D ECOUPLING E XPLORATION AND E XPLOITATION FOR M ETA -R EINFORCEMENT L EARNING WITHOUT S ACRIFICES
Anonymous authors
Paper under double-blind review
A BSTRACT
The goal of meta-reinforcement learning (meta-RL) is to build agents that can quickly learn new tasks by leveraging prior experience on related tasks.
Learning a new task often requires both exploring to gather task-relevant information and exploiting this information to solve the task.
In principle, optimal exploration and exploitation can be learned end-to-end by simply maximizing task performance.
However, such meta-RL approaches struggle with local optima due to a chickenand-egg problem: learning to explore requires good exploitation to gauge the exploration’s utility, but learning to exploit requires information gathered via exploration.
Optimizing separate objectives for exploration and exploitation can avoid this problem, but prior meta-RL exploration objectives yield suboptimal policies that gather information irrelevant to the task.
We alleviate both concerns by constructing an exploitation objective that automatically identiﬁes task-relevant information and an exploration objective to recover only this information.
This avoids local optima in end-to-end training, without sacriﬁcing optimal exploration.
Empirically, D REAM substantially outperforms existing approaches on complex meta-RL problems, such as sparse-reward 3D visual navigation.
I NTRODUCTION
A general-purpose agent should be able to perform multiple related tasks across multiple related environments.
Our goal is to develop agents that can perform a variety of tasks in novel environments, based on previous experience and only a small amount of experience in the new environment.
For example, we may want a robot to cook a meal (a new task) in a new kitchen (the environment) after it has learned to cook other meals in other kitchens.
To adapt to a new kitchen, the robot must both explore to ﬁnd the ingredients, and use this information to cook.
Existing meta-reinforcement learning (meta-RL) methods can adapt to new tasks and environments, but, as we identify in this work, struggle when adaptation requires complex exploration.
In the meta-RL setting, the agent is presented with a set of meta-training problems, each in an environment (e.g., a kitchen) with some task (e.g., make pizza); at meta-test time, the agent is given a new, but related environment and task.
It is allowed to gather information in a few initial (exploration) episodes, and its goal is to then maximize returns on all subsequent (exploitation) episodes, using this information.
A common meta-RL approach is to learn to explore and exploit end-to-end by training a policy and updating exploration behavior based on how well the policy later exploits using the information discovered from exploration (Duan et al., 2016;
Wang et al., 2016a; Stadie et al., 2018;
Zintgraf et al., 2019; Humplik et al., 2019).
With enough model capacity, such approaches can express optimal exploration and exploitation, but they create a chicken-and-egg problem that leads to bad local optima and poor sample efﬁciency: Learning to explore requires good exploitation to gauge the exploration’s utility, but learning to exploit requires information gathered via exploration; therefore, with only ﬁnal performance as signal, one cannot be learned without already having learned the other.
For example, a robot chef is only incentivized to explore and ﬁnd the ingredients if it already knows how to cook, but the robot can only learn to cook if it can already ﬁnd the ingredients by exploration.
To avoid the chicken-and-egg problem, we propose to optimize separate objectives for exploration and exploitation by leveraging the problem ID —an easy-to-provide unique one-hot for each training meta- training task and environment.
Such a problem ID can be realistically available in real-world meta-RL tasks: e.g., in a robot chef factory, each training kitchen (problem) can be easily assigned a unique
ID, and in a recommendation system that provides tailored recommendations to each user, each user (problem) is typically identiﬁed by a unique username.
Some prior works (Humplik et al., 2019; Kamienny et al., 2020) also use these problem IDs, but not in a way that avoids the chicken-and-egg problem.
Others (Rakelly et al., 2019; Zhou et al., 2019b; Gupta et al., 2018; Gurumurthy et al., 2019; Zhang et al., 2020) also optimize separate objectives, but their exploration objectives learn suboptimal policies that gather task-irrelevant information (e.g., the color of the walls).
Instead, we propose an exploitation objective that automatically identiﬁes task-relevant information, and an exploration objective to recover only this information.
We learn an exploitation policy without the need for exploration, by conditioning on a learned representation of the problem ID, which provides taskrelevant information (e.g., by memorizing the locations of the ingredients for each ID / kitchen).
We also apply an information bottleneck to this representation to encourage discarding of any information not required by the exploitation policy (i.e., task-irrelevant information).
Then, we learn an exploration policy to only discover task-relevant information by training it to produce trajectories containing the same information as the learned ID representation (Section 4).
Crucially, unlike prior work, we prove that our separate objectives are consistent : optimizing them yields optimal exploration and exploitation, assuming expressive-enough policy classes and enough meta-training data (Section 5.1).
Overall, we present two core contributions: (i) we articulate and formalize a chicken-and-egg coupling problem between optimizing exploration and exploitation in meta-RL (Section 4.1); and (ii) we overcome this with a consistent decoupled approach, called D REAM :
D ecoupling explo R ation and
E xploit A tion in M eta-RL (Section 4.2).
Theoretically, in a simple tabular example, we show that addressing the coupling problem with D REAM provably improves sample complexity over existing end-to-end approaches by a factor exponential in the horizon (Section 5).
Empirically, we stress test
D REAM ’s ability to learn sophisticated exploration strategies on 3 challenging, didactic benchmarks and a sparse-reward 3D visual navigation benchmark.
On these, D REAM learns to optimally explore and exploit, achieving 90% higher returns than existing state-of-the-art approaches (P EARL , E-RL 2 ,
I MPORT , V ARI BAD), which struggle to learn an effective exploration strategy (Section 6).
R ELATED W ORK
We draw on a long line of work on learning to adapt to related tasks (Schmidhuber, 1987;
Thrun &
Pratt, 2012; Naik & Mammone, 1992; Bengio et al., 1991; 1992; Hochreiter et al., 2001; Andrychowicz et al., 2016; Santoro et al., 2016).
Many meta-RL works focus on adapting efﬁciently to a new task from few samples without optimizing the sample collection process, via updating the policy parameters (Finn et al., 2017; Agarwal et al., 2019;
Yang et al., 2019; Houthooft et al., 2018; Mendonca et al., 2019), learning a model (Nagabandi et al., 2018; Sæmundsson et al., 2018; Hiraoka et al., 2020), multi-task learning (Fakoor et al., 2019), or leveraging demonstrations (Zhou et al., 2019a).
In contrast, we focus on problems where targeted exploration is critical for few-shot adaptation.
Approaches that speciﬁcally explore to obtain the most informative samples fall into two main categories: end-to-end and decoupled approaches.
End-to-end approaches optimize exploration and exploitation end-to-end by updating exploration behavior from returns achieved by exploitation (Duan et al., 2016;
Wang et al., 2016a; Mishra et al., 2017; Rothfuss et al., 2018; Stadie et al., 2018; Zintgraf et al., 2019; Humplik et al., 2019; Kamienny et al., 2020; Dorfman & Tamar, 2020).
These approaches can represent the optimal policy (Kaelbling et al., 1998), but they struggle to escape local optima due to a chicken-and-egg problem between learning to explore and learning to exploit (Section 4.1).
Several of these approaches (Humplik et al., 2019; Kamienny et al., 2020) also leverage the problem
ID during meta-training, but they still learn end-to-end, so the chicken-and-egg problem remains.
Decoupled approaches instead optimize separate exploration and exploitation objectives, via, e.g.,
Thompson-sampling (TS) (Thompson, 1933; Rakelly et al., 2019), obtaining exploration trajectories predictive of dynamics or rewards (Zhou et al., 2019b; Gurumurthy et al., 2019; Zhang et al., 2020), or exploration noise (Gupta et al., 2018).
While these works do not identify the chicken-and-egg problem, decoupled approaches coincidentally avoid it.
However, existing decoupled approaches, including those (Rakelly et al., 2019; Zhang et al., 2020) that leverage the problem ID, do not learn optimal exploration: TS (Rakelly et al., 2019) explores by guessing the task and executing a policy for that task, and hence cannot represent exploration behaviors that are different from exploitation (Russo et al., 2017).
Predicting the dynamics (Zhou et al., 2019b; Gurumurthy et al., 2019; Zhang et al., 2020) is inefﬁcient when only a small subset of the dynamics are relevant to solving the task.
In rr 1 rr 1 rrr
Cook pizza (gather information) ...
s...
