In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Example Input: Program verification offers a framework for ensuring program correctness and therefore systematically eliminating different classes of bugs. Inferring loop invariants is one of the main challenges behind automated verification of real-world programs which often contain many loops. In this paper, we present Continuous Logic Network (CLN), a novel neural architecture for automatically learning loop invariants directly from program execution traces. Unlike existing neural networks, CLNs can learn precise and explicit representations of formulas in Satisfiability Modulo Theories (SMT)  for loop invariants from program execution traces. We develop a new sound and complete semantic mapping for assigning SMT formulas to continuous truth values that allows CLNs to be trained efficiently. We use CLNs to implement a new inference system for loop invariants, CLN2INV, that significantly outperforms existing approaches on the popular Code2Inv dataset. CLN2INV is the first tool to solve all 124 theoretically solvable problems in the Code2Inv dataset. Moreover, CLN2INV takes only 1.1 second on average for each problem, which is 40 times faster than existing approaches. We further demonstrate that CLN2INV can even learn 12 significantly more complex loop invariants than the ones required for the Code2Inv dataset.
Example Output: We introduce the Continuous Logic Network (CLN), a novel neural architecture for automatically learning loop invariants and general SMT formulas.

Example Input: We compare the model-free reinforcement learning with the model-based approaches through the lens of the expressive power of neural networks for policies, $Q$-functions, and dynamics.   We show, theoretically and empirically, that even for one-dimensional continuous state space, there are many MDPs whose optimal $Q$-functions and policies are much more complex than the dynamics. We hypothesize many real-world MDPs also have a similar property. For these MDPs, model-based planning is a favorable algorithm, because the resulting policies can approximate the optimal policy significantly better than a neural network parameterization can, and model-free or model-based policy optimization rely on policy parameterization. Motivated by the theory, we apply a simple multi-step model-based bootstrapping planner (BOOTS) to bootstrap a weak $Q$-function into a stronger policy. Empirical results show that applying BOOTS on top of model-based or model-free policy optimization algorithms at the test time improves the performance on MuJoCo benchmark tasks.
Example Output: We compare deep model-based and model-free RL algorithms by studying the approximability of $Q$-functions, policies, and dynamics by neural networks. 

Example Input: We propose Cooperative Training (CoT) for training generative models that measure a tractable density for discrete data. CoT coordinately trains a generator G and an auxiliary predictive mediator M. The training target of M is to estimate a mixture density of the learned distribution G and the target distribution P, and that of G is to minimize the Jensen-Shannon divergence estimated through M. CoT achieves independent success without the necessity of pre-training via Maximum Likelihood Estimation or involving high-variance algorithms like REINFORCE. This low-variance algorithm is theoretically proved to be superior for both sample generation and likelihood prediction. We also theoretically and empirically show the superiority of CoT over most previous algorithms in terms of generative quality and diversity, predictive generalization ability and computational cost.
Example Output:
We proposed Cooperative Training, a novel training algorithm for generative modeling of discrete data.

