Fast and Sample-Efficient Domain Adaptation for Autoencoder-Based End-to-End Communication
Authors: Jayaram Raghuram
Yijing Zeng
Dolores Garcia
Somesh Jha
Suman Banerjee
Joerg Widmer
Rafael Ruiz
Venue: ICLR 2022 Submitted (2022)
Keywords: domain adaptation
small target dataset
communication autoencoders
mixture density networks
wireless channel

Abstract: The problem of domain adaptation conventionally considers the setting where a source domain has plenty of labeled data, and a target domain (with a different data distribution) has plenty of unlabeled data but none or very limited labeled data. In this paper, we address the setting where the target domain has only limited labeled data from a distribution that is expected to change frequently. We first propose a fast and light-weight method for adapting a Gaussian mixture density network (MDN) using only a small set of target domain samples. This method is well-suited for the setting where the distribution of target data changes rapidly (e.g., a wireless channel), making it challenging to collect a large number of samples and retrain. We then apply the proposed MDN adaptation method to the problem of end-of-end learning of a communication autoencoder, which jointly learns the encoder, decoder, and a channel networks to minimize the decoding error rate. However, the error rate of an autoencoder trained on a particular (source) channel distribution can degrade as the channel distribution changes frequently, not allowing enough time for data collection and retraining of the autoencoder to the target channel distribution. We propose a method for adapting the autoencoder without modifying the encoder and decoder neural networks, and adapting only the MDN model of the channel. The method utilizes feature transformations at the decoder to compensate for changes in the channel distribution, and effectively present to the decoder samples close to the source distribution. Experimental evaluation on simulated datasets and real mmWave wireless channels demonstrate that the proposed methods can adapt the MDN model using very limited number of samples, and improve or maintain the error rate of the autoencoder under changing channel conditions.

Under review as a conference paper at ICLR 2022
FAST AND SAMPLE-EFFICIENT DOMAIN ADAPTATION FOR AUTOENCODER-BASEN E2E COMMUNICATION
Anonymous authors Paper under double-blind review
ABSTRACT
The problem of end-to-end learning of communication system using an autoen- coder has recently been shown to be promising approach. We focus on the problem of test-time domain adaptation for such an autoencoder system whose channel is generatively moodeled using mixture density network (MDN). Different from the setting of conventional training-tima (unsupervised or semi-supervised) domain adaptation, here have fully tully-trained channel model and autoencoder from a source domain, that we would like to adapt to a target domain using only a small labeled dataset (and no unlabeled data). Moreover, since the distribution of the channel is expected to change frequently (e.g., a wireless link), the error rate of the autoencoder can degrade quickly, making it challenging to collect sufficient data for frequent retraining of the autoencoder. To address this, we propose a fast and sample-efficient method for adapting the autoencoder without modifying the encoder and decoder neural networks, and adapting only the MDN channel model. The method utilizes feature transformations the decoder to compensate for changes in the channel distribution, and effectively present to the decoder samples close to the source distribution. Experimental evaluation on simulated datasets and real mm Wave wireless channels demonstrate that the proposed method can adapt the MDN channel using very limited number of samples, and improve or maintain the error rate of the autoencoder under changing channel conditions.
INTRODUCTION
End-to-end (e2e) learning of communication systems using autoencoders has been recently shown to be a promising approach for designing the next generation of wireless networks (0 (O'Shea & Hoydis
2017: Dörner al. 2018 Aoudia & Hoydis 2019 O Shea al., 2019 Ye et al. 2018 Wang et
2017). This new paradigm is viable alternative to optimize communication for diverse applications, hardware, and environments (Hoydís et al.) 2021). is particularly promising for dense deployments of low-cost transceivers where there is interference between devices and hardware imperfections that are difficult to model. The key idea of e2e learning for communication system to use an autoencoder architecture to model and learn the transmitter and receiver jointly using neural networks in order to minimize an <2e performance metric such as the block error rate (BLER) (O Shha & Hoydis 2017). The channel (the propagation medium and transceiver imperfections) can be represented as a stochastic transfer function that transforms its input x E Rd to an output y E IRd, can be regarded as a black-box that is non-lineat and non-differentiable due to hardware imperfections (e.g., quantization and amplifiers). Since autoencoders are trained using stochastic gradient descent (SGD)-based optimization (O'Shea & Hoydis] 2017), is challenging to work with black-box channel that is not differentiable. One approach to address this problem is by using a known mathematical model of the channel (e.g., additive Gaussian noise). Use of such models enables the computation of gradients of the loss function with respect to the autoencoder parameters via backpropagation. However, such standard channel models do not capture well the realistic channel effects, as shown in (Aoudia & Hoydis 2018). Alternatively, recent works have proposed to learn the channel using deep generative models that approximate p(y x). the conditional probability density of the channel output y given the channel input x, using generative adversarial networks (GANs) (O SheShea et al. 2019 Ye et al. 2018), mixture density networks (MDNs) (García Martí et al. 2020), and conditional variational autoencoders (VAEs) (Xia et al., 2020). The use of a differentiable generative model of the channel enables SGD-based training of the autoencoder, while also capturing realistic channel effects better than standard models.

