C AUSAL C URIOSITY : RL A GENTS D ISCOVERING S ELF - SUPERVISED E XPERIMENTS FOR C AUSAL R EPRESENTATION L EARNING
Anonymous authors
Paper under double-blind review
A BSTRACT
Humans show an innate ability to learn the regularities of the world through interaction.
By performing experiments in our environment, we are able to discern the causal factors of variation and infer how they affect the dynamics of our world.
Analogously, here we attempt to equip reinforcement learning agents with the ability to perform experiments that facilitate a categorization of the rolled-out trajectories and to subsequently infer the causal factors of the environment in a hierarchical manner.
We introduce a novel intrinsic reward, called causal curiosity, and show that it allows our agents to learn optimal sequences of actions and to discover causal factors in the dynamics.
The learned behavior allows the agent to infer a binary quantized representation for the ground-truth causal factors in every environment.
Additionally, we ﬁnd that these experimental behaviors are semantically meaningful (e g., to differentiate between heavy and light blocks, our agents learn to lift them) and are learnt in a self-supervised manner with approximately 2.5 times less data than conventional supervised planners.
We show that these behaviors can be re-purposed and ﬁne-tuned (e g., from lifting to pushing or other downstream tasks).
Finally, we show that the knowledge of causal factor representations aids zero-shot learning for more complex tasks
I NTRODUCTION
Discovering causation in environments an agent might encounter remains an open and challenging problem for causal reinforcement learning (Schölkopf (2015), Bengio et al.
Schölkopf (2019)).
Most approaches take the form of BAMDPs (Bayes Adaptive Markov Decision Processes) (Zintgraf et al.
or Hi-Param MDP (Hidden Parameter MDPs)
(Doshi-Velez & Konidaris (2016); Yao et al.
; Killian et al.
; Perez et al.
(2020)) which condition the transition p ( s t +1 | s t , a t ; H ) and/or reward function R ( r t +1 | s t , a t , s t +1 ; H ) of each environment on hidden parameters (also referred to as causal factors in some of the above studies).
Let s ∈ S , a ∈ A , r ∈ R , H ∈ H where S , A , R , and H are the set of states, actions, rewards and feasible hidden parameters.
In the physical world and in the case of mechanical systems, examples of the parameter h j ∈ H include gravity, coefﬁcients of friction, masses and sizes of objects.
Typically, H is treated as a latent variable for which an embedding is learned during training, using variational methods (Kingma et al.
Ilse et al.
Let s
T be the entire state trajectory of length T .
Similarly, a 0: T is the sequence of actions applied during that trajectory by the agent that results in s 0: T .
In an environment parameterized by these causal factors, these latent variable approaches deﬁne a probability distribution over the entire sequence of (rewards, states, actions) conditioned on a latent z as p ( r 0: T , s 0: T , a 0: T − 1 ; z ) that factorizes as
[Equation]
due to the Markov assumption.
At test time, the agent infers the causal factor associated with its environment by observing the trajectories produced by its initial actions that can be issued by any policy such as model-based reinforcement learning.
In practice, however, discovering causal factors in a physical environment is prone to various challenges that are caused by the disjointed nature of the inﬂuence of these factors on the produced trajectories.
More speciﬁcally, at each time step, the transition function is affected by a subset of global causal factors.
This subset is implicitly deﬁned on the basis of the current state and the action taken.
For example, if a body in an environment loses contact with the ground, the coefﬁcient of friction between the body and the ground no longer affects the outcome of any action that is taken.
Likewise, the outcome of an upward force applied by the agent to a body on the ground is unaffected by the friction coefﬁcient.
We can therefore take advantage of this natural discontinuity to discern causal factors.
Without knowledge of how independent causal mechanisms affect the outcome of a particular action in a given state in an environment, it becomes impossible for the agent to conclude where the variation it encountered came from.
Unsurprisingly, Hi-Param and BAMDP approaches fail to learn a disentangled embedding for the causal factors, making their behaviors uninterpretable (Perez et al.