Cook soup sss T exploration episode
Trial... ss 0 s 1 ssr [Figure] [Figure] maximize returns) (
[Figure]
[Figure]
contrast, we propose a separate mutual information objective for exploration, which both avoids the chicken-and-egg problem and yields optimal exploration when optimized (Section 5).
Past work (Gregor et al., 2016; Houthooft et al., 2016;
Eysenbach et al., 2018; Warde-Farley et al., 2018) also optimize mutual information objectives, but not for meta-RL.
Exploration in general RL.
The general RL setting (i.e., learning from scratch) also requires targeted exploration to gather informative samples that enables learning a policy to solve the problem.
In contrast to exploration algorithms for general RL (Bellemare et al., 2016; Pathak et al., 2017;
Burda et al., 2018; Leibfried et al., 2019), which must visit many novel states to ﬁnd regions with high reward, exploration in meta-RL can be even more targeted by leveraging prior experience from different problems during meta-training.
As a result, D REAM can learn new tasks in just two episodes (Section 6), while learning from scratch can require thousands or even millions of episodes.
P RELIMINARIES
Meta-reinforcement learning.
The meta-RL setting considers a family of Markov decision processes (MDPs) (cid:104)S , A , R µ , T µ (cid:105) with states S , actions A , rewards R µ , and dynamics T µ , indexed by a one- hot problem ID µ ∈ M , drawn from a distribution p ( µ ) .
Colloquially, we refer to the dynamics as the environment , the rewards as the task , and the entire MDP as the problem .
Borrowing terminology from
Duan et al.
(2016), meta-training and meta-testing both consist of repeatedly running trials .
Each trial consists of sampling a problem ID µ ∼ p ( µ ) and running N + 1 episodes on the corresponding problem.
Following prior evaluation settings (Finn et al., 2017; Rakelly et al., 2019; Rothfuss et al., 2018; Fakoor et al., 2019), we designate the ﬁrst episode in a trial as an exploration episode consisting of T steps for gathering information, and deﬁne the goal as maximizing the returns in the subsequent
N exploitation episodes (Figure 1).
Following Rakelly et al.
; Humplik et al.
; Kamienny et al.
(2020), the easy-to-provide problem ID is available for meta-training, but not meta-testing trials.
We formally express the goal in terms of an exploration policy π exp used in the exploration episode and an exploitation policy π task used in exploitation episodes, but these policies may be the same or share parameters.
Rolling out π exp in the exploration episode produces an exploration trajectory τ exp =
( s 0 , a 0 , r 0 , . . .
, s T ) , which contains information discovered via exploration.
The exploitation policy π task may then condition on τ exp and optionally, its history across all exploitation episodes in a trial, to maximize exploitation episode returns.
The goal is therefore to maximize:
[Equation]
where V task ( τ exp ; µ ) is the expected returns of π task conditioned on τ exp , summed over the N exploitation episodes in a trial with problem ID µ .
End-to-end meta-RL.
A common meta-RL approach (Wang et al., 2016a; Duan et al., 2016; Rothfuss et al., 2018; Zintgraf et al., 2019; Kamienny et al., 2020; Humplik et al., 2019) is to learn to explore and exploit end-to-end by directly optimizing J in (1), updating both from rewards achieved during exploitation.
These approaches typically learn a single recurrent policy π ( a t | s t , τ : t ) for both exploration and exploitation (i.e., π task =
π exp = π ), which takes action a t given state s t and history of experiences spanning all episodes in a trial τ : t = ( s 0 , a 0 , r 0 , . . .
, s t − 1 , a t − 1 , r t − 1 ) .
Intuitively, this policy is learned by rolling out a trial, producing an exploration trajectory τ exp and, conditioned on τ exp and the exploitation experiences so far, yielding some exploitation episode returns.
Then, credit is assigned to both exploration (producing τ exp ) and exploitation by backpropagating the exploitation returns through the recurrent policy.
Directly optimizing the objective J this way can learn optimal exploration and exploitation strategies, but optimization is challenging, which we show in Section 4.1.
[Figure]
D ECOUPLING E XPLORATION AND E XPLOITATION 4. T HE P ROBLEM WITH C OUPLING E XPLORATION AND E XPLOITATION
We begin by showing that end-to-end optimization struggle with local optima due to a chicken-andegg problem.
Figure 2a illustrates this.
Learning π exp relies on gradients passed through π task .
If π task cannot effectively solve the task, then these gradients will be uninformative.
However, to learn to efﬁciently solve the task, π task needs good exploration data (trajectories τ exp ) from a good exploration policy π exp .
This results in bad local optima as follows: if our current (suboptimal) π task obtains low rewards with a good informative trajectory τ expgood , the low reward would cause π exp to learn to not generate τ expgood .
This causes π exp to instead generate trajectories τ expbad that lack information required to obtain high reward, further preventing the exploitation policy π task from learning.
Typically, early in training, both π exp and π task are suboptimal and hence will likely reach this local optimum.
In
Section 5.2, we illustrate how this local optimum can cause sample inefﬁciency in a simple example.
4. D REAM : D ECOUPLING E XPLORATION AND E XPLOITATION IN M ETA -L EARNING
While we can sidestep the local optima of end-to-end training by optimizing separate objectives for exploration and exploitation, the challenge is to construct objectives that yield the same optimal solution as the end-to-end approach.
We now discuss how we can use the easy-to-provide problem
IDs during meta-training to do so.
A good exploration objective should encourage discovering task-relevant distinguishing attributes of the problem (e.g., ingredient locations), and ignoring taskirrelevant attributes (e.g., wall color).
To create this objective, the key idea behind D REAM is to learn to extract only the task-relevant information from the problem ID, which encodes all information about the problem.
Then, D REAM ’s exploration objective seeks to recover this task-relevant information.
Concretely, D REAM extracts only the task-relevant information from the problem ID µ via a stochastic encoder F ψ ( z | µ ) .
To learn this encoder, we train an exploitation policy π task to maximize rewards, conditioned on samples z from F ψ ( z | µ ) , while simultaneously applying an information bottleneck to z to discard information not needed by π task (i.e., task-irrelevant information).
Then, D REAM learns an exploration policy π exp to produce trajectories with high mutual information with z .
In this approach, the exploitation policy π task no longer relies on effective exploration from π exp to learn, and once F ψ ( z | µ ) is learned, the exploration policy also learns independently from π task , decoupling the two optimization processes.
During meta-testing, when µ is unavailable, the two policies easily combine, since the trajectories generated by π exp are optimized to contain the same information as the encodings z ∼ F ψ ( z | µ ) that the exploitation policy π task trained on (overview in Figure 2b).
Learning the problem ID encodings and exploitation policy.
We begin with learning a stochastic encoder F ψ ( z | µ ) parametrized by ψ and exploitation policy π task θ parametrized by θ , which conditions on z .
We learn F ψ jointly with π task θ by optimizing the following objective
[Equation]
where V π task θ ( z ; µ ) is the expected return of π task θ on problem µ given and encoding z .
The information bottleneck term encourages discarding any (task-irrelevant) information from z that does not help maximize reward.
Importantly, both terms are independent of the exploration policy π exp .
This objective is derived from forming the Lagrangian of a constrained optimization problem, where λ −is the dual variable, detailed in Appendix E.
We minimize the mutual information I ( z ; µ ) by minimizing a variational upper bound on it, E µ [ D KL ( F ψ ( z | µ )
|| r ( z ))] , where r is any prior and z is distributed as p ψ ( z ) = (cid:82) µ F ψ ( z | µ ) p ( µ ) dµ .
Learning an exploration policy given problem ID encodings.
Once we’ve obtained an encoder
F ψ ( z | µ ) to extract only the necessary task-relevant information required to optimally solve each task, we can optimize the exploration policy π exp to produce trajectories that contain this same information by maximizing their mutual information
I ( τ exp ; z ) .
We slightly abuse notation to use π exp to denote the probability distribution over the trajectories τ exp .
Then, the mutual information I ( τ exp ; z ) can be efﬁciently maximized by maximizing a variational lower bound (Barber & Agakov, 2003) as follows:
[Equation]
where q ω is any distribution parametrized by ω .
We maximize the above expression over ω to learn q ω that approximates the true conditional distribution p ( z | τ exp ) , which makes this bound tight.
In addition, we do not have access to the problem µ at test time and hence cannot sample from F ψ ( z | µ ) .
Therefore, q ω serves as a decoder that generates the encoding z from the exploration trajectory τ exp .
Recall, our goal is to maximize (3) w.r.t., trajectories τ exp from the exploration policy π exp .