---

Under review as a conference paper at ICLR 2022
Although this e2e optimization with real channels learned from data can improve the physical layer design for communication systems, in reality, channels often change, requiring collection of large number of samples and retraining the channel model and autoencoder frequently. For this reason, adapting the learned conditional probability density of the channel as often as possible using only a small number of samples is required for good communication performance. Prior works have (to be best of our knowledge) not addressed the adaptation problem for autoencoder-basse which is crucial for the real-time deployment of such a system under Trequendy-change conditions. In this paper, we study the problem of domain adaptation (DA) of autoencoders using an MDN as the channel model. In contrast to the conventional DA setting, where one has access to large unlabeled dataset and none or a small labeled dataset from the target domain (Tiang] [2008] BBen-aavid et al. 2006). here we consider DA where we only have access to a small labeled dataset from the target domain. This setting applies to our problem since the channel distribution changes frequently, and we only get to collect a small number of samples at a time from the target domain.
e2e learning, channel
We make the following contributions: 1) a generative MDN (used for modeling the chamnel) based on the properties of Gaussian mixtures. 2) Based the MDN adaptation, we propose efficient trasfrrrasooonnttion methods at the decoder that compensate for changes in the class-conditional channel distribution, and decrease or maintain the error rate of the autoencoder without requiring any retraining of the encoder and decoder networks. [
We propose a fast and sample-efficient method for adapting
2 RELATED WORK
Mixture Density Networks. MDNs were first introduced by (Bishop 1994), providing a new
framework for modeling complex conditional densities using neural networks. Recently, (Garcia Marti ct al. 2020) proposed use an MDN to learn the wireless channel since Gaussian mixtures, with their strong approximation capability (Kostantinos) 2000), can accurately capture the channel distribution given sufficient parametric complexity and data. García Martí et al. (2020) also proposed to adapt the MDN model to changing channel conditions by fine-tuning the MDN with a small set of samples from a new distribution. To the best of our knowledge, (Li et al. 2020) is the only other work to study the problem of adapting an MDN. They address the speaker identification problem, and propose gradient- based meta- leta-lea algorithm for MDN that learns to transfer knowledge from an existing set of speakers to new speaker using a small number of labeled samples.
Domain Adaptation, Transfer Learning, and Few-Shot Learning. Recent approaches for DA such DANN (Ganin et al., 2016) based on adversarial learning of shared representation between the source and target domains (Ganin & Lempitsky/ (2155] Ganin et al.) 2016 Long et al., 2018 (aito
eet al. 2018 Zhao et al.) 2019 Johansson al.] 2019) have achieved much success computer vision and natural language processing tasks. Their high-level idea is to adversarially learn a shared feature representation for which inputs from the source and target distributions are nearly indistinguishable to a domain discriminator DNN, such that a label predictor DNN using this representation and trained using labeled data from only the source domain also generalizes well to the target domain. Adversarial DA methods are not suitable for our problem because of the high imbalance in the number of source and target domain samples (hard to learn a good domain discriminator). Also, adversarial DA methods being heavy on computational and data requirement, are not well-suited for fast and frequent test-time DA. Related frameworks such as transfer learning (Long et al.. 2015) 2016 model-agnostic meta-learning (Finn al., (017), domain-adaptive few-shot learning (Zhao et al. 2021 Sun et al. 2019), and supervised DA (Motiaan et al. 2017a by also deal with the problem of frequent adaptation based on a small number of samples. Most of them are not directly applicable to our problem because they primarily address novel classes (with potentially different distributions) and knowledge transfer from existing our work since they also deal with a target domain that only has a small labeled dataset and has the same set of classes (label space). The key difference that (Motian et al.) (017a) address the training-time supervised DA problem, while we focus on the test-time supervised DA problem. In the test-time setting, frequent and fast adaptation of a trained source-domain classifier (here the decoder) is required. It can be computationally challenging to adopt the adversarial DA method Motilan et al. 701 that would have to be retrained on every new batch of target domain samples.
novel tasks. MMotiaa et al. (2017a) is closely related to
'Code hase for work: anonynous wtw.r/uuuuuuuo.oere http:/..ciencee//eeeee..cc httpa//aciaaaaa//////oaaaaaaaaaa 7CO0/
2