For example, if, in an environment, a body remains stationary under a particular force, the
Hi-Param or BAMDP agent may apply a higher force to achieve its goal of perhaps moving the body, but will be unable to conclude whether the "un-movability" was caused by high friction or high mass of the body.
Additionally, these approaches require human-supervised reward engineering, making it difﬁcult to apply them outside of the simulated environments they are tested in.
Our goal is, instead of focusing on maximizing reward for some particular task, to allow agents to discover causal processes through exploratory interaction.
During training, our agents discover self-supervised experimental behaviors which they apply to a set of training environments.
These behaviors allow them to learn about the various causal mechanisms that govern the transitions in each environment.
During inference in a novel environment, they perform these discovered behaviors sequentially and use the outcome of each behavior to infer the embedding for a single causal factor (Figure 1).
The main challenge while learning a disentangled representation for the causal factors of the world is that several causal factors may affect the outcome of behaviors in each environment.
For example, when pushing a body on the ground, the outcome, i.e., whether the body moves, or how far the body is pushed, depends on several factors, e.g., mass, shape and size, frictional coefﬁcients, etc.
However, if, instead of pushing on the ground, the agent executes a perfect grasp-and-lift behavior, only mass will affect whether the body is lifted off the ground or not.
Thus, it is clear that not all experimental behaviors are created equal and that the outcomes of some behaviors are caused by fewer causal factors than others.
Our agents learn these behaviors without supervision using causal curiosity , an intrinsic reward.
The outcomes of these experimental behaviors are then used to infer a binary quantized embedding describing the single isolated causal factor.
Even though causal factors of variation in a physical world are easily identiﬁable to humans, a concrete deﬁnition is required to back up our proposed method.
We conjecture that the causality of a factor of variation depends on the available actions to the agent.
If the set of actions that an agent can take is very limited, there is no way for it to discern a diverse set of causal factors in the environment.
Deﬁnition 1 (Causal factors) .
Consider the MDP ( S , A , p, r) with state space S , action space A , the transition function p , and the reward function r .
Let s 0: T ∈ S T denotes a trajectory of states and T be the length of such trajectories.
Let d ( · , · ) : S T × S T → R + be a distance function deﬁned on the space of trajectories of length T .
The set H = { h 1 , h 2 , . . .
, h k } is called a set of (cid:15) − causal factors if for every h j ∈ H , there exists a unique sequence of actions that clusters the state trajectories into two sets S and S (cid:48)
such that
[Equation]
Intuitively, a factor of variation affecting a set of environments is called causal if there exists a sequence of actions available to the agent where the resultant trajectories are clustered into two or more sets (for simplicity here we assume binary clusters).
This is analogous to the human ability to conclude whether objects are heavy or light, big or small.
For a gentle introduction to the intuition about this deﬁnition, we refer the reader to Appendix C.
Using the above, we propose an intrinsic reward, which allows our agents to discover experimental behaviors which are semantically meaningful and can be used to re-train for downstream tasks, resulting in high sample efﬁciency.
Our work, therefore, forms an important link between structured representation learning and skill discovery, two largely disjoint ﬁelds in RL, which stand to beneﬁt from each other.
The contributions of the work are as follows:
• We equip agents with the ability to perform experiments and behave meaningfully in a set of environments in an unsupervised manner.
These behaviors can expose or obfuscate speciﬁc independent causal mechanisms that occur in the world of the agent, allowing the agent to learn about each in the absence of the others, an important human behavioral trait.
• We introduce an intrinsic reward, causal curiosity , which allows our agents to discover these behaviors without human-engineered rewards.
The outcomes of the experiments are used to learn a disentangled quantized binary representation for the causal factors of the environment, analogous to the human ability to conclude whether objects are light/heavy, big/small etc.
• Through extensive experiments, we conclude that knowledge of the causal factors aids sample efﬁciency in two ways - ﬁrst, that the knowledge of the causal factors aids transfer learning across multiple environments, and, second, that the experimental behaviors acquired can be repurposed for downstream tasks.
