Temporal difference learning | EPFL Graph Search

Temporal difference (TD) learning refers to a class of model-free reinforcement learning methods which learn by bootstrapping from the current estimate of the value function. These methods sample from the environment, like Monte Carlo methods, and perform updates based on current estimates, like dynamic programming methods. While Monte Carlo methods only adjust their estimates once the final outcome is known, TD methods adjust predictions to match later, more accurate, predictions about the future before the final outcome is known. This is a form of bootstrapping, as illustrated with the following example: Suppose you wish to predict the weather for Saturday, and you have some model that predicts Saturday's weather, given the weather of each day in the week. In the standard case, you would wait until Saturday and then adjust all your models. However, when it is, for example, Friday, you should have a pretty good idea of what the weather would be on Saturday – and thus be able to change, say, Saturday's model before Saturday arrives. Temporal difference methods are related to the temporal difference model of animal learning. The tabular TD(0) method is one of the simplest TD methods. It is a special case of more general stochastic approximation methods. It estimates the state value function of a finite-state Markov decision process (MDP) under a policy . Let denote the state value function of the MDP with states , rewards and discount rate under the policy : We drop the action from the notation for convenience. satisfies the Hamilton-Jacobi-Bellman Equation: so is an unbiased estimate for . This observation motivates the following algorithm for estimating . The algorithm starts by initializing a table arbitrarily, with one value for each state of the MDP. A positive learning rate is chosen. We then repeatedly evaluate the policy , obtain a reward and update the value function for the old state using the rule: where and are the old and new states, respectively. The value is known as the TD target.

Understanding Deep Neural Function Approximation in Reinforcement Learning via ϵ-Greedy Exploration

Volkan Cevher, Fanghui Liu, Luca Viano

This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the ϵ-greedy exploration under the online setting. This problem setting is motivated by the successful deep Q-networks (DQN) framework that falls in this regime. In this work, we provide an initial attempt on theoretical understanding deep RL from the perspective of function class and neural networks architectures (e.g., width and depth) beyond the ``linear'' regime. To be specific, we focus on the value based algorithm with the ϵ-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces, respectively, which aims at approximating an α-smooth Q-function in a d-dimensional feature space. We prove that, with T episodes, scaling the width

m = \widetilde{\mathcal{O}}(T^{\frac{d}{2\alpha + d}})

and the depth

L=\mathcal{O}(\log T)

of the neural network for deep RL is sufficient for learning with sublinear regret in Besov spaces. Moreover, for a two layer neural network endowed by the Barron space, scaling the width

\Omega(\sqrt{T})

is sufficient. To achieve this, the key issue in our analysis is how to estimate the temporal difference error under deep neural function approximation as the ϵ-greedy exploration is not enough to ensure "optimism". Our analysis reformulates the temporal difference error in an

L^2(\mathrm{d}\mu)

-integrable space over a certain averaged measure μ, and transforms it to a generalization problem under the non-iid setting. This might have its own interest in RL theory for better understanding

\epsilon

-greedy exploration in deep RL.

2022

Models of Reward-Modulated Spike-Timing-Dependent Plasticity

Nicolas Frémaux

How do animals learn to repeat behaviors that lead to the obtention of food or other “rewarding” objects? As a biologically plausible paradigm for learning in spiking neural networks, spike-timing dependent plasticity (STDP) has been shown to perform well in unsupervised learning tasks such as receptive field development. However, STDP fails to take behavioral relevance into account, and as such is inadequate to explain a vast range of learning tasks in which the final outcome, conditioned on the prior execution of a series of actions, is signaled to an animal through sparse rewards. In this thesis, I show that the addition of a third, global, reward-based factor to the pre- and postsynaptic factors of STDP is a promising solution to this problem, consistent with experimental findings. One one hand, dopamine is a neuromodulator which has been shown to encode reward signals in the brain. On the other hand, STDP has been shown to be affected by dopamine, even though the precise nature of the interaction is unclear. Moreover, the theoretical framework of reinforcement learning provides strong foundation for the analysis of these learning rules. After studying existing examples of such rules in a navigation task, I derive simple functional requirements for reward-modulated learning rules, and illustrate these in a motor learning task. One of those functional requirements is the existence a “critic” structure, constantly evaluating the potential for rewarding events. The implication of the existence of such a critic on the interpretation of psychophysical experiments are also discussed. Finally, I propose a biologically plausible implementation of such a structure, that performs motor or navigational tasks. This is based on a generalization of temporal difference learning, a well-known reinforcement learning framework, to continuous time, well suited to an implementation with spiking neurons. These result provide a unified picture of reward-modulated learning rules: even though different rules have been proposed, these can be reduced to a single model at the synaptic level, with variations in the computation of the neuromodulatory signal enabling switching between different learning rules.

EPFL2013

Astrocytes generate Na+-mediated metabolic waves

Pierre Magistretti, Yann Bernardinelli, Jean-Yves Chatton

Glutamate-evoked Na+ increase in astrocytes has been identified as a signal coupling synaptic activity to glucose consumption. Astrocytes participate in multicellular signaling by transmitting intercellular Ca2+ waves. Here we show that intercellular Na+ waves are also evoked by activation of single cultured cortical mouse astrocytes in parallel with Ca2+ waves; however, there are spatial and temporal differences. Indeed, maneuvers that inhibit Ca2+ waves also inhibit Na+ waves; however, inhibition of the Na+/glutamate cotransporters or enzymatic degradation of extracellular glutamate selectively inhibit the Na+ wave. Thus, glutamate released by a Ca2+ wave-dependent mechanism is taken up by the Na+/glutamate cotransporters, resulting in a regenerative propagation of cytosolic Na+ increases. The Na+ wave gives rise to a spatially correlated increase in glucose uptake, which is prevented by glutamate transporter inhibition. Therefore, astrocytes appear to function as a network for concerted neurometabolic coupling through the generation of intercellular Na+ and metabolic waves.

2004