Given an $n$-point metric space $(\mathcal{X},d)$ where each point belongs to one of $m=O(1)$ different categories or groups and a set of integers $k_1, \ldots, k_m$, the fair Max-Min diversification problem is to select $k_i$ points belonging to category $i\in [m]$, such that the minimum pairwise distance between selected points is maximized. The problem was introduced by Moumoulidou et al. [ICDT 2021] and is motivated by the need to down-sample large data sets in various applications so that the derived sample achieves a balance over diversity, i.e., the minimum distance between a pair of selected points, and fairness, i.e., ensuring enough points of each category are included. We prove the following results: 1. We first consider general metric spaces. We present a randomized polynomial time algorithm that returns a factor $2$-approximation to the diversity but only satisfies the fairness constraints in expectation. Building upon this result, we present a $6$-approximation that is guaranteed to satisfy the fairness constraints up to a factor $1-\epsilon$ for any constant $\epsilon$. We also present a linear time algorithm returning an $m+1$ approximation with exact fairness. The best previous result was a $3m-1$ approximation. 2. We then focus on Euclidean metrics. We first show that the problem can be solved exactly in one dimension. For constant dimensions, categories and any constant $\epsilon>0$, we present a $1+\epsilon$ approximation algorithm that runs in $O(nk) + 2^{O(k)}$ time where $k=k_1+\ldots+k_m$. We can improve the running time to $O(nk)+ poly(k)$ at the expense of only picking $(1-\epsilon) k_i$ points from category $i\in [m]$. Finally, we present algorithms suitable to processing massive data sets including single-pass data stream algorithms and composable coresets for the distributed processing.
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Federated learning (FL) aims to minimize the communication complexity of training a model over heterogeneous data distributed across many clients. A common approach is local methods, where clients take multiple optimization steps over local data before communicating with the server (e.g., FedAvg). Local methods can exploit similarity between clients' data. However, in existing analyses, this comes at the cost of slow convergence in terms of the dependence on the number of communication rounds R. On the other hand, global methods, where clients simply return a gradient vector in each round (e.g., SGD), converge faster in terms of R but fail to exploit the similarity between clients even when clients are homogeneous. We propose FedChain, an algorithmic framework that combines the strengths of local methods and global methods to achieve fast convergence in terms of R while leveraging the similarity between clients. Using FedChain, we instantiate algorithms that improve upon previously known rates in the general convex and PL settings, and are near-optimal (via an algorithm-independent lower bound that we show) for problems that satisfy strong convexity. Empirical results support this theoretical gain over existing methods.
We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.
We consider the question of adaptive data analysis within the framework of convex optimization. We ask how many samples are needed in order to compute $\epsilon$-accurate estimates of $O(1/\epsilon^2)$ gradients queried by gradient descent, and we provide two intermediate answers to this question. First, we show that for a general analyst (not necessarily gradient descent) $\Omega(1/\epsilon^3)$ samples are required. This rules out the possibility of a foolproof mechanism. Our construction builds upon a new lower bound (that may be of interest of its own right) for an analyst that may ask several non adaptive questions in a batch of fixed and known $T$ rounds of adaptivity and requires a fraction of true discoveries. We show that for such an analyst $\Omega (\sqrt{T}/\epsilon^2)$ samples are necessary. Second, we show that, under certain assumptions on the oracle, in an interaction with gradient descent $\tilde \Omega(1/\epsilon^{2.5})$ samples are necessary. Our assumptions are that the oracle has only \emph{first order access} and is \emph{post-hoc generalizing}. First order access means that it can only compute the gradients of the sampled function at points queried by the algorithm. Our assumption of \emph{post-hoc generalization} follows from existing lower bounds for statistical queries. More generally then, we provide a generic reduction from the standard setting of statistical queries to the problem of estimating gradients queried by gradient descent. These results are in contrast with classical bounds that show that with $O(1/\epsilon^2)$ samples one can optimize the population risk to accuracy of $O(\epsilon)$ but, as it turns out, with spurious gradients.
Data collection and research methodology represents a critical part of the research pipeline. On the one hand, it is important that we collect data in a way that maximises the validity of what we are measuring, which may involve the use of long scales with many items. On the other hand, collecting a large number of items across multiple scales results in participant fatigue, and expensive and time consuming data collection. It is therefore important that we use the available resources optimally. In this work, we consider how a consideration for theory and the associated causal/structural model can help us to streamline data collection procedures by not wasting time collecting data for variables which are not causally critical for subsequent analysis. This not only saves time and enables us to redirect resources to attend to other variables which are more important, but also increases research transparency and the reliability of theory testing. In order to achieve this streamlined data collection, we leverage structural models, and Markov conditional independency structures implicit in these models to identify the substructures which are critical for answering a particular research question. In this work, we review the relevant concepts and present a number of didactic examples with the hope that psychologists can use these techniques to streamline their data collection process without invalidating the subsequent analysis. We provide a number of simulation results to demonstrate the limited analytical impact of this streamlining.
