Neural networks rely on convolutions to aggregate spatial information. However, spatial convolutions are expensive in terms of model size and computation, both of which grow quadratically with respect to kernel size. In this paper, we present a parameter-free, FLOP-free "shift" operation as an alternative to spatial convolutions. We fuse shifts and point-wise convolutions to construct end-to-end trainable shift-based modules, with a hyperparameter characterizing the tradeoff between accuracy and efficiency. To demonstrate the operation's efficacy, we replace ResNet's 3x3 convolutions with shift-based modules for improved CI-FAR10 and CIFAR100 accuracy using 60% fewer parameters; we additionally demonstrate the operation's resilience to parameter reduction on ImageNet, outperforming ResNet family members. We finally show the shift operation's applicability across domains, achieving strong performance with fewer parameters on classification, face verification and style transfer.
Residual neural networks can be viewed as the forward Euler discretization of an Ordinary Differential Equation (ODE) with a unit time step. This has recently motivated researchers to explore other discretization approaches and train ODE based networks. However, an important challenge of neural ODEs is their prohibitive memory cost during gradient backpropogation. Recently a method proposed in [8], claimed that this memory overhead can be reduced from O(LNt), where Nt is the number of time steps, down to O(L) by solving forward ODE backwards in time, where L is the depth of the network. However, we will show that this approach may lead to several problems: (i) it may be numerically unstable for ReLU/non-ReLU activations and general convolution operators, and (ii) the proposed optimize-then-discretize approach may lead to divergent training due to inconsistent gradients for small time step sizes. We discuss the underlying problems, and to address them we propose ANODE, an Adjoint based Neural ODE framework which avoids the numerical instability related problems noted above, and provides unconditionally accurate gradients. ANODE has a memory footprint of O(L) + O(Nt), with the same computational cost as reversing ODE solve. We furthermore, discuss a memory efficient algorithm which can further reduce this footprint with a trade-off of additional computational cost. We show results on Cifar-10/100 datasets using ResNet and SqueezeNext neural networks.
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