M ETHOD
Consider a set of N environments E with e ( i
) ∈ E where e ( i ) denotes the i th environment.
The letter H is overloaded.
While H is a set of global causal factors (as deﬁned in Def. 1) such that h j ∈ H , each causal factor h j is itself a random variable which assumes a particular value for every instantiation of an environment.
Thus every environment e ( i ) is represented by a set of causal factors { h ( i ) j ∀ j } .
For each environment e ( i ) , ( z ( i ) (0) , z ( i ) (1) ...
z ( i ) ( K − 1) ) represents the disentangled embedding vector, such that z ( i ) ( j ) encodes h ( i ) j .
2. L EARNING TO PERFORM EXPERIMENTS
To learn about causal processes through interaction, the agent must produce a sequence of actions a 0: T − 1 that we call experimental behavior , which, when applied to environment e ( i ) ∈ E , produces a sequence of states (observation)
O ( i ) =
[ s ( i ) 0 , s ( i ) 1 ..
s ( i ) T ] , which is then used to infer the value of the embedding for a single causal factor
z ( i ) ( j ) .
According to Def. 1, a causal factor is a parameter in the environment whose value, when intervened on (i.e. varied) over a set of values, results in trajectories of states that are divisible into disjoint clusters under a particular sequence of actions.
These clusters represent the quantized values of the causal factor.
For example, mass, which is a causal factor of a body, under an action sequence of a grasping and lifting motion, results in 2 clusters, liftable (low mass) and not liftable (high mass).
1: Initialize j =2: Initialize training environment set Envs 3: for iteration m to M do 4:
Sample experimental behavior a 0: T ∼ CEM( · ) 5: for i th env in Envs do 6:
Apply a 0: T to env 7:
Collect O ( i ) =
s ( i ) 0: T 8:
Reset env 9:
Calculate − L ( O | M ) given that M is bimodal clustering model 10:
Update CEM( · ) distribution with highest reward trajectories 11:
Use learnt q M ( z | O ) for cluster assignment of each env in Envs i.e. z ( i ) j = q M ( z | O ( i ) )
12: Update j = j +13:
Repeat from step 2, ﬁrst setting Envs = { e ( i ) :
z ( i ) j − 1 = 0 } and then, setting
Envs = { e ( i ) :
z ( i ) j − 1 = 1 }
However, such an action sequence is not known in advance.
Therefore, discovering a causal factor in the environment boils down to ﬁnding a sequence of actions that makes the effect of that factor prominent by producing clustered trajectories for different values of that environmental factor.
We motivate this using model selection criterion.
Normally in model selection applications, the observations are ﬁxed and the goal is to ﬁnd a model M ∗ that is closest to reality, as represented by:
[Equation]
where L ( · ) is the description length.
However, here, the situation is reversed.
A simple bi-modal clustering model is ﬁxed, motivated by Deﬁnition 1.
Then, the agent is motivated to produce actions that result in observations that are best explained by this model.
These discovered action sequences are the experimental behaviors we desire.
[Equation]
where each observed trajectory O
= O ( a 0: T ) is a function of the action sequence.
As mentioned earlier, the model is ﬁxed in this formulation; hence, the ﬁrst term L ( M ) is constant and not a function of the actions.
− L ( O | M ) that is fed back to the RL agent as a reward function to maximize.
We regard this reward function as causal curiosity .
Note that since each causal factor has its own independent causal mechanism that causes O , the
MDL of O will be higher if multiple causal factors cause O .
On the contrary, if the agent produces actions which result in an O that is easily explained by a low-capacity bi-modal model M , then it will imply that O is caused by fewer causal factors.
Consequently, the causal curiosity reward for such an action sequence, − L ( O | M ) , will be high.
Therefore, causal curiosity favors experimental behaviors that result in observations caused by few causal factors - thereby allowing us to use O to infer a representation for a single causal factor.
For details, please refer Appendix A.