Only the third term depends on τ exp , so we train π exp on rewards set to be this third term (information gain):
[Equation]
Intuitively, the exploration reward for taking action a t and transitioning to state s t +1 is high if this transition encodes more information about the problem (and hence the encoding z ∼ F ψ ( z | µ ) ) than was already present in the trajectory τ exp : t −
= ( s 0 , a 0 , r 0 , . . .
, s t − 2 , a t − 2 , r t − 2 ) .
We also include a small penalty c to encourage exploring efﬁciently in as few timesteps as possible.
This reward is attractive because (i) it is independent from the exploitation policy and hence avoids the local optima described in Section 4.1, and (ii) it is dense, so it helps with credit assignment.
It is also non-Markov, since it depends on τ exp , so we maximize it with a recurrent π exp φ ( a t | s t , τ exp : t ) , parametrized by φ .
4. A P RACTICAL I MPLEMENTATION OF D REAM
Altogether, D REAM learns four separate neural network components, which we detail below.
Encoder F ψ ( z | µ ) :
For simplicity, we parametrize the stochastic encoder by learning a deterministic encoding f ψ ( µ ) and apply Gaussian noise, i.e., F ψ ( z | µ ) =
N ( f ψ ( µ ) , ρ 2 I ) .
We choose a convenient prior r ( z ) to be a unit Gaussian with same variance ρ 2 I , which makes the information bottleneck take the form of simple (cid:96) 2 -regularization
(cid:107) f
ψ
( µ
(cid:107) 22 .
Decoder q ω ( z | τ exp ) :
Similarly, we parametrize the decoder q ω ( z | τ exp ) as a Gaussian centered around a deterministic encoding g ω ( τ exp ) with variance ρ 2 I .
Then, q ω maximizes
E µ,z ∼ F ψ ( z | µ ) (cid:104) (cid:107)
z − g ω ( τ exp ) (cid:107) 2 2 (cid:105) w.r.t.
, ω (Equation 3), and the exploration rewards take the form r
[Equation]
[Equation]
Exploitation policy π task θ and 4.
Exploration policy π exp φ : We learn both policies with double deep Q-learning (van Hasselt et al., 2016), treating ( s, z ) as the state for π task θ .
For convenience, we jointly learn all components in an EM-like fashion, where in the exploration episode, we assume f ψ and π task θ are ﬁxed.
There is no chicken-and-egg effect in this joint training because the exploitation policy (along with the stochastic encoder) are trained independently from the exploration policy; the training of the exploration policy only uses the stochastic encodings.
Appendix A includes all details and a summary (Algorithm 1).
A NALYSIS OF D REAM 5. T HEORETICAL C ONSISTENCY OF THE D REAM O BJECTIVE
A key property of D REAM is that it is consistent : maximizing our decoupled objective also maximizes expected returns (Equation 1).
This contrasts prior decoupled approaches (Zhou et al., 2019b;
Rakelly et al., 2019; Gupta et al., 2018; Gurumurthy et al., 2019; Zhang et al., 2020), which also decouple exploration from exploitation, but do not recover the optimal policy even with inﬁnite data.
Formally,
[Figure]
Proposition 1.
Assume (cid:104)S , A , R µ , T µ (cid:105) is ergodic for all problems µ ∈ M .
Let V ∗ ( µ ) be the maximum expected returns achievable by any exploitation policy with access to the problem ID µ ,
i
[Figure]
e., with complete information.
Let π task (cid:63) , π exp (cid:63) , F (cid:63) and q (cid:63) ( z | τ exp ) be the optimizers of the D REAM objective.
Then, if the function classes D REAM optimizes over are well-speciﬁed, there exists a ﬁnite
T such that if the length of the exploration episode is at least T ,
[Equation]
Optimizing D REAM ’s objective achieves the maximal returns V ∗ ( µ ) even without access to µ during meta-testing (proof in Appendix C.1).
We can remove the ergodicity assumption by increasing the number of exploration episodes, and D REAM performs well on non-ergodic MDPs in our experiments.
5. A N E XAMPLE I LLUSTRATING THE I MPACT OF C OUPLING ON S AMPLE C OMPLEXITY
With enough capacity, end-to-end approaches can also learn the optimal policy, but can be highly sample inefﬁcient due to the coupling problem in Section 4.1.
We highlight this in a simple tabular example to remove the effects of function approximation: Each episode is a one-step bandit problem with action space A .
Taking action a (cid:63) in the exploration episode leads to a trajectory τ exp (cid:63) that reveals the problem ID µ ; all other actions
a reveal no information and lead to τ exp a .
The ID µ identiﬁes a unique action that receives reward 1 during exploitation; all other actions get reward 0 .
Therefore, taking a (cid:63) during exploration is necessary and sufﬁcient to obtain optimal reward 1 .
We now study the number of samples required for RL 2 (the canonical end-to-end approach) and D REAM to learn the optimal exploration policy with (cid:15) -greedy tabular Q-learning.
We precisely describe a more general setup in Appendix C.2 and prove that D REAM learns the optimal exploration policy in Ω( |A| H |M| ) times fewer samples than RL 2 in this simple setting with horizon H .
Figure 3a empirically validates this result and we provide intuition below.
In the tabular analog of RL 2 , the exploitation Q-values form targets for the exploration Q-values: ˆ Q exp ( a ) ← ˆ V task ( τ exp a ) := max a (cid:48)
ˆ Q task ( τ exp a , a (cid:48) ) .
We drop the ﬁxed initial state from notation.
This creates the local optimum in Section 4.1.
Initially ˆ V task ( τ exp (cid:63) ) is low, as the exploitation policy has not learned to achieve reward, even when given τ exp (cid:63) .
This causes ˆ Q exp ( a (cid:63) ) to be small and therefore arg max a ˆ Q exp ( a ) (cid:54) = a (cid:63) (Figure 3b), which then prevents ˆ V task ( τ exp (cid:63) ) from learning (Figure 3c) as τ exp (cid:63) is roughly sampled only once per |A| (cid:15) episodes.
This effect is mitigated only when ˆ Q exp ( a (cid:63) ) becomes higher than ˆ Q exp ( a ) for the other uninformative a ’s (the dot in Figure 3b-d).
Then, learning both the exploitation and exploration Q-values accelerates, but getting there takes many samples.
In D REAM , the exploration Q-values regress toward the decoder ˆ q : ˆ Q exp ( a ) ← log ˆ q ( µ | τ exp ( a )) .
This decoder learns much faster than ˆ Q task , since it does not depend on the exploitation actions.
Consequently, D REAM ’s exploration policy quickly becomes optimal (dot in Figure 3b), which enables quickly learning the exploitation Q-values and achieving high reward (Figures 3c and 3d).
E XPERIMENTS
Many real-world problem distributions (e.g., cooking) require exploration (e.g., locating ingredients) that is distinct from exploitation (e.g., cooking these ingredients).
Therefore, we desire benchmarks that require distinct exploration and exploitation to stress test aspects of exploration in meta-RL, such as if methods can: (i) efﬁciently explore, even in the presence of distractions; (ii) leverage informative objects (e.g., a map) to aid exploration; (iii) learn exploration and exploitation strategies that generalize to unseen problems; (iv) scale to challenging exploration problems with high-dimensional visual observations.
Existing benchmarks (e.g., MetaWorld (Yu et al., 2019) or MuJoCo tasks like HalfCheetahVelocity
(Finn et al., 2017; Rothfuss et al., 2018)) were not designed to test exploration and are unsuitable for answering these questions.
These benchmarks mainly vary the rewards (e.g., the speed to run at) across problems, so naively exploring by exhaustively trying different exploitation behaviors (e.g., running at different speeds) is optimal.
They further don’t include visual states, distractors, or informative objects, which test if exploration is efﬁcient.
We therefore design new benchmarks meeting the above criteria, testing (i-iii) with didactic benchmarks, and (iv) with a sparse-reward 3D visual navigation benchmark, based on Kamienny et al.
(2020), that combines complex exploration with highdimensional visual inputs.
To further deepen the exploration challenge, we make our benchmarks goal-conditioned.
This requires exploring to discover information relevant to any potential goal, rather than just a single task (e.g., locating all ingredients for any meal vs. just the ingredients for pasta).
Comparisons.
