User response prediction is a crucial component for personalized information retrieval and filtering scenarios, such as recommender system and web search. The data in user response prediction is mostly in a multi-field categorical format and transformed into sparse representations via one-hot encoding. Due to the sparsity problems in representation and optimization, most research focuses on feature engineering and shallow modeling. Recently, deep neural networks have attracted research attention on such a problem for their high capacity and end-to-end training scheme. In this paper, we study user response prediction in the scenario of click prediction. We first analyze a coupled gradient issue in latent vector-based models and propose kernel product to learn field-aware feature interactions. Then we discuss an insensitive gradient issue in DNN-based models and propose Product-based Neural Network (PNN) which adopts a feature extractor to explore feature interactions. Generalizing the kernel product to a net-in-net architecture, we further propose Product-network In Network (PIN) which can generalize previous models. Extensive experiments on 4 industrial datasets and 1 contest dataset demonstrate that our models consistently outperform 8 baselines on both AUC and log loss. Besides, PIN makes great CTR improvement (relatively 34.67%) in online A/B test.Many machine learning models are leveraged or proposed to work on such a problem, including linear models, latent vector-based models, tree models, and DNN-based models. Linear models, such as Logistic Regression (LR) [25] and Bayesian Probit Regression [14], are easy to implement and with high efficiency. A typical latent vector-based model is Factorization Machine (FM) [36]. FM uses weights and latent vectors to represent categories. According to their parametric representations, LR has a linear feature extractor, and FM has a bi-linear 2 feature extractor. The prediction of LR and FM are simply based on the sum over weights, thus their classifiers are linear. FM works well on sparse data, and inspires a lot of extensions, including Field-aware FM (FFM) [21]. FFM introduces field-aware latent vectors, which gain FFM higher capacity and better performance. However, FFM is restricted by space complexity. Inspired by FFM, we find a coupled gradient issue of latent vector-based models and refine feature interactions 3 as field-aware feature interactions. To solve this issue as well as saving memory, we propose kernel product methods and derive Kernel FM (KFM) and Network in FM (NIFM).Trees and DNNs are potent function approximators. Tree models, such as Gradient Boosting Decision Tree (GBDT) [6], are popular in various data science contests as well as industrial applications. GBDT explores very high order feature combinations in a non-parametric way, yet its exploration ability is restricted when feature space becomes extremely high-dimensional and sparse. DNN has also been preliminarily studied in information system literature [8,33,40,51]. In [51], FM supported Neura...