---

Under review as a conference paper at ICLR 2022
3 FAST ADAPTATION OF THE MDN CHANNEL MODEL
Notations and Problem Setup. We denote vectors and matrices by boldface symbols. We define (c) to be an indicator function that takes the value (0) when predicate c is true (false). For any integer > 1, we define [n] = {1.... .n.. We denote the one-hot-coded vector of all zeros except a at position i E [m]] by 1, E (0, 1 The probability density function (pdr) of a multivariate Gaussian with mean vector μl and covariance matrix Σ is denoted by N. fe, Σ). We consider a single-input, single-output autoencoder- based communication system as shown in Fig.
7 in Appendix [A] The transmitter encoder part of the autoencoder is multi-layer, forward neural network (NN) that takes as input the one-hot-coded representation 1, of a message 8 E S == {1, 2, ... .m). and produces an encoded symbol vector Eg, (1s) E Rd. The receiver decoder part is also a multi-layer, feed-forward NN that takes the channel output E IR as its input and predicts a probability distribution over the m messages. The input-output mapping of the decoder NN can be defined as De, (y) == [Po., (1 yy). Po. (m yy]" The message is equivalent to class label and the encoded symbol vector
Eg. (1,) like a representation vector for label s. Table in the Appendix provides quick reference for the notations used in the paper.
Channel Modeling using MDN. In this work, we use an MDN with Gaussian components to learn the channel conditional density P(y [x). MDNs can model complex conditional densities by combining feed-forward neural network with standard parametrie mixture model (Bishop) 1994). The MDN learns to predict the parameters of the mixture model (xx) function of the channel input x. This can be expressed as ((x) = M9. (x. where O2 is the parameter vector the neural network. The parameters of the mixture model defined by the MDN are concatenation of the parameters from the l density components, i.e., ((x)) [Ø1 (x)""... Φr (x)"). A more detailed
background on MDNs and autoenccderr-ased end-to-end learning given in Appendix A
Consider the setting where the channel state (and therefore its conditional distribution) is changing over time due to e.g. environmental factors. Let P(y x) denote the (unknown) source channel distribution underlying the dataset 'Dd used to train the MDN Mo. (x). With a sufficiently large dataset and a suit- able choice of k the Gaussian mixture learned by the MDN Po. x) can closely approximate P(y x).
Moture parameter
Acieptation
Adaepted paramoters
channel
Layer
(x(x)
d(x)
2(x)
M, (x)
Figure 1: Proposed MDN adaptation overview.
Let D = [xx "y""), n n= 1, 1..' denote small set of iid samples (N < (De)) from an unknown target channel distribution p(t) x)), which is potentially different from P(y x) but not by a large deviation. Our goal is to adapt the MDN (and therefore the underlying mixture density)
using D such that it closely approximates p(*) (y x). Note that the space of inputs to the MDN is
the finite set of modulated symbols X' (Ee. (1,), s E S} (referred to as a constellation), with each symbol corresponding to unique message E S.
3.1 TRANSFORMATION BETWEEN GAUSSIAN MIXTURES
(1) Consider the Gaussian mixtures corre- sponding to the source and target channel conditional densities, where A. and θ. are (2) the parameter vectors of the original and adapted MDN. Here (x(x)) E Rd is the mean vector, of (x) (x) E RÉ is the variance vector, and (x) E [0, is the prior probability (weight) of component i for the original mix- ture. We have assumed that the Gaussian components have diagonal covariance matrix, with (x) being the diagonal elements The mixture weights are parameterized using the softmax
Pa. (ylx)
x) Σ
(x)x N (y (x(x))
k 0 (yl x) = Σ (x(x) (y (a)x)))))))))))))())) [x(x)),,((x))
1=1
eax(x) E ea)(x), Vi. The MDN simply predicts the un-normalized weights (x) E R or the prior logits. The parameter vector of component i is defined as (x(x)T = (x). (x) (x)"), and the MDN output Φ(x) has dimension 1: (2d + 1). The
function as (x)
adapted MDN predicts the parameters of the target Gaussian mixture Φ(x) M2. (x) shown
in Fig. D The parameters of the adapted MDN and Gaussian mixture are similarly defined, with
"The diagonal covariance assumption does not imply conditional independence of y as long as h 1.
3

