亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

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.

In optical fiber communication, due to the random variation of the environment, the state of polarization (SOP) fluctuates randomly with time leading to distortion and performance degradation. The memory-less SOP fluctuations can be regarded as a two-by-two random unitary matrix. In this paper, for what we believe to be the first time, the capacity of the polarization drift channel under an average power constraint with imperfect channel knowledge is characterized. An achievable information rate (AIR) is derived when imperfect channel knowledge is available and is shown to be highly dependent on the channel estimation technique. It is also shown that a tighter lower bound can be achieved when a unitary estimation of the channel is available. However, the conventional estimation algorithms do not guarantee a unitary channel estimation. Therefore, by considering the unitary constraint of the channel, a data-aided channel estimator based on the Kabsch algorithm is proposed, and its performance is numerically evaluated in terms of AIR. Monte Carlo simulations show that Kabsch outperforms the least-square error algorithm. In particular, with complex, Gaussian inputs and eight pilot symbols per block, Kabsch improves the AIR by 0:2 to 0:35 bits/symbol throughout the range of studied signal-to-noise ratios.

Stochastic gradient methods (SGMs) are predominant approaches for solving stochastic optimization. On smooth nonconvex problems, a few acceleration techniques have been applied to improve the convergence rate of SGMs. However, little exploration has been made on applying a certain acceleration technique to a stochastic subgradient method (SsGM) for nonsmooth nonconvex problems. In addition, few efforts have been made to analyze an (accelerated) SsGM with delayed derivatives. The information delay naturally happens in a distributed system, where computing workers do not coordinate with each other. In this paper, we propose an inertial proximal SsGM for solving nonsmooth nonconvex stochastic optimization problems. The proposed method can have guaranteed convergence even with delayed derivative information in a distributed environment. Convergence rate results are established to three classes of nonconvex problems: weakly-convex nonsmooth problems with a convex regularizer, composite nonconvex problems with a nonsmooth convex regularizer, and smooth nonconvex problems. For each problem class, the convergence rate is $O(1/K^{\frac{1}{2}})$ in the expected value of the gradient norm square, for $K$ iterations. In a distributed environment, the convergence rate of the proposed method will be slowed down by the information delay. Nevertheless, the slow-down effect will decay with the number of iterations for the latter two problem classes. We test the proposed method on three applications. The numerical results clearly demonstrate the advantages of using the inertial-based acceleration. Furthermore, we observe higher parallelization speed-up in asynchronous updates over the synchronous counterpart, though the former uses delayed derivatives. Our source code is released at //github.com/RPI-OPT/Inertial-SsGM

We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. To address this, we propose modifications and new ideas, with an emphasis on requiring the constraints are satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null space basis matrix for a "wide" matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive when solving a sequence of related problems. Numerical experiments using problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches.

Stochastic gradient descent with backpropagation is the workhorse of artificial neural networks. It has long been recognized that backpropagation fails to be a biologically plausible algorithm. Fundamentally, it is a non-local procedure -- updating one neuron's synaptic weights requires knowledge of synaptic weights or receptive fields of downstream neurons. This limits the use of artificial neural networks as a tool for understanding the biological principles of information processing in the brain. Lillicrap et al. (2016) propose a more biologically plausible "feedback alignment" algorithm that uses random and fixed backpropagation weights, and show promising simulations. In this paper we study the mathematical properties of the feedback alignment procedure by analyzing convergence and alignment for two-layer networks under squared error loss. In the overparameterized setting, we prove that the error converges to zero exponentially fast, and also that regularization is necessary in order for the parameters to become aligned with the random backpropagation weights. Simulations are given that are consistent with this analysis and suggest further generalizations. These results contribute to our understanding of how biologically plausible algorithms might carry out weight learning in a manner different from Hebbian learning, with performance that is comparable with the full non-local backpropagation algorithm.

We present and analyze a momentum-based gradient method for training linear classifiers with an exponentially-tailed loss (e.g., the exponential or logistic loss), which maximizes the classification margin on separable data at a rate of $\widetilde{\mathcal{O}}(1/t^2)$. This contrasts with a rate of $\mathcal{O}(1/\log(t))$ for standard gradient descent, and $\mathcal{O}(1/t)$ for normalized gradient descent. This momentum-based method is derived via the convex dual of the maximum-margin problem, and specifically by applying Nesterov acceleration to this dual, which manages to result in a simple and intuitive method in the primal. This dual view can also be used to derive a stochastic variant, which performs adaptive non-uniform sampling via the dual variables.

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.

Self-training algorithms, which train a model to fit pseudolabels predicted by another previously-learned model, have been very successful for learning with unlabeled data using neural networks. However, the current theoretical understanding of self-training only applies to linear models. This work provides a unified theoretical analysis of self-training with deep networks for semi-supervised learning, unsupervised domain adaptation, and unsupervised learning. At the core of our analysis is a simple but realistic ``expansion'' assumption, which states that a low-probability subset of the data must expand to a neighborhood with large probability relative to the subset. We also assume that neighborhoods of examples in different classes have minimal overlap. We prove that under these assumptions, the minimizers of population objectives based on self-training and input-consistency regularization will achieve high accuracy with respect to ground-truth labels. By using off-the-shelf generalization bounds, we immediately convert this result to sample complexity guarantees for neural nets that are polynomial in the margin and Lipschitzness. Our results help explain the empirical successes of recently proposed self-training algorithms which use input consistency regularization.

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.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.