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In high-dimensional reinforcement learning settings with sparse rewards, performing
effective exploration to even obtain any reward signal is an open challenge.
While model-based approaches hold promise of better exploration via planning, it
is extremely difficult to learn a reliable enough Markov Decision Process (MDP)
in high dimensions (e.g., over 10^100 states).
In this paper, we propose learning
an abstract MDP over a much smaller number of states (e.g., 10^5), which we can
plan over for effective exploration.
We assume we have an abstraction function
that maps concrete states (e.g., raw pixels) to abstract states (e.g., agent position,
ignoring other objects).
In our approach, a manager maintains an abstract
MDP over a subset of the abstract states, which grows monotonically through targeted
exploration (possible due to the abstract MDP).
Concurrently, we learn a
worker policy to travel between abstract states; the worker deals with the messiness
of concrete states and presents a clean abstraction to the manager.
On three of
the hardest games from the Arcade Learning Environment (Montezuma's,
Pitfall!, and Private Eye), our approach outperforms the previous
state-of-the-art by over a factor of 2 in each game.
In Pitfall!, our approach is
the first to achieve superhuman performance without demonstrations.
Exploration is a key bottleneck in high-dimensional, sparse-reward reinforcement learning tasks.
Random exploration (e.g., via epsilon-greedy) suffices when rewards are abundant BID24 , but when rewards are sparse, it can be difficult for an agent starting out to even find any positive reward needed to bootstrap learning.
For example, the infamously difficult game MON-TEZUMA'S REVENGE from the Arcade Learning Environment (ALE) BID6 contains over 10 100 states and requires the agent to go thousands of timesteps without receiving reward.
Performing effective exploration in this setting is thus an open problem; without demonstrations, even state-of-the-art intrinsically-motivated RL agents BID5 BID44 achieve only about one-tenth the score of an expert human .In this paper, we investigate model-based reinforcement learning BID17 as a potential solution to the exploration problem.
The hope is that with a model of the state transitions and rewards, one can perform planning under the model to obtain a more informed exploration strategy.
However, as the model being learned is imperfect, errors in the model compound BID42 BID43 when planning over many time steps.
Furthermore, even if a perfect model were known, in high-dimensional state spaces (e.g. over 10 100 states), planning-computing the optimal policy (e.g. via value iteration)-is intractable.
As a result, model-based RL has had limited success in high-dimensional settings BID42 .
To address this, some prior work has focused on learning more accurate models by using more expressive function approximators BID25 , and learning local models BID21 BID50 .
Others have attempted to robustly use imperfect models by conditioning on, instead of directly following, model-based rollouts BID49 , frequently replanning, and combining model-based with model-free approaches BID0 BID40 .
However, none of these techniques offer a fundamental solution.
Instead of directly learning a model over the concrete state space, we propose an approach inspired by hierarchical reinforcement learning (HRL) BID41 BID46 Figure 1 : (a) Illustration of the abstract MDP on MONTEZUMA'S REVENGE.
We have superimposed a white grid on top of the original game.
At any given time, the agent is in one of the grid cellseach grid cell is an abstract state.
In this example, the agent starts at the top of a ladder (yellow dot).
The worker then navigates transitions between abstract states (green arrows) to follow a plan made by the manager (red dots).
(b) Circles represent abstract states.
Shaded circles represent states within the known set.
The manager navigates the agent to the fringe of the known set (s 3 ), then randomly explores with π d to discover new transitions near s 3 (dotted box).
(c) The worker extends the abstract MDP by learning to navigate to the newly discovered abstract states (dotted arrows).
2017), and learn a model over a much smaller abstract state space.
Specifically, we assume we have a state abstraction function BID22 BID36 BID9 , which maps a highdimensional concrete state (e.g. all pixels on the screen) to a low-dimensional abstract state (e.g. the position of the agent).
We then aim to learn an (abstract) Markov Decision Process (MDP) over this abstract state space as follows: A manager maintains an (abstract) MDP over a subset of all possible abstract states which we call the known set, which is grown over time.
The crucial property we enforce is that this abstract MDP is highly accurate and near deterministic on the known set, so we can perform planning without suffering from compounding errors, and do it efficiently since we are working with a much smaller number of abstract states.
Concurrently, we learn a worker policy that the manager uses to transition between abstract states.
The worker policy has access to the concrete states; its goal is to hide the messy details of the real world from the manager (e.g., jumping over monsters) so that the manager has a much simpler planning problem (e.g., traversing between two locations).
In our implementation, the worker keeps an inventory of skills (i.e., options BID41 ), each of which is driven by a deep neural network; the worker assigns an appropriate skill for each transition between abstract states.
In this way, the worker does not "forget" BID20 , and we ensure monotonic progress in learning the abstract MDP.
This abstract MDP, which enables us to efficiently explore via planning, is a key difference between our work and previous HRL work (e.g., BID4 BID46 ), which also learn skills and operate on latent abstract state spaces but without forming an MDP.We evaluate our approach on three of the most challenging games from the ALE BID6 : MONTEZUMA'S REVENGE, PITFALL!, and PRIVATE EYE.
In all three domains, our approach achieves more than 2x the reward of prior non-demonstration state-of-the-art approaches.
In PITFALL!
, we are the first to achieve superhuman performance without demonstrations, surpassing the prior state-of-the-art by over 100x.
Additionally, since our approach is model-based, we can generalize to new rewards without re-training, as long as the reward function is a function of the abstract states.
When evaluated on a new reward function never seen during training, our approach achieves over 3x the reward of prior state-of-the-art methods explicitly trained on the new rewards.
We assume the world is an unknown episodic finite-horizon MDP with (concrete) states x ∈ X and actions a ∈ A.
We further assume we have a simple predefined state abstraction function mapping concrete states x to abstract states s = φ(x).
In MONTEZUMA'S REVENGE, for instance, a concrete state contains the pixels on the screen, while the corresponding abstract state contains the agent's position and inventory (Figure 1 ).
