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.
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Reinforcement Learning and recently Deep Reinforcement Learning are popular methods for solving sequential decision making problems modeled as Markov Decision Processes. RL modeling of a problem and selecting algorithms and hyper-parameters require careful considerations as different configurations may entail completely different performances. These considerations are mainly the task of RL experts; however, RL is progressively becoming popular in other fields where the researchers and system designers are not RL experts. Besides, many modeling decisions, such as defining state and action space, size of batches and frequency of batch updating, and number of timesteps are typically made manually. For these reasons, automating different components of RL framework is of great importance and it has attracted much attention in recent years. Automated RL provides a framework in which different components of RL including MDP modeling, algorithm selection and hyper-parameter optimization are modeled and defined automatically. In this article, we explore the literature and present recent work that can be used in automated RL. Moreover, we discuss the challenges, open questions and research directions in AutoRL.
We study the dynamics of a neural network in function space when optimizing the mean squared error via gradient flow. We show that in the underparameterized regime the network learns eigenfunctions of an integral operator $T_{K^\infty}$ determined by the Neural Tangent Kernel (NTK) at rates corresponding to their eigenvalues. For example, for uniformly distributed data on the sphere $S^{d - 1}$ and rotation invariant weight distributions, the eigenfunctions of $T_{K^\infty}$ are the spherical harmonics. Our results can be understood as describing a spectral bias in the underparameterized regime. The proofs use the concept of "Damped Deviations", where deviations of the NTK matter less for eigendirections with large eigenvalues due to the occurence of a damping factor. Aside from the underparameterized regime, the damped deviations point-of-view can be used to track the dynamics of the empirical risk in the overparameterized setting, allowing us to extend certain results in the literature. We conclude that damped deviations offers a simple and unifying perspective of the dynamics when optimizing the squared error.
Federated learning has generated significant interest, with nearly all works focused on a "star" topology where nodes/devices are each connected to a central server. We migrate away from this architecture and extend it through the network dimension to the case where there are multiple layers of nodes between the end devices and the server. Specifically, we develop multi-stage hybrid federated learning (MH-FL), a hybrid of intra- and inter-layer model learning that considers the network as a multi-layer cluster-based structure. MH-FL considers the topology structures among the nodes in the clusters, including local networks formed via device-to-device (D2D) communications, and presumes a semi-decentralized architecture for federated learning. It orchestrates the devices at different network layers in a collaborative/cooperative manner (i.e., using D2D interactions) to form local consensus on the model parameters and combines it with multi-stage parameter relaying between layers of the tree-shaped hierarchy. We derive the upper bound of convergence for MH-FL with respect to parameters of the network topology (e.g., the spectral radius) and the learning algorithm (e.g., the number of D2D rounds in different clusters). We obtain a set of policies for the D2D rounds at different clusters to guarantee either a finite optimality gap or convergence to the global optimum. We then develop a distributed control algorithm for MH-FL to tune the D2D rounds in each cluster over time to meet specific convergence criteria. Our experiments on real-world datasets verify our analytical results and demonstrate the advantages of MH-FL in terms of resource utilization metrics.
Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the problem can be formulated as a regularised convex optimisation over a positive semidefinite cone and solved by iterative algorithms. Classic methods require an explicit convex function to reflect generic topological priors, e.g. the $\ell_1$ penalty for enforcing sparsity, which limits the flexibility and expressiveness in learning rich topological structures. We propose to learn a mapping from node data to the graph structure based on the idea of learning to optimise (L2O). Specifically, our model first unrolls an iterative primal-dual splitting algorithm into a neural network. The key structural proximal projection is replaced with a variational autoencoder that refines the estimated graph with enhanced topological properties. The model is trained in an end-to-end fashion with pairs of node data and graph samples. Experiments on both synthetic and real-world data demonstrate that our model is more efficient than classic iterative algorithms in learning a graph with specific topological properties.
Fairness has emerged as a critical problem in federated learning (FL). In this work, we identify a cause of unfairness in FL -- \emph{conflicting} gradients with large differences in the magnitudes. To address this issue, we propose the federated fair averaging (FedFV) algorithm to mitigate potential conflicts among clients before averaging their gradients. We first use the cosine similarity to detect gradient conflicts, and then iteratively eliminate such conflicts by modifying both the direction and the magnitude of the gradients. We further show the theoretical foundation of FedFV to mitigate the issue conflicting gradients and converge to Pareto stationary solutions. Extensive experiments on a suite of federated datasets confirm that FedFV compares favorably against state-of-the-art methods in terms of fairness, accuracy and efficiency.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
Federated learning has been showing as a promising approach in paving the last mile of artificial intelligence, due to its great potential of solving the data isolation problem in large scale machine learning. Particularly, with consideration of the heterogeneity in practical edge computing systems, asynchronous edge-cloud collaboration based federated learning can further improve the learning efficiency by significantly reducing the straggler effect. Despite no raw data sharing, the open architecture and extensive collaborations of asynchronous federated learning (AFL) still give some malicious participants great opportunities to infer other parties' training data, thus leading to serious concerns of privacy. To achieve a rigorous privacy guarantee with high utility, we investigate to secure asynchronous edge-cloud collaborative federated learning with differential privacy, focusing on the impacts of differential privacy on model convergence of AFL. Formally, we give the first analysis on the model convergence of AFL under DP and propose a multi-stage adjustable private algorithm (MAPA) to improve the trade-off between model utility and privacy by dynamically adjusting both the noise scale and the learning rate. Through extensive simulations and real-world experiments with an edge-could testbed, we demonstrate that MAPA significantly improves both the model accuracy and convergence speed with sufficient privacy guarantee.
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.
This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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