Support vector machines (SVMs) can solve structured multioutput learning problems such as multilabel classification, multiclass classification, and vector regression. SVM training is expensive, especially for large and high-dimensional data sets. The bottleneck of the SVM training often lies in the kernel value computation. In many real-world problems, the same kernel values are used in many iterations during the training, which makes the caching of kernel values potentially useful. The majority of the existing studies simply adopt the least recently used (LRU) replacement strategy for caching kernel values. However, as we analyze in this article, the LRU strategy generally achieves high hit ratio near the final stage of the training but does not work well in the whole training process. Therefore, we propose a new caching strategy called EFU (less frequently used), which replaces the EFU kernel values that enhance least frequently used (LFU). Our experimental results show that EFU often has 20% higher hit ratio than LRU in the training with the Gaussian kernel. To further optimize the strategy, we propose a caching strategy called hybrid caching for the SVM training (HCST), which has a novel mechanism to automatically adapt the better caching strategy in different stages of the training. We have integrated the caching strategy into ThunderSVM, a recent SVM library on many-core processors. Our experiments show that HCST adaptively achieves high hit ratios with little runtime overhead among different problems including multilabel classification, multiclass classification, and regression problems. Compared with other existing caching strategies, HCST achieves 20% more reduction in training time on average.