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This paper investigates the transmission power control in over-the-air federated edge learning (Air-FEEL) system. Different from conventional power control designs (e.g., to minimize the individual mean squared error (MSE) of the over-the-air aggregation at each round), we consider a new power control design aiming at directly maximizing the convergence speed. Towards this end, we first analyze the convergence behavior of Air-FEEL (in terms of the optimality gap) subject to aggregation errors at different communication rounds. It is revealed that if the aggregation estimates are unbiased, then the training algorithm would converge exactly to the optimal point with mild conditions; while if they are biased, then the algorithm would converge with an error floor determined by the accumulated estimate bias over communication rounds. Next, building upon the convergence results, we optimize the power control to directly minimize the derived optimality gaps under both biased and unbiased aggregations, subject to a set of average and maximum power constraints at individual edge devices. We transform both problems into convex forms, and obtain their structured optimal solutions, both appearing in a form of regularized channel inversion, by using the Lagrangian duality method. Finally, numerical results show that the proposed power control policies achieve significantly faster convergence for Air-FEEL, as compared with benchmark policies with fixed power transmission or conventional MSE minimization.

Federated learning is a distributed learning paradigm where multiple agents, each only with access to local data, jointly learn a global model. There has recently been an explosion of research aiming not only to improve the accuracy rates of federated learning, but also provide certain guarantees around social good properties such as total error. One branch of this research has taken a game-theoretic approach, and in particular, prior work has viewed federated learning as a hedonic game, where error-minimizing players arrange themselves into federating coalitions. This past work proves the existence of stable coalition partitions, but leaves open a wide range of questions, including how far from optimal these stable solutions are. In this work, we motivate and define a notion of optimality given by the average error rates among federating agents (players). First, we provide and prove the correctness of an efficient algorithm to calculate an optimal (error minimizing) arrangement of players. Next, we analyze the relationship between the stability and optimality of an arrangement. First, we show that for some regions of parameter space, all stable arrangements are optimal (Price of Anarchy equal to 1). However, we show this is not true for all settings: there exist examples of stable arrangements with higher cost than optimal (Price of Anarchy greater than 1). Finally, we give the first constant-factor bound on the performance gap between stability and optimality, proving that the total error of the worst stable solution can be no higher than 9 times the total error of an optimal solution (Price of Anarchy bound of 9).

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an estimator of $x_\star$ with bias $\delta$, variance $O(\log(1/\delta))$, and an expected sampling cost of $O(\log(1/\delta))$ stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau-Yoshida envelope of any Lipschitz convex function, allowing us to perform dimension-free randomized smoothing. We demonstrate the potential of our estimator through four applications. First, we develop a method for minimizing the maximum of $N$ functions, improving on recent results and matching a lower bound up logarithmic factors. Second and third, we recover state-of-the-art rates for projection-efficient and gradient-efficient optimization using simple algorithms with a transparent analysis. Finally, we show that an improved version of our estimator would yield a nearly linear-time, optimal-utility, differentially-private non-smooth stochastic optimization method.

We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and "drift". Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solution with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high order Taylor's method with extended precision). Then, we find an invariant curve for the return map using recent algorithms that take advantage of the geometric features of the problem. This method is based on a rapidly convergent Newton's method which is guaranteed to converge if the initial error is small enough. So, it is very suitable for a continuation algorithm. The resulting algorithm is quite efficient. We only need to deal with a one dimensional function. If this function is discretized in $N$ points, the algorithm requires $O(N \log N) $ operations and $O(N) $ storage. The most costly step (the numerical integration of the equation along a turn) is trivial to parallelize. The main goal of the paper is to present the algorithms, implementation details and several sample results of runs. We also present both a rigorous and a numerical comparison of the results of averaged and not averaged models.

Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minima. On the other hand, Langevin dynamics (LD) can explore the state space and find global minima, but in order to give accurate estimates, LD needs to run with a small discretization step size and weak stochastic force, which in general slow down its convergence. This paper shows that these two algorithms and their non-swapping variants. can ``collaborate" through a simple exchange mechanism, in which they swap their current positions if LD yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. We also study non-swapping variants of the algorithm, which achieve similar performance. We further verify our theoretical results through some numerical experiments and observe superior performance of the proposed algorithm over running GD or LD alone.

