Local feature extraction consists of keypoint detection and local descriptor extraction. Firstly, in keypoint detector learning, existing covariance constraint loss functions cannot constrain the probability distribution shapes in local probability maps that surround keypoints. And existing auxiliary peak loss functions, which are used to alleviate the problem, impair the performance of local feature methods. To solve this problem, we propose a novel Covariant Peak constraint Loss (CP Loss) which is defined as the expectations of local probability maps' position errors. Minimizing our CP Loss can make local probability maps accurately peak at reliable keypoints. Secondly, in descriptor learning, the Neural Reprojection Error (NRE) aims at constraining dense descriptor maps of images. But we argue that only those descriptors of keypoints need to be constrained. Thus, we propose a novel Conditional Neural Reprojection Error (CNRE) that is only conditioned on keypoints. Compared with the NRE, our CNRE can achieve much higher efficiency and produce more keypoint-specific descriptors with better matching performance. We use our CP Loss and CNRE to train a local feature network named as CPCN-Feat. Experimental results show that our CPCN-Feat achieves state-of-the-art performance on four challenging benchmarks.