We consider a distributed learning problem over multiple access channel (MAC) using a large wireless network. The computation is made by the network edge and is based on received data from a large number of distributed nodes which transmit over a noisy fading MAC. The objective function is a sum of the nodes' local loss functions. This problem has attracted a growing interest in distributed sensing systems, and more recently in federated learning. We develop a novel Gradient-Based Multiple Access (GBMA) algorithm to solve the distributed learning problem over MAC. Specifically, the nodes transmit an analog function of the local gradient using common shaping waveforms and the network edge receives a superposition of the analog transmitted signals used for updating the estimate. GBMA does not require power control or beamforming to cancel the fading effect as in other algorithms, and operates directly with noisy distorted gradients. We analyze the performance of GBMA theoretically, and prove that it can approach the convergence rate of the centralized gradient descent (GD) algorithm in large networks. Specifically, we establish a finite-sample bound of the error for both convex and strongly convex loss functions with Lipschitz gradient. Furthermore, we provide energy scaling laws for approaching the centralized convergence rate as the number of nodes increases. Finally, experimental results support the theoretical findings, and demonstrate strong performance of GBMA using synthetic and real data.
We consider a distributed learning problem in a wireless network, consisting of N distributed edge devices and a parameter server (PS). The objective function is a sum of the edge devices' local loss functions, who aim to train a shared model by communicating with the PS over multiple access channels (MAC). This problem has attracted a growing interest in distributed sensing systems, and more recently in federated learning, known as over-the-air computation. In this paper, we develop a novel Accelerated Gradient-descent Multiple Access (AGMA) algorithm that uses momentum-based gradient signals…
Imperial College London（伦敦帝国理工学院）；Bilkent University（比尔肯特大学）
We study wireless collaborative machine learning (ML), where mobile edge devices, each with its own dataset, carry out distributed stochastic gradient descent (DSGD) over-the-air with the help of a wireless access point acting as the parameter server (PS). At each iteration of the DSGD algorithm wireless devices compute gradient estimates with their local datasets, and send them to the PS over a wireless fading multiple access channel (MAC). Motivated by the additive nature of the wireless MAC, we propose an analog DSGD scheme, in which the devices transmit scaled versions of their gradient es…
Deep neural networks are often trained in the over-parametrized regime (i.e. with far more parameters than training examples), and understanding why the training converges to solutions that generalize remains an open problem. Several studies have highlighted the fact that the training procedure, i.e. mini-batch Stochastic Gradient Descent (SGD) leads to solutions that have specific properties in the loss landscape. However, even with plain Gradient Descent (GD) the solutions found in the over-parametrized regime are pretty good and this phenomenon is poorly understood. We propose an analysis o…
Isomorphic Functionalities between Ant Colony and Ensemble Learning: Part III -- Gradient Descent, Neural Plasticity, and the Emergence of Deep Intelligence
School of Mathematics and Statistics, Rochester Institute of Technology（罗彻斯特理工学院数学与统计学院）；Gosnell School of Life Sciences, Rochester Institute of Technology（罗彻斯特理工学院戈斯内尔生命科学学院）；Center for Imaging Science, Rochester Institute of Technology（罗彻斯特理工学院成像科学中心）
Tokyo University of Science（东京理科大学）；Tsuda University（津田塾大学）
This paper proposes a communication strategy for decentralized learning on wireless systems. Our discussion is based on the decentralized parallel stochastic gradient descent (D-PSGD), which is one of the state-of-the-art algorithms for decentralized learning. The main contribution of this paper is to raise a novel open question for decentralized learning on wireless systems: there is a possibility that the density of a network topology significantly influences the runtime performance of D-PSGD. In general, it is difficult to guarantee delay-free communications without any communication deteri…
Boston University（波士顿大学）；Northeastern University（东北大学）
In this work, we describe a generic approach to show convergence with high probability for stochastic convex optimization. In previous works, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution as opposed to the domain diameter. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations.