This paper considers decentralized minimization of $N:=nm$ smooth non-convex cost functions equally divided over a directed network of $n$ nodes. Specifically, we describe a stochastic first-order gradient method, called GT-SARAH, that employs a SARAH-type variance reduction technique and gradient tracking (GT) to address the stochastic and decentralized nature of the problem. We show that GT-SARAH, with appropriate algorithmic parameters, finds an $\epsilon$-accurate first-order stationary point with $O\big(\max\big\{N^{\frac{1}{2}},n(1-\lambda)^{-2},n^{\frac{2}{3}}m^{\frac{1}{3}}(1-\lambda)^{-1}\big\}L\epsilon^{-2}\big)$ gradient complexity, where ${(1-\lambda)\in(0,1]}$ is the spectral gap of the network weight matrix and $L$ is the smoothness parameter of the cost functions. This gradient complexity outperforms that of the existing decentralized stochastic gradient methods. In particular, in a big-data regime such that ${n = O(N^{\frac{1}{2}}(1-\lambda)^{3})}$, this gradient complexity furthers reduces to ${O(N^{\frac{1}{2}}L\epsilon^{-2})}$, independent of the network topology, and matches that of the centralized near-optimal variance-reduced methods. Moreover, in this regime GT-SARAH achieves a non-asymptotic linear speedup, in that, the total number of gradient computations at each node is reduced by a factor of $1/n$ compared to the centralized near-optimal algorithms that perform all gradient computations at a single node. To the best of our knowledge, GT-SARAH is the first algorithm that achieves this property. In addition, we show that appropriate choices of local minibatch size balance the trade-offs between the gradient and communication complexity of GT-SARAH. Over infinite time horizon, we establish that all nodes in GT-SARAH asymptotically achieve consensus and converge to a first-order stationary point in the almost sure and mean-squared sense.

In this paper, we consider an online optimization problem over $T$ rounds where at each step $t\in[T]$, the algorithm chooses an action $x_t$ from the fixed convex and compact domain set $\mathcal{K}$. A utility function $f_t(\cdot)$ is then revealed and the algorithm receives the payoff $f_t(x_t)$. This problem has been previously studied under the assumption that the utilities are adversarially chosen monotone DR-submodular functions and $\mathcal{O}(\sqrt{T})$ regret bounds have been derived. We first characterize the class of strongly DR-submodular functions and then, we derive regret bounds for the following new online settings: $(1)$ $\{f_t\}_{t=1}^T$ are monotone strongly DR-submodular and chosen adversarially, $(2)$ $\{f_t\}_{t=1}^T$ are monotone submodular (while the average $\frac{1}{T}\sum_{t=1}^T f_t$ is strongly DR-submodular) and chosen by an adversary but they arrive in a uniformly random order, $(3)$ $\{f_t\}_{t=1}^T$ are drawn i.i.d. from some unknown distribution $f_t\sim \mathcal{D}$ where the expected function $f(\cdot)=\mathbb{E}_{f_t\sim\mathcal{D}}[f_t(\cdot)]$ is monotone DR-submodular. For $(1)$, we obtain the first logarithmic regret bounds. In terms of the second framework, we show that it is possible to obtain similar logarithmic bounds with high probability. Finally, for the i.i.d. model, we provide algorithms with $\tilde{\mathcal{O}}(\sqrt{T})$ stochastic regret bound, both in expectation and with high probability. Experimental results demonstrate that our algorithms outperform the previous techniques in the aforementioned three settings.

We study the problem of boosting the accuracy of a weak learner in the (distribution-independent) PAC model with Massart noise. In the Massart noise model, the label of each example $x$ is independently misclassified with probability $\eta(x) \leq \eta$, where $\eta<1/2$. The Massart model lies between the random classification noise model and the agnostic model. Our main positive result is the first computationally efficient boosting algorithm in the presence of Massart noise that achieves misclassification error arbitrarily close to $\eta$. Prior to our work, no non-trivial booster was known in this setting. Moreover, we show that this error upper bound is best possible for polynomial-time black-box boosters, under standard cryptographic assumptions. Our upper and lower bounds characterize the complexity of boosting in the distribution-independent PAC model with Massart noise. As a simple application of our positive result, we give the first efficient Massart learner for unions of high-dimensional rectangles.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.