Barbara Barabasz

Syed Asad Alam, Andrew Anderson, Barbara Barabasz et al.

Convolutional neural networks (CNNs) have dramatically improved the accuracy of tasks such as object recognition, image segmentation and interactive speech systems. CNNs require large amounts of computing resources because ofcomputationally intensive convolution layers. Fast convolution algorithms such as Winograd convolution can greatly reduce the computational cost of these layers at a cost of poor numeric properties, such that greater savings in computation exponentially increase floating point errors. A defining feature of each Winograd convolution algorithm is a set of real-value points where polynomials are sampled. The choice of points impacts the numeric accuracy of the algorithm, but the optimal set of points for small convolutions remains unknown. Existing work considers only small integers and simple fractions as candidate points. In this work, we propose a novel approach to point selection using points of the form {-1/c , -c, c, 1/c } using the full range of real-valued numbers for c. We show that groups of this form cause cancellations in the Winograd transform matrices that reduce numeric error. We find empirically that the error for different values of c forms a rough curve across the range of real-value numbers helping to localize the values of c that reduce error and that lower errors can be achieved with non-obvious real-valued evaluation points instead of integers or simple fractions. We study a range of sizes for small convolutions and achieve reduction in error ranging from 2% to around 59% for both 1D and 2D convolution. Furthermore, we identify patterns in cases when we select a subset of our proposed points which will always lead to a lower error. Finally we implement a complete Winograd convolution layer and use it to run deep convolution neural networks on real datasets and show that our proposed points reduce error, ranging from 22% to 63%.

Barbara Barabasz

The problem how to speed up the convolution computations in Deep Neural Networks is widely investigated in recent years. The Winograd convolution algorithm is a common used method that significantly reduces time consumption. However, it suffers from a problem with numerical accuracy particularly for lower precisions. In this paper we present the application of base change technique for quantized Winograd-aware training model. We show that we can train the $8$ bit quantized network to nearly the same accuracy (up to 0.5% loss) for tested network (Resnet18) and dataset (CIFAR10) as for quantized direct convolution with few additional operations in pre/post transformations. Keeping Hadamard product on $9$ bits allow us to obtain the same accuracy as for direct convolution.

Barbara Barabasz, David Gregg

Winograd convolution is widely used in deep neural networks (DNNs). Existing work for DNNs considers only the subset Winograd algorithms that are equivalent to Toom-Cook convolution. We investigate a wider range of Winograd algorithms for DNNs and show that these additional algorithms can significantly improve floating point (FP) accuracy in many cases. We present results for three FP formats: fp32, fp16 and bf16 (a truncated form of fp32) using 2000 inputs from the ImageNet dataset. We found that in fp16 this approach gives us up to 6.5 times better image recognition accuracy in one important case while maintaining the same number of elementwise multiplication operations in the innermost loop. In bf16 the convolution can be computed using 5% fewer innermost loop multiplications than with currently used Winograd algorithms while keeping the accuracy of image recognition the same as for direct convolution method.

Barbara Barabasz, Andrew Anderson, Kirk M. Soodhalter et al.

Popular deep neural networks (DNNs) spend the majority of their execution time computing convolutions. The Winograd family of algorithms can greatly reduce the number of arithmetic operations required and is present in many DNN software frameworks. However, the performance gain is at the expense of a reduction in floating point (FP) numerical accuracy. In this paper, we analyse the worst case FP error and prove the estimation of norm and conditioning of the algorithm. We show that the bound grows exponentially with the size of the convolution, but the error bound of the \textit{modified} algorithm is smaller than the original one. We propose several methods for reducing FP error. We propose a canonical evaluation ordering based on Huffman coding that reduces summation error. We study the selection of sampling "points" experimentally and find empirically good points for the most important sizes. We identify the main factors associated with good points. In addition, we explore other methods to reduce FP error, including mixed-precision convolution, and pairwise summation across DNN channels. Using our methods we can significantly reduce FP error for a given block size, which allows larger block sizes and reduced computation.