We assume that reward only depends on the abstract states: taking any action transitioning from concrete state x to x leads to reward R(φ(x), φ(x )).Model-based approaches promise better exploration via planning, but struggle in high-dimensional state spaces due to compounding errors and computational intractability.
To avoid these problems, we propose to construct and operate on a low-dimensional representation of the world consisting of abstract states, which we call the abstract MDP (we refer to the original MDP, the world, as the concrete MDP).
Then we plan in the abstract MDP.
Concretely, the abstract MDP consists of:• The state space is a subset of all abstract states, which we call the known set S, consisting of abstract states that can be reliably reached from the initial abstract state via actions in the action set.
Over time, we monotonically grow the known set, which initially only contains the starting abstract state.• The action set comprises of calling the worker policy π w (a|x, (s, s )) on transitions (s, s ) from the current abstract state s to a nearby abstract state s .
When called on a transition (s, s ), the worker navigates from s to s by taking concrete actions a conditioned on the current concrete state x.
The worker abstracts away the messiness of the underlying concrete MDP so that all other parts of the system can operate on the abstract MDP by calling the worker.
We denote calling the worker on transition (s, s ) as the action go(s, s ).•
The transition dynamics of calling action go(s, s ) at abstract state s are defined by the original MDP: i.e., if the worker takes a concrete trajectory x 0 , a 0 , x 1 , a 1 , · · · , x T , the resulting abstract state is φ(x T ).
The rewards for transitioning from s to s are the rewards in the concrete MDP R(s, s ), which only depend on the abstract states by assumption.
The core idea behind our approach is to construct the abstract MDP (Section 3) by growing the action set (training the worker), which in turn grows the known set.
At each point in time, the manager maintains the known set and (accurate, to avoid compounding errors) estimates of the reward and transition dynamics of the abstract MDP.
With these dynamics estimates, the manager can solve the abstract MDP at all times (e.g., by value iteration), since the abstract MDP is small.
As the abstract MDP grows, it captures more and more of the concrete MDP, enabling the manager to recover a better and better policy via planning.
Ultimately, the known set of the abstract MDP contains all abstract states, enabling the manager to recover a high-reward policy.
As the manager constructs the abstract MDP, the abstract MDP maintains two key properties:• The action set of the abstract MDP consists of only reliable actions, actions go(s, s ) that transition from abstract state s to abstract state s with probability at least 1 − δ for some small δ to avoid compounding uncertainty.
This enables the manager to reach any abstract state in the known set with high probability.
To simplify notation, the manager estimates the success rate P (s, s ) of action go(s, s ) instead of the full dynamics P (•|go(s, s ), s), treating the (vanishingly small) fraction of failures equally.• The action set and known set of the abstract MDP grow monotonically.
Since the action set only contains reliable transitions, a key danger is if learning new reliable transitions (adding new actions) causes the worker to forget already learned transitions (removing actions), stalling progress.
We opt for a non-parametric approach, where the worker learns a skill (neural subpolicy) for each transition, reusing skills when possible.
When a worker learns to reliably traverse a transition, it freezes the corresponding skill's parameters.
The manager's goal is to fully construct the abstract MDP so that the known set contains all abstract states.
Then, it can compute a high-reward policy on the concrete MDP via planning on the abstract MDP.
To construct the abstract MDP, the manager adds new actions to the abstract MDP: training the worker to reliably traverse new transitions (driving the transition success rates toward 1).
Concretely, the manager discovers new transitions, trains the worker on these transitions, and updates its dynamics estimates using Algorithm 1.
On each episode, the manager either chooses to discover new transitions via randomized exploration (Section 3.1) or trains the worker.
This is done by constructing a prioritized list of exploration goals, where each goal is either a transition to learn Score all candidates and select highest priority candidate c 5:Compute a plan (s0, s1), (s1, s2), · · · , (sT −1, sT = s) with model 6:for t = 1 to T do 7:Call worker to navigate transition (st−1, st) 8:if c is a transition (s, s ) to learn then 9:LEARNWORKER(s, s ) 10:else c is an abstract state s to explore 11:DISCOVERTRANSITIONS () or an abstract state to find nearby transitions (Section 3.2).
Upon selecting the highest-priority goal (e.g., the transition (s, s )), the manager navigates to the relevant abstract state (e.g., s) by planning with its dynamics models (e.g., the plan go(s 0 , s 1 ), go(s 1 , s 2 ), · · · , go(s T −1 , s T = s)), executing the plan, and then calling the worker on or randomly exploring from the selected goal.
Finally, the manager updates its dynamics models.
It uses a sliding window estimate of the past N transition worker attempts of traversing transition (s, s ) for the transition dynamics, and updates its reward estimate of a transition (s, s ) as the reward accumulated by the first successful traversal of (s, s ).
When a transition (s, s ) becomes reliable (i.e., the dynamics T (s, s ) exceeds the threshold 1 − δ), the manager adds go(s, s ) to the action set of the abstract MDP and adds s to the known set.
For the manager to train the worker on new transitions, it must first discover new transitions.
To discover new transitions, the manager navigates to an exploration candidate: an abstract state s.
Then, it performs randomized exploration to discover new transitions (s, s ) to nearby abstract states s .
As exploration candidates, the manager simply chooses the abstract states in the known set that have been explored fewer than N visit times: i.e., n(s) < N visit , where n(s) is the number of times the manager has explored abstract state s for nearby transitions.
Effectively, the manager assumes that when n(s) ≥ N visit , all nearby transitions (s, s ) have been already discovered.
Concretely, at an abstract state s, the manager finds nearby transitions by following a simple policy π d (a t |x 0:t , a 0:t−1 ) for T d timesteps (Algorithm 3).
The policy π d outputs randomized concrete actions a t conditioned on the past concrete states x 0:t and past concrete actions a 0:t−1 , where φ(x 0 ) is the abstract state s it was initially invoked.
