FAVAE: S EQUENCE D ISENTANGLEMENT USING I NFORMATION B OTTLENECK P RINCIPLE
Anonymous authors
Paper under double-blind review
A BSTRACT
We propose the factorized action variational autoencoder (FAVAE), a state-of-theart generative model for learning disentangled and interpretable representations from sequential data via the information bottleneck without supervision.
The purpose of disentangled representation learning is to obtain interpretable and transferable representations from data.
We focused on the disentangled representation of sequential data since there is a wide range of potential applications if disentanglement representation is extended to sequential data such as video, speech and stock price data.
Sequential data is characterized by dynamic factors and static factors: dynamic factors are time dependent, and static factors are independent of time.
Previous works succeed in disentangling static factors and dynamic factors by explicit modeling the priors of latent variables to distinguish between static and dynamic factors.
However, this model cannot disentangle representations between dynamic factors, such as disentangling ”picking” and ”throwing” in robotic tasks.
In this paper, we propose new model that can disentangle multiple dynamic factors.
Since our method does not require modeling priors, it is capable of disentangling ”between” dynamic factors.
In experiments, we show that FAVAE can extract the disentangled dynamic factors
I NTRODUCTION
Representation learning is one of the most fundamental problems in machine learning.
A real world data distribution can be regarded as a low-dimensional manifold in a high-dimensional space (Bengio et al., 2013).
Generative models in deep learning, such as the variational autoencoder (VAE) (Kingma & Welling, 2013) and the generative adversarial network (GAN) (Goodfellow et al., 2014), are able to learn low-dimensional manifold representation (factor) as a latent variable.
The factors are fundamental components such as position, color, and degree of smiling in an image of a human face (Liu et al., 2015).
Disentangled representation is deﬁned as a single factor being represented by a single latent variable (Bengio et al., 2013).
Thus, if in a model of learned disentangled representation, shifting one latent variable while leaving the others ﬁxed generates data showing that only the corresponding factor was changed.
This is called latent traversals (a good demonstration of which was given by Higgins et al.
(2016a) 1 ).
There are two advantages of disentangled representation.
First, latent variables are interpretable.
Second, the disentangled representation is generalizable and robust against adversarial attacks (Alemi et al., 2016).
We focus on the disentangled representation learning of sequential data.
Sequential data is characterized by dynamic factors and static factors: dynamic factors are time dependent, and static factors are independent of time.
With disentangled representation learning from sequential data, we should be able to extract dynamic factors that cannot be extracted by disentangled representation learning models for non-sequential data such as β -VAE (Higgins et al., 2016a;b) and InfoGAN (Chen et al., 2016).
The concept of disentangled representation learning for sequential data is illustrated in Fig.
Consider that the pseudo-dataset of the movement of a submarine has a dynamic factor: the trajectory shape.
The disentangled representation learning model for sequential data can extract this shape.
On the other hand, since the disentangled representation learning model for non-sequential data does not consider the sequence of data, it merely extracts the x-position and y-position.
[Figure]
There is a wide range of potential applications if we extend disentanglement representation to sequential data such as speech, video, and stock market data.
For example, disentangled representation learning for stock price data can extract the fundamental trend of a given stock price.
Another application is the reduction of action space in reinforcement learning.
Extracting dynamic factors would enable the generation of macro-actions (Durugkar et al., 2016) which are sets of sequential actions that represent the fundamental factors of the actions.
Thus, disentangled representation learning for sequential data opens the door to new areas of research.
Very recent related work (Hsu et al., 2017; Li & Mandt, 2018)
separated factors of sequential data into dynamic and static factors.
The factorized hierarchical variational autoencoder (FHVAE) (Hsu et al., 2017) is based on a graphical model using latent variables with different time dependencies.
By maximizing the variational lower bound of the graphical model, the FHVAE separates the different time dependent factors such as the dynamic and static factors.
The VAE architecture developed by Li & Mandt (2018) is the same as the FHVAE in terms of the time dependencies of the latent variables.
