In 1-bit compressed sensing, the aim is to estimate a $k$-sparse unit vector $x\in S^{n-1}$ within an $\epsilon$ error (in $\ell_2$) from minimal number of linear measurements that are quantized to just their signs, i.e., from measurements of the form $y = \mathrm{Sign}(\langle a, x\rangle).$ In this paper, we study a noisy version where a fraction of the measurements can be flipped, potentially by an adversary. In particular, we analyze the Binary Iterative Hard Thresholding (BIHT) algorithm, a proximal gradient descent on a properly defined loss function used for 1-bit compressed sensing, in this noisy setting. It is known from recent results that, with $\tilde{O}(\frac{k}{\epsilon})$ noiseless measurements, BIHT provides an estimate within $\epsilon$ error. This result is optimal and universal, meaning one set of measurements work for all sparse vectors. In this paper, we show that BIHT also provides better results than all known methods for the noisy setting. We show that when up to $\tau$-fraction of the sign measurements are incorrect (adversarial error), with the same number of measurements as before, BIHT agnostically provides an estimate of $x$ within an $\tilde{O}(\epsilon+\tau)$ error, maintaining the universality of measurements. This establishes stability of iterative hard thresholding in the presence of measurement error. To obtain the result, we use the restricted approximate invertibility of Gaussian matrices, as well as a tight analysis of the high-dimensional geometry of the adversarially corrupted measurements.
Motivated by the need for communication-efficient distributed learning, we investigate the method for compressing a unit norm vector into the minimum number of bits, while still allowing for some acceptable level of distortion in recovery. This problem has been explored in the rate-distortion/covering code literature, but our focus is exclusively on the "high-distortion" regime. We approach this problem in a worst-case scenario, without any prior information on the vector, but allowing for the use of randomized compression maps. Our study considers both biased and unbiased compression methods and determines the optimal compression rates. It turns out that simple compression schemes are nearly optimal in this scenario. While the results are a mix of new and known, they are compiled in this paper for completeness.
The logistic regression model is one of the most popular data generation model in noisy binary classification problems. In this work, we study the sample complexity of estimating the parameters of the logistic regression model up to a given $\ell_2$ error, in terms of the dimension and the inverse temperature, with standard normal covariates. The inverse temperature controls the signal-to-noise ratio of the data generation process. While both generalization bounds and asymptotic performance of the maximum-likelihood estimator for logistic regression are well-studied, the non-asymptotic sample complexity that shows the dependence on error and the inverse temperature for parameter estimation is absent from previous analyses. We show that the sample complexity curve has two change-points (or critical points) in terms of the inverse temperature, clearly separating the low, moderate, and high temperature regimes.
We propose a first-order method for convex optimization, where instead of being restricted to the gradient from a single parameter, gradients from multiple parameters can be used during each step of gradient descent. This setup is particularly useful when a few processors are available that can be used in parallel for optimization. Our method uses gradients from multiple parameters in synergy to update these parameters together towards the optima. While doing so, it is ensured that the computational and memory complexity is of the same order as that of gradient descent. Empirical results demonstrate that even using gradients from as low as \textit{two} parameters, our method can often obtain significant acceleration and provide robustness to hyper-parameter settings. We remark that the primary goal of this work is less theoretical, and is instead aimed at exploring the understudied case of using multiple gradients during each step of optimization.
Federated Learning (FL) has gained increasing interest in recent years as a distributed on-device learning paradigm. However, multiple challenges remain to be addressed for deploying FL in real-world Internet-of-Things (IoT) networks with hierarchies. Although existing works have proposed various approaches to account data heterogeneity, system heterogeneity, unexpected stragglers and scalibility, none of them provides a systematic solution to address all of the challenges in a hierarchical and unreliable IoT network. In this paper, we propose an asynchronous and hierarchical framework (Async-HFL) for performing FL in a common three-tier IoT network architecture. In response to the largely varied delays, Async-HFL employs asynchronous aggregations at both the gateway and the cloud levels thus avoids long waiting time. To fully unleash the potential of Async-HFL in converging speed under system heterogeneities and stragglers, we design device selection at the gateway level and device-gateway association at the cloud level. Device selection chooses edge devices to trigger local training in real-time while device-gateway association determines the network topology periodically after several cloud epochs, both satisfying bandwidth limitation. We evaluate Async-HFL's convergence speedup using large-scale simulations based on ns-3 and a network topology from NYCMesh. Our results show that Async-HFL converges 1.08-1.31x faster in wall-clock time and saves up to 21.6% total communication cost compared to state-of-the-art asynchronous FL algorithms (with client selection). We further validate Async-HFL on a physical deployment and observe robust convergence under unexpected stragglers.
One-bit compressed sensing (1bCS) is an extremely quantized signal acquisition method that has been proposed and studied rigorously in the past decade. In 1bCS, linear samples of a high dimensional signal are quantized to only one bit per sample (sign of the measurement). Assuming the original signal vector to be sparse, existing results in 1bCS either aim to find the support of the vector, or approximate the signal allowing a small error. The focus of this paper is support recovery, which often also computationally facilitate approximate signal recovery. A {\em universal} measurement matrix for 1bCS refers to one set of measurements that work for all sparse signals. With universality, it is known that $\tilde{\Theta}(k^2)$ 1bCS measurements are necessary and sufficient for support recovery (where $k$ denotes the sparsity). To improve the dependence on sparsity from quadratic to linear, in this work we propose approximate support recovery (allowing $\epsilon>0$ proportion of errors), and superset recovery (allowing $\epsilon$ proportion of false positives). We show that the first type of recovery is possible with $\tilde{O}(k/\epsilon)$ measurements, while the later type of recovery, more challenging, is possible with $\tilde{O}(\max\{k/\epsilon,k^{3/2}\})$ measurements. We also show that in both cases $\Omega(k/\epsilon)$ measurements would be necessary for universal recovery. Improved results are possible if we consider universal recovery within a restricted class of signals, such as rational signals, or signals with bounded dynamic range. In both cases superset recovery is possible with only $\tilde{O}(k/\epsilon)$ measurements. Other results on universal but approximate support recovery are also provided in this paper. All of our main recovery algorithms are simple and polynomial-time.