2. U SING OUTCOMES OF EXPERIMENTS TO LEARN CAUSAL REPRESENTATION
By maximizing the causal curiosity reward it is possible to achieve behaviors which result in trajectories of states only caused by a single hidden parameter.
However, we wish to use the outcome of performing these experimental behaviors in each environment to infer a representation for the causal factor isolated by the experiment in question.
We achieve this through cluster membership.
After training the Model Predictive Control Planner (Camacho & Alba (2013)), we sample from an action sequence a 0: T and apply it to each of the training environments.
The learnt clustering model M is then used to infer a representation for each environment using the collected outcome O ( i ) obtained by applying a 0: T to each environment.
[Equation]
This corresponds to Step 11 of Algorithm (1).
The representation learnt is binary in nature corresponding to the quantization of the continuous spectrum of values a causal factor takes in the training set into high and low values.
Note however that a binary quantized embedding is not a necessary part of our method.
A dense embedding may alternatively be learnt here similar to (Perez et al.
Zintgraf et al.
using approximate variational inference.
However, performing interventions on a dense embedding (Section 2.3) increases the computational complexity exponentially.
Balancing space and time complexity, we report results using the quantized binary form of Equation (6).
We discuss the implications of increasing the complexity of z ( i ) j in the discussion.
2. I NTERVENTIONS ON BELIEFS
Having learnt about the effects of a single causal factor of the environment we wish to learn such experimental behaviors for each of the remaining hidden parameters that may vary in an environment.
To achieve this, in an ideal setting, the agent would require access to the generative mechanism of the environments it encounters.
Ideally, it would hold the values of the causal factor already learnt about constant i.e. do ( h j = constant ) , and intervene over (vary the value of) another causal factor over a set of values K i.e. do ( h j = k ) such that k ∈ K .
For example, if a human scientist were to study the effects of a causal factor, say mass of a body, she would hold the values of all causal factors constant, (interact with cubes of the same size and external texture) and vary only mass to see how it affects the outcome of speciﬁc behaviors she applies to each body.
However, in the real-world the agent does not have access to the generative mechanism of the environments it encounters, but merely has the ability to act in them.
Thus, it can intervene on the representations of a causal factor of the environment i.e. do ( z i = constant ) .
For, example having learnt about gravity, the agent picks all environments it believes have low gravity, and uses them to learn about a separate causal factor say, friction.
This corresponds to Step 13 of Algorithm (1).
Thus, to learn about the j th causal factor, we repeat steps 3 onwards on each of the clusters obtained for the j − 1 th .
[Equation]
This process continues in the form of a tree (Figure 4), where for each cluster of environments, a new experiment learns to split the cluster into 2 sub-clusters depending on the value of another hidden parameter.
At level n , the agent produces 2 n experiments and inference models, having already intervened on the binary quantized representations of n causal factors.
R ELATED W ORK
Doshi-Velez & Konidaris (2016) deﬁne a class Markov Decision Processes where transition probabilities p ( s t +1 | s t , a t ; θ ) depend on a hidden parameter θ , whose value is not observed, but its effects are felt.
Killian et al.
(2017) and Yao et al.
(2018) utilize these Hidden Parameter MDPs (Markov
Decision Processes) to enable efﬁcient policy transfer, assuming that transition probabilities across states are a function of hidden parameters.
Perez et al.
(2020) relax this assumption, allowing both transition probabilities and reward functions to be functions of hidden parameters.
Zintgraf et al.
(2019) approach the problem from a Bayes-optimal policy standpoint, deﬁning transition probabilities and reward functions to be dependent on a hidden parameter characteristic of the MDP in consideration.
We utilize this setup to deﬁne causal factors.
Substantial attempts have been made at unsupervised disentanglement, most notably, the β -VAE
Higgins et al.
Burgess et al.
(2018), where a combination of factored priors and the information bottleneck force disentangled representations.
Kim & Mnih (2018) enforce explicit factorization of the prior without compromising on the mutual information between the data and latent variables, a shortcoming of the β -VAE.