During those T d timesteps, the manager records the transitions and rewards it observes: (φ(x 0 ), r 0 , φ(x 1 )), · · · (φ(x T −1 ), r T −1 , φ(x T )), using the rewards to update its rewards model and the transitions as candidates for the worker to learn.
Additionally, if it ends in another exploration candidate (i.e., n(s) < N visit ) after exploring for T d timesteps, it simply continues exploring for another T d timesteps.
The simplest possible policy for the π d is to uniformly sample a concrete action at each timestep.
However, we found that this inadequately discovered new transitions, because it would often perform useless action sequences (e.g., left, right, left, right).
Instead, we use a simple method for π d to commit to an exploration direction.
At each timestep, the π d uniformly samples a concrete action and a number between 1 and T repeat , and repeats the action the sampled number of times.
Exploration goals.
The manager selects an exploration goal from the set of all candidate exploration goals, consisting of exploration candidates for the transition discovery and candidate transitions for the worker to learn.
The exploration candidates are just the abstract states in the known set with n(s) < N transition .
The candidate transitions are the transitions discovered by the manager.
In addition, the worker imposes a heuristic on its transition learning process in order to preserve the Markov property of the abstract MDP (Section 4), which sometimes makes it impossible for the worker to learn a transition.
To avoid getting stuck, as candidates for the worker, the manager also considers "long-distance" transitions: (s, s ) pairs for which the manager did not directly transition from s to s , but indirectly did so through a sequence of intermediate states DISPLAYFORM0 Input: a transition (s, s ) to learn, called at concrete state x0 with φ(x0) = s 1: Set worker horizon DISPLAYFORM1 Observe xt 5:Compute worker intrinsic reward rt = R (s,s ) (xt|s ) 6:Update worker on (xt−1, at−1, rt, xt) 7:Choose DISPLAYFORM2 Freeze worker's skill π I(s,s ) DISPLAYFORM3 be the length of the shortest such path, the manager considers all pairs (s, DISPLAYFORM4 Priority scores of the exploration goals.
The manager must choose exploration goals in some order.
Our theoretical results (Section 5) hold for any priority function that eventually chooses all exploration goals.
In our implementation, the manager prioritizes the easiest exploration goals (enabling the fastest growth of the action set), and the most useful goals (goal that either lead to more reward or enable the worker to learn new transitions).Concretely, the manager heuristically computes the easiness of a learning transition (s, s ) as DISPLAYFORM5 where n succ is the number of times the worker has successfully traversed (s, s ) and n f ail is the number of times the worker has failed in traversing (s, s ).
Intuitively, both 1) succeeding more and failing less and 2) shorter transitions, requiring fewer timesteps indicate easier to learn transitions.
Similarly, for an abstract state s, the manager computes the easiness of discovering new neighboring transitions as e(s) = −n(s) since abstract states that have been explored less are more likely to have undiscovered transitions.
The manager heuristically computes the usefulness of a learning a transition (s, s ) as u(s, s ) = λ 2 I new + R(s 0 , s), where I new is an indicator that is 1 if there is an outgoing transition (s , s ) and no current candidate transitions end in s .
R(s 0 , s) is the reward achieved by navigating from the initial abstract state s 0 to s. If I new is 1, then learning (s, s ) opens new candidate transitions for the worker to learn, indicating that learning (s, s ) is useful.
For an abstract state s, the manager computes the usefulness just as u(s) = λ 3 + R(s 0 , s).
The λ 3 constant accounts for how much more or less useful discovering new transitions is compared to learning new transitions.
To prioritize exploration goals, the manager uniformly switches between two priority functions.
The first priority function simply equals the easiness plus usefulness: e + u, and the second priority function is the same, but without the reward term in u, to avoid falling into a local reward maximum.
The worker forms the action set of the abstract MDP by learning many subtasks of the form: navigate from abstract state s to s .
It does this while maintaining the three properties: 1) the worker reliably (with high probability) traverses (s, s ) for each action go(s, s ) in the abstract MDP; 2) the action set grows monotonically, so learning new transitions never causes the worker's old transitions to become unreliable; and 3) the worker learns transitions in a way that preserves the Markov property.
While it is possible to learn a single policy for all transition subtasks, it is tricky to satisfy 2), since learning new transitions can have deleterious effects on previously learned transitions.
Instead, the worker maintains an inventory of skills (Section A.3), where each transition is learned by a single skill, sharing the same skill amongst many transitions when possible.
The worker uses these skills to form the action set of the abstract MDP following Algorithm 2: When the manager calls the worker on a transition (s, s ), the worker selects the appropriate skill from the skill inventory and begins an episode of the subtask of traversing s to s (Section 4.2).
During the skill episode, the skill receives intrinsic rewards, and is declared to have successfully completed the subtask if it meets the worker's holding heuristic, which heuristically maintains 3) by ensuring the worker can always control the abstract state.
If at the end of the skill episode, the success rate of the worker traversing (s, s ) exceeds the reliability threshold 1 − δ, the action go(s, s ) is added to the abstract MDP.
The worker's skill inventory I indexes skills so that the skill at index I(s, s ) reliably traverses transition (s, s ).
Each skill is a goal-conditioned subpolicy π I(s,s ) (a|x, s ), which produces concrete actions a conditioned on the current concrete state x and the goal abstract state s .
When the worker traverses a transition (s, s ), it calls on the corresponding skill until the transition is traversed: i.e., π w (a|x, (s, s )) = π I(s,s ) (a|x, s ).When learning a new transition (s, s ), the worker first tries to reuse its already learned skills from the skill inventory.
For each skill π i in the skill inventory, it measures the success rate of π i on the new transition (s, s ) over N transition attempts.
If the success rate exceeds the reliability threshold 1 − δ for any skill π i , it updates the skill repository to reuse the skill: I(s, s ) ← π i .
