# ONE AFTER ANOTHER: LEARNING INCREMENTAL SKILLS FOR A CHANGING WORLD


**Nur Muhammad (Mahi) Shafiullah**
mahi@cs.nyu.edu
New York University


**Lerrel Pinto**
lerrel@cs.nyu.edu
New York University


ABSTRACT

Reward-free, unsupervised discovery of skills is an attractive alternative to the
bottleneck of hand-designing rewards in environments where task supervision is
scarce or expensive. However, current skill pre-training methods, like many RL
techniques, make a fundamental assumption – stationary environments during
training. Traditional methods learn all their skills simultaneously, which makes
it difficult for them to both quickly adapt to changes in the environment, and to
not forget earlier skills after such adaptation. On the other hand, in an evolving or
expanding environment, skill learning must be able to adapt fast to new environment
situations while not forgetting previously learned skills. These two conditions make
it difficult for classic skill discovery to do well in an evolving environment. In this
work, we propose a new framework for skill discovery, where skills are learned one
after another in an incremental fashion. This framework allows newly learned skills
to adapt to new environment or agent dynamics, while the fixed old skills ensure
the agent doesn’t forget a learned skill. We demonstrate experimentally that in
both evolving and static environments, incremental skills significantly outperform
current state-of-the-art skill discovery methods on both skill quality and the ability
to solve downstream tasks. Videos for learned skills and code are made public on:
[https://notmahi.github.io/disk.](https://notmahi.github.io/disk)

1 INTRODUCTION

Modern successes of Reinforcement Learning (RL) primarily rely on task-specific rewards to learn
motor behavior (Levine et al., 2016; Schulman et al., 2017; Haarnoja et al., 2018; Andrychowicz et al.,
2020). This learning requires well behaved reward signals in solving control problems. The challenge
of reward design is further coupled with the inflexibility in the learned policies, i.e. policies trained
on one task do not generalize to a different task. Not only that, but the agents also generalize poorly
to any changes in the environment (Raileanu et al., 2020; Zhou et al., 2019). This is in stark contrast
to how biological agents including humans learn (Smith & Gasser, 2005). We continuously adapt,
explore, learn without explicit rewards, and are importantly incremental in our learning (Brandon,
2014; Corbetta & Thelen, 1996).

Creating RL algorithms that can similarly adapt and generalize to new downstream tasks has hence
become an active area of research in the RL community (Bellemare et al., 2016; Eysenbach et al.,
2018; Srinivas et al., 2020). One viable solution is unsupervised skill discovery (Eysenbach et al.,
2018; Sharma et al., 2019b; Campos et al., 2020; Sharma et al., 2020). Here, an agent gets access to a
static environment without any explicit information on the set of downstream tasks or reward functions.
During this unsupervised phase, the agent, through various information theoretic objectives (Gregor
et al., 2016), is asked to learn a set of policies that are repeatable while being different from each other.
These policies are often referred to as behavioral primitives or skills (Kober & Peters, 2009; Peters
et al., 2013; Schaal et al., 2005). Once learned, these skills can then be used to solve downstream
tasks in the same static environment by learning a high-level controller that chooses from the set of
these skills in a hierarchical control fashion (Stolle & Precup, 2002; Sutton et al., 1999; Kulkarni
et al., 2016) or by using Behavior Transfer to aid in agent exploration (Campos et al., 2021). However,
the quality of the skills learned and the subsequent ability to solve downstream tasks are dependent
on the unsupervised objective.


-----

tings (Nareyek, 2003; Koulouriotis & Xanthopoulos, 2008; Da Silva et al., 2006; Cheung et al., 2020),

Skill 1

(a) (b) (c)

Consistency

Skill 2

Diversity

Skill M-1

Skill M

Movement in x-y plane

Figure 1: We present Discovery of Incremental Skills (DISk). As illustrated in (a), DISk learns skills incrementally by having each subsequent skills be diverse from the previous skills while being consistent with itself. DISk
lets us learn skills in environment where dynamics change during training, an example of which is shown in (b):
an Ant environment where a different leg breaks every few episodes. In this environment, DISk’s learned skills’
trajectories are shown in (c) under each broken leg – each color shows a different skill.

Unfortunately, current objectives for unsupervised skill discovery make a major assumption about
the environment stationarity. Since all skills are simultaneously trained from start to finish, if the
environment changes, every skill is equally exposed to highly non-stationary experience during its
training. Moreover, all skills trying to adapt to the changed environment means that the agent may
catastrophically forget previously learned skills, which is undesirable for an agent with expanding
scope. This severely limits the application of current skill discovery methods in lifelong RL set
a simple example of which is illustrated in Figure 1(b), or real-world dynamic settings (Gupta et al.,
2018; Julian et al., 2020).

In this paper, we propose a new unsupervised model-free framework: Discovery of Incremental Skills
(DISk). Instead of simultaneously learning all desired skills, DISk learns skills one after another to
avoid the joint skill optimization problem. In contrast to prior work, we treat each individual skill as
an independent neural network policy without parameter sharing, which further decouples the learned
skills. Given a set of previously learned skills, new skills are optimized to have high state entropy with
respect to previously learned skills, which promotes skill diversity. Simultaneously, each new skill
is also required to have low state entropy within itself, which encourages the skill to be consistent.
Together, these objectives ensure that every new skill increases the diversity of the skill pool while
being controllable (Fig. 1(a)). Empirically, this property enables DISk to generate higher quality
skills and subsequently improve downstream task learning in both dynamic and static environments
compared to prior state-of-the-art skill discovery methods. Especially in evolving environments, DISk
can quickly adapt new skills to the changes in the environment while keeping old skills decoupled
and fixed to prevent catastrophic forgetting. We demonstrate this property empirically in MuJoCo
navigation environments undergoing varying amounts of change. While DISk was designed for
dynamic, evolving environments, we observe that it performs noticeably better than previous skill
learning methods in static environments like standard MuJoCo navigation environments by improving
skill quality and downstream learning metrics.

To summarize, this paper provides the following contributions: (a) we propose a new unsupervised
skill discovery algorithm (DISk) that can effectively learn skills in a decoupled manner. (b) We
demonstrate that the incremental nature of DISk lets agents discover diverse skills in evolving or
expanding environments. (c) Even in static environments, we demonstrate that DISk can learn a
diverse set of controllable skills on continuous control tasks that outperform current state-of-theart methods on skill quality, as well as in sample and computational efficiency. (d) Finally, we
demonstrate that our learned skills, both from dynamic and static environments, can be used without
any modifications in a hierarchical setting to efficiently solve downstream long-horizon tasks.

2 BACKGROUND AND PRELIMINARIES

To setup our incremental RL framework for unsupervised skill discovery, we begin by concisely
introducing relevant background and preliminaries. A detailed treatment on RL can be found in
Sutton & Barto (2018), incremental learning in Chen & Liu (2018), and skill discovery in Eysenbach
et al. (2018).


-----

**Reinforcement Learning: Given an MDP with actions a ∈** _A, states s ∈_ _S, standard model-free RL_
aims to maximize a task specific reward rT (s, a) by optimizing the parameters θ of a policy πθ. In
contrast to task-specific RL, where the rewards rT correspond to making progress on that specific
task, methods in unsupervised RL (Bellemare et al., 2016; Eysenbach et al., 2018; Srinivas et al.,
2020) instead optimize an auxiliary reward objective rI that does not necessarily correspond to the
task-specific reward rT .

**Incremental Learning: Incremental, lifelong, or continual learning covers a wide variety of problems,**
both within and outside of the field of Reinforcement Learning. Some works consider the problem
to fast initialization in a new task Tn+1 given previous tasks Ti, 1 ≤ _i ≤_ _n (Tanaka & Yamamura,_

1997; Fernando et al., 2017), while others focus on learning from a sequence of different domains
_D1, · · ·, Dt such that at time t + 1, the model does not forget what it has seen so far (Fei et al., 2016;_
Li & Hoiem, 2017). Finally, other works have considered a bidirectional flow of information, where
performance in both old and new tasks improves simultaneously with each new task (Ruvolo & Eaton,
2013; Ammar et al., 2015). In all of these cases, there is a distribution shift in the data over time,
which differentiates this problem setting from multi-task learning. Some of these methods (Tanaka
& Yamamura, 1997; Ruvolo & Eaton, 2013; Fei et al., 2016; Ruvolo & Eaton, 2013) learn a set of
independently parametrized models to address this distribution shift over time.

**Skill Discovery: Often, skill discovery is posited as learning a z-dependent policy πz(a|s; θ) (Eysen-**
bach et al., 2018; Sharma et al., 2020), where z ∈ _Z is a latent variable that represents an individual_
skill. The space Z hence represents the pool of skills available to an agent. To learn parameters of
the skill policy, several prior works (Hausman et al., 2018; Eysenbach et al., 2018; Campos et al.,
2020; Sharma et al., 2020) propose objectives that diversify the outcomes of different skills while
making the same skill produce consistent behavior.

One such algorithm, DIAYN (Eysenbach et al., 2018), does this by maximizing the following
information theoretic objective:

_F(θ) = I(S; Z) + H(A|S, Z)_ (1)

Here, I(S; Z) represents the mutual information between states and skills, which intuitively encourages the skill to control the states being visited. H(A|S, Z) represents the Shannon entropy of actions
conditioned on state and skill, which encourages maximum entropy (Haarnoja et al., 2017) skills.
Operationally, this objective is maximized by model-free RL optimization of an intrinsic reward
function that corresponds to maximizing Equation 1, i.e. rI (s, a _θ) := [ˆ](θ), where_ [ˆ](θ) is an
_|_ _F_ _F_
estimate of the objective. Note that the right part of the objective can be directly optimized using
maximum-entropy RL (Haarnoja et al., 2018; 2017).

Other prominent methods such as DADS and off-DADS (Sharma et al., 2019b; 2020) propose to
_′_
instead maximize I(S ; Z|S), the mutual information between the next state s′ and skill conditioned
on the current state. Similar to DIAYN, this objective is maximized by RL optimization of the
corresponding intrinsic reward. However, unlike DIAYN, the computation of the reward is modelbased, as it requires learning a skill-conditioned dynamics model. In both cases, the z-conditioned
skill policy πθ( ; z) requires end-to-end joint optimization of all the skills and shares parameters

_·|·_
across skills through a single neural network policy.

3 METHOD

In DIAYN and DADS, all skills are simultaneously trained from start to end, and they are not designed
to be able to adapt to changes in the environment (non-stationary dynamics). This is particularly
problematic in lifelong RL settings Nareyek (2003); Koulouriotis & Xanthopoulos (2008); Da Silva
et al. (2006); Cheung et al. (2020) or in real-world settings Gupta et al. (2018); Julian et al. (2020)
where the environment can change during training.

To address these challenges of reward-free learning and adaptability, we propose incremental policies.
Here, instead of simultaneous optimization of skills, they are learned sequentially. This way newer
skills can learn to adapt to the evolved environment without forgetting what older skills have learned.
In Section 3.1 we discuss our incremental objective, and follow it up with practical considerations
and the algorithm in Section 3.2.


