Although double-precision floating-point arithmetic currently dominates high-performance computing, there is increasing interest in smaller and simpler arithmetic types. The main reasons are potential improvements in energy efficiency and memory footprint and bandwidth. However, simply switching to lower-precision types typically results in increased numerical errors. We investigate approaches to improving the accuracy of lower-precision arithmetic types, using examples in an important domain for numerical computation in neuroscience: the solution of Ordinary Differential Equations (ODEs). The Izhikevich neuron model is used to demonstrate that rounding has an important role in producing accurate spike timings from explicit ODE solution algorithms. In particular, stochastic rounding consistently results in smaller errors compared to single-precision floatingpoint and fixed-point arithmetic with round-tonearest across a range of neuron behaviours and ODE solvers. A computationally much cheaper alternative is also investigated, inspired by the concept of dither that is a widely understood mechanism for providing resolution below the LSB in digital signal processing. These results will have implications for the solution of ODEs in other subject areas, and should also be directly relevant to the huge range of practical problems that are represented by Partial Differential Equations (PDEs).
In one of the earliest investigations of SR @cite_1 the authors investigate the effects of probabilistic rounding in backpropagation algorithms. Three different applications are shown with varying degrees of precision in the internal calculations of the backprogatation algorithms. It is demonstrated that when @math bits are used, training of the neural network starts to fail due to weight updates being too small to be captured by limited precision arithmetic, resulting in underflow in most of the updates. To allevate this, the authors apply probabilistic rounding in some of the arithmetic operations inside the backpropagation algorithm and show that the neural network can then perform well for word widths as small as 4 bits. The authors conclude that with probabilistic rounding, the 12-bit version of their system performs as well as a single precision floating-point version.
@cite_1
2006383931 1719489212 2950177356 2962998014
Abstract A key question in the design of specialized hardware for simulation of neural networks is whether fixed-point arithmetic of limited precision can be used with existing learning algorithms. Several studies of the backpropagation algorithm report a collapse of learning ability at around 12 to 16 bits of precision, depending on the details of the problem. In this paper, we investigate the effects of limited precision in the Cascade Correlation learning algorithm. As a general result, we introduce techniques for dynamic rescaling and probabilistic rounding that facilitate learning by gradient descent down to 6 bits of precision. Large multilayer neural networks trained with backpropagation have recently achieved state-of-the-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a large number of hyperparameters to the data, lack of calibrated probabilistic predictions, and a tendency to overfit the training data. In principle, the Bayesian approach to learning neural networks does not have these problems. However, existing Bayesian techniques lack scalability to large dataset and network sizes. In this work we present a novel scalable method for learning Bayesian neural networks, called probabilistic backpropagation (PBP). Similar to classical backpropagation, PBP works by computing a forward propagation of probabilities through the network and then doing a backward computation of gradients. A series of experiments on ten real-world datasets show that PBP is significantly faster than other techniques, while offering competitive predictive abilities. Our experiments also show that PBP provides accurate estimates of the posterior variance on the network weights. Large multilayer neural networks trained with backpropagation have recently achieved state-of-the-art results in a wide range of problems. However, using backprop for neural net learning still has some disadvantages, e.g., having to tune a large number of hyperparameters to the data, lack of calibrated probabilistic predictions, and a tendency to overfit the training data. In principle, the Bayesian approach to learning neural networks does not have these problems. However, existing Bayesian techniques lack scalability to large dataset and network sizes. In this work we present a novel scalable method for learning Bayesian neural networks, called probabilistic backpropagation (PBP). Similar to classical backpropagation, PBP works by computing a forward propagation of probabilities through the network and then doing a backward computation of gradients. A series of experiments on ten real-world datasets show that PBP is significantly faster than other techniques, while offering competitive predictive abilities. Our experiments also show that PBP provides accurate estimates of the posterior variance on the network weights. Recent advances in deep neural networks (DNNs) owe their success to training algorithms that use backpropagation and gradient-descent. Backpropagation, while highly effective on von Neumann architectures, becomes inefficient when scaling to large networks. Commonly referred to as the weight transport problem, each neuron's dependence on the weights and errors located deeper in the network require exhaustive data movement which presents a key problem in enhancing the performance and energy-efficiency of machine-learning hardware. In this work, we propose a bio-plausible alternative to backpropagation drawing from advances in feedback alignment algorithms in which the error computation at a single synapse reduces to the product of three scalar values, satisfying the three factor rule. Using a sparse feedback matrix, we show that a neuron needs only a fraction of the information previously used by the feedback alignment algorithms to yield results which are competitive with backpropagation. Consequently, memory and compute can be partitioned and distributed whichever way produces the most efficient forward pass so long as a single error can be delivered to each neuron. We evaluate our algorithm using standard data sets, including ImageNet, to address the concern of scaling to challenging problems. Our results show orders of magnitude improvement in data movement and @math improvement in multiply-and-accumulate operations over backpropagation. All the code and results are available under https: github.com bcrafton ssdfa.