Following the research agenda initiated by Munoz & Vassilvitskii [1] and Lykouris & Vassilvitskii [2] on learning-augmented online algorithms for classical online optimization problems, in this work, we consider the Online Facility Location problem under this framework. In Online Facility Location (OFL), demands arrive one-by-one in a metric space and must be (irrevocably) assigned to an open facility upon arrival, without any knowledge about future demands. We present an online algorithm for OFL that exploits potentially imperfect predictions on the locations of the optimal facilities. We prove that the competitive ratio decreases smoothly from sublogarithmic in the number of demands to constant, as the error, i.e., the total distance of the predicted locations to the optimal facility locations, decreases towards zero. We complement our analysis with a matching lower bound establishing that the dependence of the algorithm's competitive ratio on the error is optimal, up to constant factors. Finally, we evaluate our algorithm on real world data and compare our learning augmented approach with the current best online algorithm for the problem.
The naive importance sampling (IS) estimator generally does not work well in examples involving simultaneous inference on several targets, as the importance weights can take arbitrarily large values, making the estimator highly unstable. In such situations, alternative multiple IS estimators involving samples from multiple proposal distributions are preferred. Just like the naive IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. The selection of these proposal distributions is the focus of this article. We propose three methods: (i) a geometric space filling approach, (ii) a minimax variance approach, and (iii) a maximum entropy approach. The first two methods are applicable to any IS estimator, whereas the third approach is described in the context of Doss's (2010) two-stage IS estimator. For the first method, we propose a suitable measure of 'closeness' based on the symmetric Kullback-Leibler divergence, while the second and third approaches use estimates of asymptotic variances of Doss's (2010) IS estimator and Geyer's (1994) reverse logistic regression estimator, respectively. Thus, when samples from the proposal distributions are obtained by running Markov chains, we provide consistent spectral variance estimators for these asymptotic variances. The proposed methods for selecting proposal densities are illustrated using various detailed examples.
Linear mixed models (LMMs) are instrumental for regression analysis with structured dependence, such as grouped, clustered, or multilevel data. However, selection among the covariates--while accounting for this structured dependence--remains a challenge. We introduce a Bayesian decision analysis for subset selection with LMMs. Using a Mahalanobis loss function that incorporates the structured dependence, we derive optimal linear coefficients for (i) any given subset of variables and (ii) all subsets of variables that satisfy a cardinality constraint. Crucially, these estimates inherit shrinkage or regularization and uncertainty quantification from the underlying Bayesian model, and apply for any well-specified Bayesian LMM. More broadly, our decision analysis strategy deemphasizes the role of a single "best" subset, which is often unstable and limited in its information content, and instead favors a collection of near-optimal subsets. This collection is summarized by key member subsets and variable-specific importance metrics. Customized subset search and out-of-sample approximation algorithms are provided for more scalable computing. These tools are applied to simulated data and a longitudinal physical activity dataset, and demonstrate excellent prediction, estimation, and selection ability.
This paper considers the problem of inference in cluster randomized experiments when cluster sizes are non-ignorable. Here, by a cluster randomized experiment, we mean one in which treatment is assigned at the level of the cluster; by non-ignorable cluster sizes we mean that "large" clusters and "small" clusters may be heterogeneous, and, in particular, the effects of the treatment may vary across clusters of differing sizes. In order to permit this sort of flexibility, we consider a sampling framework in which cluster sizes themselves are random. In this way, our analysis departs from earlier analyses of cluster randomized experiments in which cluster sizes are treated as non-random. We distinguish between two different parameters of interest: the equally-weighted cluster-level average treatment effect, and the size-weighted cluster-level average treatment effect. For each parameter, we provide methods for inference in an asymptotic framework where the number of clusters tends to infinity and treatment is assigned using simple random sampling. We additionally permit the experimenter to sample only a subset of the units within each cluster rather than the entire cluster and demonstrate the implications of such sampling for some commonly used estimators. A small simulation study shows the practical relevance of our theoretical results.
In the storied Colonel Blotto game, two colonels allocate $a$ and $b$ troops, respectively, to $k$ distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game\footnote{To the best of our knowledge, the algorithm from \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} has computational complexity $O(k^{14}\max\{a,b\}^{13})$}, receiving substantial media coverage (e.g. \citep{Insider}, \citep{NSF}, \citep{ScienceDaily}). In this work, we present the first known $\epsilon$-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime $\widetilde{O}(\epsilon^{-4} k^8 \max\{a,b\}^2)$ for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are $\ell > 2$ colonels) with runtime $\widetilde{O}(\ell \epsilon^{-4} k^8 n^2 + \ell^2 \epsilon^{-2} k^3 n (n+k))$, where $n$ is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.
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