-----

3.1 DISCOVERY OF INCREMENTAL SKILLS (DISK)

To formalize our incremental discovery objective, we begin with Equation 1, under which categorical
or discrete skills can be expanded as:

(θ) := E (2)
_F_ _zm_ _p(z)[[][IG][(][S][;][ Z][ =][ z][m][) +][ H][(][A][|][S, Z][ =][ z][m][)]]_
_∼_

Here, IG refers to the information gain quantity IG(S; zm) = (S) (S _zm). Intuitively, this_
_H_ _−H_ _|_
corresponds to the reduction in entropy of the state visitation distribution when a specific skill zm is
run. With a uniform skill prior p(z) and M total discrete skills, this expands to:


(θ) := [1]
_F_ _M_


(IG(S; Z = zm) + (A _S, Z = zm))_ (3)
_H_ _|_
_m=1_

X


In incremental skill discovery, since our goal is to learn a new skill zM given M − 1 previously
trained skills (z1, z2, ..., zM _−1) we can fix the learned skills z1:M_ _−1, and reduce the objective to:_

(θM ) := IG(S; Z = zM ) + (A _S, Z = zM_ ) (4)
_F_ _H_ _|_
= (S) (S _zM_ ) + (A _S, Z = zM_ ) (5)
_H_ _−H_ _|_ _H_ _|_

The first term H(S) corresponds to maximizing the total entropy of states across all skills, which
encourages the new skill zM to produce diverse behavior different from the previous ones. The
second term (S _zM_ ) corresponds to reducing the entropy of states given the skill zM, which
_−H_ _|_
encourages the skill to produce consistent state visitations. Finally, the third term (A _S, Z = zM_ )
_H_ _|_
encourages maximum entropy policies for the skill zM . Note that this objective can be incrementally
applied for m = 1, · · ·, M, where M can be arbitrarily large. A step-by-step expansion of this
objective is presented in Appendix B.

3.2 A PRACTICAL ALGORITHM

Since we treat each skill as an independent policy, our framework involves learning the M _[th]_ skill
_πM given a set of prior previously learned skills π1,_ _, πM_ 1. Note that each πm is a stochastic
_· · ·_ _−_
policy representing the corresponding skill zm and has its own set of learned parameters θm. Let
_Srolling out the policiesm ∼_ _p(s|πm), S ∼_ [P] πm[M]m=1, 1[p][(][s][|]m[π][m][)][ denote random variables associated with the states visited by]M . Then, our objective from Equation 5 becomes:
_≤_ _≤_

(θM ) = (S) (SM ) + (A _SM_ ) (6)
_F_ _H_ _−H_ _H_ _|_

The rightmost term can be optimized used max-entropy based RL optimization (Haarnoja et al.,
2018) as done in prior skill discovery work DIAYN or Off-DADS. However, computing the first two
terms is not tractable in large continuous state-spaces. To address this, we employ three practical
estimation methods. First, we use Monte-Carlo estimates of the random variables S, Sm by rolling
out the policy πm. Next, we use a point-wise entropy estimator function to estimate the entropy
_H_
from the sampled rollouts. Finally, since distances in raw states are often not meaningful, we measure
similarity in a projected space for entropy computation. Given this estimate [ˆ](θM ) of the objective,
_F_
we can set the intrinsic reward of the skill as rI (s, a _θM_ ) := [ˆ](θM ). The full algorithm is described
_|_ _F_
in Algorithm 1, pseudocode is given in Appendix H, and the key features are described below.

**Monte-Carlo estimation with replay: For each skill, we collect a reward buffer containing N**
rollouts from each previous skills that make up our set . For _M_, we keep a small number of
_T_ _T_
states visited by our policy most recently in a circular buffer such that our TM cannot drift too much
from the distribution PπM (s). Since point-wise entropy computation of a set of size n has an O(n[2])
complexity, we subsample states from our reward buffer to estimate the entropy.

**Point-wise entropy computation: A direct, Monte-Carlo based entropy estimation method would**
require us to estimate entropy by computing − _|S[1]|_ _s∈S_ [log][ p][(][s][)][, which is intractable. Instead, we]

use a non-parametric, Nearest Neighbor (NN) based entropy estimator (Singh et al., 2003; Liu &

P

Abbeel, 2021; Yarats et al., 2021):


_|X|_

ln _xi_ NNk,X(xi) 2 (7)
_i=1_ _||_ _−_ _||_

X


¯X,k(p)
_H_ _∝_


_|X|_


-----

**Algorithm 1 Discovery of Incremental Skills: Learning the M** th Skill

**Input: Projection function σ, hyperparameter k, α, β, off-policy learning algorithm A.**
**Initialize: Learnable parameters θM for πM**, empty circular buffer BUF with size n, and replay
buffer R.
Sample trajectories = _m=1_

**while Not converged T do** _[{][s][ ∼]_ _[π][m][}][ from previous][ M][ −]_ [1][ policies on current environment.]

Collect transition (s, a, s[′]) by running πM on the environment.

[S][M] _[−][1]_

Store transition data (s, a, s[′]) into the replay buffer, R ← (s, a, s[′]).
Add the new projected state to the circular buffer, BUF ← _σ(s[′])._
**for t = 1,2,..Tupdates do**

Sample (s, a, s[′]) from R.
Sample batch _b_ .
Find σc, the k T[th]-Nearest Neighbor of ⊂T _σ(s[′]) within BUF,_
Find σd, the k[th]-Nearest Neighbor of σ(s[′]) within σ( _b)._
_T_
Set consistency penalty rc := ||σ(s[′]) − _σc||2._
Set diversity reward rd := ||σ(s[′]) − _σd||2._
Set intrinsic reward rI := −αrc + βrd.
Update θM using A using (s, a, s[′], rI ) as our transition.

**Return: θM**


where each pointto the overall entropy. We further approximate xi of the dataset X contributes an amount proportional to ln x ≈ _x −_ 1 similar to Yarats et al. ln ||xi − (NN2021k,) and setX(xi)||2
_k = 3 since any small value suffices. Additional details are discussed in Appendix B._

**Measuring similarity under projection: To measure entropy using Equation 7 on our problem, we**
have to consider nearest neighbors in an Euclidean space. A good choice of such a space will result
in more meaningful diversity in our skills. Hence, in general we estimate the entropy not directly
on raw state s, but under a projection σ(s), where σ captures the meaningful variations in our agent.
Learning such low-dimensional projections is in itself an active research area (Whitney et al., 2019;
Du et al., 2019). Hence, following prior work like Off-DADS and DIAYN that use velocity-based
states, we use a pre-defined σ that gives us the velocity of the agent. We use instantaneous agent
velocity since in locomotion tasks it is a global metric that is generally independent of how long the
agent is run for. Like Off-DADS and DIAYN, we found that a good projection is crucial for success
of an agent maximizing information theoretic diversity.

**Putting it all together: On any given environment, we can put together the above three ideas to learn**
arbitrary number of incremental skills using Algorithm 1. To learn a new skill πM in a potentially
evolved environment, we start with collecting an experience buffer T with states s ∼ _πm collected_
from previous skills. Then, we collect transitions (s, a, s[′]) by running πM on the environment. We
store this transition in the replay buffer, and also store the projected state σ(s[′]) in a circular buffer
BUF. Then, to update our policy parameter θM, on each update step, we sample (s, a, s[′]) from our
replay buffer and calculate the intrinsic reward of that sample. To calculate this, we first find the
_k[th]_ Nearest Neighbor of σ(s[′]) within BUF, called σc henceforth. Then, we sample a batch Tb ⊂T,
and find the k[th] Nearest Neighbor of σ(s[′]) within σ(Tb) = {σ(s) | s ∈Tb}, called σd henceforth.
Given these nearest neighbors, we define our consistency penalty rc := ||σ(s[′]) − _σc||2, and our_
diversity reward rd := ||σ(s[′]) − _σd||2, which yields an intrinsic reward rI := −αrc + βrd. α and β_
are estimated such that the expected value of αrc and βrd are close in magnitude, which is done by
using the mean values of ˆrc and ˆrd from the previous skill πM 1.
_−_

Training the first skill with DISk uses a modified reward function, since there is no prior skills T to
compute diversity rewards against. In that case, we simply set rI := 1 − _αrc throughout the first_
skills’ training, thus encouraging the agent to stay alive while maximizing the consistency. Exact
hyperparameter settings are provided in Appendix D.

4 EXPERIMENTS

We have presented an unsupervised skill discovery method DISk that can learn decoupled skills
incrementally without task-specific rewards. In this section we answer the following questions: (a)


-----

Does DISk learn diverse and consistent skills in evolving environments with non-stationary dynamics?
(b) How does DISk compare against traditional skill learning methods in stationary environments? (c)
Can skills learned by DISk accelerate the learning of downstream tasks? (d) How does our specific
design choices effect the performance of DISk? To answer these questions we first present our
experimental framework and baselines, and then follow it with our experimental results.

4.1 EXPERIMENTAL SETUP

To study DISk, we train skills on an agent that only has access to the environment without information
about the downstream tasks (Eysenbach et al., 2018; Sharma et al., 2019b; 2020). We use two
types of environments: for stationary tasks we use standard MuJoCo environments from OpenAI
Gym (Todorov et al., 2012; Brockman et al., 2016): HalfCheetah, Hopper, Ant, and Swimmer,
visualized in Fig. 4 (left), while for non-stationary environments we use two modified variants of the
Ant environment: one with disappearing blocks, and another with broken legs. Environment details
are provided in Appendix C.

**Baselines: In our experiments, we compare our algorithm against two previous state-of-the-art**
unsupervised skill discovery algorithms mentioned in Section 2, DIAYN (Eysenbach et al., 2018) and
Off-DADS (Sharma et al., 2020). To contextualize the performance of these algorithms, whenever
appropriate, we compare them with a collection of random, untrained policies. To make our comparison fair, we give both DIAYN and Off-DADS agents access to the same transformed state, σ(s), that
we use in DISk. Since Off-DADS is conditioned by continuous latents, for concrete evaluations we
follow the same method as the original paper by uniformly sampling N latents from the latent space,
and using them as the N skills from Off-DADS in all our experiments. Exact hyperparameter settings
are described in Appendix D, and our implementation of DISk is attached as supplimentary material.

4.2 CAN DISK ADAPTIVELY DISCOVER SKILLS IN EVOLVING ENVIRONMENTS?

One of the key hypothesis of this work is that DISk is able to discover useful skills even when the
environment evolves over time. To gauge the extent of DISk’s adaptability, we design two experiments
shown in Fig. 2 (left) and Fig. 1 (b) for judging adaptability of DISk in environments that change in
continuous and discrete manner respectively.

**Continuous Change: We create a MuJoCo environment similar to Ant with 40 blocks encircling**
the agent. The blocks initially prevent long trajectories in any direction. Then, as we train the agent,
we slowly remove the blocks at three different speeds. If T is the total training time of the agent, in
“fast”, we remove one block per [T]/40 steps, in “even” we remove two blocks per [T]/20 training steps, and
in “slow” we remove ten blocks per [T]/4 training steps. In each of the cases, the environment starts
with a completely encircled agent, and ends with an agent completely free to move in any direction;
across the three environments just the frequency and the magnitude of the evolution changes. This
environment is meant to emulate an expanding environment where new possibilities open up during
training; like a domestic robot being given access to new rooms in the house. The three different
rates of change lets us observe how robust a constant skill addition schedule is with different rates of
change in the world.

