L EARNING
M INIMAL
R EPRESENTATIONS
WITH
M ODEL I NVARIANCE
Anonymous authors
Paper under double-blind review
A BSTRACT
Sparsity has been identiﬁed as an important characteristic in learning neural networks that generalize well, forming the key idea in constructing minimal representations.
Minimal representations are ones that only encode information required to predict well on a task and nothing more.
In this paper we present a powerful approach to learning minimal representations.
Our method, called M OD I NV or model invariance, argues for learning using multiple predictors and a single representation, creating a bottleneck architecture.
Predictors’ learning landscapes are diversiﬁed by training independently and with different learning rates.
The common representation acts as a implicit invariance objective to avoid the different spurious correlations captured by individual predictors.
This in turn leads to better generalization performance.
We test M OD
I NV on both reinforcement learning and vision settings, showcasing strong performance boosts in both.
It is extremely simple to implement, does not lead to increased wall clock time while training, and can be applied across different problem settings.
I NTRODUCTION
Learning efﬁcient representations that generalize well is a long standing problem of machine learning, and particularly of deep learning (Doersch & Zisserman, 2017; Noroozi & Favaro, 2016;
Oord et al., 2018).
Algorithms that exploit structure in the real world through effective inductive biases are key to solving this problem.
Several inductive biases have been successfully used in the past, from early work using translational invariance for developing CNNs (LeCun et al., 2004; 2010), to recent data augmentations for developing multiple self-supervised algorithms (Chen et al., 2020; Grill et al., 2020; Chen & He, 2021).
A key inductive bias in multiple works is sparsity of concepts (Hoeﬂer et al., 2021).
Essentially, it refers to how objects in the real world tend to interact and affect the dynamics of other objects only in a small neighbourhood.
Consider for example picking up a pen.
This simple task only changes the dynamics of a very small part of the world, namely the pen and the hand of the person grasping it.
Exploiting such structure allows learning representations that can generalize better in the real world.
In this paper, we develop a method that exploits sparsity by learning minimal representations.
The well known saying ‘neurons that ﬁre together wire together’ points to the phenomenon that that neurons that have similar output for a given input also have strong weights between them (Hebb, 2005).
Ideally, given the same set of neurons to model certain concepts, we would want neurons that relate to a particular concept to have strengthened weights while those relating to another concept to have weakened weights (Sun et al., 2016).
Therefore, low correlation in the outputs of the representation would lead to less redundancy in modelling concepts and thus better generalization.
This is precisely the motivation behind minimality leading to sparse representations and improved generalization.
Minimal representations (Tishby et al., 2000; Shamir et al., 2010) refer to ones that have sufﬁcient but minimal information w.r.t.
the task at hand.
Therefore, it must be only possible to solve the task in hand and no other task, thus leading to better generalization for the given task.
Another viewpoint is that minimal representations also lead to a reduction in spurious correlations.
Since there is minimal noise present in the representation to cause spurious correlations, this again results in better generalization.
[Figure]
Our approach, which we call model invariance (M OD I NV ), learns minimal representations by incorporating multiple predictors over a single, common representation for a given task (see Figure 1).
Each predictor head is then trained independently (a given sample only trains one head) and with a learning rate different to other heads.
Theoretically, this can be looked at as a manifestation of the information bottleneck principle, which balances between sufﬁciency and minimality.
Intuitively, this can be looked at as providing robustness to spurious correlations.
Since each head learns a different prediction (based on its own optimization and learning on different parts of data set), each head is susceptible to different spurious correlations.
However, since there is only one common representation to support all the predictor heads, this representation must be robust to any spurious correlations as training progresses.
We evaluate our method on two separate settings; 1) Reinforcement Learning: The DeepMind
Control Suite(DMC) (Tassa et al., 2018) benchmark with distractors, which involves natural video playing in the background while the agent is controlled in the foreground and 2) Vision:
The CIFAR10 (Krizhevsky et al., 2009) and STL-10 (Coates et al., 2011) datasets, where no labels are used for learning the representations, followed by the standard linear probing evaluation protocol to test generalization.
Contributions:
We introduce M OD I NV , a simple method motivated by incorporating sparsity in representations, while showing connections to the information bottleneck principle.
M OD I NV is conceptually simple, easy to implement, and improves representation learning performance in both reinforcement learning (from pixels) and vision settings.