Since these models require different time dependencies for the latent variables, these approaches cannot be used disentangle variables with the same time dependency factor.
We address this problem by taking a different approach.
First, we analyze the root cause of disentanglement from the perspective of information theory.
As a result, the term causing disentanglement is derived from a more fundamental rule: reduce the mutual dependence between the input and output of an encoder while keeping the reconstruction of the data.
This is called the information bottleneck (IB) principle.
We naturally extend this principle to sequential data from the relationship between x and z to x t : T and z .
This enables the separation of multiple dynamic factors as a consequence of information compression.
It is difﬁcult to learn a disentangled representation of sequential data since not only the feature space but also the time space should be compressed.
We created the factorized action variational autoencoder (FAVAE) in which we implemented the concept of information capacity to stabilize learning and a ladder network to learn a disentangled representation in accordance with the level of data abstraction.
Since our model is a more general model without the restriction of a graphical model design to distinguish between static and dynamic factors, it can separate depen- dency factors occurring at the same time.
Moreover, it can separate factors into dynamic and static factors.
D ISENTANGLEMENT FOR N ON -S EQUENTIAL D ATA
β -VAE (Higgins et al., 2016a;b) is a commonly used method for learning disentangled representations based on the VAE framework (Kingma & Welling, 2013) for a generative model.
The VAE can estimate the probability density from data x.
The objective function of the VAE maximizes the evidence lower bound (ELBO) of log p ( x ) as
[Equation]
where z is latent variable, D KL is the Kullback-Leibler divergence, and q ( z | x ) is an approximated distribution of p ( z | x ) .
D KL ( q ( z | x ) || p ( z | x )) reduces to zero as the ELBO L VAE increases; thus, q ( z | x ) learns a good approximation of p ( z | x ) .
The ELBO is deﬁned as
[Equation]
where the ﬁrst term, E q ( z | x ) [log p ( x | z )] , is a reconstruction term used to reconstruct x , and the second term D KL ( q ( z | x ) || p ( z )) is a regularization term used to regularize posterior q ( z | x ) .
Encoder q ( z | x ) and decoder p ( x | z ) are learned in the VAE.
Next we will explain how β -VAE extracts disentangled representations from unlabeled data.
β -VAE is an extension of the coefﬁcient β > 1 of the regularization term D KL ( q ( z | x ) || p ( z )) in the VAE.
The objective function of β -VAE is
[Equation]
where β > 1 and p ( z ) =
N (0 , 1) .
β -VAE promotes disentangled representation learning via the
Kullback-Leibler divergence term.
As β increases, the latent variable q ( z | x ) approaches the prior p ( z ) ; therefore, each z i is pressured to learn the probability distribution of N (0 , 1) .
However, if all latent variables z i become N (0 , 1) , the model cannot reconstruct x .
As a result, as long as z reconstructs x , β -VAE reduces the information of z .
P RELIMINARY : O RIGIN OF D ISENTANGLEMENT
To clarify the origin of disentanglement, we will explain the regularization term.
The regularization term has been decomposed into three terms (Chen et al., 2018;
Kim & Mnih, 2018; Hoffman &
Johnson, 2016):
[Equation]
where z j denotes the j -th dimension of the latent variable.
The second term, which is called ”total correlation” in information theory, quantiﬁes the redundancy or dependency among a set of n random variables (Watanabe, 1960).
β -TCVAE has been experimentally shown to reduce the total correlation causing disentanglement (Chen et al., 2018).
The third term indirectly causes disentanglement by bringing q ( z | x ) close to the independent standard normal distribution p ( z ) .
The ﬁrst term is mutual information between the data variable and latent variable based on the empirical data distribution.
Minimizing the regularization term causes disentanglement but disturbs reconstruction via the ﬁrst term in Eq.
The shift C scheme was proposed (Burgess et al., 2018) as a means to solve this conﬂict:
[Equation]
where constant shift C , which is called ”information capacity,” linearly increases during training.
This shift C can be understood from the point of view of an information bottleneck (Tishby et al., 2000).