In Fig. 2 (middle), we can see how the skill learning algorithms react to the changing environment.
The comparison between Off-DADS and DISk is especially noticeable. The Off-DADS agent latches
onto the first opening of the block-circle, optimizes the skill dynamics model for motions on that
single direction, and ignores further evolution of the environment quite completely. On the other
hand, since the DISk agent incrementally initializes an independent policy per skill, it can easily
adapt to the new, changed environment during the training time of each skill. To quantitatively
measure the diversity of skills learned, we use the Hausdorff distance (Belogay et al., 1997), which is
a topological way of measuring distance between two sets of points (see appendix B.3.) We plot the
mean Hausdorff distance between all endpoints of a skill and all other endpoints in Fig. 2 (right). In
this metric as well, DISk does better than other methods across environments.

**Discrete Change: We modify the standard Ant environment to a changing dynamic environment by**
disabling one of the four Ant legs during test and train time. To make the environment dynamic, we
cycle between broken legs every 1M steps during the 10M steps of training. We show the trajectories
of the learned skills with each of the legs broken in Fig. 3.


-----

|Col1|Col2|
|---|---|
|s vanishin|g|
|||


performs well with that particular dynamic.


DIAYN Off-DADS DISk

Movement in x-y plane

Mean Hausdorff distance between skills

Figure 2: Left: Block environments that evolve during training; red highlights portion of blocks vanishing at
once. Middle: Trajectories of skills discovered by different algorithms; each unique color is a different skill.
Right: Mean Hausdorff distance of the skills discovered in these environments.

Off-DADS DISk

In this case, the Off-DADS and the DISk agent
both adapt to the broken leg and learn some
skills that travel a certain distance over time.3 **4**
However, a big difference becomes apparent if
we note under which broken leg the Off-DADS
agent is performing the best. The Off-DADS
agent, as seen in Fig. 3 (left), performs well2 **1** Movement in x-y plane
with the broken leg that it was trained on most
recently. To optimize for this recently-broken
leg, this agent forgets previously learned skills
(App. Fig. 7, 8). Conversely, each DISk skill Figure 3: Evaluation of skills learned on the broken Ant
learns how to navigate with a particular leg environment with different legs of the Ant disabled; leg
broken, and once a skill has learned to navi- disabled during evaluation in legend.
gate with some leg broken, that skill always


4.3 WHAT QUALITY OF SKILLS ARE LEARNED BY DISK IN A STATIC ENVIRONMENT?

While DISk was created to address skill learning in an evolving environment, intuitively it could
work just as well in a static environment. To qualitatively study DISk-discovered skills in static
environments, we plot the trajectories generated by them in Fig. 4 (middle). On the three hardest
environments (Hopper, Ant, and Swimmer), DISk produces skills that cover larger distances than
DIAYN and Off-DADS. Moreover, on Ant and Swimmer, we can see that skills from DISk are quite
consistent with tight packing of trajectories. On the easier HalfCheetah environment, we notice
that DIAYN produces higher quality skills. One reason for this is that both DIAYN and Off-DADS
share parameters across skills, which accelerates skill learning for less complex environments. Also,
note that the performance of DIAYN is worse on the complex environments like Swimmer and Ant.
The DIAYN agent stops learning when the DIAYN discriminator achieves perfect discriminability;
since we are providing the agent and the discriminator with extra information σ(s), in complex
environments like Swimmer or Ant DIAYN achieves this perfect discriminability and then stops
learning. While the performance of DIAYN on Ant may be surprising, it is in line with the observations
by Sharma et al. (2019b; 2020).

As shown in the quantitative results in Fig. 4 (right), for the harder Hopper, Ant, and Swimmer
tasks, DISk demonstrates substantially better performance. However, on the easier HalfCheetah task,


-----

DIAYN Off-DADS DISk

HalfCheetah

Hopper

Swimmer

Distribution of states across skills

Mean Hausdorff distance between skills

Ant

Figure 4: Left: Environments we evaluate on. Middle: Visualization of trajectories generated by the skills
learned by DISk and baselines, with each unique color denoting a specific skill. For environments that primarily
move in the x-axis (HalfCheetah and Hopper), we plot the final state reached by the skill across five runs. For

environments that primarily move in the x-y plane (Ant and Swimmer), we plot five trajectories for each skill.
Right: Hausdorff distance (higher is better) between the terminal states visited by each skill and terminal states

|DISk skills between nce|Col2|
|---|---|
|dista Hausdorff Mean ion of trajectories generated ecific skill. For environments e reached by the skill across er), we plot fvie trajectories|by the skills that primarily fvie runs. For for each skill|

of all other skills, averaged across the set of discovered skills to give a quantitative estimate of the skill diversity.

DISk underperforms DIAYN and Off-DADS. This supports our hypothesis that incremental learning
discovers higher quality skills in hard tasks, while prior works are better on easier environments.


4.4 CAN DISK SKILLS ACCELERATE DOWNSTREAM LEARNING?

A promise of unsupervised skill discovery methods is that in complicated environments the discovered
skills can accelerate learning on downstream tasks. In our experiments, we find that skills learned by
DISk can learn downstream tasks faster by leveraging the learned skills hierarchically. To examine the
potential advantage of using discovered skills, we set up an experiment similar to the goal-conditioned
hierarchical learning experiment seen in Sharma et al. (2019b). In this experiment, we initialize a
hierarchical Ant agent with the learned skills and task it with reaching an (x, y) coordinate chosen
uniformly from [−15, 15][2]. Note that this environment is the standard Ant environment with no
changing dynamics present during evaluations. To highlight the versatility of the skills learned
under evolving environments, we also evaluate the agents that were trained under our evolving Block
environments (see Section 4.2) under the same framework.

We generally find that DISk agents outperform other agents on this hierarchical learning benchmark,
by generally converging within 100-200k steps compared to DIAYN agent never converging and
Off-DADS taking around 500k steps (Fig. 5). This statement holds true regardless of the evolution
dynamics of the environment where the agents were learned, thus showing that DISk provides an
adaptive skill discovery algorithm under a variety of learning circumstances. This property makes
DISk particularly suitable for lifelong learning scenarios where the environment is continuously
evolving.


-----

Ant Blocks (slow) Blocks (even) Blocks (fast)

2.5

Algorithm

2.0 DIAYN

Off-DADS

1.5 DISk

SAC

1.0

0.5

Normalized distance to goal

0.0

0 1e5 2e5 3e5 4e5 5e5 0 1e5 2e5 3e5 4e5 5e5 0 1e5 2e5 3e5 4e5 5e5 0 1e5 2e5 3e5 4e5 5e5

Steps Steps Steps Steps


Figure 5: Downstream Hierarchical Learning. We plot the average normalized distance (lower is better), given
byplot the performance of agents that were trained in environment with no obstacles, and our three evolving blockT[1] _Tt=0[(][d][(][s][t][, g][)][/d][(][s][0][, g][))][ from our Ant agent to the target over 500K training steps. From left to right, we]_

environments. Shading shows variance over three runs, and we add a goal-based Soft Actor Critic model trainedP
non-hierarchically as a baseline. Across all settings, DISk outperforms prior work.

4.5 ABLATION ANALYSIS ON DESIGN CHOICES

We make several design choices that improve DISk. In this section we briefly discuss the two most
important ones. Firstly, our implementation of DISk shares the data collected by the previous skills by
initializing the replay buffer R of a new skill with data from the previous ones. Once initialized with
this replay, the new skill can relabel that experience with its own intrinsic rewards. This method has
shown promise in multi-task RL and hindsight relabelling (Andrychowicz et al., 2017; Li et al., 2020;
Eysenbach et al., 2020). Across our experiments, performing such relabelling improves performance
significantly for the later skills (App. Fig. 10 (left)) as it can reuse vast amounts of data seen by prior
skills. Training details for this experiment is presented in Appendix G.

Compared to prior work, DISk not only learns skills incrementally, but also learns independent
neural network policies for each skill. To check if the performance gains in static environment are
primarily due to the incremental addition of skills or the independent policy part, we run a version
of our algorithm where all the independent skills are trained in parallel (App. Fig. 10 (left)) on Ant.
Experimentally, we see that simply training independent skills in parallel does not discover useful
skills; with the parallel variant achieving far lower mean Hausdorff distance. Note that we cannot
have incremental addition of skills without independent policies, at least in the current instantiation of
DISk, without significant modification to the algorithm or the architecture. This is because it is highly
nontrivial to extend or modify a network’s input or parameter space, and the simplest extension is
just a new network which is DISk itself.

5 RELATED WORK

Our work on developing incremental skill discovery is related to several subfields of AI. In this
section we briefly describe the most relevant ones.

**Incremental Learning: Machine learning algorithms have traditionally focused on stationary, non-**
incremental learning, where a fixed dataset is presented to the algorithm (Giraud-Carrier, 2000;
Sugiyama & Kawanabe, 2012; Ditzler et al., 2015; Lee et al., 2018; 2017; Smith & Gasser, 2005).
Inspired by studies in child development (Mendelson et al., 1976; Smith & Gasser, 2005; Smith et al.,

1998; Spelke, 1979; Wertheimer, 1961), sub-areas such as curriculum learning (Bengio et al., 2009;

Kumar et al., 2010; Murali et al., 2018) and incremental SVMs (Cauwenberghs & Poggio, 2001;
Syed et al., 1999; Ruping, 2001) have emerged, where learning occurs incrementally. We note that
the general focus in these prior work is on problems where a labelled dataset is available. In this work,
we instead focus on RL problems, where labelled datasets or demonstrations are not provided during
the incremental learning. In the context of RL, some prior works consider learning for a sequence of
tasks (Tanaka & Yamamura, 1997; Wilson et al., 2007; Ammar et al., 2015) while other works (Peng
& Williams, 1994; Ammar et al., 2015; Wang et al., 2019a;b) create algorithms that can adapt to
changing environments through incremental policy or Q-learning. However, these methods operate in
the task-specific RL setting, where the agent is trained to solve one specific reward function.

**Unsupervised Learning in RL: Prominent works in this area focus on obtaining additional in-**
formation about the environment in the expectation that it would help downstream task-based RL:
Representation learning methods (Yarats et al., 2019; Lee et al., 2019; Srinivas et al., 2020; Schwarzer


-----

et al., 2020; Stooke et al., 2020; Yarats et al., 2021) use unsupervised perceptual losses to enable
better visual RL; Model-based methods (Hafner et al., 2018; 2019; Yan et al., 2020; Agrawal et al.,
2016) learn approximate dynamics models that allow for accelerated planning with downstream
rewards; Exploration-based methods (Bellemare et al., 2016; Ostrovski et al., 2017; Pathak et al.,
2017; Burda et al., 2018; Andrychowicz et al., 2017; Hazan et al., 2019; Mutti et al., 2020; Liu &
Abbeel, 2021) focus on obtaining sufficient coverage in environments. Since our proposed work is
based on skill discovery, it is orthogonal to these prior work and can potentially be combined with
them for additional performance gain. Most related to our work is the sub-field of skill discovery (Eysenbach et al., 2018; Sharma et al., 2019b; Campos et al., 2020; Gregor et al., 2016; Achiam et al.,
2018; Sharma et al., 2020). A more formal treatment of this is presented in Section 2. Since our
method makes the skill discovery process incremental, it allows for improved performance compared
to DIAYN (Eysenbach et al., 2018) and Off-DADS (Sharma et al., 2020), which is discussed in
Section 4. Finally, we note that skills can also be learned in supervised setting with demonstrations or
rewards (Kober & Peters, 2009; Peters et al., 2013; Schaal et al., 2005; Dragan et al., 2015; Konidaris
et al., 2012; Konidaris & Barto, 2009; Zahavy et al., 2021; Shankar et al., 2020; Shankar & Gupta,
2020). Our work can potentially be combined with such prior work to improve skill discovery when
demonstrations are present.

