In the neural network tutorial, I introduced the gradient descent algorithm which is used to train the weights in an artificial neural network. In reality, for deep learning and big data tasks standard gradient descent is not often used. Rather, a variant of gradient descent called stochastic gradient descent and in particular its cousin mini-batch gradient descent is used. That is the focus of this post. The gradient descent optimisation algorithm aims to minimise some cost/loss function based on that function's gradient.

In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent. We prove global nonassymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates (in L2 sense), and dual function values. We also show that the primal iterates converge at an accelerated linear rate in the L1 sense. This is the first time a linear rate is shown for the stochastic heavy ball method (i.e., stochastic gradient descent method with momentum). Under somewhat weaker conditions, we establish a sublinear convergence rate for Cesaro averages of primal iterates. Moreover, we propose a novel concept, which we call stochastic momentum, aimed at decreasing the cost of performing the momentum step. We prove linear convergence of several stochastic methods with stochastic momentum, and show that in some sparse data regimes and for sufficiently small momentum parameters, these methods enjoy better overall complexity than methods with deterministic momentum. Finally, we perform extensive numerical testing on artificial and real datasets, including data coming from average consensus problems.

Data parallelism is a popular technique used to speed up training on large mini-batches when each mini-batch is too large to fit on a GPU. Under data parallelism, a mini-batch is split up into smaller sized batches that are small enough to fit on the memory available on different GPUs on the network. Each GPU holds an identical copy of the network parameters and runs the forward and backward pass. At the end of the backward pass, each GPU sends the computed gradients to a parameter server. The parameter server aggregates the gradients and computes the updates to the network parameters using some variant of Stochastic Gradient Descent.

Stochastic Gradient Descent (SGD) is a very powerful technique, currently employed to optimize all deep learning models. However, the vanilla algorithm has many limitations, in particular when the system is ill-conditioned and could never find the global minimum. In this post, we're going to analyze how it works and the most important variations that can speed up the convergence in deep models. First of all, it's necessary to standardize the naming. In some books, the expression "Stochastic Gradient Descent" refers to an algorithm which operates on a batch size equal to 1, while "Mini-batch Gradient Descent" is adopted when the batch size is greater than 1.

Significant success has been realized recently on applying machine learning to real-world applications. There have also been corresponding concerns on the privacy of training data, which relates to data security and confidentiality issues. Differential privacy provides a principled and rigorous privacy guarantee on machine learning models. While it is common to design a model satisfying a required differential-privacy property by injecting noise, it is generally hard to balance the trade-off between privacy and utility. We show that stochastic gradient Markov chain Monte Carlo (SG-MCMC) -- a class of scalable Bayesian posterior sampling algorithms proposed recently -- satisfies strong differential privacy with carefully chosen step sizes. We develop theory on the performance of the proposed differentially-private SG-MCMC method. We conduct experiments to support our analysis and show that a standard SG-MCMC sampler without any modification (under a default setting) can reach state-of-the-art performance in terms of both privacy and utility on Bayesian learning.

In my previous post I outlined how machine learning works by demonstrating the central role that cost functions and gradient descent play in the learning process. This post builds on these concepts by exploring how neural networks and deep learning work. This post is light on explanation and heavy on code. The reason for this is that I cannot think of any way to elucidate the internal workings of a neural network more clearly that the incredible videos put together by three blue one brown -- see the full playlist here. These videos show how neural networks can be fed raw data -- such as images of digits -- and can output labels for these images with amazing accuracy.

We introduce a simple algorithm, True Asymptotic Natural Gradient Optimization (TANGO), that converges to a true natural gradient descent in the limit of small learning rates, without explicit Fisher matrix estimation. For quadratic models the algorithm is also an instance of averaged stochastic gradient, where the parameter is a moving average of a "fast", constant-rate gradient descent. TANGO appears as a particular de-linearization of averaged SGD, and is sometimes quite different on non-quadratic models. This further connects averaged SGD and natural gradient, both of which are arguably optimal asymptotically. In large dimension, small learning rates will be required to approximate the natural gradient well. Still, this shows it is possible to get arbitrarily close to exact natural gradient descent with a lightweight algorithm.

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

Modern statistical inference tasks often require iterative optimization methods to approximate the solution. Convergence analysis from optimization only tells us how well we are approximating the solution deterministically, but overlooks the sampling nature of the data. However, due to the randomness in the data, statisticians are keen to provide uncertainty quantification, or confidence, for the answer obtained after certain steps of optimization. Therefore, it is important yet challenging to understand the sampling distribution of the iterative optimization methods. This paper makes some progress along this direction by introducing a new stochastic optimization method for statistical inference, the moment adjusted stochastic gradient descent. We establish non-asymptotic theory that characterizes the statistical distribution of the iterative methods, with good optimization guarantee. On the statistical front, the theory allows for model misspecification, with very mild conditions on the data. For optimization, the theory is flexible for both the convex and non-convex cases. Remarkably, the moment adjusting idea motivated from "error standardization" in statistics achieves similar effect as Nesterov's acceleration in optimization, for certain convex problems as in fitting generalized linear models. We also demonstrate this acceleration effect in the non-convex setting through experiments.

Two major momentum-based techniques that have achieved tremendous success in optimization are Polyak's heavy ball method and Nesterov's accelerated gradient. A crucial step in all momentum-based methods is the choice of the momentum parameter $m$ which is always suggested to be set to less than $1$. Although the choice of $m < 1$ is justified only under very strong theoretical assumptions, it works well in practice even when the assumptions do not necessarily hold. In this paper, we propose a new momentum based method $\textit{ADINE}$, which relaxes the constraint of $m < 1$ and allows the learning algorithm to use adaptive higher momentum. We motivate our hypothesis on $m$ by experimentally verifying that a higher momentum ($\ge 1$) can help escape saddles much faster. Using this motivation, we propose our method $\textit{ADINE}$ that helps weigh the previous updates more (by setting the momentum parameter $> 1$), evaluate our proposed algorithm on deep neural networks and show that $\textit{ADINE}$ helps the learning algorithm to converge much faster without compromising on the generalization error.