---

Under review as a conference paper at ICLR 2022
hat in the notation. We next summarize the feature and parameter transformations required for mapping the component densities of one Gaussian mixture to another. As shown in Appendix C.T
the transformation between any two multivariate Gaussians ~ (- μe, Σ) and y ~ N- j 2) can be achieved by the transformation: y C (y 14) + A f + b, where the mean vector and covariance matrix of the two Gaussians are related as follows: ş = A + b and Σ = CΣ CTC
Affine and Inverse-Affine Affine Feature Transtomations, Applying the above result to our MDN with k components, we define the affine feature transformation for a given symbol x and component i, mapping from y N- (x)), of (x)) to ÿ ~ N (x,( (x)) as
y gxi (y) C, (y (x)) + A, (x) + b, X X, i [k].
(3)
It is straightforward to also define the inverse-affine transformation from y to as
y (g) C (8 (x) - b.) + 14(x), E X, i E [M]
(4)
For the case of diagonal covariances, we constrain C, be diagonal. These feature transformations will be used for aligning the target and source cossicondiiional distributions of the decoder input.
Parameter Transformations. The corresponding transformations between the source and target Gaussian mixture parameters for any symbol x E X' and component i E [k] are given by
(x) = A; (x) + b;, (x) = C. (x), and ā;(x) = B; (x) + % (5)
where A, E Rddd and by E TRd transform the means; C, = ((r: (Ga) is a diagonal scale matrix for the variances; and B3 E IR and % E RR. are the scale and offset for the prior logits. The vector of all adaptation parameters to be optimized is defined as v² [......... 4/ where v/ contains the affine-transformation parameters from component i. The number of adaptation parameters (dimension of +)) given by &s (d2 + 2d + 2). This is typically much smaller than the number of MDN parameters (weights and biases from all layers), even for shallow fully-connected
NNs. In Fig. I the adaptation layer mapping (xx) to (xx) basically implements the parameter transformations defined in Eq. (5)
Assumptions and Key Insight. The proposed adaptation of the MDN is based on the affine- transformation property of multivariate Gaussians, i.e., one can transform between any two multivari- ate Gaussians through an affine transformation.
Assumption 1: The source and target Gaussian mixtures have the same number of components. This is practical assumption
make in order to not have change the architecture of the MDN. Adding or removing components would require a change to the output layer of the MDN, Also, this assumption can be practically justified when & is chosen to be sufficiently large. Assumption 2: The two mixtures have a
correspondence between the components.
This assumption makes in mathemattcally convenient to derive a closed-form expression for the KL-divergence between two Gaussian mixtures, which would not be possible in the general case.
Based on the above assumptions, can formulate the MDN adaptation as an equivalent problem of finding the optimal set of affine transformations (one per-component) from the source to the target Gaussian mixture. This is a much smaller problem compared to optimizing the weights of all the MDN layers. Moreover, the affine transformations are bijective, allowing the feature and parameter mapping to be applied in the inverse direction. To reduce the possibility of the adapted MDN finding bad solutions due to the ssamle regime, we include a regularization term based on the KL-divergence (KLD) in the adaptation objective.
3.2 DIVERGENCE BETWEEN THE SOURCE AND TARGET DISTRIBUTIONS
For the pair of Gaussian mixtures (Eqs. I and based on Assumption 2 we can derive closed-form expression for the Kullback-Leibler divergence between them, given by
Po. (y, K [x) log 0. (y, 1x)
(x() log (x())
D(Po, Pa. Epoo
Σ p(x) XEX
(x()
4 xx(x) DKL (N (- (f))) NN (-11
+
Σ XEX
p(x)
(6)
1=1
4

---

Under review as a conference paper at ICLR 2022
A detailed derivation of this result and the final expression for the KLD function of 2/ are given in Appendix C.2 The first term in Eq. (6) is the KLD between the component prior probabilities, which simplifies into function of the parameters [31. m.... 3kx " The second term in Eq. 6 involves the KLD between two multivariate Gaussians (a standard result), which also simplifies into function of db. To make the dependence ub explicit, the KLDD is henceforth denoted by D&p P Po.
3.3 MDN ADAPTATION OBJECTIVES
We consider two scenarios for adaptation: 1) Generative adaptation of the MDN in isolation and 2) Discriminative adaptation of the MDN as the channel model for the autoencoder. In the first case, the goal of adaptation is to find good generative model for the target data distribution, while in the second case, the goal is to improve the classification performance of the autoencoder on the target data distribution. In both cases, we formulate the MDN adaptation as minimization problem with regularized negative log-likelihood objective, where the regularization term penalizes solutions with large KLD between the source and target Gaussian mixtures.
Generative Adaptation. The data-dependent term in the adaptation objective is the regularized negative conditional log-likelihood (CLL) of the target dataset:
(eb
NO Σ
-1
log Pa Oc (y (x + (Do Po.
where (x)), (x) and (x) as function of sb are given by Eq. (5). The parameters of the original mixture density a, (x), pi: (x), (x)), Vi are constants since they have no dependence on sb. The regularization constant À 2 0 controls the allowed KLD between the source and target Gaussian mixtures. Small values of À weight the CLL term more, allowing more exploration in the adaptation; large values of À impose strong regularization to constrain the space of target distributions.
Discriminative Adaptation. With the goal of improving the accuracy of the decoder in recovering the transmitted symbol from y. the data-dependent term in the adaptation objective (7) is replaced with the symbol posterior log- likelhidddood (PLL) as follows:
N(
J =
Σ
log Po. (x (y + A DD (Pe. Po.)
(8)
No
(1)
n=1
The symbol posterior Pg. (x is computed using Bayes rule, and the symbol prior {p(x Eg, (1.)), - 1, m is estimated from the source domain training set.
We observe that the adaptation objectives are smooth and nonconvex function of sb Also, com- putation of the objective and its gradient w.r.t vb are inexpensive operations since i) they do not require forward and back- prck-propagation steps through the layers of the MDN and ii) both N!! and the dimension of tb are relatively small. Therefore, we use the BFGS quasi-newton method (Nocedal & Wright 2006) for minimization, instead of SGD-based large-scale learning methods (e.g., Adam). The regularization constant À is hyper-parameter of the proposed method, and propose to set its value automatically using a validation metric based on the inverse-affine transformation from the target to the source distribution (see Appendix
ADAPTATION OF AUTOENCODER BAAOODEEE---EEDDD COMMUNICATION SYSTEM
In this section, we discuss how the proposed MDN adaptation can be combined with an autoencoder-batt communication system to adapt the decoder to changes in the channel con- ditions. Recall that the decoder is basically classifier that predicts the most-probable input message from the received channel output y.
Pe(ly))
pa ayera
Prodiced
Wineew charnal
(ala)))
ply x)
reand nebook.
When the decoder operates in a new (target) channel environment, different from the one it was trained on, its classification accuracy can degrade due to the distribution change. Specif- ically, any change in the channel conditions reflects changes in the class-conditional density of
Do (v)
Figure 2: Adapted decoder with affine transformations.
5