In reinforcement learning, M OD I NV achieves stronger than state-of-the-art performance on the DMC Suite with distractors benchmark, both with augmentations and without.
In vision, M OD I NV leads to better performance than the baseline method of S IM S IAM when using a linear predictor on both CIFAR-10 and STL-10 datasets.
P RELIMINARIES 2. I NFORMATION B OTTLENECK P RINCIPLE
The well known information bottleneck (IB) principle (Tishby et al., 2000; Shamir et al., 2010) formally describes the ideas of sufﬁciency and minimality for learning representations.
Speciﬁcally, it states that representations should contain sufﬁcient information to do well on the downstream task but also contain minimal information about the inputs, leading to better, more robust generalization.
This is in contrast to minimizing the empirical risk only, which is a sound technique in the inﬁnite sample case.
However, in the ﬁnite sample case, minimizing only empirical risk can lead to poor generalization.
A similar observation has been noted in works that build over the invariant risk minimization principle (IRM) (Arjovsky et al., 2019).
2. S ELF -S UPERVISED L EARNING IN RL
RL considers the agent’s interaction with the environment as a discrete time γ -discounted Markov
Decision Process (MDP) (Puterman, 2014)
M
=
( X , A , P, R, γ, µ 0 ) , where X denotes a ﬁnite state space and A is the action space; P ≡ P ( x (cid:48)
| x, a ) is the transition kernel; R ≡ r ( x, a ) is the reward function; γ ∈ [0 , 1) is the discount factor; and µ 0 is the initial state distribution.
The objective is to ﬁnd a policy π : X → ∆ A , where ∆ A is the set of probability distributions on A such that the value function of a policy π at a state x ∈ X , V π ( x ) ≡ E [ (cid:80) t ≥
γ t r ( x t , a t ) | x 0 = x, π ] is maximized.
Recently a lot of progress has been made in making RL over pixels as sample efﬁcient as when learning over true states.
Most approaches can be divided into two classes.
The ﬁrst involves using auxiliary losses over the representation so as to inject as much information about the downstream task as possible.
These auxiliary objectives include reconstructing observations (Hafner et al., 2020;
Ha &
Schmidhuber, 2018), or predicting next observation, reward (Schaul et al., 2015) or even encoded states (Gelada et al., 2019).
The second class borrows ideas from self-supervised learning methods and augments the observations (Yarats et al., 2021;
Laskin et al., 2020b) with techniques like random crop, cutout, color jitter etc.
Since randomly augmented samples from the same observation have the same Q values, an implicit invariance to augmentations is enforced in the representation pipeline.
In the case of hard exploration tasks, auxiliary objectives such as estimating value functions of random cumulants has shown to be useful as well.
A common baseline architecture used in a lot of methods is an actor-critic setup where a common convolutional encoder is used to compute the encoded latent state from the raw pixel observations.
For continuous control tasks, soft actor-critic (SAC, Haarnoja et al.
(2018)) is used as the actor-critic algorithm and the gradients from the actor are stopped from updating the encoded state.
Only the critic trains the representation network f θ .
This is since allowing gradients from both actor and critic leads to noisy estimates and the eventual divergence.
Moreover, some methods also add a reward prediction loss from the encoded state, a transition prediction loss, or a pixel reconstruction loss.
2. S ELF -S UPERVISED L EARNING IN V ISION
An important property of self-supervised methods in vision is avoiding collapse in representation, or trivial solutions where the representation simply outputs a constant.
These involve the use of two similar or identical networks, an online and a target network.
Particular approaches largely fall in two categories, that of contrastive (Chen et al., 2020) and non-contrastive methods (Grill et al., 2020;
Chen & He, 2021), depending on how they avoid representation collapse.
Contrastive methods create positive and negative pairs from random augmentations of data (e.g., cutout, crop, ﬂip), and enforce an objective such that positive samples are brought closer and negative samples are pulled away in representation space.
On the other hand, non-contrastive methods do away with creating positive and negative pairs and instead use the likes of assymetric architectures, stop gradient, and momentum encoders for the target network to learn non-trivial representations.
One method we use to build over later in the paper is S IM S IAM (Chen & He, 2021), which is a non-contrastive method and only incorporates the most essential components required for preventing collapse.