We compare D REAM with state-of-the-art end-to-end (E-RL 2 (Stadie et al., 2018),
V ARI BAD (Zintgraf et al., 2019), and I MPORT (Kamienny et al., 2020)) and decoupled approaches (P EARL -UB, an upper bound on the ﬁnal performance of P EARL (Rakelly et al., 2019)).
For P EARLUB, we analytically compute the expected rewards achieved by optimal Thompson sampling (TS) exploration, assuming access to the optimal problem-speciﬁc policy and true posterior problem distribution.
Like D REAM , I MPORT and P EARL also use the one-hot problem ID, during metatraining.
We also report the optimal returns achievable with no exploration as "No exploration."
Where applicable, all methods use the same architecture.
The full architecture and approach details are in Appendix B.3.
We report the average returns achieved by each approach in trials with one exploration and one exploitation episode, averaged over 3 seeds with 1-standard deviation error bars (full details in
Appendix B).
6. D IDACTIC E XPERIMENTS
[Figure]
We ﬁrst evaluate on the grid worlds shown in Figures 4a and 4b.
The state consists of the agent’s ( x, y ) - position, a one-hot indicator of the object at the agent’s position (none, bus, map, pot, or fridge), a one-hot indicator of the agent’s inventory (none or an ingredient), and the goal.
The actions are move up, down, left, or right; ride bus , which, at a bus, teleports the agent to another bus of the same color; pick up , which, at a fridge, ﬁlls the agent’s inventory with the fridge’s ingredients; and drop , which, at the pot, empties the agent’s inventory into the pot.
Episodes consist of 20 timesteps and the agent receives − 0 .
1 reward at each timestep until the goal, described below, is met (details in Appendix B.1; qualitative results in Appendix B.2).
[Figure]
Targeted exploration.
We ﬁrst test if these methods can efﬁciently explore in the presence of distractions in two versions of the benchmark in Figure 4a: distracting bus and map .
In both, there are 4 possible goals (the 4 green locations).
During each episode, a goal is randomly sampled.
Reaching the goal yields +1 reward and ends the episode.
Thecolored buses each lead to near a different potential green goal location when ridden and in different problems µ , their destinations are set to be 1 of the 4!
different permutations.
The distracting bus version tests if the agent can ignore distractions by including unhelpful gray buses, which are never needed to optimally reach any goal.
In different problems, the gray buses lead to different permutations of the gray locations.
The map version tests if the agent can leverage objects for exploration by including a map that reveals the destinations of the colored buses when touched.
Figure 5 shows the results after 1M steps.
D REAM learns to optimally explore and thus receives optimal reward in both versions: In distracting bus , it ignores the unhelpful gray buses and learns the destinations of all helpful buses by riding them.
In map , it learns to leverage informative objects,
[Figure]
a
[Figure]
R
[Figure]
t u
[Figure]
by visiting the map and ending the exploration episode.
During exploitation, D REAM immediately reaches the goal by riding the correct colored bus.
In contrast, I MPORT and E-RL 2 get stuck in a local optimum, indicative of the coupling problem (Section 4.1), which achieves the same returns as no exploration at all.
They do not explore the helpful buses or map and consequently sub-optimally exploit by just walking to the goal.
V ARI BAD learns slower, likely because it learns a dynamics model, but eventually matches the sub-optimal returns of I MPORT and RL 2 in ~3M steps (not shown).
P EARL achieves sub-optimal returns, even with inﬁnite meta-training (see line for P EARL -UB), as follows.
TS explores by sampling a problem ID from its posterior and executing its policy conditioned on this ID.
Since for any given problem (bus conﬁguration) and goal, the optimal problem-speciﬁc policy rides the one bus leading to the goal, TS does not explore optimally (i.e., explore all the buses or read the map), even with the optimal problem-speciﬁc policy and true posterior problem distribution.
Recall that D REAM tries to remove extraneous information from the problem ID with an information bottleneck that minimizes the mutual information
I ( z ; µ ) between problem IDs and the encoder
F ψ ( z | µ ) .
In distracting bus , we test the importance of the information bottleneck by ablating it from
D REAM .
As seen in Figure 5 (left), this ablation (D REAM (no bottleneck)) wastes its exploration on the gray unhelpful buses, since they are part of the problem, and consequently achieves low returns.
Generalization to new problems.
We test generalization to unseen problems in a cooking benchmark (Figure 4b).
The fridges on the right each contain 1 of 4 different (color-coded) ingredients, determined by the problem ID.
The fridges’ contents are unobserved until the agent uses the "pickup" action at the fridge.
Goals (recipes) specify placing 2 correct ingredients in the pot in the right order.
The agent receives positive reward for picking up and placing the correct ingredients, and negative reward for using the wrong ingredients.
We hold out 1 of the 4 3 = 64 problems from meta-training.
Figure 6 shows the results on training (left) and held-out (middle) problems.
Only D REAM achieves near-optimal returns on both.
During exploration, it investigates each fridge with the "pick up" action, and then directly retrieves the correct ingredients during exploitation.
E-RL 2 gets stuck in a local optimum, only sometimes exploring the fridges.
This achieves 3.8x lower returns, only slightly higher than no exploration at all.
Here, leveraging the problem ID actually hurts I MPORT compared to ERL 2 .
I MPORT successfully solves the task, given access to the problem ID, but fails without it.
As before, V ARI BAD learns slowly and TS (P EARL -UB) cannot learn optimal exploration.
6. S PARSE -R EWARD 3D V ISUAL N AVIGATION
We conclude with a challenging benchmark testing both sophisticated exploration and scalability to pixel inputs.
We modify a benchmark from Kamienny et al.
(2020) to increase both the exploration and scalability challenge by including more objects and a visual sign, illustrated in Figure 7.
In the 3 different problems, the sign on the right says “blue”, “red” or “green.”
The goals specify whether the agent should collect the key or block.
The agent receives +1 reward for collecting the correct object (color speciﬁed by the sign, shape speciﬁed by the goal), -1 reward for the wrong object, and 0 reward otherwise.
The agent begins the episode on the far side of the barrier and must walk around the barrier to visually “read”
D REAM is the only method that learns to read the sign and achieve reward (Figure 6 right).
All endto-end approaches get stuck in local optima due to the chicken-and-egg coupling problem, where they do not learn to read the sign and hence stay away from all the objects, in fear of receiving negative reward.
This achieves close to 0 returns, consistent with the results in Kamienny et al.
As before, P EARL -UB cannot learn optimal exploration.
C ONCLUSION
In summary, this work identiﬁes a chicken-and-egg problem that end-to-end meta-RL approaches suffer from, where learning good exploitation requires already having learned good exploration and vice-versa.
This creates challenging local optima, since typically neither exploration nor exploitation is good at the beginning of meta-training.
We show that appropriately leveraging simple one-hot problem IDs allows us to break this cyclic dependency with D REAM .
Consequently, D REAM has strong empirical performance on meta-RL problems requiring complex exploration, as well as substantial theoretical sample complexity improvements in the tabular setting.
Though prior works also leverage the problem ID and use decoupled objectives that avoid the chicken-and-egg problem, no other existing approaches can recover optimal exploration empirically and theoretically like D REAM .
A D REAM T RAINING D ETAILS
Algorithm 1 summarizes a practical algorithm for training D REAM .
Unlike end-to-end approaches, we choose not to make π task θ recurrent for simplicity, and only condition on z and the current state s .
We parametrize the policies as deep dueling double-Q networks (Wang et al., 2016b; van Hasselt et al., 2016), with exploration Q-values ˆ Q exp ( s, τ exp , a ; φ ) parametrized by φ (and target network parameters φ (cid:48) ) and exploitation Q-values ˆ Q task ( s, z, a ; θ ) parametrized by θ (and target network parameters θ (cid:48) ).
We train on trials with one exploration and one exploitation episode, but can test on arbitrarily many exploitation episodes, as the exploitation policy acts on each episode independently (i.e. it does not maintain a hidden state across episodes).
Using the choices for F ψ and q ω in
Section 4.3, training proceeds as follows.
We ﬁrst sample a new problem for the trial and roll-out the exploration policy, adding the roll-out to a replay buffer (lines 7-9).
Then, we roll-out the exploitation policy, adding the roll-out to a separate replay buffer (lines 10-12).
We train the exploitation policy on both stochastic encodings of the problem ID N ( f ψ ( µ ) , ρ 2 I ) and on encodings of the exploration trajectory g ω ( τ exp ) .
Next, we sample from the replay buffers and update the parameters.