---

Under review as a conference paper at ICLR 2022
the decoder's input, i.e., (P(y (s), S} changes We propose to address this, by designing transformations to the decoder's 's input that can compensate for changes in the channel distribution, and effectively present transformed imputs that are close to the source distribution decoder was trained. Our method does not require any change or adaptation to the trained encoder and decoder networks, making it fast and suitable for the small-sample setting. We next propose two such input transformation methods for the decoder.
which the
4.1 ADAPTED DECODER BASED ON AFFINE FEATURE TRANSFORMATIONS
Consider the same problem setup as $ where we observe small dataset of samples from the target channel distribution. Suppose we have adapted the MDN channel by optimizing over the parameters bb we can use the inverse-affine feature transformations (defined in Eq. ( to transform the channel output y from a component of the target Gaussian mixture to the same component of the source Gaussian mixture. However, this transformation requires knowledge of both the channel input x and the mixture component i, which are not observed (latent) at the decoder. We address this by first determining the most- poott-poobbble pair of channel input and mixture component for a given y (using the MAP rule), and applying the corresponding inverse- feature transformation
ex (y) where x², argmax
Pa. (x, ly)
xEXi,i((M]
The joint posterior over the channel input and mixture component given the channel output y is based on the adapted (target) Gaussian mixture, given by
p(x)),(( N(y (x), (x))
Ps. (x,i
Σx E3 p(x') 7 (x') N(y (x)x'), (x))
The adapted decoder based on the above affine feature transformation (see Fig. is defined as
Do, (y) = Da (xx)))))))) (y)) = Do Bxte- (y).
(10)
4.2 ADAPTED DECODER BASED ON MAP SYMBOL ESTIMATION
In the previous method, an input transformation layer is introduced at the decoder only during adap- tation, but not during training of the autoencoder. Alternatively, propose imput transformation layer at the decoder that takes the channel output and produces a best estimate of the encoded sym- bol X, which is then given as input to the decoder as shown in Fig. 3 This input transformation layer is included during the autoencoder training as a fixed non non-linear transformation that does not have any trainable parameters. Since the decoder is trained to predict using x instead of y, inherently robust to changes in the distribution of y.
Muppp derea taurs
MAP BF ayar
Donta Saver
rausage argman Pufe
Windon charnal
20
Pa(mly)
Decoder Reodwar
D2,y)
Figure 3: Adapted decoder with MAP SE.
Given generative model of the channel conditional density using Gaussian mixtures, we can estimate the plug-in Bayes posterior distribution of x given y. Po. (x yy) (ref. Eq. (7 From this, we can find the MAP estimate of x given y
X map (y) = argmax Pe. (x yy) argmax log Pa. (y x) + log)(x).
(11)
xEX
xEX
The adapted decoder based on this input transformation, referred to as the MAP symbol estimation (SE) layer, is defined as
(r,(y) = Do (x))())) Do, {y)((y),
(12)
and illustrated in Fig. 3 Whenever the MDN model is adapted to changes in the channel distribution, resulting new MDN with parameters 0.. the MAP SE layer also updated using 0... This input transformation shields the decoder from changes to the distribution of the channel output y.
'For this generative model, it is easy to see that the class-conditional density is equal to the channel- conditional density, Le., P(y s) Ply Eg, (1.)), Vs. Hence, by adapting the MDN, we are effectively also adapting the class-conditional density of the decoder's input.