In particular, it uses two networks, an online and a target network, where both use a backbone network (usually a ResNet 18 or ResNet 50 (He et al., 2016)) and a projection MLP ( hidden layer).
Moreover, only the online network has a prediction MLP attached to it, which makes the overall architecture assymetric.
The target network has a stop gradient applied to it and the overall objective is a cosine similarity loss between the outputs of the online and target networks.
It has been noted that stop-gradient is enough to avoid collapse and that slowly updating the target network weights to match those of the online network, i.e. exponential moving average (EMA), is not a necessary requirement.
Adding the EMA to the S IM S IAM architecture results in the BYOL (Grill et al., 2020) architecture.
M ETHOD
Consider a task Y , which can correspond to a variety of objectives.
For instance, it could be an auxiliary task of predicting the next state when interacting with an RL environment, or simply a classiﬁcation task on a vision benchmark.
We are interested in training a model that can do well on task Y , where the model consists of two components, a backbone representation f θ and a predictor/classiﬁer h φ which is attached over the representation.
In a lot of tasks that involve pixel input, spurious correlations can arise due to various reasons such as lack of non-iid data, irrelevant information, confounders etc.
that can result in poor generalization at test time.
An attractive property for better generalization has been the idea of sparse representations, in that the mutual information of any two dimensions of the representation must be low.
However, in order to avoid learning trivial solutions, the representation as a whole should encode enough information about the downstream task as well.
A combination of both of these objectives then leads to a model with better generalization capability.
In this paper, we balance the above two objectives using the idea of model invariance (M OD I NV ).
M OD I NV deploys multiple predictors over a single representation, while training each predictor independently, each capturing different spurious correlations.
The common representation acts as a implicit invariance loss which ensures that only the optimal representation remains at convergence.
Intuitively, each predictor can be looked at as an augmented version of the optimal
[Figure]
model, where the augmentation is over the model space and refers to particular spurious correlations arising in the model that differ it from the optimal model.
Note that each predictor head is the same in architecture, and we only diversify the learning procedure of each head (through different initialization, independent training, different learning rates).
This is so that we eventually converge to an optimal representation for a particular predictor/classiﬁer and not for all predictor/classiﬁer families, which is a much more hard to optimize in practise.
E XPERIMENTS 4. R EINFORCEMENT L EARNING
For RL over pixels, we build over a standard actor-critic setup and use M OD I NV as part of an auxiliary task that predicts the next latent state (see Figure 2).
This auxiliary task is used to train the representation which supports the actor-critic heads.
Speciﬁcally, the input is the observation x t and the representation f θ encodes it into the latent state s t .
We then concatenate the latent state and the action and pass it through a linear MLP to generate a common state-action representation (predicting the next state s t +1 requires information about both the state and action).
Since the true latent state is not known, predicting only the next state s t +1 can result in a representation collapse * .
Therefore, we ground the state prediction loss by also predicting the reward, which does have a valid grounding and thus avoids collapse.
Finally, we deploy M OD
I NV by using multiple predictor heads for the state prediction task, with each head being coupled with a corresponding reward prediction head.
4.1. I MPLEMENTATION D ETAILS
We implement SAC (Haarnoja et al., 2018) as the base agent with the actor and critic sharing a common representation.
Our actor-critic setup is similar to the one used in SAC-AE (Yarats et al., 2019), except the reward and state prediction task.
Our transition or state prediction model is deterministic in nature, involving a 3 layer encoder MLP, followed by a 3 layer decoder MLP.
The reward prediction network is a 2 layer MLP.
We do not tune for MLP sizes and widths at all in reporting our results.
Both the reward and transition losses train the representation alongside the critic.
We use three M OD I NV heads, each with a slightly different learning rate of 3 e − 3 , 3 e − 4 , and 3 e − 5 respectively.
4.1. R ESULTS
We test our method on the DMC Suite with distractors.
Particularly, we test on the six popular domains of Walker Walk, Reacher Easy, Hopper Hop, Finger Spin, Cheetah Run and Cartpole Swingup.
We compare with other methods that learn over pixels including D REAMER , CURL, DBC, and RAD.
Dreamer (Hafner et al., 2020) is a model-based method that performs pixel reconstruction and reward prediction to learn a model in the latent state space and then performs model-based updates to a base
SAC agent.
