Deep learning has made significant progress in protein structure prediction, advancing the development of computational biology. However, despite the high accuracy achieved in predicting single-chain structures, a significant number of large homo-oligomeric assemblies exhibit internal symmetry, posing a major challenge in structure determination. The performances of existing deep learning methods are limited since the symmetrical protein assembly usually has a long sequence, making structural computation infeasible. In addition, multiple identical subunits in symmetrical protein complex cause the issue of supervision ambiguity in label assignment, requiring a consistent structure modeling for the training. To tackle these problems, we propose a protein folding framework called SGNet to model protein-protein interactions in symmetrical assemblies. SGNet conducts feature extraction on a single subunit and generates the whole assembly using our proposed symmetry module, which largely mitigates computational problems caused by sequence length. Thanks to the elaborate design of modeling symmetry consistently, we can model all global symmetry types in quaternary protein structure prediction. Extensive experimental results on a benchmark of symmetrical protein complexes further demonstrate the effectiveness of our method.

Regularized empirical risk minimization (R-ERM) is an important branch of machine learning, since it constrains the capacity of the hypothesis space and guarantees the generalization ability of the learning algorithm. Two classic proximal optimization algorithms, i.e., proximal stochastic gradient descent (ProxSGD) and proximal stochastic coordinate descent (ProxSCD) have been widely used to solve the R-ERM problem. Recently, variance reduction technique was proposed to improve ProxSGD and ProxSCD, and the corresponding ProxSVRG and ProxSVRCD have better convergence rate. These proximal algorithms with variance reduction technique have also achieved great success in applications at small and moderate scales. However, in order to solve large-scale R-ERM problems and make more practical impacts, the parallel versions of these algorithms are sorely needed. In this paper, we propose asynchronous ProxSVRG (Async-ProxSVRG) and asynchronous ProxSVRCD (Async-ProxSVRCD) algorithms, and prove that Async-ProxSVRG can achieve near linear speedup when the training data is sparse, while Async-ProxSVRCD can achieve near linear speedup regardless of the sparse condition, as long as the number of block partitions are appropriately set. We have conducted experiments on a regularized logistic regression task. The results verified our theoretical findings and demonstrated the practical efficiency of the asynchronous stochastic proximal algorithms with variance reduction.