---

Under review as a conference paper at ICLR 2022
Since the MAP SE layer is also included in the autoencoder during training, the non-differentiable argmax function presents an obstacle to training the autoencoder using backpropagation. We address this by using& temperature scaled softmax approximation to the argmax, which is differentiable and provides a close approximation for small temperature values. This approximation is used only during training. whereas the exact argmax is used during inference. Details this approximation, and modified autoencoder training algorithm with temperature annealing are discussed in Appendix C.5
The proposed input transformation methods at the decoder have some similarities to equalization methods used in communication receivers (Goldsmith] 2005). However, our problem setting considers a memoryless channel, and does not deal with intersymbol interference (ISD, which is the main focus of equalization methods. A key advantage of the proposed adaptation is that it is very computationally efficient to implement at the receiver of communication system. A discussion of the computational complexity of the proposed methods is given in Appendix
Comments.
EXPERIMENTAL EVALUATION
We implemented the MDN, communication autoencoder, and the adaptation methods in Python using TensorFlow (Abadi et al. 2015) and TensorFlow Probability. We used the following setting in our experiments. The size of the message set m was fixed to 16, corresponding to bits. The dimension of the encoding (output of the encoder) d was set to 2, and the number of mixture components A: was set to 5. More details on the experimental setup, neural network architecture, and the hyper-paramette given in Appendix D.I. The generative adaptation objective 77 is used for the experiments in $5.1] where the MDN is adapted in isolation. The discriminative adaptation objective (8) is used for the experiments in and $5.3 where the MDN is adapted as part of the autoencoder.
5.1 MDN ADAPTATION ON SIMULATED CHANNELS
We evaluate the proposed adaptation method for an MDN ($ on simulated channel variations based on models commonly used for wireless communication. Specifically, we use the following channel models: i) additive white Gaussian noise (AWGN), ii) Ricean fading, and iii) Uniform or flat fading (Goldsmith) 2005). Details on these channel models and calculation of the their signal-to-non ratio (SNR) are provided in Appendix [ each case, the MDN first trained on large dataset simulated from particular type of channel model (e.g.. AWGN), referred to as the source channel. The trained MDN then adapted using a small dataset from a different type of channel model (e.g., Ricean fading), referred to the target channel. We used standard constellation corresponding to quadrature amplitude modulation of 16 symbols, referred to as 16-QAM (Goldsmith) (2005). as inputs the channel. A training of 25000 samples from the source channel is used train the MDN. The size of the adaptation dataset from the target channel is varied over a few different values- 5, 10, 15, and 20 samples per symbol, corresponding to target datasets of size 80, 160, 240, and 320.
Table 1: Log-likelinee likelihood of the MDN adaptation methods on simulated channel variations
Source channel
Target channel
#Target samples
Proposed
Transfer
Transfe-llast-layer
median
95% CI
median
95% CI
median
95% CT
80
-0.49
(-3,89, 0.45)
-6.97
(-17.92, -1.72)
-2.09
(-6.50, -0.21)
160
-0.43
(-2.32, 0.48)
-1.65
(4,441, -0.14)
-0.79
(-1.90,..1)
AWGN
Uniform fading
240
-0.58
(-1.94, 0.52)
-0.74
(-2.07, 0.25)
-0.35
(-1.32, 0.51)
320
-0.22
(-2.27, 0.63)
-0.40
(-1.35, 0.31)
-0.19
(-1.16, 0.54)
80
1.17
(0.68, 1.33)
-1.78
(-6.88, 0.22)
-0.64
(-6.86, 0.96)
160
1.26
(0.51, 1.39)
0.37
(-1.10, 0.88)
0.55
(-0.71, 1.22)
AWGN
Riccan fading
240
1.31
(-0,09, 1.39)
0.91
(0.28, 1.17)
1.00
(0.42, 1.29)
320
1.27
(0.70, 1.41)
1.07
(0.73, 1.22)
1.14
(0,86, 1.32)
80
-0,53
(-3.77, 0.49)
-11.48
(-26.06, -3.15)
-5.77
(-16.20, -2,27)
Ricean fading
Uniform fading
160
-0.10
(-3.68, 0.74)
-2.91
(-5.48, -0.91)
-1.45
(-3,72,
240
-0.59
(-5.44, 0.68)
-1.24
(-2,21, -0.22)
-0.71
(-1.43,
320
-0.41
(-3,57, 0.68)
-0.43
.1.51,0.21) -0.23
(-1.15, 0.35)
Baseline Methods. We evaluate the following two baseline methods for adapting the MDN. 1)
A new MDN is initialized using the weights of the MDN trained on the source dataset, and it is adapted using the target dataset. 2) Same as baseline 1, but only the weights of the final layer are adapted (fine-tuned) using the target dataset. The above methods are referred to as transfer and transfer-last-layer respectively. We used the Adam optimization method (Kingma & Ba 2015) for 200 epochs, with a batch size of 10 or 0.01 times the target dataset size, whichever is larger. Table compares the number of parameters being optimized by the proposed and baseline MDN adaptation
7

