Federated learning and analytics are a distributed approach for collaboratively learning models (or statistics) from decentralized data, motivated by and designed for privacy protection. The distributed learning process can be formulated as solving federated optimization problems, which emphasize communication efficiency, data heterogeneity, compatibility with privacy and system requirements, and other constraints that are not primary considerations in other problem settings. This paper provides recommendations and guidelines on formulating, designing, evaluating and analyzing federated optimization algorithms through concrete examples and practical implementation, with a focus on conducting effective simulations to infer real-world performance. The goal of this work is not to survey the current literature, but to inspire researchers and practitioners to design federated learning algorithms that can be used in various practical applications.

We develop new voting mechanisms for the case when voters and candidates are located in an arbitrary unknown metric space, and the goal is to choose a candidate minimizing social cost: the total distance from the voters to this candidate. Previous work has often assumed that only ordinal preferences of the voters are known (instead of their true costs), and focused on minimizing distortion: the quality of the chosen candidate as compared with the best possible candidate. In this paper, we instead assume that a (very small) amount of information is known about the voter preference strengths, not just about their ordinal preferences. We provide mechanisms with much better distortion when this extra information is known as compared to mechanisms which use only ordinal information. We quantify tradeoffs between the amount of information known about preference strengths and the achievable distortion. We further provide advice about which type of information about preference strengths seems to be the most useful. Finally, we conclude by quantifying the ideal candidate distortion, which compares the quality of the chosen outcome with the best possible candidate that could ever exist, instead of only the best candidate that is actually in the running.

In this work we consider general facility location and social choice problems, in which sets of agents A and facilities F are located in a metric space, and our goal is to assign agents to facilities (as well as choose which facilities to open) in order to optimize the social cost. We form new algorithms to do this in the presence of only ordinal information, i.e., when the true costs or distances from the agents to the facilities are unknown, and only the ordinal preferences of the agents for the facilities are available. The main difference between our work and previous work in this area is that while we assume that only ordinal information about agent preferences in known, we know the exact locations of the possible facilities F. Due to this extra information about the facilities, we are able to form powerful algorithms which have small distortion, i.e., perform almost as well as omniscient algorithms but use only ordinal information about agent preferences. For example, we present natural social choice mechanisms for choosing a single facility to open with distortion of at most 3 for minimizing both the total and the median social cost; this factor is provably the best possible. We analyze many general problems including matching, k-center, and k-median, and present black-box reductions from omniscient approximation algorithms with approximation factor β to ordinal algorithms with approximation factor 1 + 2β; doing this gives new ordinal algorithms for many important problems, and establishes a toolkit for analyzing such problems in the future.

We study ordinal approximation algorithms for maximum-weight bipartite matchings. Such algorithms only know the ordinal preferences of the agents/nodes in the graph for their preferred matches, but must compete with fully omniscient algorithms which know the true numerical edge weights (utilities). Ordinal approximation is all about being able to produce good results with only limited information. Because of this, one important question is how much better the algorithms can be as the amount of information increases. To address this question for forming high-utility matchings between agents in X and Y, we consider three ordinal information types: when we know the preference order of only nodes in X for nodes in Y, when we know the preferences of both X and Y, and when we know the total order of the edge weights in the entire graph, although not the weights themselves. We also consider settings where only the top preferences of the agents are known to us, instead of their full preference orderings. We design new ordinal approximation algorithms for each of these settings, and quantify how well such algorithms perform as the amount of information given to them increases.

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