First, we sample ( s t , a t , s t +1 , µ, τ exp ) -tuples from the exploration replay buffer and perform a normal DDQN update on the exploration Q-value parameters φ using rewards computed from the decoder (lines 13-15).
Concretely, we minimize the following standard DDQN loss function w.r.t.
, the parameters φ , where the rewards are computed according to Equation 4:
[Equation]
where r exp t
[Equation]
and a DDQN = arg max a
[Equation]
We perform a similar update with the exploitation Q-value parameters (lines 16-17).
We sample ( s, a, r, s (cid:48) , µ, τ exp ) -tuples from the exploitation replay buffer and perform a DDQN update from the encoding of the problem ID by minimizing the following loss:
[Equation]
where a traj = arg max a ˆ Q task ( s (cid:48) , g ω ( τ exp ) , a ; θ ) and a prob = arg max a ˆ Q task ( s (cid:48) , f ψ ( µ ) , a ; θ ) .
Finally, from the same exploitation replay buffer samples, we also update the problem ID embedder to enforce the information bottleneck (line 18) and the decoder to approximate the true conditional distribution (line 19) by minimizing the following losses respectively:
[Equation]
and L
[Equation]
E
[Equation]
Since the magnitude (cid:107) f ψ ( µ ) (cid:107) 22 partially determines the scale of the reward, we add a hyperparameter
K and only minimize the magnitude when it is larger than K .
Altogether, we minimize the following loss:
[Equation]
As is standard with deep Q-learning (Mnih et al., 2015), instead of sampling from the replay buffers and updating after each episode, we sample and perform all of these updates every 4 timesteps.
We periodically update the target networks (lines 20-22).
[Table] [Table]
[Table] [Table]
[Table] [Table]
[Table] [Table]
[Table]
(cid:13) 2 2 − c (Equation 4).
Optimize φ with DDQN update with tuple ( s t , a t , r exp t , s t +1 ) with L exp ( φ ) 16:
Sample batches of ( s, a, r, s (cid:48) , µ, τ exp ) ∼ B task-id from exploitation replay buffer.
Optimize θ and ψ with DDQN update with tuple (( s, µ ) , a, r, ( s (cid:48) , µ )) with L task-id ( θ, ψ ) 18:
Optimize ψ on L bottleneck ( ψ ) = ∇ ψ min( (cid:107) f ψ ( µ ) (cid:107) 22 , K ) 19:
Optimize ω on L decoder ( ω ) = ∇ ω (cid:80) t (cid:107) f ψ ( µ ) − g ω ( τ exp : t ) (cid:107)
(Equation 3) 20: if trial ≡ 0 (mod target freq ) then 21:
Update target parameters φ (cid:48)
=
φ , θ (cid:48)
= θ , ψ (cid:48) = ψ , ω (cid:48)
=
ω 22: end if 23: end for
[Figure]
B E XPERIMENT D ETAILS B. P ROBLEM D ETAILS
Distracting bus / map.
Riding each of the four colored buses teleports the agent to near one of the green goal locations in the corners.
In different problems, the destinations of the colored buses change, but the bus positions and their destinations are ﬁxed within each problem.
Additionally, in the distracting bus domain, the problem ID also encodes the destinations of the gray buses, which are permutations of the four gray locations on the midpoints of the sides.
More precisely, the problem ID µ ∈ { 0 , 1 , . .
× 4!
= 576 } encodes both the permutation of the colored helpful bus destinations, indexed as µ (mod 4!)
and the permutation of the gray unhelpful bus destinations as (cid:4) µ 4!
(cid:5) .
We hold out most of the problem IDs during meta-training ( 2324 × 576 = 552 are held-out for meta-training).
In the map domain, the problem µ is an integer representing which of the 4!
permutations of the four green goal locations the colored buses map to.
The states include an extra dimension, which is set to 0 when the agent is not at the map, and is set to this integer value µ when the agent is at the map.
Figure 8 displays three such examples.
[Figure]
Cooking.
In different problems, the (colorcoded) fridges contain 1 of 4 different ingredients.
The ingredients in each fridge are unknown until the goes to the fridge and uses the pickup action.
Figure 9 displays three example problems.
The problem ID µ is an integer between 0 and 4 3 , where µ = 4 2 a + 4 b + c indicates that the top right fridge has ingredient a , the middle fridge has ingredient b and the bottom right fridge has ingredient c .
The goals correspond to a recipe of placing the two correct ingredients in the pot in the right order.
Goals are tuples ( a, b ) , which indicate placing ingredient a in the pot ﬁrst, followed by ingredient b .
In a given problem, we only sample goals involving the recipes actually present in that problem.
During meta-training, we hold out a randomly selected problem µ = 11 .
We use the following reward function R µ .
The agent receives a per timestep penalty of − 0 .
1 reward and receives +0 .
25 reward for completing each of the four steps: (i) picking up the ﬁrst ingredient speciﬁed by the goal; (ii) placing the ﬁrst ingredient in the pot; (iii) picking up the second ingredient speciﬁed by the goal; and (iv) placing the second ingredient in the pot.
To prevent the agent from gaming the reward function, e.g., by repeatedly picking up the ﬁrst ingredient, dropping the ﬁrst ingredient anywhere but in the pot yields a penalty of − 0 .
25 reward, and similarly for all steps.
To encourage efﬁcient exploration, the agent also receives a penalty of − 0 .
25 reward for picking up the wrong ingredient.
Cooking without goals.
While we evaluate on goal-conditioned benchmarks to deepen the exploration challenge, forcing the agent to discover all the relevant information for any potential goal, many standard benchmarks (Finn et al., 2017; Yu et al., 2019) don’t involve goals.
We therefore include a variant of the cooking task, where there are no goals.
We simply concatenate the goal (recipe) to the problem ID µ .
Additionally, we modify the rewards so that picking up the second ingredient yields +0 . 25 and dropping it yields − 0 .
25 reward, so that it is possible to infer the recipe from the rewards.
Finally, to make the problem harder, the agent cannot pick up new ingredients unless its inventory is empty (by using the drop action), and we also increase the number of ingredients to 7 .
The results are in Section B.2.
Sparse-reward 3D visual navigation.
We implement this domain in Gym MiniWorld (ChevalierBoisvert, 2018), where the agent’s observations are 80 × 60 × 3 RGB arrays.
There are three problems µ =
(the sign says “blue”), µ = 1 (the sign says “red”), and µ = 2 (the sign says “green”).
There are two goals, represented as 0 and 1 , corresponding to picking up the key and the box, respectively.
The reward function A µ ( s, i ) is +1 for picking up the correct colored object (according to µ ) and the correct type of object (according to the goal) and − 1 for picking up an object of the incorrect color or
[Figure]
type.
Otherwise, the reward is 0.
On each episode, the agent begins at a random location on the other side of the barrier from the sign.
B. A DDITIONAL R ESULTS
Analysis of the learned policies.
Please see https://anonymouspapersubmission.
github.io/dream/ for videos and analysis of the exploration and exploitation behavior learned by D REAM and other approaches, which is described in text below.
Distracting bus / map.
Figure 10 shows the exploration policy D REAM learns on the distracting bus and map domains.
With the information bottleneck, D REAM optimally explores by riding 3 of thecolored buses and inferring the destination of the last colored bus (Figure 8a).
Without the information bottleneck, D REAM explores the unhelpful gray buses and runs out of time to explore all of the colored buses, leading to lower reward (Figure 8b).
In the map domain, D REAM optimally explores by visiting the map and terminating the exploration episode.
In contrast, the other methods (RL 2 , I MPORT ,
V ARI BAD) rarely visit the colored buses or map during exploration and consequently walk to their destination during exploitation, which requires more timesteps and therefore receives lower returns.
In Figure 11, we additionally visualize the exploration trajectory encodings g ω ( τ exp ) and problem
ID encodings f ψ ( µ ) that D REAM learns in the distracting bus domain by applying t-SNE (van der
Maaten & Hinton, 2008).
We visualize the encodings of all possible problem IDs as dots.
They naturally cluster into 4!
= 24 clusters, where the problems within each cluster differ only in the destinations of the gray distracting buses, and not the colored buses.
Problems in the support of the true posterior p ( µ | τ exp ) are drawn in green, while problems outside the support (e.g., a problem that speciﬁes that riding the green bus goes to location (0 , 1) when it has already been observed in τ exp that riding the orange bus goes to location (0 , 1) ) are drawn in red.
We also plot the encoding of the exploration trajectory τ exp so far as a blue cross and the mean of the green clusters as a black square.
