bias in the Estimation of the COVID-19 Case Fatality Rate

We study the matrix completion problem when the observation pattern is deterministic and possibly non-uniform. We propose a simple and efficient debiased projection scheme for recovery from noisy observations and analyze the error under a suitable weighted metric. We introduce a simple function of the weight matrix and the sampling pattern that governs the accuracy of the recovered matrix. We derive theoretical guarantees that upper bound the recovery error and nearly matching lower bounds that showcase optimality in several regimes. Our numerical experiments demonstrate the computational efficiency and accuracy of our approach, and show that debiasing is essential when using non-uniform sampling patterns.

Weighted Matrix Completion From Non-Random, Non-Uniform Sampling Patterns

/pdf/fedsplit-an-algorithmic-framework-for-fast-federated-10pmsmi1qg.pdf

FedSplit: an algorithmic framework for fast federated optimization

We study the covariate shift problem in the context of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We focus on two natural families of covariate shift problems defined using the likelihood ratios between the source and target distributions. When the likelihood ratios are uniformly bounded, we prove that the kernel ridge regression (KRR) estimator with a carefully chosen regularization parameter is minimax rate-optimal (up to a log factor) for a large family of RKHSs with regular kernel eigenvalues. Interestingly, KRR does not require full knowledge of likelihood ratios apart from an upper bound on them. In striking contrast to the standard statistical setting without covariate shift, we also demonstrate that a naive estimator, which minimizes the empirical risk over the function class, is strictly sub-optimal under covariate shift as compared to KRR. We then address the larger class of covariate shift problems where the likelihood ratio is possibly unbounded yet has a finite second moment. Here, we propose a reweighted KRR estimator that weights samples based on a careful truncation of the likelihood ratios. Again, we are able to show that this estimator is minimax rate-optimal, up to logarithmic factors.

Reese Pathak

Papers

FedSplit: An algorithmic framework for fast federated optimization

On Identifying and Mitigating Bias in the Estimation of the COVID-19 Case Fatality Rate

Weighted Matrix Completion From Non-Random, Non-Uniform Sampling Patterns

FedSplit: an algorithmic framework for fast federated optimization

Optimally tackling covariate shift in RKHS-based nonparametric regression