Paper:
TWO-TIMESCALE NETWORKS FOR NONLINEAR VALUE FUNCTION APPROXIMATION
A key component for many reinforcement learning agents is to learn a value function, either for policy evaluation or control. Many of the algorithms for learning values, however, are designed for linear function approximation-with a fixed basis or fixed representation. Though there have been a few sound extensions to nonlinear function approximation, such as nonlinear gradient temporal difference learning, these methods have largely not been adopted, eschewed in favour of simpler but not sound methods like temporal difference learning and Q-learning. In this work, we provide a two-timescale network (TTN) architecture that enables linear methods to be used to learn values, with a nonlinear representation learned at a slower timescale. The approach facilitates the use of algorithms developed for the linear setting, such as data-efficient least-squares methods, eligibility traces and the myriad of recently developed linear policy evaluation algorithms, to provide nonlinear value estimates. We prove convergence for TTNs, with particular care given to ensure convergence of the fast linear component under potentially dependent features provided by the learned representation. We empirically demonstrate the benefits of TTNs, compared to other nonlinear value function approximation algorithms, both for policy evaluation and control.

Review:
The paper introduces an algorithm (TTN) for non-linear online and on-policy value function approximation. The main novelty of the paper is to view non-linear value estimation as two separate components. One of representation learning from a non-linear mapping and one of linear value function estimation. The soundness of the approach stems from the rate at which each component is updated. The authors argue that if the non-linear component is updated at a slower rate than the linear component, the former can be viewed as fixed in the limit and what remains is a linear value function estimation problem for which several sound algorithms exist. TTN is evaluated on 4 domains and compared to several other value estimation methods as well as DQN on a control problem with two variations on the task's state space.
I'll start off the review by stating that I find the idea and theoretical justification of separating the non-linear and linear parts of value function estimation to be quite interesting, potentially impacting RL at large. Indeed, this view promises to reconcile latest developments in deep RL with the long-lasting work on RL with linear function approximators. However, there are a few unclear aspects that do not allow one to be fully convinced that this paper lives up to the aforementioned promise.
- For the theoretical contribution. The authors claim that the main challenge was to deal with the potentially dependent features outputted by the neural network. It is dealt with by using a projection that projects the linear parameters of the value function to a compact subset of the parameter space. Bar the appendix, there is no mention of this projection in the paper, on how this compact subset (that must include the optimal parameter) is defined and if this projection is merely a theoretical tool or if it was necessary to implement it in practice. There is a projection for the neural net weights too but I can see how for these it might not be necessary to use in practice. However, for the linear weights, as their computation potentially involves inverting ill-conditioned matrices, they can indeed blow-up relatively fast.
- I found the experimental validation to be quite rich but not done in a systematic enough manner. For instance, the experiment "utility of optimizing the MSPBE" demonstrates quite nicely the importance of each component but is only performed on a single task. As the theoretical analysis does not say anything about the improvements the representation learning can have on the linear value estimation nor if the loss used for learning the representation effectively yields better features for the MSPBE minimization, this experiment is rather important and should have been performed on more than a single domain.
Secondly, I do not find the chosen baselines to be sufficiently competitive. The authors state in Sec. 2 that nonlinear-GTD has not seen widespread use, but having this algorithm as the main competitor does not provide strong evidence that TTN will know a better fate. In the abstract, it is implied that outside of nonlinear-GTD, value function approximation methods are not sound. In approximate policy iteration algorithms such as DDPG or TRPO, there is a need in performing value estimation. It is done by essentially a fitted-Q iteration procedure which is sound. Why wasn't TTN compared to these methods? If it is because they are not online, why being online in the experiments of the paper is important? Showing that TTN is competitive with currently widespread methods for value estimated would have been more convincing than the comparison with nonlinear-GTD.
Thirdly, for the sake of reproducibility, as LSTD seems to be the method of choice for learning the linear part, it would have been adequate to provide an algorithm box for this version as is done for GTD2/TDC. LSTD is essentially a batch algorithm and there could be many ways to turn it into an online algorithm. With which algorithm were the results in the experimental section obtained?
Finally, on the control task, the authors add several modifications to their algorithm which results in an algorithm that is very close to that of Levine et al., 2017. Why was not the latter a baseline for this experiment? Especially since it was included in other experiments.