Chen et al.
(2018) factor the KL divergence into a more explicit form, highlighting an improved objective function and a classiﬁer-agnostic disentanglement metric.
Locatello et al.
(2018) show theoretically that unsupervised disentanglement (in the absence of inductive biases) is impossible and highly unstable, susceptible to random seed values.
They follow this up with Locatello et al.
(2020) where they show, both theoretically and experimentally, that pair-wise images provide sufﬁcient inductive bias to disentangle causal factors of variation.
However, these works have been applied to supervised learning problems whereas we attempt to disentangle the effects of hidden variables in dynamical environments, a relatively untouched question.
Curiosity for robotics is not a new area of research.
Schmidhuber (2006), Ngo et al.
(2012), Pathak et al.
(2017) describe curiosity as the motivation behind the behavior of an agent in an environment for which the outcome is unpredictable, i.e., an intrinsic reward that motivates the agent to explore the unseen portions of the state space (and subsequent transitions).
While causal curiosity is an intrinsic reward, it differs from these traditional deﬁnitions of curiosity in that it motivates the agent to produce structure in the outcome of its behavior.
E XPERIMENTS
Our work has 2 main thrusts - the discovered experimental behaviors and the representations obtained from the outcome of the behaviors in environments.
The experimental behaviors are tied to contributions 1 and 2 in the Introduction.
The causal factors allow us to achieve contribution 3 in the Introduction.
We visualize these learnt behaviors and verify that they are indeed semantically meaningful and interpretable.
We quantify the utility of the learned behaviors by using the behaviors as pre-training for a downstream task.
In our experimental setup, we verify that these behaviors are indeed invariant to all other causal factors except one.
We visualize the representations obtained using these behaviors and verify that they are indeed the binary quantized representations for each of the ground truth causal factors that we manipulated in our experiments.
Finally, we verify that the knowledge of the representation does indeed aid transfer learning and zero-shot generalizability in downstream tasks.
Causal World We use the Causal World Simulation (Ahmed et al.
(Under submission 2020))
based on the Pybullet Physics engine to test our approach.
The simulator consists of a 3-ﬁngered robot, with 3 joints on each ﬁnger.
We constrain each environment to consist of a single object that the agent can interact with.
The causal factors that we manipulate for each of the objects are size, shape and mass of the blocks.
The simulator allows us to capture and track the positions and velocities of each of the movable objects in an environment.
While, for most experiments, the 3D position and 3D pose of the blocks is used as the state at each time step, we perform ablation studies where less information is provided to the agent.
4. V ISUALIZING D ISCOVERED B EHAVIORS
We would like to analyze whether the discovered experimental behaviors are human interpretable, i.e., are the experimental behaviors discovered in each of the setups semantically meaningful?
We ﬁnd that our agents learn to perform several useful behaviors without any supervision.
For instance, to differentiate between objects with varying mass, we ﬁnd that they acquire a perfect grasp-and-lift behavior with an upward force.
In other random seed experiments, the agents learn to lift the blocks by using the wall of the environment for support.
To differentiate between cubes and spheres, the agent discovers a pushing behavior which gently rolls the spheres along a horizontal direction.
Qualitatively, we ﬁnd that these behaviors are stable and predictable.
See videos of discovered behaviors here (website under construction).
Concurrent with the objective they are trained on, we ﬁnd that the acquired behaviors impose structure on the outcome when applied to each of the training environments.
The outcome of each experimental behavior on the set of training environments results in dividing it into 2 subsets.
These subsets correspond to the binary quantized values of a single factor, e.g., large or small, while being invariant to the values of other causal factors of the environments.
We also perform ablation studies where instead of providing the full state vector, we provide only one coordinate (e.g., only x, y or z coordinate of the block).
We ﬁnd that causal curiosity results in behaviors that differentiate the environments based on outcomes along the direction provided.
For example, when only the x coordinate was provided, the agent learned to evaluate mass by applying a pushing behavior along the x direction.
