In artificial neural networks, learning from data is a computationally demanding task in which a large number of connection weights are iteratively tuned through stochastic-gradient-based heuristic processes over a cost-function. It is not well understood how learning occurs in these systems, in particular how they avoid getting trapped in configurations with poor computational performance. Here we study the difficult case of networks with discrete weights, where the optimization landscape is very rough even for simple architectures, and provide theoretical and numerical evidence of the existence of rare - but extremely dense and accessible - regions of configurations in the network weight space. We define a novel measure, which we call the "robust ensemble" (RE), which suppresses trapping by isolated configurations and amplifies the role of these dense regions. We analytically compute the RE in some exactly solvable models, and also provide a general algorithmic scheme which is straightforward to implement: define a cost-function given by a sum of a finite number of replicas of the original cost-function, with a constraint centering the replicas around a driving assignment. To illustrate this, we derive several powerful new algorithms, ranging from Markov Chains to message passing to gradient descent processes, where the algorithms target the robust dense states, resulting in substantial improvements in performance. The weak dependence on the number of precision bits of the weights leads us to conjecture that very similar reasoning applies to more conventional neural networks. Analogous algorithmic schemes can also be applied to other optimization problems.

Examining the Fisher for diagonal Gaussians allows us to see exactly how the natural gradient differs from the standard gradient. If a dimension has small variance, then preconditioning by the inverse Fisher makes the natural gradient smaller along the mean dimension, . Intuitively, this makes sense -- we want our optimization routine to slow down when the component variance is small because small changes to the mean correspond to big changes in KL (see the two skinny Gaussians above), which can result in chatoic looking optimization traces. When the variance along a dimension is large, the inverse Fisher elongates the standard gradient along that dimension. Again, this makes sense -- when the component variance is high we can move the mean a lot farther (in Euclidean distance) without moving that far in terms of KL (see the two wide Gaussians above).

Consequently, there is an equivalence between parameter averaging and update-based data parallelism, when parameters are updated synchronously (this last part is key). This equivalence also holds for multiple averaging steps and other updaters (not just simple SGD). Update-based data parallelism becomes more interesting (and arguably more useful) when we relax the synchronous update requirement. That is, by allowing the updates Wi,j to be applied to the parameter vector as soon as they are computed (instead of waiting for N 1 iterations by all workers), we obtain asynchronous stochastic gradient descent algorithm. These benefits are not without cost, however. By introducing asynchronous updates to the parameter vector, we introduce a new problem, known as the stale gradient problem.

Subspace learning and matrix factorization problems have a great many applications in science and engineering, and efficient algorithms are critical as dataset sizes continue to grow. Many relevant problem formulations are non-convex, and in a variety of contexts it has been observed that solving the non-convex problem directly is not only efficient but reliably accurate. We discuss convergence theory for a particular method: first order incremental gradient descent constrained to the Grassmannian. The output of the algorithm is an orthonormal basis for a $d$-dimensional subspace spanned by an input streaming data matrix. We study two sampling cases: where each data vector of the streaming matrix is fully sampled, or where it is undersampled by a sampling matrix $A_t\in \R^{m\times n}$ with $m\ll n$. We propose an adaptive stepsize scheme that depends only on the sampled data and algorithm outputs. We prove that with fully sampled data, the stepsize scheme maximizes the improvement of our convergence metric at each iteration, and this method converges from any random initialization to the true subspace, despite the non-convex formulation and orthogonality constraints. For the case of undersampled data, we establish monotonic improvement on the defined convergence metric for each iteration with high probability.

I'm curious if it's possible to use the reparametrization trick to move the stochasticity outside the model, like this: Now that the stochasticity is outside the model, you can just use vanilla stochastic gradient descent in tensorflow. What is interesting about this approach seems to be the many different gradient estimators available, and i'm wondering if these tricks can bleed back into the variational deep learning literature.

Ben Recht spoke about optimization a few days ago at the Simons Institute. His talk was a highly entertaining tour de force through about a semester of convex optimization. You should go watch it. It's easy to spend a semester of convex optimization on various guises of gradient descent alone. Simply pick one of the following variants and work through the specifics of the analysis: conjugate, accelerated, projected, conditional, mirrored, stochastic, coordinate, online.

I think that gradient descent is one of those things which suffers a lot from Dunning Kruger. People will often learning about things like gradient descent, and feel ok, I know this. When in reality, there is such a wealth beneath the surface.

Many machine learning tasks can be formulated as Regularized Empirical Risk Minimization (R-ERM), and solved by optimization algorithms such as gradient descent (GD), stochastic gradient descent (SGD), and stochastic variance reduction (SVRG). Conventional analysis on these optimization algorithms focuses on their convergence rates during the training process, however, people in the machine learning community may care more about the generalization performance of the learned model on unseen test data. In this paper, we investigate on this issue, by using stability as a tool. In particular, we decompose the generalization error for R-ERM, and derive its upper bound for both convex and non-convex cases. In convex cases, we prove that the generalization error can be bounded by the convergence rate of the optimization algorithm and the stability of the R-ERM process, both in expectation (in the order of $\mathcal{O}((1/n)+\mathbb{E}\rho(T))$, where $\rho(T)$ is the convergence error and $T$ is the number of iterations) and in high probability (in the order of $\mathcal{O}\left(\frac{\log{1/\delta}}{\sqrt{n}}+\rho(T)\right)$ with probability $1-\delta$). For non-convex cases, we can also obtain a similar expected generalization error bound. Our theorems indicate that 1) along with the training process, the generalization error will decrease for all the optimization algorithms under our investigation; 2) Comparatively speaking, SVRG has better generalization ability than GD and SGD. We have conducted experiments on both convex and non-convex problems, and the experimental results verify our theoretical findings.

The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of the paper analyzes an inexact Newton method that solves linear systems approximately using the conjugate gradient (CG) method, and that samples the Hessian and not the gradient (the gradient is assumed to be exact). We provide a complexity analysis for this method based on the properties of the CG iteration and the quality of the Hessian approximation, and compare it with a method that employs a stochastic gradient iteration instead of the CG method. We report preliminary numerical results that illustrate the performance of inexact subsampled Newton methods on machine learning applications based on logistic regression.

We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many corruptions can we tolerate) via polynomial-time algorithms is by now understood. This paper presents and analyzes a non-convex optimization approach that greatly reduces the computational complexity of the above problems, compared to the best available algorithms. In particular, in the fully observed case, with $r$ denoting rank and $d$ dimension, we reduce the complexity from $\mathcal{O}(r^2d^2\log(1/\varepsilon))$ to $\mathcal{O}(rd^2\log(1/\varepsilon))$ -- a big savings when the rank is big. For the partially observed case, we show the complexity of our algorithm is no more than $\mathcal{O}(r^4d \log d \log(1/\varepsilon))$. Not only is this the best-known run-time for a provable algorithm under partial observation, but in the setting where $r$ is small compared to $d$, it also allows for near-linear-in-$d$ run-time that can be exploited in the fully-observed case as well, by simply running our algorithm on a subset of the observations.