We study generalization properties of random features (RF) regression in high dimensions optimized by stochastic gradient descent (SGD). In this regime, we derive precise non-asymptotic error bounds of RF regression under both constant and adaptive step-size SGD setting, and observe the double descent phenomenon both theoretically and empirically. Our analysis shows how to cope with multiple randomness sources of initialization, label noise, and data sampling (as well as stochastic gradients) with no closed-form solution, and also goes beyond the commonly-used Gaussian/spherical data assumption. Our theoretical results demonstrate that, with SGD training, RF regression still generalizes well for interpolation learning, and is able to characterize the double descent behavior by the unimodality of variance and monotonic decrease of bias. Besides, we also prove that the constant step-size SGD setting incurs no loss in convergence rate when compared to the exact minimal-norm interpolator, as a theoretical justification of using SGD in practice.
相關內容
Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.
Direct policy search serves as one of the workhorses in modern reinforcement learning (RL), and its applications in continuous control tasks have recently attracted increasing attention. In this work, we investigate the convergence theory of policy gradient (PG) methods for learning the linear risk-sensitive and robust controller. In particular, we develop PG methods that can be implemented in a derivative-free fashion by sampling system trajectories, and establish both global convergence and sample complexity results in the solutions of two fundamental settings in risk-sensitive and robust control: the finite-horizon linear exponential quadratic Gaussian, and the finite-horizon linear-quadratic disturbance attenuation problems. As a by-product, our results also provide the first sample complexity for the global convergence of PG methods on solving zero-sum linear-quadratic dynamic games, a nonconvex-nonconcave minimax optimization problem that serves as a baseline setting in multi-agent reinforcement learning (MARL) with continuous spaces. One feature of our algorithms is that during the learning phase, a certain level of robustness/risk-sensitivity of the controller is preserved, which we termed as the implicit regularization property, and is an essential requirement in safety-critical control systems.
The gradient noise of Stochastic Gradient Descent (SGD) is considered to play a key role in its properties (e.g. escaping low potential points and regularization). Past research has indicated that the covariance of the SGD error done via minibatching plays a critical role in determining its regularization and escape from low potential points. It is however not much explored how much the distribution of the error influences the behavior of the algorithm. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the Multiplicative Stochastic Gradient Descent (M-SGD) algorithm as introduced by Wu et al., which has a much more general noise class than the SGD algorithm done via minibatching. We establish nonasymptotic bounds for the M-SGD algorithm mainly with respect to the Stochastic Differential Equation corresponding to SGD via minibatching. We also show that the M-SGD error is approximately a scaled Gaussian distribution with mean $0$ at any fixed point of the M-SGD algorithm. We also establish bounds for the convergence of the M-SGD algorithm in the strongly convex regime.
Motivated by the problem of online canonical correlation analysis, we propose the \emph{Stochastic Scaled-Gradient Descent} (SSGD) algorithm for minimizing the expectation of a stochastic function over a generic Riemannian manifold. SSGD generalizes the idea of projected stochastic gradient descent and allows the use of scaled stochastic gradients instead of stochastic gradients. In the special case of a spherical constraint, which arises in generalized eigenvector problems, we establish a nonasymptotic finite-sample bound of $\sqrt{1/T}$, and show that this rate is minimax optimal, up to a polylogarithmic factor of relevant parameters. On the asymptotic side, a novel trajectory-averaging argument allows us to achieve local asymptotic normality with a rate that matches that of Ruppert-Polyak-Juditsky averaging. We bring these ideas together in an application to online canonical correlation analysis, deriving, for the first time in the literature, an optimal one-time-scale algorithm with an explicit rate of local asymptotic convergence to normality. Numerical studies of canonical correlation analysis are also provided for synthetic data.
While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
The Transformer is widely used in natural language processing tasks. To train a Transformer however, one usually needs a carefully designed learning rate warm-up stage, which is shown to be crucial to the final performance but will slow down the optimization and bring more hyper-parameter tunings. In this paper, we first study theoretically why the learning rate warm-up stage is essential and show that the location of layer normalization matters. Specifically, we prove with mean field theory that at initialization, for the original-designed Post-LN Transformer, which places the layer normalization between the residual blocks, the expected gradients of the parameters near the output layer are large. Therefore, using a large learning rate on those gradients makes the training unstable. The warm-up stage is practically helpful for avoiding this problem. On the other hand, our theory also shows that if the layer normalization is put inside the residual blocks (recently proposed as Pre-LN Transformer), the gradients are well-behaved at initialization. This motivates us to remove the warm-up stage for the training of Pre-LN Transformers. We show in our experiments that Pre-LN Transformers without the warm-up stage can reach comparable results with baselines while requiring significantly less training time and hyper-parameter tuning on a wide range of applications.
We investigate how the final parameters found by stochastic gradient descent are influenced by over-parameterization. We generate families of models by increasing the number of channels in a base network, and then perform a large hyper-parameter search to study how the test error depends on learning rate, batch size, and network width. We find that the optimal SGD hyper-parameters are determined by a "normalized noise scale," which is a function of the batch size, learning rate, and initialization conditions. In the absence of batch normalization, the optimal normalized noise scale is directly proportional to width. Wider networks, with their higher optimal noise scale, also achieve higher test accuracy. These observations hold for MLPs, ConvNets, and ResNets, and for two different parameterization schemes ("Standard" and "NTK"). We observe a similar trend with batch normalization for ResNets. Surprisingly, since the largest stable learning rate is bounded, the largest batch size consistent with the optimal normalized noise scale decreases as the width increases.
We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activiation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both gradient descent and stochastic gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under mild assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by (stochastic) gradient descent produces a sequence of iterates that stay inside a small perturbation region centering around the initial weights, in which the empirical loss function of deep ReLU networks enjoys nice local curvature properties that ensure the global convergence of (stochastic) gradient descent. Our theoretical results shed light on understanding the optimization of deep learning, and pave the way to study the optimization dynamics of training modern deep neural networks.
Asynchronous distributed machine learning solutions have proven very effective so far, but always assuming perfectly functioning workers. In practice, some of the workers can however exhibit Byzantine behavior, caused by hardware failures, software bugs, corrupt data, or even malicious attacks. We introduce \emph{Kardam}, the first distributed asynchronous stochastic gradient descent (SGD) algorithm that copes with Byzantine workers. Kardam consists of two complementary components: a filtering and a dampening component. The first is scalar-based and ensures resilience against $\frac{1}{3}$ Byzantine workers. Essentially, this filter leverages the Lipschitzness of cost functions and acts as a self-stabilizer against Byzantine workers that would attempt to corrupt the progress of SGD. The dampening component bounds the convergence rate by adjusting to stale information through a generic gradient weighting scheme. We prove that Kardam guarantees almost sure convergence in the presence of asynchrony and Byzantine behavior, and we derive its convergence rate. We evaluate Kardam on the CIFAR-100 and EMNIST datasets and measure its overhead with respect to non Byzantine-resilient solutions. We empirically show that Kardam does not introduce additional noise to the learning procedure but does induce a slowdown (the cost of Byzantine resilience) that we both theoretically and empirically show to be less than $f/n$, where $f$ is the number of Byzantine failures tolerated and $n$ the total number of workers. Interestingly, we also empirically observe that the dampening component is interesting in its own right for it enables to build an SGD algorithm that outperforms alternative staleness-aware asynchronous competitors in environments with honest workers.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191