For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable and the number of cutting planes invoked by traditional remedies is difficult to bound, leading to convergences guarantees that are sublinear relative to the cumulative number of gradient evaluations. We instead propose a multipoint generalization of the gradient descent iteration for local op…
Generalized Gradient Descent is a Hypergraph Functor
Department of Computer and Information Science and Engineering（计算机与信息科学与工程系）；University of Florida（佛罗里达大学）；Information Directorate（信息总局）；Air Force Research Lab（空军研究实验室）
Institute of Informatics, University of Warsaw（华沙大学信息学研究所）
We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of…
University of Cambridge（剑桥大学）；Harvard University（哈佛大学）
Finding parameters that minimise a loss function is at the core of many machine learning methods. The Stochastic Gradient Descent algorithm is widely used and delivers state of the art results for many problems. Nonetheless, Stochastic Gradient Descent typically cannot find the global minimum, thus its empirical effectiveness is hitherto mysterious. We derive a correspondence between parameter inference and free energy minimisation in statistical physics. The degree of undersampling plays the role of temperature. Analogous to the energy-entropy competition in statistical physics, wide but shal…
The (stochastic) gradient descent and the multiplicative update method are probably the most popular algorithms in machine learning. We introduce and study a new regularization which provides a unification of the additive and multiplicative updates. This regularization is derived from an hyperbolic analogue of the entropy function, which we call hypentropy. It is motivated by a natural extension of the multiplicative update to negative numbers. The hypentropy has a natural spectral counterpart which we use to derive a family of matrix-based updates that bridge gradient methods and the multipli…
Redwood Center for Theoretical Neuroscience（雷德伍德理论神经科学中心）；University of California at Berkeley（加州大学伯克利分校）
The natural gradient allows for more efficient gradient descent by removing dependencies and biases inherent in a function's parameterization. Several papers present the topic thoroughly and precisely. It remains a very difficult idea to get your head around however. The intent of this note is to provide simple intuition for the natural gradient and its use. We review how an ill conditioned parameter space can undermine learning, introduce the natural gradient by analogy to the more widely understood concept of signal whitening, and present tricks and specific prescriptions for applying the na…
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）
We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be exactly equivalent to continuous-time mirror descent by Amid and Warmuth 2020, but theory f…
Consider the problem of minimizing functions that are Lipschitz and strongly convex, but not necessarily differentiable. We prove that after $T$ steps of stochastic gradient descent, the error of the final iterate is $O(\log(T)/T)$ with high probability. We also construct a function from this class for which the error of the final iterate of deterministic gradient descent is $Ω(\log(T)/T)$. This shows that the upper bound is tight and that, in this setting, the last iterate of stochastic gradient descent has the same general error rate (with high probability) as deterministic gradient descent.…
We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this norm, the unit ball in the space for $L$-layer networks has low Rademacher complexity and thus favorable generalization properties. Functions in these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates. The key to this work is a new way of representing functions in some form of expectations, motivated by mult…
Convergence of the gradient descent algorithm has been attracting renewed interest due to its utility in deep learning applications. Even as multiple variants of gradient descent were proposed, the assumption that the gradient of the objective is Lipschitz continuous remained an integral part of the analysis until recently. In this work, we look at convergence analysis by focusing on a property that we term as concavifiability, instead of Lipschitz continuity of gradients. We show that concavifiability is a necessary and sufficient condition to satisfy the upper quadratic approximation which i…
Much of studies on neural computation are based on network models of static neurons that produce analog output, despite the fact that information processing in the brain is predominantly carried out by dynamic neurons that produce discrete pulses called spikes. Research in spike-based computation has been impeded by the lack of efficient supervised learning algorithm for spiking networks. Here, we present a gradient descent method for optimizing spiking network models by introducing a differentiable formulation of spiking networks and deriving the exact gradient calculation. For demonstration,…
We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.
Gradient descent is a fundamental algorithm in both theory and practice for continuous optimization. Identifying its quantum counterpart would be appealing to both theoretical and practical quantum applications. A conventional approach to quantum speedups in optimization relies on the quantum acceleration of intermediate steps of classical algorithms, while keeping the overall algorithmic trajectory and solution quality unchanged. We propose Quantum Hamiltonian Descent (QHD), which is derived from the path integral of dynamical systems referring to the continuous-time limit of classical gradie…
Predicting fading channels is a classical problem with a vast array of applications, including as an enabler of artificial intelligence (AI)-based proactive resource allocation for cellular networks. Under the assumption that the fading channel follows a stationary complex Gaussian process, as for Rayleigh and Rician fading models, the optimal predictor is linear, and it can be directly computed from the Doppler spectrum via standard linear minimum mean squared error (LMMSE) estimation. However, in practice, the Doppler spectrum is unknown, and the predictor has only access to a limited time s…
When a channel model is available, learning how to communicate on fading noisy channels can be formulated as the (unsupervised) training of an autoencoder consisting of the cascade of encoder, channel, and decoder. An important limitation of the approach is that training should be generally carried out from scratch for each new channel. To cope with this problem, prior works considered joint training over multiple channels with the aim of finding a single pair of encoder and decoder that works well on a class of channels. As a result, joint training ideally mimics the operation of non-coherent…
Princeton University（普林斯顿大学）；Imperial College London（伦敦帝国理工学院）
We study federated machine learning at the wireless network edge, where limited power wireless devices, each with its own dataset, build a joint model with the help of a remote parameter server (PS). We consider a bandwidth-limited fading multiple access channel (MAC) from the wireless devices to the PS, and propose various techniques to implement distributed stochastic gradient descent (DSGD). We first propose a digital DSGD (D-DSGD) scheme, in which one device is selected opportunistically for transmission at each iteration based on the channel conditions; the scheduled device quantizes its …
In this paper, we consider federated learning (FL) over a noisy fading multiple access channel (MAC), where an edge server aggregates the local models transmitted by multiple end devices through over-the-air computation (AirComp). To realize efficient analog federated learning over wireless channels, we propose an AirComp-based FedSplit algorithm, where a threshold-based device selection scheme is adopted to achieve reliable local model uploading. In particular, we analyze the performance of the proposed algorithm and prove that the proposed algorithm linearly converges to the optimal solution…
Analog Physical Systems Can Exhibit Double Descent
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA（物理学与天文学系，宾夕法尼亚大学，费城，PA，USA）；Dayhoff Labs, Inc. Cambridge, MA, USA（Dayhoff实验室公司，剑桥，MA，USA）；Santa Fe Institute, Santa Fe, NM, USA（圣菲研究所，圣菲，NM，USA）
In this paper, we consider the problem of wireless federated learning based on sign stochastic gradient descent (signSGD) algorithm via a multiple access channel. When sending locally computed gradient's sign information, each mobile device requires to apply precoding to circumvent wireless fading effects. In practice, however, acquiring perfect knowledge of channel state information (CSI) at all mobile devices is infeasible. In this paper, we present a simple yet effective precoding method with limited channel knowledge, called sign-alignment precoding. The idea of sign-alignment precoding is…