The VAE can be derived by maximizing the ELBO, but β -VAE can no longer be interpreted as an ELBO once this scheme has been applied.
The objective function of β -VAE is derived from the information bottleneck (Alemi et al., 2016; Achille & Soatto, 2018; Tishby et al., 2000; Chechik et al., 2005).
[Equation]
where C is the information capacity and ˆ x is the empirical distribution.
Solving this equation by using Lagrange multipliers drives the objective function of β -VAE Eq.
(5) with β as the Lagrange multiplier (details in Appendix B of (Alemi et al., 2016)).
In Eq.
(5), information capacity C prevents I (ˆ x, z ) from becoming zero.
In the information bottleneck literature, y typically stands for a classiﬁcation task; however, the formulation can be related to the autoencoding objective (Alemi et al., 2016).
Therefore, the objective function of β -VAE can be understood using the information bottleneck principle.
P ROPOSED M ODEL : D ISENTANGLEMENT FOR S EQUENTIAL D ATA
Our proposed FAVAE model learns disentangled and interpretable representations from sequential data without supervision.
We consider sequential data x 1: T ≡ {
[Equation]
[Equation] [Equation]
x
[Equation]
} generated from a latent variable model,
[Equation]
For sequential data, we replace x with ( x 1: T ) in Eq.
The objective function of the FAVAE model is
[Equation]
where p ( z ) = N (0 , 1) .
The variational recurrent neural network (Chung et al., 2015) and stochastic recurrent neural network (Fraccaro et al., 2016) extend the VAE model to a recurrent framework.
The priors of both networks are dependent on time.
The time dependent prior experimentally improves the ELBO.
In contrast, the prior of our model is independent of time like those of the stochastic recurrent network (Bayer & Osendorfer, 2014) and the Deep Recurrent Attentive Writer (DRAW) neural network architecture (Gregor et al., 2015); this is because FAVAE is disentangled representation learning rather than density estimation.
For better understanding, consider FAVAE from the perspective of IB.
As with β -VAE, FAVAE can be understood from the information bottleneck principle.
[Equation]
where ˆ x 1: T follows an empirical distribution.
These principles make the representation of z compact while reconstruction of the sequential data is represented by x 1: T (see Appendix A).
4. L ADDER N ETWORK
An important extension to FAVAE is a hierarchical representation scheme inspired by the VLAE (Zhao et al., 2017).
Encoder q ( z | x 1: T ) within a ladder network is deﬁned as
[Equation]
where l is a layer index, h 0 ≡ x 1: T , and f is a time convolution network, which is explained in the next section.
Decoder p ( x 1: T | z ) within the ladder network is deﬁned as
[Equation] [Equation]
[Equation]
14) where g l is the time deconvolution network with l = 1 , · · · , L − 1 , and r is a distribution family parameterized by g 0 (˜ z 0 ) .
The gate computes the Hadamard product of its learnable parameter and
[Figure]
input tensor.
We set r as a ﬁxed-variance factored Gaussian distribution with the mean given by µ t : T = g
(˜ z 0 ) .
Fig.
(2) shows the architecture of the proposed model.
The difference between each ladder network in the model is the number of convolution networks through which data passes.
The abstract expressions should differ between ladders since the time convolution layer abstracts sequential data.
Without the ladder network, the proposed method can disentangle only the representations at the same level of abstraction; with the ladder network, it can disentangle representations at different levels of abstraction.
4. H OW TO ENCODE AND DECODE
There are several mainstream neural network models designed for sequential data, such as the long short-term memory model (Hochreiter & Schmidhuber, 1997), the gated recurrent unit model (Chung et al., 2014), and the quasi-recurrent neural network QRNN (Bradbury et al., 2016).
However, the VLAE has a hierarchical structure created by abstracting a convolutional neural network, so it is simple to add the time convolution of the QRNN to our model.
The input data are x t,i , where t is the time index and i is the dimension of the feature vector index.
The time convolution considers the dimensions of feature vector j as a convolution channel and performs convolution in the time direction:
[Equation]
(15) where j is the channel index.