CURL (Laskin et al., 2020a) uses a contrastive loss similar to that in CPC (Oord et al., 2018) to train the representation network, using data augmentations to generate positive and negative samples.
RAD (Laskin et al., 2020b) is another method that simply augments the samples in the replay buffer without adding any other loss function to the base SAC setup.
Finally, note that both
RAD and DrQ (Kostrikov et al., 2020) are similar in performance as they both use data augmentations in the replay buffer samples.
Since data augmentations explicitly remove the actual distractions in the background, a direct comparison with data augmentations is not fair.
Therefore, we also compare our method when augmentations are added alongside M OD I NV (denoted by M OD I NV +A UG ).
We evaluate all methods at both 100K and 500K environment steps, which is the standard number of steps in this benchmark.
Our results (Table 1) show that M OD I NV clearly outperforms all 4 methods (D REAMER , CURL,
DBC) at the 500K step mark.
Note that since all methods use the same number of gradient steps,
M OD I NV is at a slight disadvantage as the total number of gradient updates are divided amongst the
K predictor heads.
4. V ISION
[Figure]
the other heads.
For vision, we adopt the base framework of S IMS IAM , a non-contrastive self-supervised method for learning representations.
S IM S IAM has three model components, the backbone network or the representaion, a projection MLP and a prediction MLP.
We deploy M OD
I NV in the projection
MLP, with a single backbone representation and a single prediction MLP (see Figure 3).
At each training iteration, the same M OD I NV projection head is sampled for both the online and target networks and trained using the original S IMS IAM loss function.
Unlike the reinforcement learning case, where any of the modules after the representation (state and reward prediction heads) are not used actively in the algorithm, in the vision case the projector and predictor both are used to align the backbone representation (Tian et al., 2021).
Therefore, ensuring that each projection head in M OD I NV is trained sufﬁciently (as in the base case) is important.
4.2. I MPLEMENTATION D ETAILS
We test our approach on the CIFAR-10 and STL-10 datasets.
As data augmentations, we use random crop, color jitter, horizontal ﬂip, random grayscale for both datasets and also use blur for STL-10.
Furthermore, we deploy three M OD
I NV heads for the projection MLP, with the learning rates of 0 . 03 , 0 . 3 , 0 .
003 for each head respectively.
The learning rate of all other components in the model is set to 0 .
03 , with a weight decay of 0 .
0004 , momentum 0 .
We run two sets of experiments, one with a linear prediction MLP and one with the standard two layer prediction MLP.
The rest of the architecture is the same as that of S IM S IAM , with the output dimension of the projector and the predictor MLP set to 2048.
We use a batch size of 512 for CIFAR-10 and 128 for STL-10.
Linear vs Non-linear Predictor .
For the SSL setting we deploy M OD
I NV in two different conﬁgurations.
The ﬁrst involves a linear predictor while the second involves a standard 2-layer predictor.
The 2-layer predictor has been speculated to be using a lot of ‘lucky’ initialization that lead to alignment between the backbone representation and the prediction MLP (Tian et al., 2021).
However, the linear predictor does not have this property and thus testing in this setting offers a much more robust evaluation option.
Unless speciﬁed otherwise, our ablations for vision in Section 5 use the linear predictor setting for M OD I NV .
[Table] [Table]
R ESULTS
We see that with a linear predictor, the base S IM S IAM version fails for CIFAR-10 and only achieves 41.3% † accuracy while the M OD I NV version achieves an accuracy of 82.5% at 330 epochs, similar to that of the 2-layer predictor case.
A similar gap in performance is observed for earlier training epochs as well (Table 2).
A careful reader might attribute this gap to the use of different learning rates in M OD I NV , since a relatively slower learning rate for the non-predictor layers can correspond to improved performance, especially when EMA is not used in the target network (Tian et al., 2021).
To test this, we train the base S IM S IAM model individually with the different projector learning rates used in the M OD I NV version.
We see that all three learning rates fail to the same performance of around 40% accuracy.
This clearly shows that M OD I NV contributes much more to better generalization than using slower or faster learning rates.
A BLATIONS 5. A BLATION ON N UMBER OF H EADS
In our experiments, we observe that only K = 3 heads are sufﬁcient to provide good performance boosts compared to when not using M OD I NV (i.e. K = 1 ) across the RL and SSL settings.
Furthermore, with everything else the same, adding an extra head after K = 3 leads to diminishing performance gains.