**RL with evolving dynamics: Creating RL algorithms that can adapt and learn in environments with**
changing dynamics is a longstanding problem (Nareyek, 2003; Koulouriotis & Xanthopoulos, 2008;
Da Silva et al., 2006; Cheung et al., 2020). Recently, a variant of this problem has been studied in
the sim-to-real transfer community (Peng et al., 2018; Chebotar et al., 2019; Tan et al., 2018), where
minor domain gaps between simulation and reality can be bridged through domain randomization.
Another promising direction is online adaptation (Rakelly et al., 2019; Hansen et al., 2020; Yu et al.,
2017) and Meta-RL (Finn et al., 2017; Nagabandi et al., 2018a;b), where explicit mechanisms to
infer the environment are embedded in policy execution. We note that these methods often focus on
minor variations in dynamics during training or assume that there is an underlying distribution of
environment variability, while our framework does not.

6 DISCUSSION, LIMITATIONS, AND SCOPE FOR FUTURE WORK

We have presented DISk, an unsupervised skill discovery method that takes the first steps towards
learning skills incrementally. Although already powerful, we still have significant scope for future
work before such methods can be applied in real-world settings. For instance, we notice that on easy
tasks such as HalfCheetah, DISk underforms prior works, which is partly due to its inability to share
parameters from previously learned skills. Sharing parameters efficiently can allow improved learning
and a reduction of the total parameters. Next, to measure similarity in states, we use a fixed projection
function. Being able to simultaneously learn this projection function is likely to further improve
performance. Also, we use a fixed schedule based on skill convergence to add new skills to our
repertoire. A more intelligent schedule for skill addition would help DISk be even more optimized.
Finally, to apply such skill discovery methods in the real-world, we may have to bootstrap from
supervised data or demonstrations. Such bootstrapping can provide an enormous boost in settings
where humans can interact with the agent.


-----

REFERENCES

Joshua Achiam, Harrison Edwards, Dario Amodei, and Pieter Abbeel. Variational option discovery
algorithms. arXiv preprint arXiv:1807.10299, 2018. 10, 25, 26

Pulkit Agrawal, Ashvin V Nair, Pieter Abbeel, Jitendra Malik, and Sergey Levine. Learning to
poke by poking: Experiential learning of intuitive physics. In Advances in Neural Information
_Processing Systems, pp. 5074–5082, 2016. 10_

Haitham Bou Ammar, Eric Eaton, Paul Ruvolo, and Matthew E Taylor. Unsupervised cross-domain
transfer in policy gradient reinforcement learning via manifold alignment. In Twenty-Ninth AAAI
_Conference on Artificial Intelligence, 2015. 3, 9_

Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob
McGrew, Josh Tobin, Pieter Abbeel, and Wojciech Zaremba. Hindsight experience replay. NIPS,
2017. 9, 10

OpenAI: Marcin Andrychowicz, Bowen Baker, Maciek Chociej, Rafal Jozefowicz, Bob McGrew,
Jakub Pachocki, Arthur Petron, Matthias Plappert, Glenn Powell, Alex Ray, et al. Learning
dexterous in-hand manipulation. The International Journal of Robotics Research, 39(1):3–20,
2020. 1

Marc G. Bellemare, Sriram Srinivasan, Georg Ostrovski, Tom Schaul, David Saxton, and Remi Munos.
Unifying count-based exploration and intrinsic motivation. arXiv preprintarXiv:1606.01868, 2016.

1, 3, 10

E Belogay, C Cabrelli, U Molter, and R Shonkwiler. Calculating the hausdorff distance between
curves. Information Processing Letters, 64(1), 1997. 6

Yoshua Bengio, Jérôme Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In
_ICML, 2009. 9_

Robert N Brandon. Adaptation and environment, volume 1040. Princeton University Press, 2014. 1

Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and
Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016. 6, 20

Yuri Burda, Harrison Edwards, Amos Storkey, and Oleg Klimov. Exploration by random network
distillation. arXiv preprint arXiv, 2018. 10

Víctor Campos, Alexander Trott, Caiming Xiong, Richard Socher, Xavier Giro-i Nieto, and Jordi Torres. Explore, discover and learn: Unsupervised discovery of state-covering skills. In International
_Conference on Machine Learning, pp. 1317–1327. PMLR, 2020. 1, 3, 10_

Víctor Campos, Pablo Sprechmann, Steven Stenberg Hansen, Andre Barreto, Steven Kapturowski,
Alex Vitvitskyi, Adria Puigdomenech Badia, and Charles Blundell. Beyond fine-tuning: Transferring behavior in reinforcement learning. In ICML 2021 Workshop on Unsupervised Reinforcement
_Learning, 2021. 1_

Gert Cauwenberghs and Tomaso Poggio. Incremental and decremental support vector machine
learning. Advances in neural information processing systems, pp. 409–415, 2001. 9

Yevgen Chebotar, Ankur Handa, Viktor Makoviychuk, Miles Macklin, Jan Issac, Nathan Ratliff, and
Dieter Fox. Closing the sim-to-real loop: Adapting simulation randomization with real world
experience. In 2019 International Conference on Robotics and Automation (ICRA), pp. 8973–8979.
IEEE, 2019. 10

Zhiyuan Chen and Bing Liu. Lifelong machine learning. Synthesis Lectures on Artificial Intelligence
_and Machine Learning, 12(3):1–207, 2018. 2_

Wang Chi Cheung, David Simchi-Levi, and Ruihao Zhu. Reinforcement learning for non-stationary
markov decision processes: The blessing of (more) optimism. In International Conference on
_Machine Learning, pp. 1843–1854. PMLR, 2020. 2, 3, 10_


-----

Daniela Corbetta and Esther Thelen. The developmental origins of bimanual coordination: a dynamic
perspective. Journal of Experimental Psychology: Human Perception and Performance, 22(2):502,
1996. 1

Bruno C Da Silva, Eduardo W Basso, Ana LC Bazzan, and Paulo M Engel. Dealing with nonstationary environments using context detection. In Proceedings of the 23rd international confer_ence on Machine learning, pp. 217–224, 2006. 2, 3, 10_

Gregory Ditzler, Manuel Roveri, Cesare Alippi, and Robi Polikar. Learning in nonstationary environments: A survey. IEEE Computational Intelligence Magazine, 10(4):12–25, 2015. 9

Anca D Dragan, Katharina Muelling, J Andrew Bagnell, and Siddhartha S Srinivasa. Movement
primitives via optimization. In ICRA, 2015. 10

Simon Du, Akshay Krishnamurthy, Nan Jiang, Alekh Agarwal, Miroslav Dudik, and John Langford.
Provably efficient rl with rich observations via latent state decoding. In International Conference
_on Machine Learning, pp. 1665–1674. PMLR, 2019. 5_

Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. Diversity is all you
[need: Learning skills without a reward function. CoRR, abs/1802.06070, 2018. URL http:](http://arxiv.org/abs/1802.06070)
[//arxiv.org/abs/1802.06070. 1, 2, 3, 6, 10, 18, 22, 24](http://arxiv.org/abs/1802.06070)

Benjamin Eysenbach, Xinyang Geng, Sergey Levine, and Ruslan Salakhutdinov. Rewriting history
with inverse rl: Hindsight inference for policy improvement. arXiv preprint arXiv:2002.11089,
2020. 9

Geli Fei, Shuai Wang, and Bing Liu. Learning cumulatively to become more knowledgeable. In
_Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and_
_Data Mining, pp. 1565–1574, 2016. 3_

Chrisantha Fernando, Dylan Banarse, Charles Blundell, Yori Zwols, David Ha, Andrei A Rusu,
Alexander Pritzel, and Daan Wierstra. Pathnet: Evolution channels gradient descent in super neural
networks. arXiv preprint arXiv:1701.08734, 2017. 3

Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of
deep networks. In International Conference on Machine Learning, pp. 1126–1135. PMLR, 2017.

10

Scott Fujimoto, Herke Hoof, and David Meger. Addressing function approximation error in actorcritic methods. In International Conference on Machine Learning, pp. 1587–1596. PMLR, 2018.

17

Christophe Giraud-Carrier. A note on the utility of incremental learning. Ai Communications, 13(4):
215–223, 2000. 9

Karol Gregor, Danilo Jimenez Rezende, and Daan Wierstra. Variational intrinsic control. arXiv
_preprint arXiv:1611.07507, 2016. 1, 10_

Abhinav Gupta, Adithyavairavan Murali, Dhiraj Prakashchand Gandhi, and Lerrel Pinto. Robot
learning in homes: Improving generalization and reducing dataset bias. Advances in Neural
_Information Processing Systems, 31:9094–9104, 2018. 2, 3_

Tuomas Haarnoja, Haoran Tang, Pieter Abbeel, and Sergey Levine. Reinforcement learning with
deep energy-based policies. In International Conference on Machine Learning, pp. 1352–1361.
PMLR, 2017. 3

Tuomas Haarnoja, Aurick Zhou, Kristian Hartikainen, George Tucker, Sehoon Ha, Jie Tan, Vikash
Kumar, Henry Zhu, Abhishek Gupta, Pieter Abbeel, et al. Soft actor-critic algorithms and
applications. arXiv preprint arXiv:1812.05905, 2018. 1, 3, 4, 17

Danijar Hafner, Timothy Lillicrap, Ian Fischer, Ruben Villegas, David Ha, Honglak Lee, and James
Davidson. Learning latent dynamics for planning from pixels. arXiv preprint arXiv:1811.04551,
2018. 10


-----

Danijar Hafner, Timothy Lillicrap, Jimmy Ba, and Mohammad Norouzi. Dream to control: Learning
behaviors by latent imagination. arXiv preprint arXiv:1912.01603, 2019. 10

Nicklas Hansen, Rishabh Jangir, Yu Sun, Guillem Alenyà, Pieter Abbeel, Alexei A Efros, Lerrel
Pinto, and Xiaolong Wang. Self-supervised policy adaptation during deployment. arXiv preprint
_arXiv:2007.04309, 2020. 10_

Karol Hausman, Jost Tobias Springenberg, Ziyu Wang, Nicolas Heess, and Martin Riedmiller.
Learning an embedding space for transferable robot skills. In International Conference on Learning
_[Representations, 2018. URL https://openreview.net/forum?id=rk07ZXZRb. 3](https://openreview.net/forum?id=rk07ZXZRb)_

Elad Hazan, Sham M. Kakade, Karan Singh, and Abby Van Soest. Provably efficient maximum
entropy exploration, 2019. 10

Ryan Julian, Benjamin Swanson, Gaurav S Sukhatme, Sergey Levine, Chelsea Finn, and Karol
Hausman. Never stop learning: The effectiveness of fine-tuning in robotic reinforcement learning.
_arXiv e-prints, pp. arXiv–2004, 2020. 2, 3_

Sham M Kakade. A natural policy gradient. In Advances in neural information processing systems,
pp. 1531–1538, 2002. 17

Jens Kober and Jan Peters. Learning motor primitives for robotics. In ICRA, 2009. 1, 10

George Konidaris and Andrew Barto. Skill chaining: Skill discovery in continuous domains. In the
_Multidisciplinary Symposium on Reinforcement Learning, Montreal, Canada, 2009. 10_