The proposed FAVAE model has a network similar to the VAE one regarding time convolution and a loss function similar to the β -VAE one (Eq.
We used the batch normalization (Hinton et al., 2012) and ReLU as activation functions though other variations are possible.
For example, 1d convolutional neural networks use a ﬁlter size of 3 and a stride ofand do not use a pooling layer.
M EASURING D ISENTANGLEMENT
While latent traversals are useful for checking the success or failure of disentanglement, quantiﬁcation of the disentanglement is required for reliably evaluating the model.
Various disentanglement quantiﬁcation methods have been reported (Eastwood & Williams, 2018;
Chen et al., 2018; Kim &
Mnih, 2018; Higgins et al., 2016b;a), but there is no standard method.
We use the mutual information gap (MIG) (Chen et al., 2018) as the metric for disentanglement.
The basic idea of MIG is measuring the mutual information between latent variables
z j and a ground truth factor v k .
Higher mutual information means that z j contains more information regarding v k .
MIG ≡
[Equation]
K (cid:80)
Kk = 1 H ( v k ) (cid:0) I (cid:0)
z j ( k ) ; v k (cid:1) − max j (cid:54) =
j ( k ) I ( z j ; v k ) (cid:1) , where j ( k ) ≡ arg max j I ( z j ; v k ) , and H ( v k ) is entropy for normalization.
In our evaluation we experimentally measure disentanglement with MIG.
[Table]
R ELATED W ORK
Several recently reported models (Hsu et al., 2017; Li & Mandt, 2018) graphically disentangle static and dynamic factors in sequential data such as speech data and video data (Garofolo et al., 1993;
Pearce & Picone, 2002).
In contrast, our model performs disentanglement by using a loss function (see Eq.
The advantage of the graphical models is that they can control the interpretable factors by controlling the priors time dependency.
Since dynamic factors have the same time dependency
[Table]
these models cannot disentangle dynamic factors.
A loss function model can disentangle sets of dynamic factors as well as disentangle static and dynamic factors.
E VALUATION
We evaluated our model experimentally using three sequential datasets: 2D Reaching, 2D Wavy
Reaching
[Table]
and Gripper.
We used a batch size of 128 and the Adam (Kingma & Ba, 2014) optimizer with a learning rate of 10 − 3 .
7. 2D R EACHING
To determine the differences between FAVAE and β -VAE
[Table]
we used a bi-dimensional space reaching dataset.
Starting from point (0, 0), the point travels to goal position (-0.1, +1) or (+0.1, +1).
There are ten possible trajectories to each goal; ﬁve are curved inward, and the other ﬁve are curved outward.
The degree of curvature for all ﬁve trajectories is different.
The number of factor combinations was thus 20 (2x2x5).
The trajectory length was 1000, so the size of one trajectory was [1000x2].
We compared the performances of β -VAE and FAVAE trained on the 2D Reaching dataset.
The results of latent traversal are transforming one dimension of latent variable z into another value and reconstructing something from the traversed latent variables.
β -VAE, which is only able to learn from every point of a trajectory separately, encodes data points into latent variables that are parallel to the x and y axes (3b, 3c).
In contrast, FAVAE learns through one entire trajectory and can encode disentangled representation effectively so that feasible trajectories are generated from traversed latent variables (3d, 3e).
7. 2D W AVY R EACHING
To conﬁrm the effect of disentanglement through information bottleneck, we evaluated the validity of our model under more complex factors by adding more factors to the 2D Reaching dataset.
Five factors in total generated data compared to the three factors that generate data in 2D Reaching.
This modiﬁed dataset differed in that four out of the ﬁve factors affect only part of the trajectory: two of them affect the ﬁrst half, and the other two affect the second half.
This means that the model should be able to focus on a certain part of the whole trajectory and be able to extract factors related to that part.
A detailed explanation of these factors is given in Appendix B.
We compared various models on the basis of MIG to demonstrate the validity of our proposed model in comparison of a time convolution AE in which a loss function is used only for the autoencoder ( β = 0 ), FAVAE without the ladder network and information capacity C , and FAVAE with the ladder network and information capacity C .
