Y-axis: Cross Entropy Loss on validation set. Figure 1 shows the validation loss changes with respect to training time for 50k steps as supplementary results for the experiments in Section 5. In general, Skyformer converges faster and finishes 50k steps earlier than vanilla Attention and Kernelized Attention over all tasks. We further remark that on Text Classification, all models quickly fall into over-fitting, and thus the validation losses rise quickly. On Pathfinder, due to the difficulty of training, in the trial shown in the figure vanilla Attention fails to reach the best long-time limit under a certain setting. Figure 2 shows the singular value distribution of attention output from the second layer of a trained vanilla transformer.

Transformers are expensive to train due to the quadratic time and space complexity in the self-attention mechanism. On the other hand, although kernel machines suffer from the same computation bottleneck in pairwise dot products, several approximation schemes have been successfully incorporated to considerably reduce their computational cost without sacrificing too much accuracy. In this work, we leverage the computation methods for kernel machines to alleviate the high computational cost and introduce Skyformer, which replaces the softmax structure with a Gaussian kernel to stabilize the model training and adapts the Nyström method to a non-positive semidefinite matrix to accelerate the computation. We further conduct theoretical analysis by showing that the matrix approximation error of our proposed method is small in the spectral norm. Experiments on Long Range Arena benchmark show that the proposed method is sufficient in getting comparable or even better performance than the full self-attention while requiring fewer computation resources.

Transformers are expensive to train due to the quadratic time and space complexity in the self-attention mechanism. On the other hand, although kernel machines suffer from the same computation bottleneck in pairwise dot products, several approximation schemes have been successfully incorporated to considerably reduce their computational cost without sacrificing too much accuracy. In this work, we leverage the computation methods for kernel machines to alleviate the high computational cost and introduce Skyformer, which replaces the softmax structure with a Gaussian kernel to stabilize the model training and adapts the Nyström method to a non-positive semidefinite matrix to accelerate the computation. We further conduct theoretical analysis by showing that the matrix approximation error of our proposed method is small in the spectral norm. Experiments on Long Range Arena benchmark show that the proposed method is sufficient in getting comparable or even better performance than the full self-attention while requiring fewer computation resources.