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Register a new userAccelerating SGD with momentum for over-parameterized learning
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Nesterov SGD is widely used for training modern neural networks and other machine learning models. Yet, its advantages over SGD have not been theoretically clarified. Indeed, as we show in our paper, both theoretically and empirically, Nesterov SGD with any parameter selection does not in general provide acceleration over ordinary SGD. Furthermore, Nesterov SGD may diverge for step sizes that ensure convergence of ordinary SGD. This is in contrast to the classical results in the deterministic scenario, where the same step size ensures accelerated convergence of the Nesterovs method over optimal gradient descent. To address the non-acceleration issue, we introduce a compensation term to Nesterov SGD. The resulting algorithm, which we call MaSS, converges for same step sizes as SGD. We prove that MaSS obtains an accelerated convergence rates over SGD for any mini-batch size in the linear setting. For full batch, the convergence rate of MaSS matches the well-known accelerated rate of the Nesterovs method. We also analyze the practically important question of the dependence of the convergence rate and optimal hyper-parameters on the mini-batch size, demonstrating three distinct regimes: linear scaling, diminishing returns and saturation. Experimental evaluation of MaSS for several standard architectures of deep networks, including ResNet and convolutional networks, shows improved performance over SGD, Nesterov SGD and Adam.
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We consider training over-parameterized two-layer neural networks with Rectified Linear Unit (ReLU) using gradient descent (GD) method. Inspired by a recent line of work, we study the evolutions of network prediction errors across GD iterations, which can be neatly described in a matrix form. When the network is sufficiently over-parameterized, these matrices individually approximate {em an} integral operator which is determined by the feature vector distribution $rho$ only. Consequently, GD method can be viewed as {em approximately} applying the powers of this integral operator on the underlying/target function $f^*$ that generates the responses/labels. We show that if $f^*$ admits a low-rank approximation with respect to the eigenspaces of this integral operator, then the empirical risk decreases to this low-rank approximation error at a linear rate which is determined by $f^*$ and $rho$ only, i.e., the rate is independent of the sample size $n$. Furthermore, if $f^*$ has zero low-rank approximation error, then, as long as the width of the neural network is $Omega(nlog n)$, the empirical risk decreases to $Theta(1/sqrt{n})$. To the best of our knowledge, this is the first result showing the sufficiency of nearly-linear network over-parameterization. We provide an application of our general results to the setting where $rho$ is the uniform distribution on the spheres and $f^*$ is a polynomial. Throughout this paper, we consider the scenario where the input dimension $d$ is fixed.
Federated learning (FL) has emerged as a prominent distributed learning paradigm. FL entails some pressing needs for developing novel parameter estimation approaches with theoretical guarantees of convergence, which are also communication efficient, differentially private and Byzantine resilient in the heterogeneous data distribution settings. Quantization-based SGD solvers have been widely adopted in FL and the recently proposed SIGNSGD with majority vote shows a promising direction. However, no existing methods enjoy all the aforementioned properties. In this paper, we propose an intuitively-simple yet theoretically-sound method based on SIGNSGD to bridge the gap. We present Stochastic-Sign SGD which utilizes novel stochastic-sign based gradient compressors enabling the aforementioned properties in a unified framework. We also present an error-feedback variant of the proposed Stochastic-Sign SGD which further improves the learning performance in FL. We test the proposed method with extensive experiments using deep neural networks on the MNIST dataset and the CIFAR-10 dataset. The experimental results corroborate the effectiveness of the proposed method.
Communication overhead hinders the scalability of large-scale distributed training. Gossip SGD, where each node averages only with its neighbors, is more communication-efficient than the prevalent parallel SGD. However, its convergence rate is reversely proportional to quantity $1-beta$ which measures the network connectivity. On large and sparse networks where $1-beta to 0$, Gossip SGD requires more iterations to converge, which offsets against its communication benefit. This paper introduces Gossip-PGA, which adds Periodic Global Averaging into Gossip SGD. Its transient stage, i.e., the iterations required to reach asymptotic linear speedup stage, improves from $Omega(beta^4 n^3/(1-beta)^4)$ to $Omega(beta^4 n^3 H^4)$ for non-convex problems. The influence of network topology in Gossip-PGA can be controlled by the averaging period $H$. Its transient-stage complexity is also superior to Local SGD which has order $Omega(n^3 H^4)$. Empirical results of large-scale training on image classification (ResNet50) and language modeling (BERT) validate our theoretical findings.
While over-parameterization is widely believed to be crucial for the success of optimization for the neural networks, most existing theories on over-parameterization do not fully explain the reason -- they either work in the Neural Tangent Kernel regime where neurons dont move much, or require an enormous number of neurons. In practice, when the data is generated using a teacher neural network, even mildly over-parameterized neural networks can achieve 0 loss and recover the directions of teacher neurons. In this paper we develop a local convergence theory for mildly over-parameterized two-layer neural net. We show that as long as the loss is already lower than a threshold (polynomial in relevant parameters), all student neurons in an over-parameterized two-layer neural network will converge to one of teacher neurons, and the loss will go to 0. Our result holds for any number of student neurons as long as it is at least as large as the number of teacher neurons, and our convergence rate is independent of the number of student neurons. A key component of our analysis is the new characterization of local optimization landscape -- we show the gradient satisfies a special case of Lojasiewicz property which is different from local strong convexity or PL conditions used in previous work.
Adversarial training is a popular method to give neural nets robustness against adversarial perturbations. In practice adversarial training leads to low robust training loss. However, a rigorous explanation for why this happens under natural conditions is still missing. Recently a convergence theory for standard (non-adversarial) supervised training was developed by various groups for {em very overparametrized} nets. It is unclear how to extend these results to adversarial training because of the min-max objective. Recently, a first step towards this direction was made by Gao et al. using tools from online learning, but they require the width of the net to be emph{exponential} in input dimension $d$, and with an unnatural activation function. Our work proves convergence to low robust training loss for emph{polynomial} width instead of exponential, under natural assumptions and with the ReLU activation. Key element of our proof is showing that ReLU networks near initialization can approximate the step function, which may be of independent interest.
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