Otherwise, if no already learned skill can reliably traverse the new transition, the worker creates a new skill and trains it to navigate the transition by optimizing intrinsic rewards during skill episodes (Section 4.2).
Given a transition (s, s ), the worker's subtask is to navigate from abstract state s to abstract state s .
Each episode of this subtask consists of d(s, s ) × H worker timesteps (longer transitions need more timesteps to traverse), where the reward at each timestep is R (s,s ) (x t ) = 1 if the skill has successfully reached the end of the transition (φ(x t ) = s ) and 0 otherwise.
These episodes additionally terminate if the main episode terminates or if the manager receives negative environment reward.
When solving these subtasks to construct the action set of the abstract MDP, the worker must be careful not to violate the Markov property.
In particular, the concrete state may contain some historydependent information lost due to the state abstraction function.
For example, consider the task of jumping over a dangerous hole, consisting of three abstract states: s 1 (the cliff before the hole), s 2 (the air above the hole), and s 3 (the solid ground on the other side of the hole).
The worker might incorrectly assume that it can reliably traverse from s 1 to s 2 by simply walking off the cliff.
But adding this as a reliable transition to the abstract MDP causes a problem: there is now no way to successfully traverse from s 2 to s 3 due to missing history-dependent information in the abstract state (i.e., the way the worker navigated s 1 to s 2 ), violating the Markov property.
On navigating a transition (s, s ), the worker avoids this problem by navigating to s and then checking for history-dependent consequences with the holding heuristic.
The worker assumes that if its navigation of (s, s ) changed unobserved parts of the state, then those changes would eventually cause the abstract state to change (e.g., in the example, the worker would eventually hit the bottom and die).
Consequently, if the worker can stay in s for many timesteps, then it did not significantly unobserved parts of the state.
This corresponds to only declaring the episode as a success if the worker accumulates at least R hold reward (equivalent to being in s for R hold timesteps).Any RL algorithm can be used to represent and train the skills to perform this subtask.
We choose to represent each skill as a Dueling DDQN (van Hasselt et al., 2016; BID47 .
For faster training, the skills use self-imitation BID29 to more quickly learn from previous successful episodes, and count-based exploration similar to BID5 to more quickly initially discover skill reward.
Since the skill inventory can contain many skills, we save parameters by occasionally using pixel-blind skills.
Appendix A.3 fully describes our skill training and architecture.
We are interested in the sample complexity BID16 , the number of samples required to learn a policy that achieves reward close to the optimal policy, with high probability.
Standard results in the tabular setting (e.g., MBIE-EB BID38 ) guarantee learning a nearoptimal policy, but require a number of timesteps polynomial in the size of the state space, which is effectively vacuous in the deep RL setting, where state spaces can be exponentially large (e.g., > 10 100 states).
In contrast, our approach is able to use time and space polynomial in the size of the abstract state space, which is exponentially smaller, by operating on the abstract MDP.Formally, assuming that our neural network policy class is rich enough to represent all necessary skills, with high probability, our approach can learn a near-optimal policy on a subclass of MDPs in time and space polynomial in the size of the abstract MDP (details in Appendix C).
The key intuition is that instead of learning a single task where the time horizon is the length of the game, our approach learns many subtasks where the time horizon is the number of steps required to navigate from one abstract state to another.
This is critical, as many deep RL algorithms (e.g. -greedy) require a number of samples exponential in the time horizon to solve a task.
Following BID2 , we empirically evaluate our approach on three of the most challenging games from the ALE BID6 : MONTEZUMA'S REVENGE, PITFALL!, and PRIVATE EYE.
We do not evaluate on simpler games (e.g., Breakout), because they are already solved by prior state-of-the-art methods BID13 and do not require sophisticated exploration.
We use the standard ALE setup (Appendix A) and end the episode when the agent loses a life.
We report rewards from periodic evaluations every 4000 episodes, where the manager plans for optimal reward in the currently constructed abstract MDP.
We average our approach over 4 seeds and report 1 standard deviation error bars in the training curves.
Our experiments use the same set of hyperparameters (Appendix A.1) across all three games, where the hyperparameters were exclusively and minimally tuned on MONTEZUMA'S REVENGE.In all three games, the state abstraction function uses the RAM state, available through the ALE simulator, to extract the bucketed location of the agent and the agent's current inventory.
Roughly, this distinguishes states where the agent is in different locations or has picked up different items, but doesn't distinguish states where other details differ (e.g. monster positions or obstacle configurations).
Notably, the abstract state function does not specify what each part of the abstract state means, and the agent does not know the entire abstract state space beforehand.
We describe the exact abstract states in Appendix A.2.
Among the many deep RL approaches, in each game, we compare with the prior non-demonstration state-of-the-art approach, which use prior knowledge comparable to our RAM information:• In MONTEZUMA'S REVENGE, we compare with SmartHash BID44 , a countbased exploration approach which estimates state visit counts with a hash-based density model and provides intrinsic reward to revisit states with low visit counts.
Like our approach, SmartHash also uses RAM state information.
It hashes each state to a hand-selected subset of the RAM state and maintains visit counts on the hashes.
• In PITFALL!
, we compare with SOORL BID18 , a planning approach which requires prior knowledge to extract the objects on the screen.
SOORL is the only prior nondemonstration approach to achieve positive reward in PITFALL!, but requires extensive engineering (much stronger than RAM state info) to identify and extract all objects.
Once SOORL has access to the objects, it learns in very few frames since data from similar objects can be pooled in learning.
Consequently, in our training curves, we report its final average performance over 100 runs, as well as its final best performance over 100 runs, • In PRIVATE EYE, we compare with another count-based exploration method, DQNPixelCNN , which uses a pixel-based density model to estimate state visitation counts.
We compare with the results reported in .
DQNPixelCNN uses less prior knowledge than our approach, but we compare with it because it achieves the previous non-demonstration state-of-the-art results.• In all three games, we compare with AbstractStateHash, which performs count-based exploration identical to SmartHash, but uses the same RAM information as our approach.