This can be most clearly seen for S IM S IAM + M OD I NV , where we see increasing gains with increasing K (see Table 3).
Note that although relative gains saturate after K = 3 , it is possible that further diversiﬁcation in predictor training can lead to even better results when K > 3 .
[Table]
5. C ORRELATION IN D IMENSIONS
As mentioned, our initial motivation stems from sparsity in concepts.
This refers to low mutual information between any two dimensions in the representation f θ .
A useful metric to check for such a characteristic is therefore the mean correlation between any two dimensions.
We plot this metric during training with and without M OD I NV .
Ideally, we would hope to get lower mean correlation when using M OD I NV as opposed to when not.
Our experiments show that this is indeed the case, as the mean correlation decreases much more steadily when we use M OD I NV .
Moreover, the decrease in this metric has a strong correlation with the performance of the agent.
[Equation]
where C ij =
[Equation]
z
[Equation]
z
[Equation]
Based on this observation, we also run an experiment where we take the baseline architecture (i.e. without M OD I NV ) and add the mean correlation between dimensions of f θ as an auxiliary loss.
Equation 1 describes the exact loss, where the summation is over the batch b .
The ﬁrst term is the state prediction error from Section 4.1 while the second term is the exact loss from Zbontar et al.
Our results show that optimizing directly for this loss leads to much better performance in the
Reacher Easy task, while much worse performance in the Walker Walk (Table 4).
We choose to test on these two domains since they are hard to solve for most methods that we compare to.
This is an interesting observation since similar losses that have been used successfully in vision recently (Bardes et al., 2021) might offer strong performance improvements in RL as well.
[Table]
5. I MPORTANCE OF D IFFERENT L EARNING R ATES
[Table]
When using multiple heads in M OD
I NV
[Table]
it is important to diversify the learning of heads so they capture different spurious correlations.
Using different learning rates is a vital component for achieving this, besides different model initialization and independent training.
We test this by evaluating performance with multiple sets of learning rates for K = 3 predictor heads.
We see that performance improves when the learning rates are diverse as compared to when all heads use the same learning rate (Table 5).
Moving on to diversifying learning of predictor heads through different optimizers is left for future work (see Section 7).
5. A BLATION ON O NLY P REDICTING R EWARD
The RL application of M OD I NV involves both state prediction and reward prediction.
We now test how the performance might vary if we only have the reward prediction task.
It is worth noting that M OD I NV can be applied to the actor-critic heads as well, however optimization in that case becomes difﬁcult as the critic learning loss does not have a stable ground truth.
In Table 6 we see that performance deteriorate in general for the three tasks, especially for Cheetah Run.
This may suggest that state prediction provides a slightly more stable ground truth to optimize the representation.
5. A BLATION WITH D ATA A UGMENTATIONS
[Table]
Our results show that M OD I NV acts complimentarily to augmentations.
In our experiments in the RL setting, we observe that adding data augmentations over M OD I NV leads to better performance than when only using M OD I NV .
‘Crop’ is the only augmentation used here, which is standard for RL environments.
Essentially, cropping removes a lot of background information which is irrelevant to the task (top-down approach), thus aiding in better or more robust estimates for next state and critic losses.
However, not all augmentations have a similar effect (e.g. ﬂip), and thus end up with similar performance as the baseline.
On the other hand, M OD I NV avoids spurious correlations by enforcing minimality, resulting in a bottom-up approach to robust reward and critic loss optimization.
R ELATED W ORK AND D ISCUSSION 6. D ECODABLE I NFORMATION B OTTLENECK
The IB principle does not take into account the classiﬁer/predictor family which is attached over the representation and thus can be considered too restrictive.
Moreover, the minimality term is hard to approximate for practical losses.
To remedy this, the decodable information bottleneck (Dubois et al., 2020) recently introduced notions of sufﬁciency and minimality for a given predictor/classiﬁer family.
They also provided a practical method which has a similar structure to the M OD I NV approach.
In particular, multiple predictor heads are attached to a common representation, where one head (sufﬁciency head) minimizes the standard empirical risk while all other heads (minimality heads) are learnt such that they cannot predict arbitrary relabellings of the same data.
This is done by providing different random relabellings and then reversing the gradients for all the minimality heads.
This ensures that the common representation contains information required to predict only the correct labels and no other.