George Konidaris, Scott Kuindersma, Roderic Grupen, and Andrew Barto. Robot learning from
demonstration by constructing skill trees. IJRR, 2012. 10

Dimitris E Koulouriotis and A Xanthopoulos. Reinforcement learning and evolutionary algorithms
for non-stationary multi-armed bandit problems. Applied Mathematics and Computation, 196(2):
913–922, 2008. 2, 3, 10

Tejas D Kulkarni, Karthik R Narasimhan, Ardavan Saeedi, and Joshua B Tenenbaum. Hierarchical
deep reinforcement learning: Integrating temporal abstraction and intrinsic motivation. arXiv
_preprint arXiv:1604.06057, 2016. 1_

M Pawan Kumar, Benjamin Packer, and Daphne Koller. Self-paced learning for latent variable
models. In Advances in Neural Information Processing Systems, pp. 1189–1197, 2010. 9

A. X. Lee, A. Nagabandi, P. Abbeel, and S. Levine. Stochastic latent actor-critic: Deep reinforcement
learning with a latent variable model. arXiv e-prints, 2019. 9

Kimin Lee, Honglak Lee, Kibok Lee, and Jinwoo Shin. Training confidence-calibrated classifiers for
detecting out-of-distribution samples. arXiv preprint arXiv:1711.09325, 2017. 9

Kimin Lee, Kibok Lee, Honglak Lee, and Jinwoo Shin. A simple unified framework for detecting
out-of-distribution samples and adversarial attacks. arXiv preprint arXiv:1807.03888, 2018. 9

Sergey Levine, Chelsea Finn, Trevor Darrell, and Pieter Abbeel. End-to-end training of deep
visuomotor policies. JMLR, 2016. 1

Alexander C Li, Lerrel Pinto, and Pieter Abbeel. Generalized hindsight for reinforcement learning.
_arXiv preprint arXiv:2002.11708, 2020. 9_

Zhizhong Li and Derek Hoiem. Learning without forgetting. IEEE transactions on pattern analysis
_and machine intelligence, 40(12):2935–2947, 2017. 3_

Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa,
David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv
_preprint arXiv:1509.02971, 2015. 17_

Hao Liu and Pieter Abbeel. Unsupervised active pre-training for reinforcement learning. openreview,
[2021. URL https://openreview.net/forum?id=cvNYovr16SB. 4, 10](https://openreview.net/forum?id=cvNYovr16SB)


-----

Morton J Mendelson, Marshall M Haith, and James J Gibson. The relation between audition and
vision in the human newborn. Monographs of the Society for Research in Child Development, pp.
1–72, 1976. 9

Adithyavairavan Murali, Lerrel Pinto, Dhiraj Gandhi, and Abhinav Gupta. Cassl: Curriculum
accelerated self-supervised learning. In 2018 IEEE International Conference on Robotics and
_Automation (ICRA), pp. 6453–6460. IEEE, 2018. 9_

Mirco Mutti, Lorenzo Pratissoli, and Marcello Restelli. A policy gradient method for task-agnostic
exploration, 2020. 10

Anusha Nagabandi, Ignasi Clavera, Simin Liu, Ronald S Fearing, Pieter Abbeel, Sergey Levine, and
Chelsea Finn. Learning to adapt in dynamic, real-world environments through meta-reinforcement
learning. arXiv preprint arXiv:1803.11347, 2018a. 10

Anusha Nagabandi, Chelsea Finn, and Sergey Levine. Deep online learning via meta-learning:
Continual adaptation for model-based rl. arXiv preprint arXiv:1812.07671, 2018b. 10

Alexander Nareyek. Choosing search heuristics by non-stationary reinforcement learning. In
_Metaheuristics: Computer decision-making, pp. 523–544. Springer, 2003. 2, 3, 10_

Georg Ostrovski, Marc G. Bellemare, Aaron van den Oord, and Remi Munos. Count-based exploration
with neural density models. arXiv preprint arXiv, 2017. 10

Deepak Pathak, Pulkit Agrawal, Alexei A. Efros, and Trevor Darrell. Curiosity-driven exploration by
self-supervised prediction. arXiv preprint arXiv, 2017. 10

Jing Peng and Ronald J Williams. Incremental multi-step q-learning. In Machine Learning Proceed_ings 1994, pp. 226–232. Elsevier, 1994. 9_

Xue Bin Peng, Marcin Andrychowicz, Wojciech Zaremba, and Pieter Abbeel. Sim-to-real transfer of
robotic control with dynamics randomization. In 2018 IEEE international conference on robotics
_and automation (ICRA), pp. 3803–3810. IEEE, 2018. 10_

Jan Peters, Jens Kober, Katharina Mülling, Oliver Krämer, and Gerhard Neumann. Towards robot
skill learning: From simple skills to table tennis. In Joint European Conference on Machine
_Learning and Knowledge Discovery in Databases, pp. 627–631. Springer, 2013. 1, 10_

Antonin Raffin, Ashley Hill, Maximilian Ernestus, Adam Gleave, Anssi Kanervisto, and Noah
[Dormann. Stable baselines3. https://github.com/DLR-RM/stable-baselines3,](https://github.com/DLR-RM/stable-baselines3)
2019. 21

Roberta Raileanu, Max Goldstein, Arthur Szlam, and Rob Fergus. Fast adaptation via policy-dynamics
value functions. arXiv preprint arXiv:2007.02879, 2020. 1

Kate Rakelly, Aurick Zhou, Chelsea Finn, Sergey Levine, and Deirdre Quillen. Efficient off-policy
meta-reinforcement learning via probabilistic context variables. In International conference on
_machine learning, pp. 5331–5340. PMLR, 2019. 10_

Stefan Ruping. Incremental learning with support vector machines. In Proceedings 2001 IEEE
_International Conference on Data Mining, pp. 641–642. IEEE, 2001. 9_

Paul Ruvolo and Eric Eaton. Ella: An efficient lifelong learning algorithm. In International
_Conference on Machine Learning, pp. 507–515. PMLR, 2013. 3_

Stefan Schaal, Jan Peters, Jun Nakanishi, and Auke Ijspeert. Learning movement primitives. In RR,
2005. 1, 10

John Schulman, Sergey Levine, Pieter Abbeel, Michael I Jordan, and Philipp Moritz. Trust region
policy optimization. In ICML, pp. 1889–1897, 2015. 17

John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy
optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. 1


-----

Max Schwarzer, Ankesh Anand, Rishab Goel, R Devon Hjelm, Aaron Courville, and Philip Bachman.
Data-efficient reinforcement learning with momentum predictive representations. arXiv preprint
_arXiv:2007.05929, 2020. 9_

Tanmay Shankar and Abhinav Gupta. Learning robot skills with temporal variational inference. In
_International Conference on Machine Learning, pp. 8624–8633. PMLR, 2020. 10_

Tanmay Shankar, Shubham Tulsiani, Lerrel Pinto, and Abhinav Gupta. Discovering motor programs
by recomposing demonstrations. In International Conference on Learning Representations, 2020.
[URL https://openreview.net/forum?id=rkgHY0NYwr. 10](https://openreview.net/forum?id=rkgHY0NYwr)

Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman. Dynamics-aware
[unsupervised discovery of skills. CoRR, abs/1907.01657, 2019a. URL http://arxiv.org/](http://arxiv.org/abs/1907.01657)
[abs/1907.01657. 24](http://arxiv.org/abs/1907.01657)

Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman. Dynamics-aware
unsupervised discovery of skills. arXiv preprint arXiv:1907.01657, 2019b. 1, 3, 6, 7, 8, 10, 22

Archit Sharma, Michael Ahn, Sergey Levine, Vikash Kumar, Karol Hausman, and Shixiang Gu.
Emergent real-world robotic skills via unsupervised off-policy reinforcement learning. arXiv
_preprint arXiv:2004.12974, 2020. 1, 3, 6, 7, 10, 23_

Harshinder Singh, Neeraj Misra, Vladimir Hnizdo, Adam Fedorowicz, and Eugene Demchuk. Nearest
neighbor estimates of entropy. American journal of mathematical and management sciences, 23
(3-4):301–321, 2003. 4, 19

Linda Smith and Michael Gasser. The development of embodied cognition: Six lessons from babies.
_Artificial life, 11(1-2):13–29, 2005. 1, 9_

Linda B Smith, Alexandra L Quittner, Mary Joe Osberger, and Richard Miyamoto. Audition and
visual attention: the developmental trajectory in deaf and hearing populations. Developmental
_psychology, 34(5):840, 1998. 9_

Elizabeth S Spelke. Perceiving bimodally specified events in infancy. Developmental psychology, 15
(6):626, 1979. 9

Aravind Srinivas, Michael Laskin, and Pieter Abbeel. Curl: Contrastive unsupervised representations
for reinforcement learning. arXiv preprint arXiv:2004.04136, 2020. 1, 3, 9

Martin Stolle and Doina Precup. Learning options in reinforcement learning. In International
_Symposium on abstraction, reformulation, and approximation, pp. 212–223. Springer, 2002. 1_

Adam Stooke, Kimin Lee, Pieter Abbeel, and Michael Laskin. Decoupling representation learning
from reinforcement learning. arXiv preprint arXiv, 2020. 10

Masashi Sugiyama and Motoaki Kawanabe. Machine learning in non-stationary environments:
_Introduction to covariate shift adaptation. MIT press, 2012. 9_

Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018. 2

Richard S Sutton, Doina Precup, and Satinder Singh. Between mdps and semi-mdps: A framework
for temporal abstraction in reinforcement learning. Artificial intelligence, 112(1-2):181–211, 1999.

1

Nadeem Ahmed Syed, Syed Huan, Liu Kah, and Kay Sung. Incremental learning with support vector
machines. 1999. 9

Jie Tan, Tingnan Zhang, Erwin Coumans, Atil Iscen, Yunfei Bai, Danijar Hafner, Steven Bohez, and
Vincent Vanhoucke. Sim-to-real: Learning agile locomotion for quadruped robots. arXiv preprint
_arXiv:1804.10332, 2018. 10_

Fumihide Tanaka and Masayuki Yamamura. An approach to lifelong reinforcement learning through
multiple environments. In 6th European Workshop on Learning Robots, pp. 93–99, 1997. 3, 9

Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control.
In IROS, 2012. 6


-----

Hado Van Hasselt, Arthur Guez, and David Silver. Deep reinforcement learning with double qlearning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 30, 2016.