Comparison between our worst performing seed with the best performing seed of the prior-state-of-the-art.
Our worst performing seed outperforms the prior best on MONTEZUMA'S REVENGE and PITFALL! and performs comparably to the prior best on PRIVATE EYE.
Our best performing seed achieves new peak rewards.
FIG0 shows the main results.
AbstractStateHash matches the prior state-of-the-art on MON-TEZUMA'S REVENGE, but performs relatively poorly on PRIVATE EYE and PITFALL!. This suggests both that prior state-of-the-art methods do not effectively leverage the state abstraction function, and that the state abstraction function does not trivialize the learning problem.
In MONTEZUMA'S REVENGE, after 2B training frames, our approach achieves a final average reward of 11020, more than doubling the average reward of SmartHash: 5001.
Our approach achieves higher average reward than SmartHash at every point along the training curves and continues to learn even late into training, while SmartHash plateaus (Appendix B.3 presents more results on the ability of our approach to continue to learn without plateauing).Our approach is the first non-demonstration approach to achieve superhuman performance on PIT-FALL!, achieving a final average reward of 9959.6 after 2B frames of training, compared to average human performance: 6464 BID33 .
In addition, our approach achieves more than double the reward of SOORL, which achieves a maximum reward of 4000 over 100 seeds and a mean reward of 80.52, and even significantly outperforms Ape-X DQfD BID33 , which uses high-scoring expert demonstrations during training to achieve a final mean reward of 3997.5.In PRIVATE EYE, our approach achieves a mean reward of 35636.1, more than double the reward of DQN-PixelCNN, which achieves 15806.5.
Our approach performs even better, approaching human performance, if we change a single hyperparameter (Appendix B.4).Stability.
Recent work BID12 has drawn attention to the instability of deep RL results.
To highlight the stability of our results, we compare our worst performing seed against the prior state-of-the-art's best performing seed in TAB3 .
Even our worst seed outperforms the mean performance of the prior state-of-the-art approaches.
In addition, our worst seed is competitive with the highest previously reported rewards in each of the games, significantly outperforming the previous high in MONTEZUMA'S REVENGE and PITFALL!, and narrowly performing worse than the previous high in PRIVATE EYE.
Even in PRIVATE EYE, while DQN-PixelCNN achieves 39000 reward on its best single episode across all seeds, none of its seed consistently achieves more than 15806.5 reward over many episodes.
In contrast, our worst seed consistently obtains 36200 reward.
Furthermore, our best seeds achieve new peak performances in each of the games.
By using the abstract MDP, our approach can quickly generalize to new tasks in the same environment that were not seen during training.
It does this by revisiting the transitions in the abstract MDP to update the rewards model with the newly observed reward.
We study the ability of our approach to do this.
After our approach completes training on the original reward function on MONTEZUMA'S REVENGE, we evaluate its performance on three new reward functions, allowing our approach to interact with the environment for an additional 1M frames to observe each new reward function.
We compare with SmartHash, trained from scratch directly on the new reward functions for 1B frames.
Even when evaluated on an unseen reward function, our approach achieves about 3x as much reward as SmartHash, which is directly trained on the new reward function.
This suggests that our approach can generalize to new tasks with the same dynamics.
We describe the details in Appendix B.2.
We additionally evaluate the performance of our approach on the recommended BID23 form of ALE stochasticity (sticky actions 25% of the time) on PRIVATE EYE (selected because it requires the fewest frames for training).
Figure 3 compares the performance of our method on the stochastic version of PRIVATE EYE with the performance of our method on the deterministic version of PRIVATE EYE.
Performance degrades slightly on the stochastic version, because the worker's skills become harder to learn.
However, both versions outperform the prior state-of-the-art DQNPixelCNN, and the worker is able to successfully abstract away stochasticity from the manager in the stochastic version of the game so that the abstract MDP remains near-deterministic.
To see how easy it is to create a high-performing state abstraction function on new tasks, we study the robustness of our approach to state abstraction functions of varying degrees of coarseness on PRIVATE EYE.
Our state abstraction function buckets the agent's (x, y) coordinates.
We vary the coarseness of the abstraction function by varying the bucketing size: increasing the bucketing size results in fewer, coarser abstract states.
We report results in Figure 4 on five different bucket sizes obtained by scaling the original bucket size by 1 2 , 2 3 , 1, 3 2 , and 2.
To adjust for the updated bucket sizes, we also scale the worker's skill episode horizon H worker by the same value.
Our method outperforms the prior state-of-the-art approach DQN-PixelCNN across the entire range of bucket sizes, suggesting that our approach does not require a highly tuned state abstraction function.
Exploration in tabular settings is well-understood via optimism in the face of uncertainty (OFU) BID7 BID39 ) and posterior sampling BID31 BID30 .
OFU methods BID7 BID37 Jaksch et al., 2010) achieve provably near-optimal policies in polynomial time by providing reward bonuses for exploring regions of uncertainty.
Nevertheless, despite recent stronger optimality bounds BID8 , these methods do not scale to the deep RL setting, where the state space is prohibitively large.
BID5 ; BID44 ; similarly apply optimism reward bonuses in the case of high-dimensional state spaces, empirically improving exploration.
However, these methods no longer guarantee optimality, and can suffer from insufficient exploration because they reactively seek infrequently visited states, whereas our manager proactively seeks new abstract states.
While model-based RL succeeds in the tabular setting BID39 ) and in tasks with relatively small (e.g., < 100 dimensions) state spaces BID25 , it has little success on tasks with exponentially large state spaces (e.g., pixel inputs).
Prior work BID27 that learns models in these large state spaces suffers from compounding errors BID42 , where longterm predictions become extremely inaccurate.
To make matters worse, while prior model-based works BID49 BID11 ) succeed on relatively dense reward tasks, model-based planning methods (e.g., value iteration) can be computationally intractable in longhorizon, sparse-reward tasks, even when a perfect model is known.