As shown in Table 1, FAVAE with the ladder network and C had the highest MIG scores for 2D Reaching and 2D Wavy Reaching.
This indicates that this model learned a disentangled representation best.
Note that for 2D Reaching, the best value for C was small, meaning that there was little effect from adding C (since the dataset was simple, this task can be solved even if the amount of information of z is small).
[Figure]
We evaluated reconstruction loss and the MIG score for various values of β using three ladder networks (Fig.
6) with a different value of C for each ladder: C =
[LowerLadder , MiddleLadder , HigherLadder] .
One setting used was C =
[0 , 0 , 0] , meaning that
C was not used; another setting was C =
[20 , 1 , 5] , meaning that C was adjusted on the basis of values of KL-Divergence for β = 1 and C = [0 , 0 , 0] .
When C was not used, the model could not reconstruct data when β was high; thus, disentangled representation was not learned well when β was high.
When C was used, the MIG score increased with β while reconstruction loss was suppressed.
The latent traversal results for 2D Wavy Reaching are plotted in Fig.
Even though not all learned representations are perfectly disentangled, the visualization shows that all ﬁve generation factors were learned from ﬁve latent variables; the other latent variables did not learn any meaningful factors, indicating that the factors could be expressed as a combination of ﬁve ”active” latent variables.
We tested our model for β = 300 .
[Figure]
The use of a ladder network in our model improved disentangled representation learning and minimized the reconstruction loss.
The graph in Fig.
6 shows the MIG scores for networks with different numbers of ladders.
The error bars represent the standard deviation for ten repetitive trials.
Using all three ladders resulted in the minimum reconstruction loss with the highest MIG score (green curve) except for ”Higher Ladder One”. ”
Higher Ladder One” has a large reconstruction error.
7. S PRITES DATASET .
To evaluate the effectiveness of the video dataset, we trained our model with the Sprites dataset which is used in (Li & Mandt, 2018).
This dataset has sequential length = 8 RGB video data with 3 × 64 × 64 .
This data set consists of static factor and dynamic factor.
We note that the motion is not created with the combination of dynamic factors, and each motion exists individually (detail is E.2).
Table 4 show the factors used in our experiment.
We executed disentangled representation learning by using the FAVAE model with β = 20 , C = [0 . 3 , 0 . 17 , 0 . 06] and network architecture used for this training is explained in Section F.1.
Fig.
7 shows the results of latent traversal, and we chose two z values to change from z = 3 to 3 .
Since this dataset is composed of discrete factors, we show two z values at a time.
The latent variables in the 1st ladder extract expressions of motion, pants color, direction of character and the shirts color.
The latent variables in the 2nd ladder extract expressions of the hair color and the body color.
FHVAE can extract the disentangled representations between static factors and dynamic factor in high dimension dataset.
S UMMARY AND F UTURE W ORK
Our factorized action variational autoencoder (FAVAVE) generative model learns disentangled and interpretable representations via the information bottleneck from sequential data.
Evaluation using three sequential datasets demonstrated that it can learn disentangled representations.
Future work includes extending the time convolution part to a sequence-to-sequence model (Sutskever et al., 2014) and applying the model to actions of reinforcement learning to reduce the pattern of actions.
A I NFORMATION B OTTLENECK P RINCIPLE FOR S EQUENTIAL D ATA .
Here the aim is to show the relationship between FAVAE and the information bottleneck for sequential data.
Consider the information bottleneck object:
[Equation]
is expanded from Alemi et al.
(2016) to sequential data.
We need to distinguish between ˆ x 1: T and x 1: T , where x 1: T is the true distribution and ˆ x 1: T is the empirical distribution created by sampling from the true distribution.
We maximize the mutual information of true data x 1: T and z while constraining the information contained in the empirical data distribution.
We do this by using Lagrange multiplier:
[Equation]
where β is a constant.
For the ﬁrst term,
[Equation]
where H ( x 1: T ) is entropy, which can be neglected in optimization.
The last line is Monte Carlo approximation.
