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1. Deep Policy Gradient for Reactive Power Control in Distribution Systems

   Q. Yang, A. Sadeghi (January 2020, Proceedings of IEEE Smartgridcom Conference)

   null (Ed.)

   Pronounced variability due to the growth of renewable energy sources, flexible loads, and distributed generation is challenging residential distribution systems. This context, motivates well fast, efficient, and robust reactive power control. Optimal reactive power control is possible in theory by solving a non-convex optimization problem based on the exact model of distribution flow. However, lack of high-precision instrumentation and reliable communications, as well as the heavy computational burden of non-convex optimization solvers render computing and implementing the optimal control challenging in practice. Taking a statistical learning viewpoint, the input-output relationship between each grid state and the corresponding optimal reactive power control (a.k.a., policy) is parameterized in the present work by a deep neural network, whose unknown weights are updated by minimizing the accumulated power loss over a number of historical and simulated training pairs, using the policy gradient method. In the inference phase, one just feeds the real-time state vector into the learned neural network to obtain the ‘optimal’ reactive power control decision with only several matrix-vector multiplications. The merits of this novel deep policy gradient approach include its computational efficiency as well as robustness to random input perturbations. Numerical tests on a 47-bus distribution network using real solar and consumption data corroborate these practical merits. 

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2. A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks

   Simsekli, Umut; Sagun, Levent; Gurbuzbalaban, Mert (June 2019, Proceedings of Machine Learning Research)

   The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the classical central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a more general context and invoke the generalized CLT (GCLT), which suggests that the GN converges to a heavy-tailed -stable random variable. Accordingly, we propose to analyze SGD as an SDE driven by a Lévy motion. Such SDEs can incur ‘jumps’, which force the SDE transition from narrow minima to wider minima, as proven by existing metastability theory. To validate the -stable assumption, we conduct extensive experiments on common deep learning architectures and show that in all settings, the GN is highly non-Gaussian and admits heavy-tails. We further investigate the tail behavior in varying network architectures and sizes, loss functions, and datasets. Our results open up a different perspective and shed more light on the belief that SGD prefers wide minima. 

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3. Beyond Point Prediction: Capturing Zero-Inflated & Heavy-Tailed Spatiotemporal Data with Deep Extreme Mixture Models

   https://doi.org/10.1145/3534678.3539464  

   Wilson, Tyler; McDonald, Andrew; Galib, Asadullah Hill; Tan, Pang-Ning; Luo, Lifeng (August 2022, KDD '22: Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining)

   Zhang, Aidong; Rangwala, Huzefa (Ed.)

   Zero-inflated, heavy-tailed spatiotemporal data is common across science and engineering, from climate science to meteorology and seismology. A central modeling objective in such settings is to forecast the intensity, frequency, and timing of extreme and non-extreme events; yet in the context of deep learning, this objective presents several key challenges. First, a deep learning framework applied to such data must unify a mixture of distributions characterizing the zero events, moderate events, and extreme events. Second, the framework must be capable of enforcing parameter constraints across each component of the mixture distribution. Finally, the framework must be flexible enough to accommodate for any changes in the threshold used to define an extreme event after training. To address these challenges, we propose Deep Extreme Mixture Model (DEMM), fusing a deep learning-based hurdle model with extreme value theory to enable point and distribution prediction of zero-inflated, heavy-tailed spatiotemporal variables. The framework enables users to dynamically set a threshold for defining extreme events at inference-time without the need for retraining. We present an extensive experimental analysis applying DEMM to precipitation forecasting, and observe significant improvements in point and distribution prediction. All code is available at https://github.com/andrewmcdonald27/DeepExtremeMixtureModel. 

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4. Stochastic algorithms with geometric step decay converge linearly on sharp functions

   Davis, Damek; Drusvyatskiy, Dmitriy; Charisopoulos, Vasileios (September 2023, Mathematical programming)

   Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called geometric step decay and proceeds by halving the step size after every few epochs. In recent work, geometric step decay was shown to improve exponentially upon classical sublinear rates for the class of sharp convex functions. In this work, we ask whether geometric step decay similarly improves stochastic algorithms for the class of sharp weakly convex problems. Such losses feature in modern statistical recovery problems and lead to a new challenge not present in the convex setting: the region of convergence is local, so one must bound the probability of escape. Our main result shows that for a large class of stochastic, sharp, nonsmooth, and nonconvex problems a geometric step decay schedule endows well-known algorithms with a local linear (or nearly linear) rate of convergence to global minimizers. This guarantee applies to the stochastic projected subgradient, proximal point, and prox-linear algorithms. As an application of our main result, we analyze two statistical recovery tasks—phase retrieval and blind deconvolution—and match the best known guarantees under Gaussian measurement models and establish new guarantees under heavy-tailed distributions. 

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5. Robustness to Unbounded Smoothness of Generalized SignSGD

   Crawshaw, Michael; Liu, Mingrui; Orabona, Francesco; Zhang, Wei; Zhuang, Zhenxun (November 2022, Advances in neural information processing systems)

   Oh, Alice H.; Agarwal, Alekh; Belgrave, Danielle; Cho, Kyunghyun (Ed.)

   Traditional analyses in non-convex optimization typically rely on the smoothness assumption, namely requiring the gradients to be Lipschitz. However, recent evidence shows that this smoothness condition does not capture the properties of some deep learning objective functions, including the ones involving Recurrent Neural Networks and LSTMs. Instead, they satisfy a much more relaxed condition, with potentially unbounded smoothness. Under this relaxed assumption, it has been theoretically and empirically shown that the gradient-clipped SGD has an advantage over the vanilla one. In this paper, we show that clipping is not indispensable for Adam-type algorithms in tackling such scenarios: we theoretically prove that a generalized SignSGD algorithm can obtain similar convergence rates as SGD with clipping but does not need explicit clipping at all. This family of algorithms on one end recovers SignSGD and on the other end closely resembles the popular Adam algorithm. Our analysis underlines the critical role that momentum plays in analyzing SignSGD-type and Adam-type algorithms: it not only reduces the effects of noise, thus removing the need for large mini-batch in previous analyses of SignSGD-type algorithms, but it also substantially reduces the effects of unbounded smoothness and gradient norms. To the best of our knowledge, this work is the first one showing the benefit of Adam-type algorithms compared with non-adaptive gradient algorithms such as gradient descent in the unbounded smoothness setting. We also compare these algorithms with popular optimizers on a set of deep learning tasks, observing that we can match the performance of Adam while beating others. 

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