We propose a general Variational Embedding Learning Framework (VELF) for alleviating the severe cold-start problem in CTR prediction. VELF addresses the cold start problem via alleviating over-fits caused by data-sparsity in two ways: learning probabilistic embedding, and incorporating trainable and regularized priors which utilize the rich side information of cold start users and advertisements (Ads). The two techniques are naturally integrated into a variational inference framework, forming an end-to-end training process. Abundant empirical tests on benchmark datasets well demonstrate the advantages of our proposed VELF. Besides, extended experiments confirmed that our parameterized and regularized priors provide more generalization capability than traditional fixed priors.