17

Zhi Wang, Chunlin Chen, Han-Xiong Li, Daoyi Dong, and Tzyh-Jong Tarn. Incremental reinforcement learning with prioritized sweeping for dynamic environments. IEEE/ASME Transactions on
_Mechatronics, 24(2):621–632, 2019a. 9_

Zhi Wang, Han-Xiong Li, and Chunlin Chen. Incremental reinforcement learning in continuous
spaces via policy relaxation and importance weighting. IEEE transactions on neural networks and
_learning systems, 31(6):1870–1883, 2019b. 9_

Michael Wertheimer. Psychomotor coordination of auditory and visual space at birth. Science, 134
(3491):1692–1692, 1961. 9

William Whitney, Rajat Agarwal, Kyunghyun Cho, and Abhinav Gupta. Dynamics-aware embeddings.
_arXiv preprint arXiv:1908.09357, 2019. 5_

Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement
learning. Machine learning, 8(3-4):229–256, 1992. 17

Aaron Wilson, Alan Fern, Soumya Ray, and Prasad Tadepalli. Multi-task reinforcement learning: a
hierarchical bayesian approach. In Proceedings of the 24th international conference on Machine
_learning, pp. 1015–1022. ACM, 2007. 9_

Wilson Yan, Ashwin Vangipuram, Pieter Abbeel, and Lerrel Pinto. Learning predictive representations
for deformable objects using contrastive estimation. arXiv preprint arXiv:2003.05436, 2020. 10

[Denis Yarats and Ilya Kostrikov. Soft actor-critic (sac) implementation in pytorch. https://](https://github.com/denisyarats/pytorch_sac)
[github.com/denisyarats/pytorch_sac, 2020. 21, 22](https://github.com/denisyarats/pytorch_sac)

Denis Yarats, Amy Zhang, Ilya Kostrikov, Brandon Amos, Joelle Pineau, and Rob Fergus. Improving sample efficiency in model-free reinforcement learning from images. arXiv preprint
_arXiv:1910.01741, 2019. 9_

Denis Yarats, Rob Fergus, Alessandro Lazaric, and Lerrel Pinto. Reinforcement learning with
prototypical representations. arXiv preprint arXiv:2102.11271, 2021. 4, 5, 10

Wenhao Yu, C Karen Liu, and Greg Turk. Preparing for the unknown: Learning a universal policy
with online system identification. arXiv preprint arXiv:1702.02453, 2017. 10

Tom Zahavy, Brendan O’Donoghue, Andre Barreto, Volodymyr Mnih, Sebastian Flennerhag, and
Satinder Singh. Discovering diverse nearly optimal policies withsuccessor features. arXiv preprint
_arXiv:2106.00669, 2021. 10_

Wenxuan Zhou, Lerrel Pinto, and Abhinav Gupta. Environment probing interaction policies. arXiv
_preprint arXiv:1907.11740, 2019. 1_

Brian D Ziebart, Andrew Maas, J Andrew Bagnell, and Anind K Dey. Maximum entropy inverse
reinforcement learning. 2008. 17

APPENDIX

A REINFORCEMENT LEARNING

In our continuous-control RL setting, an agent receives a state observation st from the environment and applies an action at according to policy π. In our setting, where the policy is ∈S
stochastic, the policy returns a distribution ∈A _π(st), and we sample a concrete action at_ _π(st). The_
environment returns a reward for every action rt. The goal of the agent is to maximize expected ∼
cumulative discounted reward Es0:T,a0:T −1,r0:T −1 _Tt=0 −1_ _[γ][t][r][t]_ for discount factor γ and horizon
length T . hP i


-----

On-policy RL (Schulman et al., 2015; Kakade, 2002; Williams, 1992) optimizes π by iterating
between data collection and policy updates. It hence requires new on-policy data every iteration,
which is expensive to obtain. On the other hand, off-policy reinforcement learning retains past
experiences in a replay buffer and is able to re-use past samples. Thus, in practice, off-policy
algorithms have been found to achieve better sample efficiency (Lillicrap et al., 2015; Haarnoja et al.,
2018). For our experiments we use SAC (Haarnoja et al., 2018) as our base RL optimizer due to its
implicit maximization of action distribution entropy, sample efficiency, and fair comparisons with
baselines that also build on top of SAC. However, we note that our framework is compatible with any
standard off-policy RL algorithm that maximizes the entropy of the action distribution π(·) either
implicitly or explicitly.

**Soft Actor-Critic** The Soft Actor-Critic (SAC) (Haarnoja et al., 2018) is an off-policy model-free
RL algorithm that instantiates an actor-critic framework by learning a state-action value function Qθ, a
stochastic policy πθ and a temperature α over a discounted infinite-horizon MDP (X _, A, P, R, γ, d0)_
by optimizing a γ-discounted maximum-entropy objective (Ziebart et al., 2008). With a slight abuse
of notation, we define both the actor and critic learnable parameters by θ. SAC parametrizes the actor
policy πθ(at|xt) via a tanh-Gaussian defined as at = tanh(µθ(xt)+ _σθ(xt)ϵ), where ϵ ∼N_ (0, 1),
_µθ and σθ are parametric mean and standard deviation. The SAC’s critic Qθ(xt, at) is parametrized_
as an MLP neural network.

The policy evaluation step learns the critic Qθ(xt, at) network by optimizing the one-step soft
Bellman residual:

_Q(_ ) = E(xt,at,xt+1) [(Qθ(xt, at) _yt)[2]] and_
_L_ _D_ **_at+1∼π(·|xt+1∼D)_** _−_

_yt = R(xt, at) + γ[Qθ′_ (xt+1, at+1) _α log πθ(at+1_ **_xt+1)],_**
_−_ _|_

where D is a replay buffer of transitions, θ[′] is an exponential moving average of θ as done in (Lillicrap
et al., 2015). SAC uses clipped double-Q learning (Van Hasselt et al., 2016; Fujimoto et al., 2018),
which we omit from our notation for simplicity but employ in practice.

The policy improvement step then fits the actor πθ(at **_st) network by optimizing the following_**
_|_
objective:

_π(_ ) = Ext [DKL(πθ( **_xt)_** exp [1]
_L_ _D_ _∼D_ _·|_ _||_ _{_ _α_ _[Q][θ][(][x][t][,][ ·][)][}][)]][.]_

Finally, the temperature α is learned with the loss:

_α(_ ) = E **_xt_** [ _α log πθ(at_ **_xt)_** _α_ [¯]],
_L_ _D_ **_at∼πθ∼D(·|xt)_** _−_ _|_ _−_ _H_

where _H ∈[¯]_ R is the target entropy hyper-parameter that the policy tries to match, which in practice is
set to _H[¯] = −|A|. The overall optimization objective of SAC equals to:_

SAC( ) = _π(_ ) + _Q(_ ) + _α(_ ).
_L_ _D_ _L_ _D_ _L_ _D_ _L_ _D_

B FURTHER MATHEMATICAL DETAILS

B.1 EXPANSION AND DERIVATION OF OBJECTIVES

The objective function F(θ), as defined in Equation 1, is the source from where we derive our
incremental objective function, reproduced here.

_F(θ) = I(S; Z) + H(A|S, Z)_

We can expand the first term in Equation 1 as

_I(S; Z) ≡_ _H(S) −_ _H(S | Z)_


-----

by definition of mutual information. Now, once we assume Z is a discrete variable, the second part
of this equation becomes


_H(S | Z) ≡_


_p(zm)_ (S _Z = zm)_
_H_ _|_
_zm_

X

E
_zm_ _p(zm)_ [[][H][(][S][ |][ Z][ =][ z][m][)]]
_∼_


And thus we have

_I(S; Z) =_ (S) E
_H_ _−_ _zm_ _p(zm)_ [[][H][(][S][ |][ Z][ =][ z][m][)]]
_∼_

= E
_zm_ _p(zm)_ [[][H][(][S][)][ −H][(][S][ |][ Z][ =][ z][m][)]]
_∼_

But the term inside the expectation is the definition of information gain (not to be confused with KL
divergence), defined by


_IG(S; Z = zm)_ (S) (S _Z = zm)_
_≡H_ _−H_ _|_

_I(S; Z) =_ E
_zm_ _p(zm)_ [[][IG][(][S][;][ Z][ =][ z][m][)]]
_∼_


Thus, we arrive at


Similarly, by definition of conditional entropy, we can expand the second part of the Equation 1


_H(A | S, Z) ≡_

=

Thus, we can convert Equation 1 into


_p(zm)_ (A _S, Z = zm)_
_H_ _|_
_zm_

X

E
_zm_ _p(zm)_ [[][H][(][A][ |][ S, Z][ =][ z][m][)]]
_∼_


_F(θ) = I(S; Z) + H(A|S, Z)_
= E E
_zm_ _p(zm)_ [[][IG][(][S][;][ Z][ =][ z][m][)] +] _zm_ _p(zm)_ [[][H][(][A][ |][ S, Z][ =][ z][m][)]]
_∼_ _∼_

= E
_zm_ _p(zm)_ [[][IG][(][S][;][ Z][ =][ z][m][) +][ H][(][A][ |][ S, Z][ =][ z][m][)]]
_∼_

If we assume a uniform prior over our skills, which is another assumption made by Eysenbach et al.
(2018), and also assume we are trying to learn M skills in total, we can further expand Equation 1
into:


(θ) = [1]
_F_ _M_



[IG(S; Z = zm) + (A _S, Z = zm)]_
_H_ _|_
_m=1_

X


Ignoring the number of skills term (which is constant over a single skills’ learning period) gives us
exactly Equation 3, which was:


(θ) := [IG(S; Z = zm) + (A _S, Z = zm)]_
_F_ _H_ _|_

_m=1_

X


Now, under our framework, we assume that skills 1, 2, · · ·, M − 1 has been learned and fixed, and we
are formulating an objective for the M th skill. As a result, we can ignore the associated Information
Gain and action distribution entropy terms from skills 1, 2, · · ·, M − 1, and simplify our objective to
be:

(θ) := IG(S; Z = zM ) + (A _S, Z = zM_ )
_F_ _H_ _|_
= (S) (S _Z = zM_ ) + (A _S, Z = zM_ )
_H_ _−H_ _|_ _H_ _|_

which is exactly the same objective we defined in Equation 5.


-----

B.2 POINT-BASED NEAREST NEIGHBOR ENTROPY ESTIMATION

In our work, we use an alternate approach, first shown by Singh et al. (2003), to estimate the entropy
of a set of points. This method gives us a non-parametric Nearest Neighbor (NN) based entropy
estimator:


Hˆ _k,X_ (p) =
_−_ _N[1]_


_N_

ln _[k][Γ(][q/][2 + 1)]_ + Ck,
_i=1_ _Nπ[q/][2]Ri,k,[q]_ **_X_**

X


where Γ is the gamma function, Ck = ln k − [Γ]Γ([′][(]k[k])[)] [is the bias correction term, and][ R][i,k,][X][ =]

**_X∥x, defined asi −_** NNk,X ( NNxi)k,∥ **_Xis the Euclidean distance between(xi)._** **_xi and its k[th]_** nearest neighbor from the dataset

The term inside the sum can be simplified as


ln _Nπ[k][Γ(][q/][q/][2][2 + 1)]Ri,k,[q]_ **_X_** = ln _[k][Γ(]Nπ[q/][2 + 1)][q/][2]_ _−_ ln Ri,k,[q] **_X_**

= ln _[k][Γ(][q/][2 + 1)]_ _q ln Ri,k,X_

_Nπ[q/][2]_ _−_


= ln _[k][Γ(][q/][2 + 1)]_ _q ln_ **_xi_** NNk,X (xi) _._

_Nπ[q/][2]_ _−_ _∥_ _−_ _∥_

Here, ln _[k][Γ(][q/][2 + 1)]_ is a constant term independent of xi. If we ignore the this term and the

_Nπ[q/][2]_
bias-correction term Ck and the constant, we get


Hˆ _k,X_ (p) ∝


ln **_xi_** NNk,X (xi) _._
_i=1_ _∥_ _−_ _∥_

X


Which is the formulation we use in this work. This estimator is shown to be asymptotically unbiased
and consistent in Singh et al. (2003).