---

Under review as a conference paper at ICLR 2022
methods for the architecture in Table 4 We observe that the number of parameters optimized by the proposed method is orders of magnitude smaller.
Results and Inference. Since the MDN is generative model, we evaluate the conditional log- likelihood (CLL) of the adapted Gaussian mixture on an unseen test set of 25000 samples from the target channel. Table T compares the median and 95% confidence interval (CI) of the CLL for three (source, target) channel pairs. For each pair, the methods are run on 50 randomly generated training, adaptation, and test datasets. The training dataset is sampled from the source channel, while the adaptation and test datasets are sampled from the target channel. The SNR of the source and target channels are independently and randomly selected from the range 10 dB 20 dB for each trial. We observe that the proposed method has a higher median CLL in a majority of the cases (particularly for low sample sizes), and has comparable median CLLL for higher sample sizes. We also observe that the baseline methods have more negatively-skeww skeeed CLL at the smaller sample sizes, suggesting that they often found poor adaptation solutions.
We performed additional experiments where the source and target channel distributions are Gaus- sian mixtures with different parameters that are randomly sampled (in a controlled way). We also evaluated the performance of the methods in the special cases where i) the source and tar- get distributions are the same, and ii) where the number of components in the two mixtures is mis-matched. These results in Table 5 in Ap- pendix support the strong adaptation performance of the proposed method.
Table 2: Number of parameters being optimized by the MDN adaptation methods.
Adaptation method
parameters (specific)
parameters
ny (Mi, + d 2)
Transfer
+ +k + 1) (m,+1) 1)
12925
Tra-slerrlst-llyer Proposed
((2d + 1) (ma 1)
2525
[dd + 2d +2)
50
5.2
AUTOENCODER ADAPTATION ON SIMULATED CHANNELS
3.0
0.00
6.002
6000
1851
(a) AWGN to Ricean fading.
(b) AWGN to Uniform fading.
(c) Ricean fading to Uniform fading.
Figure 4: Results of affine transformation based adaptation on simulated channels.
1.04
0.09
0.07
60001
Sytatto-obss stte 1531
Sigial Wolio Sob 130
Sigsstip 24 red0
(a) AWGN to Ricean fading. Figure 5: Results of MAP SE based adaptation on simulated channels.
(b) AWGN to Uniform fading. (c) Ricean fading to Uniform fading.
We evaluate the proposed decoder adaptation methods on different pairs of simulated source and target channel distributions. The setup for this experiment for adapting from a source channel A to target channel B is as follows. The autoencoder is initially trained using data from the source channel A at an SNR of dB. Details of how the SNR is related to the distribution parameters of the simulated channels is discussed in Appendix D The MDN and the decoder are adapted using small dataset from the target channel B for different fixed SNRs varied over 8 dB to 20 dB in steps of 2 dB. For each SNR, the adaptation is repeated over 10 mample datasets from the target channel, and the average block error rate (BLER) values are calculated on a large held-out test dataset (specific to each SNR). The size of training dataset (from channel A) and test dataset (from channel
8