To circumvent these problems, our work learns an abstract MDP consisting of an exponentially smaller abstract state space and learned skills.
BID28 similarly learns a model over abstract states and skills, but uses manually engineered skills, whereas ours are learned.
Our work relates to prior work on hierarchical reinforcement learning (HRL), which also operates on abstract states BID10 BID22 with learned skills or subgoals BID35 BID36 BID46 BID4 .
However, a key difference is that our work constructs the abstract MDP, enabling us to perform targeted exploration via planning and rely on the Markov property to avoid exponentially large state histories.
In contrast, the abstract states and skills in these other works do not meet such useful structural properties, and consequently can be difficult to learn with.
BID34 similarly constructs an abstract MDP like ours.
However, due to critical design decisions, our approach outperforms theirs by nearly an order of magnitude.
Whereas our approach monotonically grows the known set by saving worker parameters as transitions become reliable, BID34 uses the same worker parameters to simultaneously learn many transitions.
This causes catastrophic forgetting, as training on a new transition causes the worker to fail a previously learned transition, and prevents growth of the abstract MDP.Only imitation learning methods BID2 BID33 outperform our method on the ALE's hardest exploration games.
However, using demonstrations sidesteps the exploration problem our approach seeks to solve because following demonstrations leads to high reward.
This work presents a framework for tackling long-horizon, sparse-reward, high-dimensional tasks by using abstraction to decrease the dimensionality of the state space and to address compounding model errors.
Empirically, this framework performs well in hard exploration tasks, and theoretically guarantees near-optimality.
However, this work has limitations as well.
First, our approach relies on some prior knowledge in the state abstraction function, although we compare against state-ofthe-art methods using a similar amount of prior knowledge in our experiments.
This information is readily available in the ALE, which exposes the RAM, and in many robotics tasks, which expose the underlying state (e.g., joint angles and object positions).
Still, future work could attempt to automatically learn the state abstraction or extract the abstraction directly from the visible pixels.
One potential method might be to start with a coarse represention, and iteratively refine the representation by splitting abstract states whenever reward is discovered.
Another limitation of our work is that our simple theoretical guarantees require relatively strong assumptions.
Fortunately, even when these assumptions are not satisfied, our approach can still perform well, as in our experiments.
A EXPERIMENT DETAILS Following Mnih et al. (2015) , the pixel concrete states are downsampled and cropped to 84 by 84 and then are converted to grayscale.
To capture velocity information, the worker receives as input the past four frames stacked together.
Every action is repeated 4 times.
In addition, MONTEZUMA'S REVENGE and PITFALL! are deterministic by default.
As a result, the manager deterministically navigates to the fringes of the known set by calling on the worker's deterministic, saved skills.
To minimize wallclock training time, we save the states at the fringes of the known set and enable the worker to teleport to those states, instead of repeatedly re-simulating the entire trajectory.
When the worker teleports, we count all the frames it would have had to simulate as part of the training frames.
Importantly, this only affects wallclock time, and does not benefit or change the agent in any way.
Notably, this does not apply to PRIVATE EYE, where the initial state is stochastically chosen from two similar possible states.
A.1 HYPERPARAMETERS All of our hyperparameters are only tuned on MONTEZUMA'S REVENGE.
Our skills are trained with the Adam optimizer BID19 with the default hyperparameters.
TAB5 describes all hyperparameters and the values used during experiments (bolded), as well as other values that we tuned over (non-bolded).
Most of our hyperparameters were selected once and never tuned.
In MONTEZUMA'S REVENGE, each abstract state is a (bucketed agent x-coordinate, bucketed agent y-coordinate, agent room number, agent inventory, current room objects, agent inventory history) tuple.
These are given by the RAM state at indices 42 (bucketed by 20), 43 (bucketed by 20), 3, 65, and 66 respectively.
The agent inventory history is a counter of the number of times the current room objects change (the room objects change when the agent picks up an object).In PITFALL!, each abstract state is a (bucketed agent x-coordinate, bucketed agent y-coordinate, agent room number, items that the agent has picked up) tuple.
These are given by the RAM state at indices 97 (bucketed by 20), 105 (bucketed by 20), 1, and 113 respectively.
In PRIVATE EYE, each abstract state is a (bucketed agent x-coordinate, bucketed agent y-coordinate, agent room number, agent inventory, agent inventory history, tasks completed by the agent) tuple.
These are given by the RAM state at indices 63 (bucketed by 40), 86 (bucketed by 20), 92, 60, 72, and 93 respectively.
Architecture.
Our skills are represented as Dueling DDQNs BID45 BID47 , which produce the state-action value Q (s,s ) (x, a) = A (s,s ) (x, a) + V (s,s ) (x), where A (s,s ) (x, a) is the advantage and V (s,s ) (x) is the state-value function.
The skills recover a policy π K(s,s ) (a|x, (s, s )) by greedily selecting the action with the highest Q-value at each concrete state x.
The skill uses the standard architecture BID24 to represent A (s,s ) (x, a) and V (s,s ) (x) with a small modification to also condition on the transition (s, s ).
First, after applying the standard ALE pre-processing, the skill computes the pixel embedding e x of the pixel state x by applying three square convolutional layers with (filters, size, stride) equal to (32, 8, 4), (64, 4, 2), and (64, 4, 2) respectively with rectifier non-linearities BID26 , and applying a final rectified linear layer with output size 512.
Next, the skill computes the transition embedding e (s,s ) by concatenating [e r ; e dif f ] and applying a final rectified linear layer with output size 64, where:• e r is computed as the cumulative reward received by the skill during the skill episode, represented as one-hot, and passed through a single rectified linear layer of output size 32.• e dif f is computed as s − s passed through a single rectified linear layer of output size 96.Finally, e x and e (s,s ) are concatenated and passed through a final linear layer to obtain A (s,s ) (x, a) and V (s,s ) (x).To prevent the skill from changing rapidly as it begins to converge on the optimal policy, we keep a sliding window estimate of its success rate p success .
