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Title: TD Learning with Constrained Gradients

Paper content:
Temporal Difference Learning with function approximation is known to be unstable.
Previous work like \citet{sutton2009fast} and \citet{sutton2009convergent} has presented alternative objectives that are stable to minimize.
However, in practice, TD-learning with neural networks requires various tricks like using a target network that updates slowly \citep{mnih2015human}.
In this work we propose a constraint on the TD update that minimizes change to the target values.
This constraint can be applied to the gradients of any TD objective, and can be easily applied to nonlinear function approximation.
We validate this update by applying our technique to deep Q-learning, and training without a target network.
We also show that adding this constraint on Baird's counterexample keeps Q-learning from diverging.
Temporal Difference learning is one of the most important paradigms in Reinforcement Learning (Sutton & Barto) .
Techniques based on nonlinear function approximators and stochastic gradient descent such as deep networks have led to significant breakthroughs in the class of problems that these methods can be applied to BID9 BID13 BID12 .
However, the most popular methods, such as TD(λ), Q-learning and Sarsa, are not true gradient descent techniques BID2 and do not converge on some simple examples BID0 .
BID0 and BID1 propose residual gradients as a way to overcome this issue.
Residual methods, also called backwards bootstrapping, work by splitting the TD error over both the current state and the next state.
These methods are substantially slower to converge, however, and BID16 show that the fixed point that they converge to is not the desired fixed point of TD-learning methods.
BID16 propose an alternative objective function formulated by projecting the TD target onto the basis of the linear function approximator, and prove convergence to the fixed point of this projected Bellman error is the ideal fixed point for TD methods.
BID5 extend this technique to nonlinear function approximators by projecting instead on the tangent space of the function at that point.
Subsequently, BID11 has combined these techniques of residual gradient and projected Bellman error by proposing an oblique projection, and BID8 has shown that the projected Bellman objective is a saddle point formulation which allows a finite sample analysis.However, when using deep networks for approximating the value function, simpler techniques like Q-learning and Sarsa are still used in practice with stabilizing techniques like a target network that is updated more slowly than the actual parameters BID10 .In
this work, we propose a constraint on the update to the parameters that minimizes the change to target values, freezing the target that we are moving our current predictions towards. Subject
to this constraint, the update minimizes the TD-error as much as possible. We show
that this constraint can be easily added to existing techniques, and works with all the techniques mentioned above.We validate our method by showing convergence on Baird's counterexample and a gridworld domain. On the
gridworld domain we parametrize the value function using a multi-layer perceptron, and show that we do not need a target network.
In this paper we introduce a constraint on the updates to the parameters for TD learning with function approximation.
This constraint forces the targets in the Bellman equation to not move when the update is applied to the parameters.
We enforce this constraint by projecting the gradient of the TD error with respect to the parameters for state s t onto the orthogonal space to the gradient with respect to the parameters for state s t+1 .We
show in our experiments that this added constraint stops parameters in Baird's counterexample from exploding when we use TD-learning. But
since we do not allow changes to target parameters, this also keeps Residual Gradients from converging to the true values of the Markov Process.On a Gridworld domain we demonstrate that we can perform TD-learning using a 2-layer neural network, without the need for a target network that updates more slowly. We
compare the solution obtained with DQN and show that it is closer to the solution obtained by tabular policy evaluation. Finally
, we also show that constrained DQN can learn faster and with less variance on the classical Cartpole domain.For future work, we hope to scale this approach to larger problems such as the Atari domain BID4 . We would
also like to prove convergence of TD-learning with this added constraint.

assistant: text: We show that adding a constraint to TD updates stabilizes learning and allows Deep Q-learning without a target network