B.3 HAUSDORFF DISTANCE

In our work, to compare between two algorithms learning skills on the same environment, we used
a metric based on Hausdorff distance. Hausdorff distance, also known as the Hausdorff metric or
the Pompeiu–Hausdorff distance, is a metric that measures the distance between two subsets of a
metric space. Informally, we think of two sets in a metric space as close in the Hausdorff distance if
every point of either set is close to some point of the other set. The Hausdorff distance is the longest
distance one can force you to travel by choosing a point adversarially in one of the two sets, from
which you have to travel to the other set. Put simply, it is the greatest of all the distances from a point
in one set to the nearest point in the other.

Mathematically, given two subsets A and B of a metric space (M, d) we define Hausdorff distance
_dH_ (A, B) as:

_dH_ (A, B) = max sup _d(a, B), sup_ _d(b, A)_
a∈A _b∈B_ 

Where sup represents the supremum, d(x, Y ) = inf _y_ _Y d(x, y) is the distance between a point and_
_∈_
another set, and inf represents the infimum.

Given a set of skills, we calculate the diversity of one skill over all other skills by calculating the
Hausdorff distance between that skill’s trajectory end (x, y) location, and the terminal (x, y) locations
of all other trajectories. Intuitively, a skill has high Hausdorff distance if the end states it generates is
far away from other skills’ endpoints. Similarly, a high average Hausdorff distance for skills from
an algorithm means that the algorithm’s generated skills on average have a high distance from each
other, which is a desirable property for an algorithm which needs to generate diverse skills.


-----

C ENVIRONMENTS

**Gym Experiments** We derive all environments used in the experiments in this paper from OpenAI
Gym (Brockman et al., 2016) MuJoCo tasks. Namely, we use the HalfCheetah-v3 and Hopperv3 environments for 2d locomotion tasks and Swimmer-v3 and Ant-v3 environments for the 3d
locomotion tasks (see Figure 2 for the agent morphologies).

Since we aim to train primitives, we want policies that perform well regardless of the global states of
the agent (global position etc.), only depending on local states (join angles etc.). Thus, we train our
agents and each of our baselines with a maximum episode length of 100 (200 for Swimmer only),
while we test them with a maximum episode length of 500 for static or the block environments and
200 for the broken leg environments.

As our projection function σ, we measured the x velocity of the agent in 2d environments, and the
(x, y) velocity of the agent in 3d environments. We made σ(s) available to the intrinsic reward
calculation functions of both our methods and the baselines.

**Block Experiments** For our block experiment set, we implemented the blocks as immovable
spheres of radius 3 at a distance 10 from origin. We dynamically added 40 blocks at the environment
creation, and deleted them with the MuJoCo interface available in Gym. The blocks were all added
before the agent took the first step in the environment, and removed over the agents’ lifetime as
described in Section 4.3. The blocks were always removed counter-clockwise, following the trajectory
of (cos [2]T[πt] _[,][ sin][ 2]T[πt]_ [)][ over][ t][ ∈] [[0][, T] []][, where][ t][ is the current timestep and][ T][ is the total timestep for]

training.

**Broken Leg Experiments** For our broken leg experiment set, we implemented a broken leg as a
leg where no actions have any effect. We switch which leg is broken every 1M steps, and train all
skills for a total of 10M steps in both Off-DADS and DISk.

Leg 1 Leg 2 Leg 3 Leg 4

40

20

0

20

40

40 20 0 20 40 40 20 0 20 40 40 20 0 20 40 40 20 0 20 40


Figure 6: Skills learned by DISk, evaluated with each of the legs broken. The legs are numbered such that the
final leg is numbered #4

Figure 7: Skills learned by Off-DADS at the end of 10M steps, evaluated with each of the legs broken. The legs
are numbered such that the final leg is numbered #4,

**Hierarchical Experiments** For the hierarchical environments, we use the Ant-v3 environment in
a goal-conditioned manner. The goals are sampled from [−15, 15][2] uniformly, and the hierarchical
agent can take 100 steps to reach as close to the goal as possible. At each step of the hierarchical


-----

Figure 8: Skills learned by Off-DADS at the end of 9.3M, or 0.3M steps after breaking the final leg, evaluated
with each of the legs broken. Compared to this agent, the agent in Fig. 7 performs worse on all broken leg but
the last one, which shows that Off-DADS suffers from an instance of catastrophic forgetting.

agent, it chooses a skill, which is then executed for 10 timesteps in the underlying environment.
So, in total, the agent has 1000 timesteps to reach the goal. At every timestep, the agent is given a
dense reward of _x_ _g_ 2, where x is the current location of the agent, and g is the location of the
_−∥_ _−_ _∥_
goal. On each step, the hierarchical agent gets the sum of the goal conditioned reward from the 10
timesteps in the underlying environment.

All the hierarchical agents were trained with the stable-baselines3 package (Raffin et al.,
2019). We used their default PPO agent for all the downstream set of skills, and trained the hierarchical
agent for a total 500 000 environment steps.

D IMPLEMENTATION DETAILS AND HYPERPARAMETERS

We base our implementation on the PyTorch implementation of SAC by Yarats & Kostrikov (2020).
On top of the implementation, we add a reward buffer class that keeps track of the intrinsic reward.
We provide the pseudocode for our algorithm in Appendix H.

D.1 ARCHITECTURE

For all of our environments, we used MLPs with two hidden layers of width 256 as our actor and
critic networks. We used ReLU units as our nonlinearities. For our stochastic policies, the actor
networks generated a mean and variance value for each coordinate of the action, and to sample an
action we sampled each dimension from a Gaussian distribution defined by those mean and variance
values. For stability, we clipped the log standard deviation between [−5, 2], similar to Yarats &
Kostrikov (2020).

D.2 REWARD NORMALIZATION

In learning all skills except the first, we set α and β such that both the diversity reward and the
consistency penalty have a similar magnitude. We do so by keeping track of the average of the
previous skills’ diversity reward and consistency penalty, and using the inverse of that as β and α
respectively, which is largely successful at keeping the two terms to similar orders of magnitude.

Another trick that helps the DISk learn is using a tanh-shaped annealing curve to scale up α, which
starts at 0 and ends at its full value. This annealing is designed to keep only the diversity reward
term relevant at the beginning of training, and then slowly introduce a consistency constraint into
the objective over the training period. This annealing encourages the policy to explore more in the
beginning. Then, as the consistency term kicks in, the skill becomes more and more consistent while
still being diverse. As a note, while DISk is more efficient with this annealing, its success is not
dependent on it.


-----

Table 1: DISks list of hyper-parameters.

|Parameter|Setting|
|---|---|
|Replay buffer capacity (static env) Replay buffer capacity (changing env) Seed steps Per-skill collected steps Minibatch size Discount (γ) Optimizer Learning rate Critic target update frequency Critic target EMA momentum (τ ) Q Actor update frequency Actor log stddev bounds Encoder target update frequency Encoder target EMA momentum (τ ) enc SAC entropy temperature Number of samples in Tb Size of circular buffer BUF k in NN|2 000 000 4 000 000 5000 10 000 256 0.99 Adam 3 10−4 × 2 0.01 2 [ 5, 2] − 2 0.05 0.1 256 50 3|


Parameter Setting

Replay buffer capacity (static env) 2 000 000

Replay buffer capacity (changing env) 4 000 000

Seed steps 5000

Per-skill collected steps 10 000

Minibatch size 256

Discount (γ) 0.99

Optimizer Adam

Learning rate 3 × 10[−][4]

Critic target update frequency 2

Critic target EMA momentum (τQ) 0.01

Actor update frequency 2

Actor log stddev bounds [−5, 2]

Encoder target update frequency 2

Encoder target EMA momentum (τenc) 0.05

SAC entropy temperature 0.1

Number of samples in _b_ 256
_T_

Size of circular buffer BUF 50

_k in NN_ 3


Table 2: DISks number of steps per skill.

|Environment|(Average) steps per skill|Number of skills|Total number of steps|
|---|---|---|---|
|HalfCheetah-v3 Hopper-v3 Swimmer-v3 Ant-v3 Ant-v3 (blocks) Ant-v3 (broken)|125 000 50 000 250 000 500 000 500 000 500 000|20 50 10 10 10 20|2 500 000 2 500 000 2 500 000 5 000 000 5 000 000 10 000 000|


Environment (Average) steps per skill Number of skills Total number of steps

HalfCheetah-v3 125 000 20 2 500 000

Hopper-v3 50 000 50 2 500 000

Swimmer-v3 250 000 10 2 500 000

Ant-v3 500 000 10 5 000 000

Ant-v3 (blocks) 500 000 10 5 000 000

Ant-v3 (broken) 500 000 20 10 000 000


D.3 FULL LIST OF HYPER-PARAMETERS

D.4 LEARNING SCHEDULE FOR ANT-V3 ON DISK

We noticed that on the static Ant environment, not all learned skills have the same complexity. For
example, the first two skills learned how to sit in place, and flip over. On the other hand, later skills
which just learns how to move in a different direction could reuse much of the replay buffers of the
earlier, complex skills which learned to move, and thus they did not need as many training steps
as the earlier complex skills. As a result, they converged at different speed. A schedule that most
closely fits this uneven convergence time is what we used: 5 000 000 environment steps unevenly
between the skills, with the skills getting 250 000, 250 000, 1 000 000, 1 000 000, 500 000, 500 000,
500 000, 500 000, 250 000, 250 000 steps respectively. On all other static environments, we use a
regular learning schedule with equal number of steps per skill since they are simple enough and
converge in roughly equal speed.

D.5 BASELINES

**DIAYN** We created our own implementation PyTorch based on Yarats & Kostrikov (2020), following the official implementation of Eysenbach et al. (2018) method to the best of our ability. We
used the same architecture as DISk for actor and critic network. For the discriminator, we used an
MLP with two hidden layers of width 256 that was optimized with a cross-entropy loss. To make the
comparison fair with our method, the discriminator was given access to the σ(s) values in all cases.
Each of the DIAYN models were trained with the same number of skills and steps as DISk, as shown
in Table 2.

While the performance of DIAYN in complex environments like Ant may seem lackluster, we found
that to be in line with the work by Sharma et al. (2019b) (see their Fig. 6, top right.) We believe a


-----

big part of the seeming gap is because of presenting them in a [−125, 125][2] grid. We believe this
presentation is fair since we want to showcase the abilities of Off-DADS and DISk.

**Off-DADS** We used the official implementation of Sharma et al. (2020), with their standard set
of parameters and architectures. We provide 5 000 000 steps to each environment except otherwise
specified, and to evaluate, we sample the same number of skills as DISk from the underlying skill
latent distribution. For the hierarchical experiments, we sample the skills and fix them first, and then
we create a hierarchical agent with those fixed skills.

D.6 COMPUTE DETAILS

All of our experiments were run between a local machine with an AMD Threadripper 3990X CPU
and two NVIDIA RTX 3080 GPUs running Ubuntu 20.04, and a cluster with Intel Xeon CPUs and
NVIDIA RTX 8000 GPUs, on a Ubuntu 18.04 virtual image. Each job was restricted to maximum of
one GPU, with often up to three jobs sharing a GPU.

E CONSISTENCY OF SKILLS

Figure 9: Mean normalized variance over skills of different agents on four different environments (lower is
better). On average, the initial position s0 has high normalized variance since the agent has little control over it,
and the subsequent steps has lower normalized variance as the skills lead to consistent states.