The idea in M OD I NV is similar but we do not deploy reverse gradients or relabel data, which in practise can lead to instability issues.
Therefore, M OD I NV can be seen as a speciﬁc instantiation of the DBC framework, following similar theoretical guarantees while being much easier to implement and resulting in better performance.
6. D ECORRELATION IN N ON -C ONTRASTIVE SSL
Recently, B ARLOW T WINS (Zbontar et al., 2021) was introduced as a new non-contrastive selfsupervised learning method which does not use an asymmetric architecture while still avoids collapse in representation.
This was achieved by enforcing the off-diagonal elements of the cross-correlation matrix of the outputs of the representation to be zero.
Essentially, such a loss ensures that each output dimension of the representation is decorrelated with each other while minimizing the standard loss of enforcing invariance to augmentations.
Interestingly, the B ARLOW T WINS loss can be traced back as an approximation to the information bottleneck principle again, hence connecting decorrelation in representation dimensions as an indicator for better generalization (as noted similarly in the introduction).
We showed that M OD I NV also leads to more decorrelation in the representation dimensions, thus supporting this observation.
However, we also showed that directly optimizing for this loss might not be a good idea always, with such a variant leading to worse empirical performance than M OD I NV in the RL setting.
Nevertheless, it is important to note that the M OD I NV and B ARLOW
T WINS objectives are primarily complimentary in nature.
This is since M OD I NV always requires a sufﬁciency objective to start with, which could be provided by any SSL loss which avoids collapse, be it S IM S IAM with the use of a predictor and stop gradient or be it B ARLOW T WINS with the use of the decorrelation loss.
In similar ﬂavor to B ARLOW T WINS , VICR EG (Bardes et al., 2021) also uses the decorrelation loss from B ARLOW T WINS and adds a variance regularization loss to prevent collapse.
Finally, Hua et al.
(2021) introduced a technique called decorrelated batch normalization (DBN) and also showed that collapse can be avoided with no predictor.
6. A UXILIARY V ALUE P REDICTION IN RL
To improve representation learning in RL, a prominent idea has been that of predicting multiple value functions from a common representation.
Sutton et al.
(2011) use random cumulants to deﬁne the different value functions.
The value improvement path (Dabney et al., 2020) idea uses a mixture of value functions of past policies.
Although these works have a similar looking structure to M OD I NV , there are a couple of critical differences.
First, the multiple heads in M OD I NV are predictors of the same object while the above papers use different objects (different value functions).
Second, the training of each head is not independent.
These two differences dissociate the above works from the
IB idea, thus making them complimentary to M OD I NV .
F UTURE W ORK
M OD I NV shows promising improvements in both the RL and SSL settings, thus hinting at the beneﬁts offered by the IB principle in practice.
Nevertheless, M OD I NV is still one instantiation of this idea.
There remain quite a few directions to explore in future.
In particular, analysing how further diversiﬁcation in the K predictors can be achieved is a promising idea.
For example, we can introduce different optimizers (Adam, SGD) for each predictor, while also varying parameters such as momentum, weight decay etc.
Another possibility could be to update a subset of predictors for a given sample, instead of only updating one predictor at each training iteration.
Finally, since data augmentations change the input data, each predictor could be trained on a different augmentation scheme, thus ensuring further diversiﬁcation in optimizing different predictors.
On a separate axis, analysing how similar M OD I NV for vision is with other SSL methods such as
VICR EG is interesting.
For the RL case, M OD I NV can also potentially be applied to the Q functions directly (each predictor is a separate Q head over a common representation) instead of the reward and transition modules.
This could turn out to be hard to optimize for certain environments though, since
Q functions can be less developed for sparse reward settings.
C ONCLUSION
We introduce a general representation learning method motivated from the ideas of sparsity and minimal-sufﬁcient representations.
Our method, M OD I NV , uses multiple predictors over a common representation, diversifying the training of each predictor such that the common representation acts as an implicit invariance mechanism to spurious correlations.
We show instances of this idea in both reinforcement learning and vision settings.
Overall, M OD I NV leads to more robust generalization across different evaluations, leading us to believe that with further analysis, we can get further gains in representation generalization.
M OD I NV can be viewed as a speciﬁc instantiation of the information bottleneck principle, one that is simple in implementation and in concept.
Appendix H YPERPARAMETERS
[Table]