This monograph presents the main complexity theorems in convex optimization and their corresponding algorithms. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by Nesterov’s seminal book and Nemirovski’s lecture n...

Randomized algorithms that base iteration-level decisions on samples from some pool are ubiquitous in machine learning and optimization. Examples include stochastic gradient descent and randomized coordinate descent. This paper makes progress at theoretically evaluating the difference in performance between sampling withand without-replacement in such algorithms. Focusing on least means squares...

We revisit the choice of SGD for training deep neural networks by reconsidering the appropriate geometry in which to optimize the weights. We argue for a geometry invariant to rescaling of weights that does not affect the output of the network, and suggest Path-SGD, which is an approximate steepest descent method with respect to a path-wise regularizer related to max-norm regularization. Path-S...

The method of conjugate gradients provides a very effective way to optimize large, deterministic systems by gradient descent. In its standard form, however, it is not amenable to stochastic approximation of the gradient. Here we explore a number of ways to adopt ideas from conjugate gradient in the stochastic setting, using fast Hessian-vector products to obtain curvature information cheaply. I...

Recent work shows that decentralized parallel stochastic gradient decent (D-PSGD) can outperform its centralized counterpart both theoretically and practically. While asynchronous parallelism is a powerful technology to improve the efficiency of parallelism in distributed machine learning platforms and has been widely used in many popular machine learning softwares and solvers based on centrali...

In 1963, Polyak proposed a simple condition that is sufficient to show a global linear convergence rate for gradient descent. This condition is a special case of the Lojasiewicz inequality proposed in the same year, and it does not require strong convexity (or even convexity). In this work, we show that this much-older PolyakLojasiewicz (PL) inequality is actually weaker than the main condition...

In this paper, we consider a general stochastic optimization problem which is often at the core of supervised learning, such as deep learning and linear classification. We consider a standard stochastic gradient descent (SGD) method with a fixed, large step size and propose a novel assumption on the objective function, under which this method has the improved convergence rates (to a neighborhoo...

Stochastic gradient descent (SGD) is an immensely popular approach for online learningin settings where data arrives in a stream or data sizes are very large. However, despite anever-increasing volume of work on SGD, much less is known about the statistical inferentialproperties of SGD-based predictions. Taking a fully inferential viewpoint, this paper introducesa novel proc...

Recently, we have demonstrated a new and efficient method to simultaneously reconstruct two unknown interfering wavefronts. A three-dimensional interference pattern was analyzed and then Zernike polynomials and the stochastic parallel gradient descent algorithm were used to expand and calculate wavefronts. In this paper, as one of the applications of this method, the reflected wavefronts from t...

Big Data problems in Machine Learning have large number of data points or large number of features, or both, which make training of models difficult because of high computational complexities of single iteration of learning algorithms. To solve such learning problems, Stochastic Approximation offers an optimization approach to make complexity of each iteration independent of number of data poin...