At each timestep, with probability 1 − p success , we sample a batch of (x, a, r, x ) tuples for transition (s, s ) from the replay buffer and update the policy according the DDQN loss function: L = ||Q (s,s ) (x, a) − target|| 2 2 , where target = (r +Q target (x , arg max a ∈A Q (s,s ) (x , a ))).
Additionally, since the rewards are intrinsically given, the optimal Q-value is known to be between 0 and R hold .
We increase stability by clipping target between these values.
Pixel blindness.
In addition, some skills are easy to learn (e.g. move a few steps to the left) and don't require pixel inputs to learn at all.
To prevent the skills from unnecessarily using millions of parameters for these easy skills, the worker first attempts to learn pixel-blind skills for simple transitions (s, s ) with d(s, s ) = 1 (i.e. (s, s ) was directly observed by the manager).
The pixelblind skills only compute e (s,s ) and pass this through a final layer to compute the advantage and value functions (they do not compute or concatenate with e x ).
If the worker fails to learn a pixelblind skill, (e.g. if the skill actually requires pixel inputs, such as jumping over a monster)
it will later try to learn a pixel-aware skill instead.
Epsilon schedule.
The skills use epsilon-greedy exploration, where at each timestep, with probability , a random action is selected instead of the one produced by the skill's policy BID48 .
Once a skill becomes frozen, is permanently set to 0.The number of episodes required to learn each skill is not known in advance, since some skills require many episodes to learn (e.g. traversing a difficult obstacle), while other skills learn in few episodes (e.g. moving a little to the left).
Because of this, using an epsilon schedule that decays over a fixed number of episodes, which is typical for many RL algorithms, is insufficient.
If epsilon is decayed over too many episodes, the simple skills waste valuable training time making exploratory actions, even though they've already learned near-optimal behavior.
In contrast, if epsilon is decayed over too few episodes, the most difficult skills may never observe reward, and may consequently fail to learn.
To address this, we draw motivation from the doubling trick in online learning BID1 to create an epsilon schedule, which accomodates skills requiring varying number of episodes to learn.
Instead of choosing a fixed horizon, we decay epsilon over horizons of exponentially Figure 5 : The saw-tooth epsilon schedule used by our skills increasing length, summarized in Figure 5 .
This enables skills that learn quickly to achieve low values of epsilon early on in training, while skills that learn slowly will later explore with high values of epsilon over many episodes.
Count-based exploration.
Our skill additionally use count-based exploration similar to BID44 ; BID5 to learn more quickly.
Each skill maintains a count of the number of times visit(s) it has visited each abstract state s.
Then, the skill provides itself with additional intrinsic reward to motivate itself to visit novel states, equal to .
Self-imitation.
When learning to traverse difficult obstacles (e.g. jumping over a disappearing floor), the skill may observe a few successes long before successfully learning a policy to reliably traverse the difficult obstacle.
We use a variant of the self-imitation described in BID29 to decrease this time.
Whenever a skill successfully traverses a transition, it adds the entire successful trajectory to a separate replay buffer and performs imitation learning on the successful trajectories.
These successful trajectories are actually optimal skill trajectories because the skill episode uses undiscounted reward, so all successful trajectories are equally optimal.
To update on these skills, the skill periodically samples from this replay buffer and updates on an imitation loss function DISPLAYFORM0 , where θ is the skill's parameters, and L 1 and L 2 are defined as below:• Let G t = T i=t r t be the reward to-go for a successful trajectory a) on the reward to-go of the successful trajectory, because G t is actually the optimal Q-value on successful trajectories (all successful trajectories are equally optimal): i.e., L 1 = ||G t − Q (s,s ) (x t , a t )|| 2 .
DISPLAYFORM1 • We use the margin-loss from DISPLAYFORM2 Intuitively, L 2 encourages the skill to replay the actions that led to successful trajectories over other actions.
We use λ = 0.5, which was chosen with no hyperparameter tuning.
B ADDITIONAL RESULTS
The worker learns skills that successfully apply in to many similar transitions.
Figure 6 depicts the number of different transitions each skill is used on in MONTEZUMA'S REVENGE, PITFALL!, and PRIVATE EYE.
The simplest skills (e.g. move to the left) enjoy the highest number of reuses, while more esoteric skills (e.g. jump over a particular monster) are only useful in few scenarios.
Figure 7 provides an example of a skill in MONTEZUMA'S REVENGE with relatively high reuse.
The arrows denote the movement of the agent when it executes the skill.
The same skill that jumps over a monster in the first room (Figure 7(a) ) can also climb up ladders.
In Figure 7(b) , the skill appears to know how to climb up all parts of the ladder except for this middle.
This occurs because the spider occasionally blocks the middle of the ladder, and a different special skill must be used to avoid the spider.
However, the skill reuse is not perfect.
For example, in Figure 7 (a), the skill can climb up the top half of ladders, but a separate skill climbs the bottom half of the ladders.
To evaluate the ability of our approach to generalize to new reward functions, we train our approach on the basic MONTEZUMA'S REVENGE reward function and then test it on three challenging new reward functions (illustrated in FIG4 ), not seen during training:• Get key: the agent receives 1000 reward for picking up the key in room 14 (6 rooms away from the start).
In addition, the agent receives -100 reward for picking up any other objects or opening any other doors.
Figure 9 : Training curves of SmarthHash on alternate tasks in MONTEZUMA'S REVENGE, compared with the performance of our approach generalizing to the new task.
Our approach only receives 1M frames with the new task.• Kill spider: the agent receives 1000 reward for killing the spider in room 13 (5 rooms away from the start).
To kill the spider, the agent must first pick up the sword in room 6 (3 rooms away from the start) and save the sword for the spider.
