Bellman equation

About: Bellman equation is a research topic. Over the lifetime, 5884 publications have been published within this topic receiving 135589 citations.

Learning to trade via direct reinforcement

Abstract: We present methods for optimizing portfolios, asset allocations, and trading systems based on direct reinforcement (DR). In this approach, investment decision-making is viewed as a stochastic control problem, and strategies are discovered directly. We present an adaptive algorithm called recurrent reinforcement learning (RRL) for discovering investment policies. The need to build forecasting models is eliminated, and better trading performance is obtained. The direct reinforcement approach differs from dynamic programming and reinforcement algorithms such as TD-learning and Q-learning, which attempt to estimate a value function for the control problem. We find that the RRL direct reinforcement framework enables a simpler problem representation, avoids Bellman's curse of dimensionality and offers compelling advantages in efficiency. We demonstrate how direct reinforcement can be used to optimize risk-adjusted investment returns (including the differential Sharpe ratio), while accounting for the effects of transaction costs. In extensive simulation work using real financial data, we find that our approach based on RRL produces better trading strategies than systems utilizing Q-learning (a value function method). Real-world applications include an intra-daily currency trader and a monthly asset allocation system for the S&P 500 Stock Index and T-Bills.

Learning continuous control policies by stochastic value gradients

Abstract: We present a unified framework for learning continuous control policies using backpropagation. It supports stochastic control by treating stochasticity in the Bellman equation as a deterministic function of exogenous noise. The product is a spectrum of general policy gradient algorithms that range from model-free methods with value functions to model-based methods without value functions. We use learned models but only require observations from the environment instead of observations from model-predicted trajectories, minimizing the impact of compounded model errors. We apply these algorithms first to a toy stochastic control problem and then to several physics-based control problems in simulation. One of these variants, SVG(1), shows the effectiveness of learning models, value functions, and policies simultaneously in continuous domains.

Leader-Based Optimal Coordination Control for the Consensus Problem of Multiagent Differential Games via Fuzzy Adaptive Dynamic Programming

Abstract: In this paper, a new online scheme is presented to design the optimal coordination control for the consensus problem of multiagent differential games by fuzzy adaptive dynamic programming, which brings together game theory, generalized fuzzy hyperbolic model (GFHM), and adaptive dynamic programming. In general, the optimal coordination control for multiagent differential games is the solution of the coupled Hamilton-Jacobi (HJ) equations. Here, for the first time, GFHMs are used to approximate the solutions (value functions) of the coupled HJ equations, based on policy iteration algorithm. Namely, for each agent, GFHM is used to capture the mapping between the local consensus error and local value function. Since our scheme uses the single-network architecture for each agent (which eliminates the action network model compared with dual-network architecture), it is a more reasonable architecture for multiagent systems. Furthermore, the approximation solution is utilized to obtain the optimal coordination control. Finally, we give the stability analysis for our scheme, and prove the weight estimation error and the local consensus error are uniformly ultimately bounded. Further, the control node trajectory is proven to be cooperative uniformly ultimately bounded.