The brain is easily able to process and categorize complex time-varying signals. For example, the two sentences, "It is cold in London this time of year" and "It is hot in London this time of year," have different meanings, even though the words hot and cold appear several seconds before the ends of the two sentences. Any network that can tell these sentences apart must therefore have a long temporal memory. In other words, the current state of the network must depend on events that happened several seconds ago. This is a difficult task, as neurons are dominated by relatively short time constants--tens to hundreds of milliseconds. Nevertheless, it was recently proposed that randomly connected networks could exhibit the long memories necessary for complex temporal processing. This is an attractive idea, both for its simplicity and because little tuning of recurrent synaptic weights is required. However, we show that when connectivity is high, as it is in the mammalian brain, randomly connected networks cannot exhibit temporal memory much longer than the time constants of their constituent neurons.

Sutton, Szepesvári and Maei (2009) recently introduced the first temporal-difference learning algorithm compatible with both linear function approximation and off-policy training, and whose complexity scales only linearly in the size of the function approximator. Although their gradient temporal difference (GTD) algorithm converges reliably, it can be very slow compared to conventional linear TD (on on-policy problems where TD is convergent), calling into question its practical utility. In this paper we introduce two new related algorithms with better convergence rates. The first algorithm, GTD2, is derived and proved convergent just as GTD was, but uses a different objective function and converges significantly faster (but still not as fast as conventional TD). The second new algorithm, linear TD with gradient correction, or TDC, uses the same update rule as conventional TD except for an additional term which is initially zero. In our experiments on small test problems and in a Computer Go application with a million features, the learning rate of this algorithm was comparable to that of conventional TD. This algorithm appears to extend linear TD to off-policy learning with no penalty in performance while only doubling computational requirements.

We introduce the first temporal-difference learning algorithms that converge with smooth value function approximators, such as neural networks. Conventional temporal-difference (TD) methods, such as TD($\lambda$), Q-learning and Sarsa have been used successfully with function approximation in many applications. However, it is well known that off-policy sampling, as well as nonlinear function approximation, can cause these algorithms to become unstable (i.e., the parameters of the approximator may diverge). Sutton et al (2009a,b) solved the problem of off-policy learning with linear TD algorithms by introducing a new objective function, related to the Bellman-error, and algorithms that perform stochastic gradient-descent on this function. In this paper, we generalize their work to nonlinear function approximation. We present a Bellman error objective function and two gradient-descent TD algorithms that optimize it. We prove the asymptotic almost-sure convergence of both algorithms for any finite Markov decision process and any smooth value function approximator, under usual stochastic approximation conditions. The computational complexity per iteration scales linearly with the number of parameters of the approximator. The algorithms are incremental and are guaranteed to converge to locally optimal solutions.

We introduce the first temporal-difference learning algorithm that is stable with linear function approximation and off-policy training, for any finite Markov decision process, target policy, and exciting behavior policy, and whose complexity scales linearly in the number of parameters. We consider an i.i.d.\ policy-evaluation setting in which the data need not come from on-policy experience. The gradient temporal-difference (GTD) algorithm estimates the expected update vector of the TD(0) algorithm and performs stochastic gradient descent on its L_2 norm. Our analysis proves that its expected update is in the direction of the gradient, assuring convergence under the usual stochastic approximation conditions to the same least-squares solution as found by the LSTD, but without its quadratic computational complexity. GTD is online and incremental, and does not involve multiplying by products of likelihood ratios as in importance-sampling methods.