The optimization problem behind neural networks is highly non-convex. Training with stochastic gradient descent and variants requires careful parameter tuning and provides no guarantee to achieve the global optimum. In contrast we show under quite weak assumptions on the data that a particular class of feedforward neural networks can be trained globally optimal with a linear convergence rate. Up to our knowledge this is the first practically feasible method which achieves such a guarantee. While the method can in principle be applied to deep networks, we restrict ourselves for simplicity in this paper to one- and two hidden layer networks. Our experiments confirms that these models are already rich enough to achieve good performance on a series of real-world datasets.

The amount of data available in the world is growing faster than our ability to deal with it. However, if we take advantage of the internal structure, data may become much smaller for machine learning purposes. In this paper we focus on one of the fundamental machine learning tasks, empirical risk minimization (ERM), and provide faster algorithms with the help from the clustering structure of the data. We introduce a simple notion of raw clustering that can be efficiently computed from the data, and propose two algorithms based on clustering information. Our accelerated algorithm ClusterACDM is built on a novel Haar transformation applied to the dual space of the ERM problem, and our variance-reduction based algorithm ClusterSVRG introduces a new gradient estimator using clustering. Our algorithms outperform their classical counterparts ACDM and SVRG respectively.

Stochastic gradient-based Monte Carlo methods such as stochastic gradient Langevin dynamics are useful tools for posterior inference on large scale datasets in many machine learning applications. These methods scale to large datasets by using noisy gradients calculated using a mini-batch or subset of the dataset. However, the high variance inherent in these noisy gradients degrades performance and leads to slower mixing. In this paper, we present techniques for reducing variance in stochastic gradient Langevin dynamics, yielding novel stochastic Monte Carlo methods that improve performance by reducing the variance in the stochastic gradient. We show that our proposed method has better theoretical guarantees on convergence rate than stochastic Langevin dynamics. This is complemented by impressive empirical results obtained on a variety of real world datasets, and on four different machine learning tasks (regression, classification, independent component analysis and mixture modeling). These theoretical and empirical contributions combine to make a compelling case for using variance reduction in stochastic Monte Carlo methods.

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tackle this issue, we develop fast stochastic algorithms that provably converge to a stationary point for constant minibatches. Furthermore, using a variant of these algorithms, we obtain provably faster convergence than batch proximal gradient descent. Our results are based on the recent variance reduction techniques for convex optimization but with a novel analysis for handling nonconvex and nonsmooth functions. We also prove global linear convergence rate for an interesting subclass of nonsmooth nonconvex functions, which subsumes several recent works.

We study the stochastic optimization of canonical correlation analysis (CCA), whose objective is nonconvex and does not decouple over training samples. Although several stochastic gradient based optimization algorithms have been recently proposed to solve this problem, no global convergence guarantee was provided by any of them. Inspired by the alternating least squares/power iterations formulation of CCA, and the shift-and-invert preconditioning method for PCA, we propose two globally convergent meta-algorithms for CCA, both of which transform the original problem into sequences of least squares problems that need only be solved approximately. We instantiate the meta-algorithms with state-of-the-art SGD methods and obtain time complexities that significantly improve upon that of previous work. Experimental results demonstrate their superior performance.

Word embeddings are a powerful approach to capturing semantic similarity among terms in a vocabulary. In this paper, we develop exponential family embeddings, which extends the idea of word embeddings to other types of high-dimensional data. As examples, we studied several types of data: neural data with real-valued observations, count data from a market basket analysis, and ratings data from a movie recommendation system. The main idea is that each observation is modeled conditioned on a set of latent embeddings and other observations, called the context, where the way the context is defined depends on the problem. In language the context is the surrounding words; in neuroscience the context is close-by neurons; in market basket data the context is other items in the shopping cart. Each instance of an embedding defines the context, the exponential family of conditional distributions, and how the embedding vectors are shared across data. We infer the embeddings with stochastic gradient descent, with an algorithm that connects closely to generalized linear models. On all three of our applications—neural activity of zebrafish, users’ shopping behavior, and movie ratings—we found that exponential family embedding models are more effective than other dimension reduction methods. They better reconstruct held-out data and find interesting qualitative structure.

Stochastic gradient methods for machine learning and optimization problems are usually analyzed assuming data points are sampled *with* replacement. In contrast, sampling *without* replacement is far less understood, yet in practice it is very common, often easier to implement, and usually performs better. In this paper, we provide competitive convergence guarantees for without-replacement sampling under several scenarios, focusing on the natural regime of few passes over the data. Moreover, we describe a useful application of these results in the context of distributed optimization with randomly-partitioned data, yielding a nearly-optimal algorithm for regularized least squares (in terms of both communication complexity and runtime complexity) under broad parameter regimes. Our proof techniques combine ideas from stochastic optimization, adversarial online learning and transductive learning theory, and can potentially be applied to other stochastic optimization and learning problems.

Although artificial neural networks have shown great promise in applications including computer vision and speech recognition, there remains considerable practical and theoretical difficulty in optimizing their parameters. The seemingly unreasonable success of gradient descent methods in minimizing these non-convex functions remains poorly understood. In this work we offer some theoretical guarantees for networks with piecewise affine activation functions, which have in recent years become the norm. We prove three main results. Firstly, that the network is piecewise convex as a function of the input data. Secondly, that the network, considered as a function of the parameters in a single layer, all others held constant, is again piecewise convex. Finally, that the network as a function of all its parameters is piecewise multi-convex, a generalization of biconvexity. From here we characterize the local minima and stationary points of the training objective, showing that they minimize certain subsets of the parameter space. We then analyze the performance of two optimization algorithms on multi-convex problems: gradient descent, and a method which repeatedly solves a number of convex sub-problems. We prove necessary convergence conditions for the first algorithm and both necessary and sufficient conditions for the second, after introducing regularization to the objective. Under the squared error objective, we show that by varying the training data, a single rectifier neuron admits local minima arbitrarily far apart, both in objective value and parameter space.

In a "purist" implementation of SGD, your mini-batch size would be set to 1. However, we often uses mini-batches that are 1. Typical values include 32, 64, 128, and 256. To start, using batches 1 helps reduce variance in the parameter update, ultimately leading to a more stable convergence. Secondly, optimized matrix operation libraries are often more efficient when the input matrix size is a power of 2. In general, the mini-batch size is not a hyperparameter that you should worry much about. You basically determine how many training examples will fit on your GPU/main memory and then use the nearest power of 2 as the batch size.

Gradient descent is one of the most popular algorithms to perform optimization and by far the most common way to optimize neural networks. At the same time, every state-of-the-art Deep Learning library contains implementations of various algorithms to optimize gradient descent (e.g. These algorithms, however, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by. This blog post aims at providing you with intuitions towards the behaviour of different algorithms for optimizing gradient descent that will help you put them to use. We are first going to look at the different variants of gradient descent.