A back-propagation algorithm following a gradient descent approach is used for training deep neural networks. Since it iteratively performs a large number of matrix operations to compute the gradients, GPUs (Graphics Processing Units) are efficient especially for the training phase. Thus, a cluster of computers each of which equips multiple GPUs can significantly accelerate the training phase. Although the gradient computation is still a major bottleneck of the training, gradient aggregation and parameter optimization impose both communication and computation overheads, which should also be reduced for further shortening the training time. To address this issue, in this paper, multiple GPUs are interconnected with a PCI Express (PCIe) over 10 Gbit Ethernet (10GbE) technology. Since these remote GPUs are interconnected via network switches, gradient aggregation and optimizers (e.g., SGD, Adagrad, Adam, and SMORMS3) are offloaded to an FPGA-based network switch between a host machine and remote GPUs; thus, the gradient aggregation and optimization are completed in the network. Evaluation results using four remote GPUs connected via the FPGA-based 10GbE switch that implements the four optimizers demonstrate that these optimization algorithms are accelerated by up to 3. 0x and 1. 25x compared to CPU and GPU implementations, respectively. Also, the gradient aggregation throughput by the FPGA-based switch achieves 98.3% of the 10GbE line rate.