Similarly, a lifting behavior was obtained when only the z coordinate was supplied to the curiosity module (Figure 2).
4. U TILITY OF LEARNED BEHAVIORS FOR DOWNSTREAM TASKS
While the behaviors acquired are semantically meaningful, we would like to quantify their utility as pre-training for downstream tasks.
We analyze the performance on Lifting where the agent must grasp and lift a block to a predetermined height and Travel , where the agent must impart a velocity to the block along a predetermined direction.
We re-train the learnt planner using an external reward for these tasks.
We implement a baseline vanilla Cross Entropy Method optimized
Model Predictive Control Planner (De Boer et al.
trained using the identical reward function and compare the rewards per trajectory during training.
We ﬁnd high zero-shot generalizability and quicker convergence as compared to the vanilla CEM planner (Figure 3).
4. V ISUALIZATION OF HIERARCHICAL BINARY LATENT SPACE
Our agents discover a disentangled latent space such that they are able to isolate the sources of causation of the variability they encounters in their environments.
For every environment, they learn a disentangled embedding vector which describes each of the causal factors.
To show this, we use 3 separate experimental setups - Mass , SizeMass and ShapeSizeMass where each of the causal factors are allowed to vary over a range of discrete values.
During Mass , the agent is allowed access to 5 environments with objects having the same shape (cuboids) and size but differing only in mass.
During SizeMass , the agent has access to 30 environments with cuboids having masses and sizes ranging over 5 and 6 values respectively.
Finally, during ShapeSizeMass , the agent has access to 60 environments with objects having masses, sizes and shapes ranging over 5, 6, and 2 values respectively.
During training, the agent discovers a hierarchical binary latent space (Figure 4), where each level of hierarchy corresponds to a single causal factor.
The binary values at each level of hierarchy correspond to the high/low values of the causal factor in question.
To our knowledge, we obtain the ﬁrst interpretable latent space describing the various causal processes in the environment of an agent.
This implies that it learns to quantify each physical attribute of the blocks it encounters in a completely unsupervised manner.
4. K NOWLEDGE OF CAUSAL FACTORS AIDS TRANSFER
Next, we test whether knowledge of the causal factors does indeed aid transfer and zero-shot generalizability.
To this end, we supply the representations obtained by the agent during the experimental behavior phase as input to a policy network in addition to the state of the simulator, and train it for a place-and-orient downstream task (Figure 1).
We deﬁne 2 experimental setups - TransferMass and TransferSizeMass .
In Mass , the agent is given access to 10 environments, with 10 varying values of mass.
In TransferSizeMass , the agent is allowed access to 10 environments, withand 2 values of mass and size respectively.
In both setups, the agent learns about the varying causal mechanisms by optimizing causal curiosity.
Subsequently, using the causal representation along with the state for each environment, it is trained to maximize external reward.
After training, the agents are exposed to a set of unseen test environments, where we analyze their zero-shot generalizability.
These test environments consist of unseen masses and sizes and their unseen combinations.
This corresponds to "Strong Generalization" as deﬁned by Perez et al.
We report results averaged over 10 random seeds.
For each setup, we train a PPO-optimized Actor-Critic Policy (referred to as Causally-curious agent ) with access to the causal representations and a 56 dimensional state vector from the environment (thus, a total of 57 dimensional input for TransferMass , and a 58 dimensional for
TransferSizeMass ).
Similar to Perez et al.
, we implement 2 baselines - the Generalist and the Specialist .
The Specialist consists of an agent with identical architecture as Causallycurious agent , but without access to causal representations (i.e., receives a 56 dimensional state vector).
It is initialized randomly and is trained only on the test environments, serving as a benchmark for complexity of the test tasks.
It performs poorly, indicating that the test tasks are complex.
The architecture of the Generalist is identical to the Specialist .
Like the Specialist , the Generalist also does not have access to the causal representations, but is trained on the same set of training environments that the Causally-curious agent is trained on.
The poor performance of the generalist indicates that the tasks distribution of training and test tasks differs signiﬁcantly and that memorization of behaviors does not yield good transfer.