Compressed sensing has been a very successful high-dimensional signal acquisition and recovery technique that relies on linear operations. However, the actual measurements of signals have to be quantized before storing or processing. 1(One)-bit compressed sensing is a heavily quantized version of compressed sensing, where each linear measurement of a signal is reduced to just one bit: the sign of the measurement. Once enough of such measurements are collected, the recovery problem in 1-bit compressed sensing aims to find the original signal with as much accuracy as possible. The recovery problem is related to the traditional "halfspace-learning" problem in learning theory. For recovery of sparse vectors, a popular reconstruction method from 1-bit measurements is the binary iterative hard thresholding (BIHT) algorithm. The algorithm is a simple projected sub-gradient descent method, and is known to converge well empirically, despite the nonconvexity of the problem. The convergence property of BIHT was not theoretically justified, except with an exorbitantly large number of measurements (i.e., a number of measurement greater than $\max\{k^{10}, 24^{48}, k^{3.5}/\epsilon\}$, where $k$ is the sparsity, $\epsilon$ denotes the approximation error, and even this expression hides other factors). In this paper we show that the BIHT algorithm converges with only $\tilde{O}(\frac{k}{\epsilon})$ measurements. Note that, this dependence on $k$ and $\epsilon$ is optimal for any recovery method in 1-bit compressed sensing. With this result, to the best of our knowledge, BIHT is the only practical and efficient (polynomial time) algorithm that requires the optimal number of measurements in all parameters (both $k$ and $\epsilon$). This is also an example of a gradient descent algorithm converging to the correct solution for a nonconvex problem, under suitable structural conditions.
To capture inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a \emph{Geometric Block Model}. The geometric block model builds on the \emph{random geometric graphs} (Gilbert, 1961), one of the basic models of random graphs for spatial networks, in the same way that the well-studied stochastic block model builds on the Erd\H{o}s-R\'{en}yi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancements in community detection. To analyze the geometric block model, we first provide new connectivity results for \emph{random annulus graphs} which are generalizations of random geometric graphs. The connectivity properties of geometric graphs have been studied since their introduction, and analyzing them has been difficult due to correlated edge formation. We then use the connectivity results of random annulus graphs to provide necessary and sufficient conditions for efficient recovery of communities for the geometric block model. We show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. For this we consider two regimes of graph density. In the regime where the average degree of the graph grows logarithmically with number of vertices, we show that our algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model in the logarithmic degree regime. We also look at the regime where the average degree of the graph grows linearly with the number of vertices $n$, and hence to store the graph one needs $\Theta(n^2)$ memory. We show that our algorithm needs to store only $O(n \log n)$ edges in this regime to recover the latent communities.
Understanding complex dynamics of two-sided online matching markets, where the demand-side agents compete to match with the supply-side (arms), has recently received substantial interest. To that end, in this paper, we introduce the framework of decentralized two-sided matching market under non stationary (dynamic) environments. We adhere to the serial dictatorship setting, where the demand-side agents have unknown and different preferences over the supply-side (arms), but the arms have fixed and known preference over the agents. We propose and analyze a decentralized and asynchronous learning algorithm, namely Decentralized Non-stationary Competing Bandits (\texttt{DNCB}), where the agents play (restrictive) successive elimination type learning algorithms to learn their preference over the arms. The complexity in understanding such a system stems from the fact that the competing bandits choose their actions in an asynchronous fashion, and the lower ranked agents only get to learn from a set of arms, not \emph{dominated} by the higher ranked agents, which leads to \emph{forced exploration}. With carefully defined complexity parameters, we characterize this \emph{forced exploration} and obtain sub-linear (logarithmic) regret of \texttt{DNCB}. Furthermore, we validate our theoretical findings via experiments.
While mixture of linear regressions (MLR) is a well-studied topic, prior works usually do not analyze such models for prediction error. In fact, {\em prediction} and {\em loss} are not well-defined in the context of mixtures. In this paper, first we show that MLR can be used for prediction where instead of predicting a label, the model predicts a list of values (also known as {\em list-decoding}). The list size is equal to the number of components in the mixture, and the loss function is defined to be minimum among the losses resulted by all the component models. We show that with this definition, a solution of the empirical risk minimization (ERM) achieves small probability of prediction error. This begs for an algorithm to minimize the empirical risk for MLR, which is known to be computationally hard. Prior algorithmic works in MLR focus on the {\em realizable} setting, i.e., recovery of parameters when data is probabilistically generated by a mixed linear (noisy) model. In this paper we show that a version of the popular alternating minimization (AM) algorithm finds the best fit lines in a dataset even when a realizable model is not assumed, under some regularity conditions on the dataset and the initial points, and thereby provides a solution for the ERM. We further provide an algorithm that runs in polynomial time in the number of datapoints, and recovers a good approximation of the best fit lines. The two algorithms are experimentally compared.