The agent receives no other reward.• Enter room 8: the agent receives 1000 reward for entering room 8 (6 rooms away from the start).
The agent receives no other reward.
In all three tasks, the episode ends when the agent completes its goal and receives positive reward.
Our approach trains on the basic MONTEZUMA'S REVENGE reward function for 2B frames, and then is allowed to observe the new reward functions for only 1M frames.
We compare with SmartHash, which trains directly on the new reward functions for 1B frames.
The results are summarized in Figure 9 .
Even when evaluated on a reward function different from the reward function it was trained on, our approach achieves about 3x as much reward as SmartHash, which is trained directly on the new reward function.
Averaged over all 3 tasks, our approach achieves an average reward of 716.7 out of an optimal reward of 1000, whereas SmartHash only achieves an average reward of 220, even when trained directly on the new reward function.
These experiments suggest that after our approach is trained in an environment on one task, it can quickly and successfully adapt to new tasks with little additional training.
Whereas many prior state-of-the-art approaches tend to plateau toward the end of training, our approach continues to make near-linear progress.
Figure 10 graphs the number of transitions learned by the worker against the number of frames in training.
In MONTEZUMA'S REVENGE and PIT-FALL!
particularly, the rate the worker learns new transitions is nearly constant throughout training.
Because of this, when we continued to train a single seed on PITFALL!, by 5B frames, it achieved a reward of 26000, and by 20B frames, it achieved a reward of 35000.
By changing a single hyperparameter, our approach can perform even better on PRIVATE EYE, exceeding human performance on 2 of 4 seeds.
Since nearby abstract states are particularly easy to discover in PRIVATE EYE, the manager needs not explore for new transitions as many times.
Consequently, if we decrease the number of times the manager explores around each abstract state N visit from the 500 (used in the main experiments for all three games) to 10, performance improves.
Figure 11 compares the performance with the decreased value of N visit with the original value of N visit reported in the main experiments.
Decreasing N visit prevents the manager from wasting frames with unnecessary exploration and consequently enables the worker to learn more transitions in fewer total frames.
With N visit set to 10, our approach achieves a final average performance of 60247 after 200M frames of training.
Additionally, the top 2 of our 4 seeds achieve rewards of 75600 and 75400, exceeding average human performance: 69571 BID33 .
In general, a hierarchical policy over skills is not guaranteed to be near-optimal, because certain optimal trajectories may be impossible to follow using the skills.
Because of this, hierarchical reinforcement learning literature typically focuses on hierarchical optimality BID10 optimality given the abstractions.
However, under the following assumptions, our approach provably achieves a near-optimal policy on the original MDP in time polynomial in the size of the abstract MDP, with high probability.
Proposition 1.
Under the assumptions, for a given input η and , π t is at most suboptimal, To prove Proposition 1, we require the following three lemmas: DISPLAYFORM0 Lemma 1.
By setting N transition to be O( log 1−(1−η ) 1 |Φ| 2 /2 2( /|Φ|HV * (x0)) 2 ), with probability 1 − η , at each timestep t, π t is near-optimal on the current abstract MDP: i.e., V πt t (s) ≥ V * t (s)− for all abstract states s ∈ Φ.Lemma 2.
If the known set is equal to the set of all abstract states (S = Φ) at timestep T , then for any policy π on the abstract MDP, π achieves the same expected reward on the abstract MDP as on the concrete MDP: i.e., V π T (φ(x 0 )) = V π (x 0 ), where x 0 is the initial concrete state.
In addition, the expected return of the optimal policy on the abstract MDP is equal to the expected return of the optimal policy on the concrete MDP: i.e., V * T (φ(x 0 )) = V * (x 0 ) where x 0 is the initial state.
Lemma 3.
With probability 1 − η, the known set grows to cover all abstract states in O |Φ| 3 (|A| + Given these lemmas, we are ready to prove Proposition 1:Proof of Proposition 1.
For simplicity, we ignore terms due to appropriately setting N transition and 1 − η from Lemma 1, but these terms are all polynomial in the size of the abstract MDP.By Lemma 3 the known set grows to cover all abstract states in T = O |Φ| 3 (|A| + log K|Φ|+log 1 η log 1 p ) + d max × H worker timesteps.
For all timesteps t ≥ T , by Lemma 1, π t is at most suboptimal on the abstract MDP.
On all those timesteps, the known set is equal to all abstract states, so by Lemma 2, π t is at most suboptimal on the concrete MDP.Proofs.
Now, we prove Lemma 1, Lemma 2, and Lemma 3.Proof of Lemma 1.
LetP (s |o, s) denote the estimated transition dynamics 3 andR(s, s ) denote the estimated reward model in the abstract MDP.For each reliable transition (s, s ) (action in the abstract MDP), the manager estimatesP (s |o, s) from N transition samples of the worker.
We bound the error in the model |P (s |o, s) − P (s |o, s)| with high probability by Hoeffding's inequality: DISPLAYFORM1 By the Assumption 2,R(s, s ) = R(s, s ) because all trajectories leading to s achieve some reward r and all trajectories leading to s achieve some reward r , so a single estimate R(s, s ) = r − r is sufficient to accurately determineR.Because the model errors are bounded, and because the abstract MDP is Markov, we can apply the simulation lemma BID17 , which states that if |P (s |o, s) − P (s |o, s)| ≤ α and |R(s, s )−R(s, s )| ≤ α, then the policy π optimizing the MDP formed byP andR is at most suboptimal: i.e., at each timestep t, V π t (s) ≥ V * t (s)− for all s ∈ S, where α is O ( /|S|HV * (x 0 )) 2 , and H is the horizon of the abstract MDP.
Since the total number of transitions is bounded by |S| 2 , substituting for N transition gives the desired result.

assistant: text: We automatically construct and explore a small abstract Markov Decision Process, enabling us to achieve state-of-the-art results on Montezuma's Revenge, Pitfall!, and Private Eye by a significant margin.