We ﬁnd that the encoding of the exploration trajectory g ω ( τ exp ) tracks the mean of the green clusters until the end of the exploration episode, when only one cluster remains, and the destinations of all the colored buses has been discovered.
Intuitively, this captures uncertainty in what the potential problem
ID may be.
More precisely, when the decoder is a Gaussian, placing g ω ( τ exp ) at the center of the encodings of problems in the support exactly minimizes Equation 3.
[Figure] [Figure]
[Figure]
Robustness to imperfections in the problem ID.
Recall that the problem ID is a simple and easyto-provide unique one-hot for each problem.
We test of D REAM is robust to imperfections in the problem ID by assigning each problem in the map benchmark 3 different problem IDs.
When a problem is sampled during meta-training, it is randomly labeled with 1 of these 3 different problem
IDs.
We ﬁnd that this imperfection in the problem ID does not impact D REAM ’s ﬁnal performance at all: it still achieves optimal exploitation returns of 0 .
8 after 1 M time steps of training.
Cooking.
Figure 12 shows the exploration policy D REAM learns on the cooking domain, which visits each of the fridges and investigates the contents with the "pickup" action.
In contrast, the other methods rarely visit the fridges during exploration
[Figure]
and instead determine the locations of the ingredients during exploitation
[Figure]
which requires more timesteps and therefore receives lower returns.
Cooking without goals.
We provide additional results in the case where the cooking domain is modiﬁed to not include goals (see Section B.1).
The results are summarized in Figure 13 and show the same trends as the results in original cooking benchmark.
D REAM learns to optimally explore by investigating the fridges, and then also optimally exploits, by directly collecting the relevant ingredients.
The next best approach E-RL 2 , only sometimes explores the fridges, again getting stuck in a local optimum, yielding only slightly higher reward than no exploration at all.
Sparse-reward 3D visual navigation.
D REAM optimally explores by walking around the barrier and reading the sign.
See https://anonymouspapersubmission.github.io/dream/ for videos.
The other methods do not read the sign at all and therefore cannot solve the problem.
B. O THER A PPROACHES AND A RCHITECTURE D ETAILS
In this section, we detail the loss functions that E-RL 2 , I MPORT , and V ARI BAD optimize, as well as the model architectures used in our experiments.
Where possible, we use the same model architecture for all methods: D REAM , E-RL 2 , I MPORT , and V ARI BAD.
All approaches are implemented in
PyTorch (Paszke et al., 2017), using a DQN implementation adapted from Liu et al.
(2020b)
and code adapted from Liu et al.
(2020a).
State and problem ID embeddings.
All approaches use the same method to embed the state and problem ID.
For these embeddings, we embed each dimension independently with an embedding matrix of output dimension 32.
Then, we concatenate the per-dimension embeddings and apply two linear layers with output dimensions 256 and 64 respectively, with ReLU activations.
In the 3D visual navigation task, we use a different embedding scheme for the states, as they are images.
We apply 3 CNN layers, each with 32 output layers and stride length 2, and with kernel sizes of 5, 5, and 4 respectively.
We apply ReLU activations between the CNN layers and apply a ﬁnal linear layer to the ﬂattened output of the CNN layers, with an output dimension of 128.
All state and problem ID embeddings below use this scheme.
Experience embeddings.
E-RL 2 , I MPORT , V ARI BAD and the exploration policy in D REAM also learn an embedding of the history of prior experiences τ exp =
( s 0 , a 0 , r 0 , s 1 , . . . )
and current state s T .
To do this, we ﬁrst separately embed each ( s t +1 , a t , r t , d t ) -tuple, where d t is an episode termination ﬂag (true if the episode ends on this experience, and false otherwise), as follows:
• Embed the state s t as e ( s t ) , using the state embedding scheme described above.
• Embed the action a t as e ( a t ) with an embedding matrix of output dimension 16.
We set a − 1 to be 0.
• For the standard meta-RL setting and during exploitation episodes, embed the rewards with a linear layer of output dimension 16.
With reward-free exploration in IMRL, the rewards are not embedded in the exploration policy of D REAM , and are embedded as 0 reward for the other approaches, since the same policy is used during both exploration and exploitation.
We set r − 1 to be 0.
• Embed the episode termination d t as e ( d t ) with an embedding matrix of output dimension 16.
Note that d is true during all episode terminations within a trial for RL 2 , I MPORT , and V ARI BAD.
Then, we apply a ﬁnal linear layer with output dimension 64 to the concatenation of the above [ e ( s t ); e ( a t ); e ( r t );
d t ] .
Finally, to obtain an embedding of the entire history τ exp , we embed each experience separately as above, and then pass an LSTM with hidden dimension 64 over the experience embeddings, where the initial hidden and cell states are set to be 0-vectors.
D REAM .
For the decoder g ω ( τ exp =
( s 0 , a 0 , r 0 , s 1 , . .
, s T )) , we embed each transition ( s t , a t , r t , s t +1 ) of the exploration trajectory τ exp using the same embedding scheme as above, except we also embed the next state s t +1 .
We do not embed the rewards in the reward-free adaptation version of IMRL.
Then, given embeddings for each transition, we embed the entire trajectory by passing an LSTM with output dimension 128 on top of the transition embeddings, followed by two linear layers of output dimension 128 and 64 with ReLU activations.
For the exploitation policy Q-values ˆ Q task θ ( a | s, z ) , we either choose z to be the decoder embedding of the exploration trajectory g ω ( τ exp ) or to be an embedding of the problem ID e θ ( µ ) , where we always use the exploration trajectory embedding g ω ( τ exp ) at meta-test time.
We embed the state with a learned embedding functions e ( s ) .
Then we apply a linear layer of output dimension 64 to the concatenation of [ e ( s ); z ] with a ReLU activation.
Finally, we apply two linear layer heads of output dimension 1 and |A| respectively to form estimates of the value and advantage functions, using the dueling Q-network parametrization.
To obtain Q-values, we add the value function to the advantage function, subtracting the mean of the advantages.
For the exploration policy Q-values ˆ Q exp φ ( a t | s t , τ exp : t ) , we embed the s t and τ exp : t according to the embedding scheme above.
Then, we apply two linear layer heads to obtain value and advantage estimates as above.
E-RL 2 .
E-RL 2 learns a policy π ( a t | s t , τ : t ) producing actions a t given the state s t and history τ : t .
Like with all approaches, we parametrize this with dueling double Q-networks, learning Q-values ˆ Q ( s t , τ : t , a t ) .
We embed the current state s t and history τ : t using the embedding scheme described above (with episode termination embeddings).
Then, we apply two ﬁnal linear layer heads to obtain value and advantage estimates.
I MPORT
I MPORT also learns a recurrent policy π ( a t | s t , z ) , but conditions on the embedding z , which is either an embedding of the problem µ or the history τ : t .
We also parametrize this policy with dueling double Q-networks, learning Q-values ˆ Q ( s t , z, a t ) .
We embed the state s t as e ( s t ) , the problem µ as e φ ( µ ) and the history τ : t as e θ ( τ : t ) using the previously described embedding schemes.
Then we alternate meta-training trials between choosing z = e φ ( µ ) and
z = e θ ( τ : t ) .
We apply a linear layer of output dimension 64 to the concatenation [ e ( s t ); z ] with ReLU activations and then apply two linear layer heads to obtain value and advantage estimates.
Additionally, I MPORT uses the following auxiliary loss function to encourage the history embedding e θ ( τ : t ) to be close to the problem embedding e φ ( µ ) (optimized only w.r.t.
, θ ):
[Equation]
(cid:34)(cid:88)
[Equation]
where τ is a trajectory from rolling out the policy on problem µ .
Following Kamienny et al.
(2020), we use β = 1 in our ﬁnal experiments, and found that performance changed very little when we experimented with other values of β .
V ARI BAD.
V ARI BAD also learns a recurrent policy π ( a t | z ) , but over a belief
state z capturing the history τ : t and current state s t .
We also parametrize this dueling double Q-networks, learning
Q-values ˆ Q ( s t , z, a t ) .
V ARI BAD encodes the belief state with an encoder enc ( z | s t , τ : t ) .
Our implementation of this encoder embeds s t and τ : t using the same experience embedding approach as above, and use the output as the mean m for a distribution.
Then, we set enc ( z | s t , τ : t ) =
N ( m, ν 2 I ) , where ν 2 = 0 .
