arXiv subjects

James Martens

Publications and source records attributed to James Martens.

24 records · Page 2Linked to original sources

arXiv2019

On the Variance of Unbiased Online Recurrent Optimization

The recently proposed Unbiased Online Recurrent Optimization algorithm (UORO, arXiv:1702.05043) uses an unbiased approximation of RTRL to achieve fully online gradient-based learning in RNNs. In this work we analyze the variance of the gradient estimate computed by UORO, and propose several possible changes to the method which reduce this variance both in theory and practice. We also contribute significantly to the theoretical and intuitive understanding of UORO (and its existing variance reduction technique), and demonstrate a fundamental connection between its gradient estimate and the one that would be computed by REINFORCE if small amounts of noise were added to the RNN's hidden units.

cs.LG↗

arXiv2018

The Mechanics of n-Player Differentiable Games

The cornerstone underpinning deep learning is the guarantee that gradient descent on an objective converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, where there are multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new techniques to understand and control the dynamics in general games. The key result is to decompose the second-order dynamics into two components. The first is related to potential games, which reduce to gradient descent on an implicit function; the second relates to Hamiltonian games, a new class of games that obey a conservation law, akin to conservation laws in classical mechanical systems. The decomposition motivates Symplectic Gradient Adjustment (SGA), a new algorithm for finding stable fixed points in general games. Basic experiments show SGA is competitive with recently proposed algorithms for finding stable fixed points in GANs -- whilst at the same time being applicable to -- and having guarantees in -- much more general games.

cs.LG↗

arXiv2016

A Kronecker-factored approximate Fisher matrix for convolution layers

Second-order optimization methods such as natural gradient descent have the potential to speed up training of neural networks by correcting for the curvature of the loss function. Unfortunately, the exact natural gradient is impractical to compute for large models, and most approximations either require an expensive iterative procedure or make crude approximations to the curvature. We present Kronecker Factors for Convolution (KFC), a tractable approximation to the Fisher matrix for convolutional networks based on a structured probabilistic model for the distribution over backpropagated derivatives. Similarly to the recently proposed Kronecker-Factored Approximate Curvature (K-FAC), each block of the approximate Fisher matrix decomposes as the Kronecker product of small matrices, allowing for efficient inversion. KFC captures important curvature information while still yielding comparably efficient updates to stochastic gradient descent (SGD). We show that the updates are invariant to commonly used reparameterizations, such as centering of the activations. In our experiments, approximate natural gradient descent with KFC was able to train convolutional networks several times faster than carefully tuned SGD. Furthermore, it was able to train the networks in 10-20 times fewer iterations than SGD, suggesting its potential applicability in a distributed setting.

stat.ML↗

arXiv2015

Adding Gradient Noise Improves Learning for Very Deep Networks

Deep feedforward and recurrent networks have achieved impressive results in many perception and language processing applications. This success is partially attributed to architectural innovations such as convolutional and long short-term memory networks. The main motivation for these architectural innovations is that they capture better domain knowledge, and importantly are easier to optimize than more basic architectures. Recently, more complex architectures such as Neural Turing Machines and Memory Networks have been proposed for tasks including question answering and general computation, creating a new set of optimization challenges. In this paper, we discuss a low-overhead and easy-to-implement technique of adding gradient noise which we find to be surprisingly effective when training these very deep architectures. The technique not only helps to avoid overfitting, but also can result in lower training loss. This method alone allows a fully-connected 20-layer deep network to be trained with standard gradient descent, even starting from a poor initialization. We see consistent improvements for many complex models, including a 72% relative reduction in error rate over a carefully-tuned baseline on a challenging question-answering task, and a doubling of the number of accurate binary multiplication models learned across 7,000 random restarts. We encourage further application of this technique to additional complex modern architectures.

stat.ML↗

arXiv2015

On the Expressive Efficiency of Sum Product Networks

Sum Product Networks (SPNs) are a recently developed class of deep generative models which compute their associated unnormalized density functions using a special type of arithmetic circuit. When certain sufficient conditions, called the decomposability and completeness conditions (or "D&C" conditions), are imposed on the structure of these circuits, marginal densities and other useful quantities, which are typically intractable for other deep generative models, can be computed by what amounts to a single evaluation of the network (which is a property known as "validity"). However, the effect that the D&C conditions have on the capabilities of D&C SPNs is not well understood. In this work we analyze the D&C conditions, expose the various connections that D&C SPNs have with multilinear arithmetic circuits, and consider the question of how well they can capture various distributions as a function of their size and depth. Among our various contributions is a result which establishes the existence of a relatively simple distribution with fully tractable marginal densities which cannot be efficiently captured by D&C SPNs of any depth, but which can be efficiently captured by various other deep generative models. We also show that with each additional layer of depth permitted, the set of distributions which can be efficiently captured by D&C SPNs grows in size. This kind of "depth hierarchy" property has been widely conjectured to hold for various deep models, but has never been proven for any of them. Some of our other contributions include a new characterization of the D&C conditions as sufficient and necessary ones for a slightly strengthened notion of validity, and various state-machine characterizations of the types of computations that can be performed efficiently by D&C SPNs.

cs.LG↗

arXiv2012

Estimating the Hessian by Back-propagating Curvature

In this work we develop Curvature Propagation (CP), a general technique for efficiently computing unbiased approximations of the Hessian of any function that is computed using a computational graph. At the cost of roughly two gradient evaluations, CP can give a rank-1 approximation of the whole Hessian, and can be repeatedly applied to give increasingly precise unbiased estimates of any or all of the entries of the Hessian. Of particular interest is the diagonal of the Hessian, for which no general approach is known to exist that is both efficient and accurate. We show in experiments that CP turns out to work well in practice, giving very accurate estimates of the Hessian of neural networks, for example, with a relatively small amount of work. We also apply CP to Score Matching, where a diagonal of a Hessian plays an integral role in the Score Matching objective, and where it is usually computed exactly using inefficient algorithms which do not scale to larger and more complex models.

cs.LG↗