For the second term,
[Equation]
As a result,
[Equation]
For convenience of calculation, we use x i sampled from mini-batch data for both the reconstruction term and the regularization term.
This is only an approximation.
If the information bottleneck principle is followed completely, it is better to use different batch data for the reconstruction and regularization terms.
B E XTRACTED FACTORS OF EACH LADDER IN FAVAE
We expect the ladder network can disentangle representations at different levels of abstraction.
In this section, We check the factor extracted in each ladder by using 2D Reaching and 2D Wavy.
[Table]
[Table] [Table]
[Table] [Table]
Table 2 shows the counting index of latent variable with the highest mutual information in each ladder network.
In Table 2, the rows represent factor and the columns represent the index of the ladder networks.
The factor 1 (goal left / goal right) in 2D Reaching and the factor 1 (goal position) in 2D wavy Reaching were extracted to the most frequently in the latent variable in 3rd ladder.
Since the latent variables have 8 dimensions for the 1st ladder, 4 dimensions for the 2nd ladder and 2 dimensions for the 3rd ladder, the 3rd ladder should be the least frequent when factors are randomly entered for each z.
In both datasets, only factor 1 represents goal positions while others represent shape of the trajectories.
Since factor 1 has different abstraction level from others, factor 1 and others result in different ladders such as ladder 3 and others.
C C OMPARING WITH FHVAE
FHVAE model is the recently proposed disentangled representation learning model.
We note that
FHVAE model uses label information to disentangle time series data, which is different setup with our FAVAE model.
Table 3 shows a comparison of MIG and reconstruction using FHVAE as the baseline.
It was not possible to disentangle in 2D Reaching and 2D wavy using FHVAE, because
LSTM used at the FHVAE can not learn data with very long sequences (sequence length of 1000).
For fair comparison with the FHVAE, we experimented with 2D Reaching (sequence length 100), 2D wavy Reaching (sequence length 100) at Table 3.
In 2D Reaching, FHVAE model has the best score, while in 2D wavy
Reaching FAVAE model with ladders and C has the best score.
Our model is slightly worse than FHVAE in 2D Reaching Dataset.
The parameters searched are α = 1 to 30 (the best α = 20 and 1 ), β = 0 to 60 (the best β = 0 .
4 and 4 in 2D Reaching and 2D wavy Reaching) and the best C =
[ c = 0 .
02] and [0 .
03 , 0 , 0 , 06] in 2D Reaching and 2D wavy Reaching.
The best C was decided by the value of KL divergence loss when we experimented to allow reconstruction with C = 0 , 0 , 0 .
D G RIPPER
To evaluate the potential application of our model to robotic tasks, we constructed a robot endeffector simulation environment based on Bullet Real-Time Physics Simulation 2 .
The environment consisted of an end-effector, two balls (red and blue), and two baskets (red and blue) in a bi-dimensional space.
The end-effector grabs one of the balls and places it into the basket with the same color as the ball.
The data factors include movement ”habits.”
For example, the end-effector could reach the targeted ball by either directly moving toward the ball obliquely or ﬁrst moving above of the ball and then lowering itself until it reached the ball (perpendicular movement).
The end-effector could drop the ball from far above the basket or place it gently in the basket.
Each factor could affect different lengths among the data; e.g., ”the plan to place the ball in the basket” factor affects different lengths per datum since the initial distance from the target ball to the corresponding basket may differ.
This means that the time required to reach the ball should differ.
Note that input is a value such as gripper’s position, not an image.
See Appendix B for a more detailed explanation.
FAVAE learned the disentangled factors of the Gripper dataset.
Example visualized latent traversals are shown in Fig.
The traversed latent variable determined which factors were disentangled, such as the initial position of blue ball and basket (b), the targeted ball (red or blue) (c), plan to reach the ball (move obliquely or move perpendicularly)
(d), and plan to place the ball in the basket (drop the ball or placing it in the basket gently) (e).
These results indicate that our model can learn generative factors such as disentangled latent variables for robotic tasks, even though the length of the data affected by each factor may differ for each datum.
