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Recently, Sidford, Wang, Wu and Ye (2018) developed an algorithm combining variance reduction techniques with value iteration to solve discounted Markov decision processes. This algorithm has a sublinear complexity when the discount factor…

Online nonparametric estimators are gaining popularity due to their efficient computation and competitive generalization abilities. An important example includes variants of stochastic gradient descent. These algorithms often take one…

Despite its empirical success, deep learning still lacks a comprehensive theoretical understanding of model fitting and generalization. This paper proposes the probability distribution (PD) learning framework to analyze the optimization and…

Optimization algorithms for solving nonconvex inverse problem have attracted significant interests recently. However, existing methods require the nonconvex regularization to be smooth or simple to ensure convergence. In this paper, we…

Deep neural networks (DNNs) have shown great success in many machine learning tasks. Their training is challenging since the loss surface of the network architecture is generally non-convex, or even non-smooth. How and under what…

High-quality data is crucial for accurate machine learning and actionable analytics, however, mislabeled or noisy data is a common problem in many domains. Distinguishing low- from high-quality data can be challenging, often requiring…

Training a neural network (NN) typically relies on some type of curve-following method, such as gradient descent (GD) (and stochastic gradient descent (SGD)), ADADELTA, ADAM or limited memory algorithms. Convergence for these algorithms…

Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon…

One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies…

The training of machine learning models is typically carried out using some form of gradient descent, often with great success. However, non-asymptotic analyses of first-order optimization algorithms typically employ a gradient smoothness…

Temporal difference (TD) learning is a popular algorithm for policy evaluation in reinforcement learning, but the vanilla TD can substantially suffer from the inherent optimization variance. A variance reduced TD (VRTD) algorithm was…

A commonly used heuristic in non-convex optimization is Normalized Gradient Descent (NGD) - a variant of gradient descent in which only the direction of the gradient is taken into account and its magnitude ignored. We analyze this heuristic…

Using gradient descent (GD) with fixed or decaying step-size is a standard practice in unconstrained optimization problems. However, when the loss function is only locally convex, such a step-size schedule artificially slows GD down as it…

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…

The adaptive moment estimation (Adam) optimizer proposed by Kingma & Ba (2014) is presumably the most popular stochastic gradient descent (SGD) optimization method for the training of deep neural networks (DNNs) in artificial intelligence…

We consider learning methods based on the regularization of a convex empirical risk by a squared Hilbertian norm, a setting that includes linear predictors and non-linear predictors through positive-definite kernels. In order to go beyond…

Given 2D point correspondences between an image pair, inferring the camera motion is a fundamental issue in the computer vision community. The existing works generally set out from the epipolar constraint and estimate the essential matrix,…

Variational data assimilation and machine-learning based super-resolution are two alternative approaches to state estimation in turbulent flows. The former is an optimisation problem featuring a time series of coarse observations, the…

Visual error metrics play a fundamental role in the quantification of perceived image similarity. Most recently, use cases for them in real-time applications have emerged, such as content-adaptive shading and shading reuse to increase…

Among the biggest challenges we face in utilizing neural networks trained on waveform data (i.e., seismic, electromagnetic, or ultrasound) is its application to real data. The requirement for accurate labels forces us to develop solutions…