To ensure that the diversity of the agent as seen from Section 4.2 are not coming from entirely random
skills, we run an experiment where we track the average consistency of the skills and compare them
to our baseline. To measure consistency, we measure the variance of the agent location normalized
by the agent’s distance from the origin for DISk, each of our baselines, and a set of random skills as
shown in the Figure 6.

We see from this experiment that our method consistently achieves much lower normalized variance
than a random skill, thus showing that our skills are consistent. Note that, on the Hopper environment,
random skills always crash down onto the ground, thus achieving a very high consistency. While
skills from the baselines sometimes achieve higher average consistency compared to DISk, we believe
that it might be caused by the higher overall diversity in the skills from DISk.

F HIERARCHICAL BENCHMARK

Note that in the fast block environment Off-DADS is able to achieve nontrivial performance; this
is because as the circle of blocks opens up fast, the agent learns to move slightly left to avoid the
last-to-be-opened block to take advantage of the first opening. In the same environment, DISk suffers
the most because the environment itself changes too much while a single skill is training, but it is still
able to learn a decent set of skills (see Fig. 2.) As a result, their plots seem similar.

G ABLATION STUDIES

In this section, we discuss the details of the ablation studies discussed on Section 4.5.


-----

G.1 REUSING THE REPLAY BUFFER

Since we use an off-policy algorithm to
learn our reinforcement learning policies
and compute the rewards given the (s, a, s[′])
tuples on the fly, it is possible for us to
reuse the replay buffer from one policy to
the next. In this ablation, we examine the
efficacy of reusing the replay buffer from
previous skills.

Intuitively, if using the replay buffer is effective, it makes more sense that the impact
of reusing the replay buffer will be more
noticable when there is a large amount
of experience stored in the replay buffer.

ery and replay skill relabelling.

Thus, for this ablation, we first train six _R_

(last 4 skills)

Mean Hausdorff Distance Mean Hausdorff distance

Figure 10: Ablation analysis of DISk on parallel skill discov
skills on an Ant agent for 4 000 000 steps
total while reusing the replay buffer, and then train four further skills for 250 000 steps each. In first
case, we reuse the replay buffers from the first six skills. On the second case, we do not reuse the
replay buffer from the first skills and instead clear the replay buffer every time we initialize a new
skill. Once the skills are trained, we measure their diversity with the mean Hausdorff distance metric.

As we have seen on Figure 10, the four skills learned while reusing the buffer are much more diverse
than the skills learned without reusing the replay buffer. This result goes to show that the ability to
reuse the replay buffer gives DISk a large advantage, and not reusing the replay buffer would make
this unsupervised learning method much worse in terms of sample complexity.

G.2 PARALLEL TRAINING WITH INDEPENDENT SKILLS

We trained our DISk skills incrementally, but also with independent neural network policies. We run
this ablation study to ensure that our performance is not because of the incremental only.

To run this experiment, we initialize 10 independent policies on the Ant environment, and train them
in parallel using our incremental objective. We initialize a separate circular buffer for each policy.
For each policy, we use the union of the other policies’ circular buffer as that policy’s T . We train
the agent for 5 000 000 total steps, with each policy trained for 500 000 steps. To train the policies in
parallel, we train them in round-robin fashion, with one environment step taken per policy per step in
the training loop.

As we have seen in Section 4.5, this training method does not work – the policies never learn anything.
We hypothesize that this failure to learn happens because the circular dependencies in the learning
objective does not give the policies a stable enough objective to learn something useful.

G.3 BASELINE METHODS WITH INDEPENDENT SKILLS

To understand the effect of training DISk with independent skills, ideally, we would run an ablation
where DISk uses a shared parameter network instead. However, it is nontrivial to do so, since DISk
is necessarily incremental and we would need to create a single network with incremental input or
output space to properly do so. Still, to understand the effect of using independent skill, we run an
ablation where we modify some of our baseline methods to use one network per skill.

Out of our baselines, Sharma et al. (2019a) uses continuous skill latent space, and thus it is not
suitable to have an independent network per skill. Moreover, its performance degrades if we use
a discrete latent space as shown in Sharma et al. (2019a) Fig. 5. So, we take our other baseline,
Eysenbach et al. (2018), which uses discrete skills, and convert it to use one policy per skill.

We see a better performance with Disjoint-DIAYN agent compared to the DIAYN agent, which
implies that some of the performance gain on DISk may indeed come from using an independent
policy per skill. The reason why disjoint DIAYN performs much better than simple DIAYN may
also stem from the fact that it is much harder for the skills in DIAYN to “coordinate" and exploit the


-----

Figure 11: Training DIAYN with one policy per skill improves its performance. On the (left) we show the
comparison of mean Hausdorff distance of DIAYN with independent policies alongside our method and other
baselines, and on the (right) we show trajectories from running skills from the Disjoint-DIAYN agent.

diversity based reward in a distributed settings where the skills only communicate through the reward
function.

G.4 VALOR-LIKE INCREMENTAL SCHEDULE FOR BASELINES

In VALOR(Achiam et al., 2018), to stabilize training options based on a variational objective, the
authors present an curriculum training method where the latents were sampled from a small initial
distribution, and this distribution was extended every time the information theoretic objective was
saturated. More concretely, if c is a variable signifying the option latent and τ a trajectory generated
by it, then every time their discriminator reached E[Pdiscrim(c | τ )] = pD, where pD is a set constant
_≈_ 0.86, they expanded the set of option latents. If there were K latents when the objective saturates,
it was updated using the following formula:

3K
_K := min_ + 1 _, Kmax_ (8)

2

  

Where Kmax is a hyperparameter that signifies the maximum number of options that can be learned.

Figure 12: Training DIAYN for 4M steps with curriculum scheduling similar to VALOR(Achiam et al., 2018)
results in better performance than vanilla DIAYN but suboptimal performance compared to Fig. 11. On the (left)
we show the trajectories from learned skills on Disjoint-DIAYN with an added curriculum training on a static
Ant environment. On the (right), we show the same on the dynamic Ant-block environment, where the agent
learned two skills that wasn’t available at the environment initialization; however all other skills are degenerate.

Adapting this schedule for Off-DADS is difficult since it has nominally infinite number of skills
throughout training. However, we implemented this scheduling for DIAYN by sampling the skill


-----

_z from the distribution Uniform(K), starting with K = 2, Kmax = 25 and updating K with Eq. 8_
whenever E[Pdiscrim(c | τ )] ≈ _pD threshold was reached, where pD is a hyperparameter. We ran_
this experiments on top of the disjoint DIAYN (Sec. G.3) baseline and searched over possible pD
values, since we found it necessary to have DIAYN converge to any meaningful skills. We chose
the pD values from the set _n+1n_

6 _n_ 8, or pD [0.83, 0 {.87], which lines up with the findings in the VALOR paper.[|][ 1][ ≤] _[n][ ≤]_ [10][}][, and found that the best performance was around]
_≤_ _≤_ _∈_

As we can see on the figures Fig. 12, in complicated static environments like Ant, adding curriculum
scheduling to DIAYN performs better than vanilla DIAYN with shared weights, but may actually
result in more degenerate skills than Disjoint-DIAYN. We hypothesize this may be caused by the
arbitrary nature of the threshold E[Pdiscrim(c | τ )], which is also mentioned by Achiam et al. (2018).

On dynamic environments like Ant-block, adding skills successively does allow DIAYN to learn one
skill that was not available at the beginning, unlike vanilla DIAYN. However, we believe because of
the downsides of curriculum scheduling, it also results in a lot of “dead” skills, which slow down the
addition of new skills, which results in few useful skills overall.


-----

H DISK PSEUDO CODE

**Algorithm 2 Pseudocode for DISk training routine for the M** th skill.

# Env: environment
# π: Stochastic policy network
# σ: Projection function
# D: State dimension
# Dσ: Dimension after projection
# M: Current skill number
# P: Number of collected states from each past skills
# max_steps: Number of steps to train the policy for
# replay_buffer: Standard replay buffer for off-policy algorithms
# buf: Circular replay buffer
# α, β: Hyperparameter for normalizing consistency penalty and diversity reward respectively

# Training loop
# At the beginning of training a skill, collect T using the learned policies

#Tfor m in {1, 2, ..., M-1}: T= []: P sample states (projected) visited from each of the M-1 previous skills, (M-1)xPxDπm, m ∈{1, · · ·, M − 1} _σ_

total_collection_step = 0
collected_projected_states = []
while total_collection_step < P:

x = Env.reset()
done = False
while not done:

total_collection_step += 1
a = mode(πm(x))
x_next, reward, done = Env.step(a)
# Add projected state to the collected states
collected_projected_states.append(σ(x_next))
x = x_next

_T .append(collected_projected_states)_

total_steps = 0
while total_steps < max_steps:

x = Env.reset()
done = False
while not done:

total_steps += 1
ax_next, reward, done = Env.step(a) ∼ _π(x)_
replay_buffer.add(x, a, x_next)
x = x_next

# Push projected state to the circular buffer
buf.push(σ(x_next))

# Update phase
for i in range(update_steps):

# sample a minibatch of B transitions without reward from the replay buffer
# (xt, at, xt+1): state (BxD), action (Bx|A|), next state (BxD)
(xt, at, xt+1) = sample(replay_buffer)
# compute entropy-based reward using the next projected state σ(xt+1)
rt = compute_rewards(σ(xt+1))
# train exploration RL agent on an augmented minibatch of B transitions (xt, at, rt, x

_t+1[)]_

update_π(xt, at, rt, xt+1) # standard state-based SAC


# Entropy-based task-agnostic reward computation
# s: state (BxDσ)
# b: diversity reward batch size
def compute_rewards(s, b = 256):

_Tb = uniform_sample(T, b) # Sample diversity reward candidates_

# Finding k-nearest neighbor for each sample in s (BxDbxDσ) _σ) over the candidates buffer Tb, (_

# Find pairwise L2 distances (Bxb) between s anddists = norm(s[:, None, :] -topk_dists, _ = topk(dists, k=3, dim=1, largest=False) Tb[None, :, :], dim=-1, p=2) Tb # compute topk distances (Bx3)
# Diversity rewards (Bx1) are defined as L2 distances to the k-nearest neighbor fromdiversity_reward = topk_dists[:, -1:] _Tb_

# Finding k-nearest neighbor for each sample in s (BxDσ) over the circular buffer buf, (|

buf|xDσ)

# Find pairwise L2 distances (Bx|buf|) between s and buf
dists = norm(s[:, None, :] - buf[None, :, :], dim=-1, p=2)
topk_dists, _ = topk(dists, k=3, dim=1, largest=False) # compute topk distances (Bx3)
# Consistency penalties (Bx1) are defined as L2 distances to the k-nearest neighbor from

buf

consistency_penalty = topk_dists[:, -1:]

reward = β*diversity_reward - α*consistency_penalty
return reward


-----

I DISK SKILLS LEARNED IN THE STATIC ANT ENVIRONMENT IN
SEQUENTIAL ORDER

Skill 1 Skill 2 Skill 3

100

50

0

50

100

Skill 4 Skill 5 Skill 6

100

50

0

50

100

Skill 7 Skill 8 Skill 9

100

50

0

50

100

Skill 10 All skills

100

50

0

50

100

100 50 0 50 100 100 50 0 50 100 100 50 0 50 100


Figure 13: All 10 skills learned by DISk in a static Ant environment in the order learned. Even though it seems
like skill 1 and 2 did not learn anything, skill 1 learns to stand perfectly still in the environment while skill 2
learns to flip over and terminate the episode.


-----