We used FAVAE in a two-ladder network withand 8 latent variables with β =1000.
E D ATASET D ETAILS E. 2D W AVY R EACHING
Five different factors generate each trajectory: (9a) Shape of trajectory from start position to intermediate point.
The ﬁrst shape is created using a cycloid function while the other one adds a sine curve on the cycloid.
(9b) Shape of trajectory from intermediate point to goal; shape is the same
[Figure]
as for (9a).
(9c) Location of goal; ﬁve different goal locations were used.
(9d) Degree of curvature from start to intermediate point.
(9e) Degree of curvature from intermediate point to goal.
E. S PRITES
Sprits dataset is video data of video game ”sprites”.
It used was used in (Li & Mandt, 2018) for conﬁrming the extraction of disentangled representation between static factors and dynamic factors.
The dataset consists of sequences with T = 8 frames of dimension 3 × 64 × 64 .
We use factors and motion of Sprites is shown in Table 4 and Fig.
E. G RIPPER
We implemented the end-effector only rather than the entire robot arm since controlling the robot arm during the picking task is easily computable by calculating the inverse kinematics and inverse dynamics.
Gripper is a 12 dimensional data set: [joint x position, joint y position, ﬁnger1 joint position(angle), ﬁnger2 joint position(angle), box1 x position, box1 y position, box2 x position, box2 y position, ball1 x position, ball1 y position, ball2 x position, ball2 y position].
Eight factors are represented in this dataset: 1) color of ball to pick up, 2) initial location of red ball, 3) initial location of blue ball, 4) initial location of blue basket, 5) initial location of red basket, 6) plan for using end effector to move to ball to pick it up [ﬁrst, moving horizontally to the x-location of ball and then descending horizontally to the y-location of ball, like the movement of the doll drawing machine (perpendicular motion); second, moving straight to the location of he ball to pick it up (oblique motion)], 7) plan for using end effector to move to point above basket after picking up ball (perpendicular or oblique motion), 8) plan for placing ball in basket (by dropping ball or descending to basket and gently placing ball in basket).
Among the four initial positions, the two balls and two baskets are placed randomly.
The movement of the robot is hard-coded on the basis of a goalposition-based script.
To reduce collision detection errors during the simulation, we used a large physical model (end effector size 1 m), which does not affect the overall validity.
The length of the data-point sequence was 400.
[Table]
F A RCHITECTURE D ETAILS F. FAVAE
We show the hyper parameters in Table 1, Fig.
7 and Table 3.
The parameters searched are β = 0 to 10000 (the best β = 1 and C =
[0 , 0 , 0] and the best β = 100000 and C = [20 , 1 , 5] in 2D Reaching and 2D wavy Reaching in Table 1)
The best C was decided by the value of KL divergence loss when we experimented to allow reconstruction with C = 0 , 0 , 0 .
In Table 3, we search the parameters from β = 0 to 60 (the best β = 0 .
4 and 4 in 2D Reaching and 2D wavy Reaching) and the best
C =
[ c = 0 .
02] and [0 .
03 , 0 , 0 , 06] in 2D Reaching and 2D wavy Reaching.
In Fig.
7.3, we add the pre-encoder and post-decoder for encoding and decoding of images at each time step.
The pre-encoding and post-decoding architecture is same as ”encode frames” and ”decode frames” in the code implementing Disentangled-Sequential-Autoencoder 3 with conv dim = 48 .
We search the parameters from β = 0 to β = 100 and the best β = 20 with C =
F. FHVAE
For the FHVAE experiments, we used the code from the implementation at Github 4 .
We used recurrent setting with LSTM encoders and decoder with unit size 256 and batch size 80.
Dimensions of latent variable z1 and z2 are 7 for each.
We used Adam optimizer with learning rate 0.001.
We applied goal position factors (2 classes for 2D Reaching and 5 classes for 2D wavy Reaching) as label inputs.
We varied α from 1.0 to 30.0, and the best α was 20.0 for 2D Reaching and 1.0 for 2D wavy reaching.