We ﬁnd that causally-curious agents signiﬁcantly outperform the both baselines indicating that indeed, knowledge of the causal representation does aid zero-shot generalizability.
C ONCLUSION
We introduce causal curiosity , an intrinsic reward that allows agents to discover binary quantized representations for the causal factors that affect environments an RL agent may encounter.
We show that optimizing causal curiosity rewards results in the agent performing self-supervised experiments.
We ﬁnd that these experiments happen to be semantically meaningful and can be used as pre-training for downstream tasks.
While our work learns binary quantized causal representations, a dense encoding may improve the amount of encoded information about the causal mechanisms of the environments.
We leave this to future work.
A I MPLEMENTATION D ETAILS FOR E XPERIMENT D ISCOVERY
A.
P LANNER
The Experiment Planner consisted of a uniform distribution planner for a horizon of 6 control signals.
The planner was trained using the Cross Entropy Method Model Predictive Control (Camacho &
Alba (2013); De Boer et al.
on the true environment.
We sampled 40 plans per iteration from the distribution initialized to uniform U ( controlLow, controlHigh ) .
Each of the sampled plans are applied to each of the training environments and the top 10% of the plans are used to update the distribution.
A. T RAINING E NVIRONMENTS
The training environments vary in each experiment.
In Section 4.3, we utilize 3 setups, Mass ,
SizeMass and ShapeSizeMass .
For Mass , we allow the agent to access 5 environments with masses varying from 0.1 kg to 0.5 kg.
In SizeMass , the agent has access to 30 environments with masses varying uniformly from 0.1 to 0.5 kg and sizes from 0.05 to 0.1 meters.
Finally, in
ShapeSizeMass , the agent has access to 60 environments, with masses varying uniformly from 0.
to 0.5 kg, sizes from 0.05 to 0.1 meters and shapes either being cubes or spheres.
During experiment discovery, in each environment, the agent has access to the position of the block in the environment along with its quaternion orientation.
A. C URIOSITY R EWARD C ALCULATION
We predetermine the minimum description length of the clustering model L ( M ) by assuming that the observations O , obtained by applying experimental behavior a 0: T are produced by a bi-modal generator distribution, where each mode corresponds to either a low or high (quantized) value of a causal factor.
This also ensures that L ( M ) is as small as possible.
The planner, eq.
(4) solves the following optimization problem:
[Equation]
the distance function d ( · , · ) in the space of trajectories is set to be Soft Dynamic Time Warping (Cuturi & Blondel (2017)).
The trajectory length T is 6 control steps long.
The objective is a modiﬁed version of the Silhouette Score (Rousseeuw (1987)).
Intuitively, Objective (7) expresses the ability of a low complexity model, assumed to be bi-modal, to encode the observations O = s 0: T .
If multiple causal factors control O , then the Minimum
Description Length of L ( O ) will be high.
Subsequently, since M is a simple model, the deviation of
O from M will be high i.e. L ( O | M ) will be high resulting in a low value of the optimization objective.
S and S (cid:48) correspond to clusters of outcomes which quantize the values of a causal factor isolated by a 0: T .
s 0: T , s (cid:48)(cid:48) 0: T ∈ S correspond to trajectories of states i.e. observations obtained by applying a 0: T to environments with say, low values of a causal factor while s (cid:48)
: T , s (cid:48)(cid:48)(cid:48) 0: T ∈ S (cid:48) correspond to trajectories of states i.e. observations obtained by applying a 0: T to environments with say, high values of the same causal factor.
Objective (7) attempts to ensure that these clusters are far apart from each other and are tight i.e. a simple model M encodes
O well.
We further motivate how this formulation allows disentanglement of causal factors.
A central assumption is that causal factors are independent, by deﬁnition, i.e. Independent Mechanisms
Assumption Peters et al.
Consider the outcome O obtained by applying an action sequence a 0: T to a set of environments.