We also tried learning the variance instead of ﬁxing it to ν 2 I by applying a linear head to the output of the experience embeddings, but found no change in performance, so stuck with the simpler ﬁxed variance approach.
Finally, after sampling z from the encoder, we also embed the current state s t as e ( s t ) and apply a linear layer of output dimension 64 to the concatenation [ e ( s t ); z ] .
Then, we apply two linear layer heads to obtain value and advantage estimates.
V ARI BAD does not update the encoder via gradients through the policy.
Instead, V ARI BAD jointly trains the encoder with state decoder ˆ T ( s (cid:48) | a, s, z ) and reward decoder ˆ R ( s (cid:48) | a, s, z ) , where z is sampled from the encoder, as follows.
Both decoders embed the action a as e ( a ) with an embedding matrix of output dimension 32 and embed the state s as e ( s ) .
Then we apply two linear layers with output dimension 128 to the concatenation [ e ( s ); e ( a ); z ] .
Finally, we apply two linear heads, one for the state decoder and one for the reward decoder and take the mean-squared error with the true next state s (cid:48) and the true rewards r respectively.
In the 3D visual navigation domain, we remove the state decoder, because the state is too high-dimensional to predict.
Note that Zintgraf et al.
(2019) found better results when removing the state decoder in all experiments.
We also tried to remove the state decoder in the grid world experiments, but found better performance when keeping the state decoder.
We also found that V ARI BAD performed better without the KL loss term, so we excluded that for our ﬁnal experiments.
B. H YPERPARAMETERS
In this section, we detail the hyperparameters used in our experiments.
Where possible, we used the default DQN hyperparameter values from Mnih et al. (2015).
and shared the same hyperparameter values across all methods for fairness.
We optimize all methods with the Adam optimizer ( ? ).
Table 1 summarizes the shared hyperparameters used by all methods and we detail any differences in hyperparameters between the methods below.
[Table]
All methods use a linear decaying (cid:15) schedule for (cid:15) -greedy exploration.
For RL 2 , I MPORT , and
V ARI BAD, we decay (cid:15) from 1 to 0 .
01 over 500000 steps.
For D REAM , we split the decaying between the exploration and exploitation policies.
We decay each policy’s (cid:15) from 1 to 0.01 over 250000 steps.
We train the recurrent policies (D REAM ’s exploration policy, RL 2 , I MPORT , and V ARI BAD) with a simpliﬁed version of the methods in Kapturowski et al.
(2019) by storing a replay buffer with up to 16000 sequences of 50 consecutive timesteps.
We decrease the maximum size from 16000 tofor the 3D visual navigation experiments in order to ﬁt inside a single NVIDIA GeForce RTX
GPU.
For D REAM ’s exploitation policy, the replay buffer stores up to 100K experiences (60K for 3D visual navigation).
For D REAM , we additionally use per timestep exploration reward penalty c = 0 .
01 , decoder and stochastic encoder variance ρ 2 = 0 .
1 , and information bottleneck weight λ = 1 .
Note that this information bottleneck weight λ could be adapted via dual gradient descent to solve the constrained optimization problem in Appendix E, but we ﬁnd that dynamically adjusting λ is not necessary for good performance.
For the MiniWorld experiments, we use c = 0 .
C A NALYSIS C. C ONSISTENCY
We restate the consistency result of D REAM (Section 5.1) and prove it below.
Proposition 1.
Assume (cid:104)S , A , R µ , T µ (cid:105) is ergodic for all problems µ ∈ M .
Let V ∗ ( µ ) be the maximum expected returns achievable by any exploitation policy with access to the problem ID µ , i.e., with complete information.
Let π task (cid:63) , π exp (cid:63) , F (cid:63) and q (cid:63) ( z | τ exp ) be the optimizers of the D REAM objective.
Then, if the function classes D REAM optimizes over are well-speciﬁed, there exists a ﬁnite
T such that if the length of the exploration episode is at least T ,
[Equation]
Proof.
Recall that π task (cid:63) and F (cid:63) ( z | µ ) are optimized with an information bottleneck according to
Equation 2 in order to solve the constrained optimization problem in Appendix E. Note that if π task (cid:63) is optimized over an expressive enough function class and λ approaches 0 , which is necessary to solve the constrained optimization problem associated with the information bottleneck (Appendix E), then π task (cid:63) achieves the desired expected returns conditioned on the stochastic encoding of the problem
F (cid:63) ( z | µ ) (i.e., has complete information):
[Equation]
where V π task (cid:63) ( z ; µ ) is the expected returns of π task (cid:63) on problem µ given embedding z .
Therefore, it sufﬁces to show that the distribution over z from the decoder q (cid:63) ( z | τ exp ) is equal to the distribution from the encoder F (cid:63) ( z | µ ) for all exploration trajectories in the support of π exp ( τ exp | µ ) 2 , for each problem µ .
Then,
[Equation]
as desired.
We show that this occurs as follows.
Given stochastic encoder F (cid:63) ( z | µ ) , exploration policy π exp (cid:63) maximizes I ( τ exp ; z )
=
H ( z ) − H
( z | τ exp )
(Equation 3) by assumption.
Since only H ( z | τ exp ) depends on π exp (cid:63) , the exploration policy outputs trajectories that minimize
[Equation]
where p ( z | τ exp ) is the true conditional distribution and H ( F (cid:63) ( z | µ ) , p ( z | τ exp )) is the crossentropy.
The cross-entropy is minimized when p ( z | τ exp ) =
F (cid:63) ( z | µ ) , which can be achieved with long enough exploration trajectories T
if (cid:104)S , A , R µ , T µ (cid:105) is ergodic (by visiting each transition sufﬁciently many times).
Optimized over an expressive enough function class, q (cid:63) ( z | τ
[Equation]
) equals the true conditional distribution p ( z | τ exp ) at the optimum of Equation 3, which equals F (cid:63) ( z | µ ) as desired.
[Equation]
T ABULAR E XAMPLE
We ﬁrst formally detail a more general form of the simple tabular example in Section 5.2, where episodes are horizon H rather than 1-step bandit problems.
Then we prove sample complexity bounds for RL 2 and D REAM , with (cid:15) -greedy tabular Q-learning with (cid:15)
= 1 , i.e., uniform random exploration.
Setting.
We construct this horizon H setting so that taking a sequence of H actions a (cid:63) (instead of a single action as before) in the exploration episode leads to a trajectory τ exp (cid:63) that reveals the problem µ ; all other action sequences a lead to a trajectory τ exp a that reveals no information.
Similarly, the problem µ identiﬁes a unique sequence of H actions a µ that receives reward 1 during exploitation, while all other action sequences receive reward 0 .
Again, taking the action sequence a (cid:63) during exploration is therefore necessary and sufﬁcient to obtain optimal reward 1 during exploitation.
We formally construct this setting by considering a family of episodic MDPs (cid:104)S , A , R µ , T µ (cid:105) parametrized by the problem ID µ ∈ M , where:
• Each exploitation and exploration episode is horizon H .
• The action space
A consists of A discrete actions { 1 , 2 , . . .
, A } .
• The space of problems M = { 1 , 2 , . .
, |A| H } and the distribution p ( µ ) is uniform.
To reveal the problem via the optimal action sequence a (cid:63) and to allow a µ to uniquely receive reward, we construct the state space and deterministic dynamics as follows.
• States s ∈ S are ( H + 2) -dimensional
and the deterministic dynamics are constructed so the ﬁrst index represents the current timestep t , the middle H indices represent the history of actions taken, and the last index reveals the problem ID if a (cid:63) is taken.
The initial state is the zero vector s
=
and we denote the state at the t -th timestep s t as ( t, a 0 , a 1 , . .
. , a t − 1 , 0 , . .
• The optimal exploration action sequence a (cid:63) is set to be taking action 1 for H timesteps.
In problem µ taking action a H − 1 = 1 at state s H − 1 = ( H − 1 , a 0 = 1 , . . .
, a H − 2 = 1 , 0 , 0) (i.e., taking the entire action sequence a (cid:63) ) transitions to the state s H =
( H, a 0 = 1 , . . .
, a H − 2 = 1 , a H − 1 = 1 , µ ) , revealing the problem µ .
• The action sequence a µ identiﬁed by the problem µ is set as the problem µ in base |A| : i.e., a µ is a sequence of H actions ( a 0 , a 1 , . .
, a H − 1 ) with (cid:80) H − 1 t =0 a t |A| t = µ .
