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Convolutional neural networks (CNNs) have dramatically improved the accuracyof tasks such as object recognition, image segmentation and interactive speechsystems. CNNs require large amounts of computing resources becauseofcomputationally intensive convolution layers. Fast convolution algorithmssuch as Winograd convolution can greatly reduce the computational cost of theselayers at a cost of poor numeric properties, such that greater savings incomputation exponentially increase floating point errors.  A defining feature of each Winograd convolution algorithm is a set ofreal-value points where polynomials are sampled. The choice of points impactsthe numeric accuracy of the algorithm, but the optimal set of points for smallconvolutions remains unknown. Existing work considers only small integers andsimple fractions as candidate points. In this work, we propose a novel approachto point selection using points of the form {-1/c , -c, c, 1/c } using the fullrange of real-valued numbers for c. We show that groups of this form causecancellations in the Winograd transform matrices that reduce numeric error. Wefind empirically that the error for different values of c forms a rough curveacross the range of real-value numbers helping to localize the values of c thatreduce error and that lower errors can be achieved with non-obvious real-valuedevaluation points instead of integers or simple fractions. We study a range ofsizes for small convolutions and achieve reduction in error ranging from 2% toaround 59% for both 1D and 2D convolution. Furthermore, we identify patterns incases when we select a subset of our proposed points which will always lead toa lower error. Finally we implement a complete Winograd convolution layer anduse it to run deep convolution neural networks on real datasets and show thatour proposed points reduce error, ranging from 22% to 63%.

Winograd Convolution for Deep Neural Networks: Efficient Point Selection