If the action sequence a 0: T results in multiple causal factors affecting the outcome O , the Kolmogorov complexity of O will be high.
The reason for this is that each causal factor has its own independent causal mechanism (Peters et al.
(2017); Parascandolo et al.
(2018)) that affects O .
Thus, given this independence, the information in O will be a sum of the information “injected” into it from the multiple causes.
Conversely, if the outcome O obtained by applying an action sequence a 0: T has a lower Kolmogorov Complexity, then O is caused by fewer causal factors.
Causal Curiosity attempts to reduce this complexity of O , by assuming a simple generative model M is sufﬁcient to encode O .
Thus for experimental behaviors which allow several causes to affect O , the “overﬂow” of O from M will be high and subsequently the causal curiosity reward will be low.
Thus, post-optimization of the objective, we arrive at an action sequence that allows for disentanglement of the causal factors.
B I MPLEMENTATION D ETAILS FOR T RANSFER
In Section 4.4, we show the utility of learning causal representations in 2 separate experimental setups.
During TransferMass , the agent has access to 10 environments during training, with masses ranging from 0.1 to 0.5 kg.
At test time, the agent is required to perform the place-and-orient task masses 2 masses - 0.7 kg and 0.75 kg.
During TransferSizeMass , the agent has access to 10 environments during training, with masses ranging from 0.1 to 0.5 kg and sizes from either 0.01 or 0.05 m. At test time the agent is asked to perform the task on 2 environments with masses 0.7 kg and 0.75 kg with sizes = 0.05 m.
We ﬁnd that testing with large and light blocks increase the chances of accidental goal completions.
Thus, during test-time, we use environments with high masses for out-of-distribution testing.
The causal representation is concatenated to the state of the environment as a contextual input and supplied to a PPO-Optimized Actor-Critic Policy.
The policy network consists of 2 hidden layers withand 128 units respectively.
The experiments are parallelized on 10 CPUs and implemented using stable baselines (Hill et al.
The agent receives a dense reward at each time step during the maximizing external reward phase (Figure 1), the negative of the distance of the block from the goal position scaled by factor of 1000.
The control signal was repeated 10 times to the actuators of the motors on each ﬁnger.
C I NTUITION FOR D EFINITION OF C AUSAL F ACTORS
We begin with a simple example of a person walking on earth.
This person experiences various physical processes while interacting in her world, for example gravity, friction, wind etc.
These physical processes affect the outcome of interactions of the person with her environment.
For example, while jumping on earth, the human experiences gravity which affects the outcome of her jump, the fact that she falls back to the ground.
Additionally, these physical processes (or causal mechanisms) are parameterized by causal factors, for example, acceleration constant due to gravity g = 9 .
8 m/son earth, or coefﬁcients of friction between her feet and the ground which assume particular numerical values.
These causal factors may vary across multiple environments.
For example, the person may walk on sand or on ice, surfaces with varying frictional values.
Thus the outcome of running on such surfaces will vary, running on sand will require signiﬁcant effort, while running on ice may result in the person slipping.
Thus the coefﬁcient of friction between the person’s feet and the surface she walks on affects the outcome of a particular behavior in said environment.
In our deﬁnition, h j are causal factors such friction with some particular coefﬁcient of friction, or gravity with acceleration constant g or other.
H is the global set containing all such causal factors.
Now we ask the question (which we subsequently answer), given multiple environments, how would a human characterize each of them depending on the value of a causal factor?
Through experimental behaviors.
The human in the above example would attempt to run in each of the environments she encountered, be it on sand, on ice, in mud etc.
If she slipped in an environment, she would characterize it as slippery.
If she didn’t, she would characterize it as non-slippery.
We attempt to equip our agent with similar logic.
The “sequence of actions” ( a 0: T ) described in our paper corresponds to the human running.
The observation O ( i ) in the environment e ( i ) consisting of the sequence of states ( s 0: T ) corresponds to the outcome of running.
O might belong to either of the clusters of outcomes S or S (cid:48) corresponding to slipping or not slipping.