The majority of online display ads are served through real-time bidding (RTB) --- each ad display impression is auctioned off in real-time when it is just being generated from a user visit. To place an ad automatically and optimally, it is critical for advertisers to devise a learning algorithm to cleverly bid an ad impression in real-time. Most previous works consider the bid decision as a static optimization problem of either treating the value of each impression independently or setting a bid price to each segment of ad volume. However, the bidding for a given ad campaign would repeatedly happen during its life span before the budget runs out. As such, each bid is strategically correlated by the constrained budget and the overall effectiveness of the campaign (e.g., the rewards from generated clicks), which is only observed after the campaign has completed. Thus, it is of great interest to devise an optimal bidding strategy sequentially so that the campaign budget can be dynamically allocated across all the available impressions on the basis of both the immediate and future rewards. In this paper, we formulate the bid decision process as a reinforcement learning problem, where the state space is represented by the auction information and the campaign's real-time parameters, while an action is the bid price to set. By modeling the state transition via auction competition, we build a Markov Decision Process framework for learning the optimal bidding policy to optimize the advertising performance in the dynamic real-time bidding environment. Furthermore, the scalability problem from the large real-world auction volume and campaign budget is well handled by state value approximation using neural networks. The empirical study on two large-scale real-world datasets and the live A B testing on a commercial platform have demonstrated the superior performance and high efficiency compared to state-of-the-art methods.
 The success of reinforcement learning in practical problems depends on the ability to combine function approximation with temporal difference methods such as value iteration. Experiments in this area have produced mixed results; there have been both notable successes and notable disappointments. Theory has been scarce, mostly due to the difficulty of reasoning about function approximators that generalize beyond the observed data. We provide a proof of convergence for a wide class of temporal difference methods involving function approximators such as k-nearest-neighbor, and show experimentally that these methods can be useful. The proof is based on a view of function approximators as expansion or contraction mappings. In addition, we present a novel view of approximate value iteration: an approximate algorithm for one environment turns out to be an exact algorithm for a different environment. A recent surge in research in kernelized approaches to reinforcement learning has sought to bring the benefits of kernelized machine learning techniques to reinforcement learning. Kernelized reinforcement learning techniques are fairly new and different authors have approached the topic with different assumptions and goals. Neither a unifying view nor an understanding of the pros and cons of different approaches has yet emerged. In this paper, we offer a unifying view of the different approaches to kernelized value function approximation for reinforcement learning. We show that, except for different approaches to regularization, Kernelized LSTD (KLSTD) is equivalent to a modelbased approach that uses kernelized regression to find an approximate reward and transition model, and that Gaussian Process Temporal Difference learning (GPTD) returns a mean value function that is equivalent to these other approaches. We also discuss the relationship between our modelbased approach and the earlier Gaussian Processes in Reinforcement Learning (GPRL). Finally, we decompose the Bellman error into the sum of transition error and reward error terms, and demonstrate through experiments that this decomposition can be helpful in choosing regularization parameters.