In problem µ with a µ = ( a 0 , a 1 , . . .
, a H − 1 ) , taking action a H − 1 at timestep H − 1 at state s H − 1 =
( H − 1 , a 0 , a 1 , . . .
, a H − 2 , 0 , 0) (i.e., taking the entire action sequence a µ ) yields R µ ( s H − 1 , a H − 1 )
= 1 .
Reward is 0 everywhere else: i.e., R µ ( s, a ) = 0 for all other states s and actions a .
• With these dynamics, the exploration trajectory τ exp a =
( s 0 , a 0 , r 0 , . . .
, s H ) is uniquely identiﬁed by the action sequence a
and the problem µ if revealed in s H .
We therefore write τ exp a =
( a , µ ) for when a = a (cid:63) reveals the problem µ , and τ exp a
=
( a , 0) , otherwise.
Uniform random exploration.
In this general setting, we analyze the number of samples required to learn the optimal exploration policy with RL 2 and D REAM via (cid:15) -greedy tabular Q-learning.
We formally analyze the simpler case where (cid:15) = 1 , i.e., uniform random exploration, but empirically ﬁnd that D REAM only learns faster with smaller (cid:15) , and RL 2 only learns slower.
In this particular tabular example with deterministic dynamics that encode the entire action history and rewards, learning a per timestep Q-value is equivalent to learning a Q-value for the entire trajectory.
Hence, we denote exploration Q-values ˆ Q exp ( a ) estimating the returns from taking the entire sequence of H actions a at the initial state s
0 and exeuction Q-values ˆ Q task ( τ exp , a ) estimating the returns from taking the entire sequence of H actions a at the initial state s 0 given the exploration trajectory τ exp .
We drop s 0 from notation, since it is ﬁxed.
Recall that RL 2 learns exploration Q-values ˆ Q exp by regressing toward the exploitation Q-values ˆ Q task .
We estimate the exploitation Q-values ˆ Q task ( τ exp , a ) as the sample mean of returns from taking actions a given the exploration trajectory τ exp and estimate the exploration Q-values ˆ Q exp ( a ) as the sample mean of the targets.
More precisely, for action sequences a (cid:54) = a (cid:63) , the resulting exploration trajectory τ exp a is deterministically ( a , 0) , so we set ˆ Q exp ( a ) =
ˆ V task ( τ exp a ) =
max a (cid:48)
ˆ Q task ( τ exp a , a (cid:48) ) .
For a (cid:63) , the resulting exploration trajectory τ exp a (cid:63) may be any of ( a (cid:63) , µ ) for µ ∈ M , so we set ˆ Q exp ( a (cid:63) ) as the empirical mean of ˆ V task ( τ exp a (cid:63) ) of observed τ exp a (cid:63) .
Recall that D REAM learns exploration Q-values ˆ Q exp by regressing toward the learned decoder log ˆ q ( µ | τ exp a ) .
We estimate the decoder ˆ q ( µ | τ exp a ) as the empirical counts of ( µ, τ exp a ) divided by the empirical counts of τ exp a and similarly estimate the Q-values as the empirical mean of log ˆ q ( µ | τ exp a ) .
We denote the exploration Q-values learned after t timesteps as ˆ Q exp t , and similarly denote the estimated decoder after t timesteps as ˆ q t .
Given this setup, we are ready to state the formal sample complexity results.
Intuitively, learning the exploitation Q-values for RL 2 is slow, because, in problem µ , it involves observing the optimal exploration trajectory from taking actions a (cid:63) and then observing the corresponding exploitation actions a µ , which only jointly happens roughly once per |A| 2 H samples.
Since RL 2 regresses the exploration Q-values toward the exploitation Q-values, the exploration Q-values are also slow to learn.
In contrast, learning the decoder ˆ q ( µ | τ exp a ) is much faster, as it is independent of the exploitation actions, and in particular, already learns the correct value from a single sample of a (cid:63) .
We formalize this intuition in the following proposition, which shows that D REAM learns in a factor of at least |A|
H |M| fewer samples than RL 2 .
Proposition 2.
Let T be the number of samples from uniform random exploration such that the greedy-exploration policy is guaranteed to be optimal (i.e., arg max a ˆ Q exp t ( a )
= a (cid:63) ) for all t ≥ T .
If ˆ Q exp is learned with D REAM , the expected value of T is O ( |A| H log |A| H ) .
If ˆ Q exp is learned with RL 2 , the expected value of T is Ω( |A| 2 H |M| log |A| H ) .
Proof.
For D REAM , ˆ Q exp T ( a (cid:63) ) >
ˆ Q exp T ( a ) for all a (cid:54) = a (cid:63) if log ˆ q T ( µ | ( a (cid:63) , µ )) > log ˆ q T ( µ | ( a , 0)) for all µ and a (cid:54) = a (cid:63) .
For all t ≥ T , ˆ Q exp t is guaranteed to be optimal, since no sequence of samples will cause log ˆ q t ( µ | ( a (cid:63) , µ ))
= 0 ≤ log ˆ q t ( µ | ( a , 0))
for any a (cid:54) = a (cid:63) .
This occurs once we’ve observed ( µ, ( a , 0)) for two distinct µ for each a (cid:54) = a (cid:63) and we’ve observed ( µ, ( a (cid:63) , µ )) for at least one µ .
We can compute an upperbound on the expected number of samples required to observe ( µ, τ exp a ) for two distinct µ for each action sequence a by casting this as a coupon collector problem, where each pair ( µ, τ exp a ) is a coupon.
There are 2 |A| H total coupons to collect.
This yields that the expected number of samples is O ( |A| H log |A| H ) .
For RL 2 , the exploration policy is optimal for all timesteps t ≥ T for some T only if the exploitation values ˆ V task T ( τ exp =
( a (cid:63) , µ ))
= 1 for all µ in M .
Otherwise, there is a small, but non-zero probability that ˆ V task t ( τ exp =
( a , 0)) will be greater at some t > T .
For the exploitation values to be optimal at all optimal exploration trajectories ˆ V task T ( τ exp =
( a (cid:63) , µ ))
= 1 for all µ ∈ M , we must jointly observe exploration trajectory τ exp =
( a (cid:63) , µ ) and corresponding action sequence a µ for each problem µ ∈ M .
We can lower bound the expected number of samples required to observe this by casting this as a coupon collector problem, where each pair ( τ exp =
( a (cid:63) , µ ) , a µ ) is a coupon.
There are
|M| · |A|
H unique coupons to collect and collecting any coupon only occurs with probability 1 |A| H in each episode.
This yields that the expected number of samples is Ω( |A| 2 H · |M| · log |A| H ) .
D E XTENDED R ELATED W ORK
Detailed comparison with I MPORT and Humplik et al.
We compare D REAM with I MPORT (Kamienny et al., 2020) and Humplik et al.
(2019) in greater detail, since they also leverage the problem ID.
The key difference between D REAM and these works is that these works still learn exploration from returns achieved during exploitation, which leads to the chicken-and-egg problem.
In contrast, D REAM learns exploration by maximizing the mutual information between exploration trajectories and learned z ’s containing only task-relevant information (Section 4.2).
More speciﬁcally, like D REAM , I MPORT also conditions on the problem ID to learn exploitation without the need for exploration, which can accelerate learning exploration, but unlike D REAM , I MPORT still learns exploration from end-to-end signal, which can make learning exploration challenging as follows.
Suppose that optimal exploitation conditioned on the problem ID is already learned, and exploration is learned from the exploitation returns.
Since initially, the exploitation has not learned to interpret the trajectories produced by the exploration policies, it may still achieve low returns given a good exploration trajectory, which would erroneously "down weight" the good exploration trajectory (i.e., the chicken-and-egg problem).
Humplik et al.
(2019) attempts to learn better representations by predicting the problem ID as an auxiliary task.
This can help learn a better policy, but does not address the chicken-and-egg coupling problem.
E I NFORMATION B OTTLENECK
The objective (Equation 2) for learning the exploitation policy and encoder is derived from minimizing the mutual information between problem IDs and the encoder’s stochastic outputs, under the constraint that the exploitation policy achieves maximal returns, i.e.: minimize
[Equation]
subject to E µ
[Equation]
We form the Lagrangian with λ − 1 as the dual variable, which yields Equation 2.
This constrained optimization problem is only satisﬁed with the Langrangian is maximized, as the dual variable tends to inﬁnity: i.e., λ approaches 0.