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Title: Near-Optimal Representation Learning for Hierarchical Reinforcement Learning

Paper content:
We study the problem of representation learning in goal-conditioned hierarchical reinforcement learning.
In such hierarchical structures, a higher-level controller solves tasks by iteratively communicating goals which a lower-level policy is trained to reach.
Accordingly, the choice of representation -- the mapping of observation space to goal space -- is crucial.
To study this problem, we develop a notion of sub-optimality of a representation, defined in terms of expected reward of the optimal hierarchical policy using this representation.
We derive expressions which bound the sub-optimality and show how these expressions can be translated to representation learning objectives which may be optimized in practice.
Results on a number of difficult continuous-control tasks show that our approach to representation learning yields qualitatively better representations as well as quantitatively better hierarchical policies, compared to existing methods.
Hierarchical reinforcement learning has long held the promise of extending the successes of existing reinforcement learning (RL) methods BID9 BID22 BID16 to more complex, difficult, and temporally extended tasks BID18 BID25 BID3 .
Recently, goal-conditioned hierarchical designs, in which higher-level policies communicate goals to lower-levels and lower-level policies are rewarded for reaching states (i.e. observations) which are close to these desired goals, have emerged as an effective paradigm for hierarchical RL BID17 ; BID14 ; Vezhnevets et al. (2017) , inspired by earlier work BID6 ; BID21 ).
In this hierarchical design, representation learning is crucial; that is, a representation function must be chosen mapping state observations to an abstract space.
Goals (desired states) are then specified by the choice of a point in this abstract space.Previous works have largely studied two ways to choose the representation: learning the representation end-to-end together with the higher-and lower-level policies (Vezhnevets et al., 2017) , or using the state space as-is for the goal space (i.e., the goal space is a subspace of the state space) BID17 BID14 .
The former approach is appealing, but in practice often produces poor results (see BID17 and our own experiments), since the resulting representation is under-defined; i.e., not all possible sub-tasks are expressible as goals in the space.
On the other hand, fixing the representation to be the full state means that no information is lost, but this choice is difficult to scale to higher dimensions.
For example, if the state observations are entire images, the higher-level must output target images for the lower-level, which can be very difficult.We instead study how unsupervised objectives can be used to train a representation that is more concise than the full state, but also not as under-determined as in the end-to-end approach.
In order to do so in a principled manner, we propose a measure of sub-optimality of a given representation.
This measure aims to answer the question: How much does using the learned representation in place of the full representation cause us to lose, in terms of expected reward, against the optimal policy?
This question is important, because a useful representation will compress the state, hopefully making the learning problem easier.
At the same time, the compression might cause the representation to lose information, making the optimal policy impossible to express.
It is therefore critical to understand how lossy a learned representation is, not in terms of reconstruction, but in terms of the ability to represent near-optimal policies on top of this representation.Our main theoretical result shows that, for a particular choice of representation learning objective, we can learn representations for which the return of the hierarchical policy approaches the return of the optimal policy within a bounded error.
This suggests that, if the representation is learned with a principled objective, the 'lossy-ness' in the resulting representation should not cause a decrease in overall task performance.
We then formulate a representation learning approach that optimizes this bound.
We further extend our result to the case of temporal abstraction, where the higher-level controller only chooses new goals at fixed time intervals.
To our knowledge, this is the first result showing that hierarchical goal-setting policies with learned representations and temporal abstraction can achieve bounded sub-optimality against the optimal policy.
We further observe that the representation learning objective suggested by our theoretical result closely resembles several other recently proposed objectives based on mutual information (van den Oord et al., 2018; BID12 , suggesting an intriguing connection between mutual information and goal representations for hierarchical RL.
Results on a number of difficult continuous-control navigation tasks show that our principled representation learning objective yields good qualitative and quantitative performance compared to existing methods.
We have presented a principled approach to representation learning in hierarchical RL.
Our approach is motivated by the desire to achieve maximum possible return, hence our notion of sub-optimality is in terms of optimal state values.
Although this notion of sub-optimality is intractable to optimize directly, we are able to derive a mathematical relationship between it and a specific form of representation learning.
Our resulting representation learning objective is practical and achieves impressive results on a suite of high-dimensional, continuous-control tasks.

assistant: text: This paper proposes a way to handle sub-optimality in the context of learning representations which refer to the sub-optimality of hierarchical polity with respect to the task reward.