---

Under review as a conference paper at ICLR 2022
B) are both set to 20, 000 samples per symbol, with 16 symbols. The size of the adaptation dataset from the target channel B is varied over 20, 40, and 60 samples per symbol.
The results of this experiment are given in Figs. 4 and 5 for three pairs of source and target channels. T to the adaptation method of 4.2 referred to MAP The plots show the BLER vs. SNR curve, with average BLER on the y-axis (log-scaled) and SNR on the x-axis. The performance of a standard 16-QAM decoder (Haykin 1988), and an autoencoder trained on the source channel without any adaptation (referred to as no poaadppt) are included as baselines, The number of samples per symbol from the target channel used by the proposed methods is shown as suffix to the method name. In Appendix D.3] we compare the proposed adaptation methods to the best-case performance of fully-retrained autoencoder.
to the adaptation method of 4. referred to as Affine, and Figure corresspns
Observations and Takeaways. 1) Both the adaptation methods significantly decrease the BLER for the cases AWGN to Uniform fading and Ricean fading to Uniform fading. 2) For the case of AWGN to Ricean fading, the adaptation methods perform at the same level or slightly worse compared to the baselines. We think this is because the distribution the two domains are not very different. 3) In general, the BLER decreases with increasing number of the target domain samples. 4) Between the two adaptation methods, MAP SE generally outperforms than the Affine method (see Appendix D.4)
5.3 AUTOENCODER ADAPTATION ON REAL FPGA TRACES
We evaluate the performance of the adaptation methods on real over-the-air wireless experiments. We use recent high-performane testbed (Lacruz et al. 2021). featuring high-end FPGA board with GHz bandwidth per-channel and GHz
mawavv
OF
10-
SIVERS antennas (SIVER-
30
40
30"
SIMA (2020). We train the MDN with a standard 16-QAM constellation using 96,000 samples. We introduce distribution changes via IQ imbalance based distortions to the symbol constellation, and gradually increase the level of imbalance the system More details the
(b) MAP SE Figure 6: Results of adaptation on the real FPGA traces.
(a) Affine
experimental setup and the source and target domains are given in Appendix D.2 We evaluate the performance of the proposed adaptation for 20, 35 and 50 samples per symbol. The BLER of the proposed adaptation methods and the baseline methods (16-QAM and no adaptation) is shown as a function of the IQ imbalance in Fig. 6 The proposed methods (both Affine and MAP SE) show an order of magnitude decrease in BLER compared to the baselines when the IQ imbalance is over 25%.
6 CONCLUSIONS
In this paper we proposed fast and light-weight method for adapting Gaussian MDN with very limited number of samples from the target distribution. We applied the MDN adaptation to an autoencoder-basee based e2e communication system, specifically by transforming the inputs to the decoder such that their Ilass-conditional distributions are close to that of the source domain. This allows for fast adaptation of both the MDN channel and the autoencoder without the need for expensive data collection and retraining. We demonstrated the effectiveness of the proposed methods through extensive experiments on both simulated channels and a real mm FPGA testbed.
Limitations & Future Work. The proposed adaptation for a Gaussian MDN is primarily targeted for oiw-dinensional problems such as the wireless channel. It can be challenging to apply on high-dimensional input domains with structure. Extensions of the proposed work to deep generative models based on normalizing flows (Dinh et al., 2017 Kingma & Dhartwal 2018 Weng 2018) is an interesting direction, which would be more suitable for high-dimensional inputs. In this work, we do not adapt the encoder network, i.e., the autoencoder constellation is not adapted to changes in the channel distribution. Adapting the encoder, decoder, and channel networks jointly would allow for more flexibility, but would likely be slower and require more data from the target distribution.
*IQ imbalance is common issue in RF communication that introduces distortions the final constellation.

---

Under review as a conference paper at ICLR 2022
ETHICS STATEMENT
This work does not raise any ethical issues to the best our knowledge and judgement.
REPRODUCIBILITY STATEMENT
The experimental results can be reproduced through our code base hittps://anonynon
4open. wwwesccceeeeeeeeeeee doma in adaptattionn7CO0 ottp:///r///coaaaaaaaaaaaaoaoaaa
REFERENCES
Martn Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro. Greg Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin. Sanjay Ghemawat, Ian Goodfellow, Andrew Harp. Geoffrey Irving. Michael Isard, Yangqing Jia, Rafal Jozefowicz. Lukasz Kaiser. Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.
URL https: / www tensorflos
org/
Fayçal Ait Aoudia and Jakob Hoydis. End-to-end learning of communications systems without a channel model. In 52nd Asilomur Conference on Signals. Systems, and Computers (ACSSC), pp. 298-303 IEEE, 2018. doi: 10. 10.1109/ACSSC.00 866.444416..
8645416. jRtttts::///// //do:::::/ddi. www.oogg//.o.ooooooooooo http:/0.1119/AACCC 2018. 8645416
Fayçal Ait Aoudia and Jakob Hoydis. Model-free training end-to-end communication systems. IEEE Journal on Selected Areas in Communication, 25:0131)5233251111 2019. doi: 10.1100//AAC.20112233399 URL
https::/dee org/ 11////0..00 10..1199/JSC.C 2019. 2933891
Shai Ben-David, John Blitzer, Koby Crammer, and Fernando Pereira. Analysis representa- tions for domain adaptation. In Advances in Neural Information Processing Systems 19. Pro-
ceedings of the Twentieth Annual Conference Neural Information Processing Systems, pp. 137- 144. MIT Press, 2006. URL
neurips...... cc/ /20.pppr//000//ahh.
Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine Learning, 790 1125115555 2010. doi: 10.1007/
$10994-009-5152-4 URL hts.s:css::///// http:////oioooog////// 1000000000000000001
Christopher M. Bishop. Mixture density networks. Technical report, Aston University, 1994.
http://
publicati ins cns ast.an.ttn.... ac. https/asttt..e...////appppp.p...
Christopher M. Bishop. Pattern recognition and machine learning, 5th Edition, chapter 5, pp. 272-277. Infor-
mation Science and Statistics. Springer, 2007. ISBN 9780387310732. URL https:/ www. wor porldcat
brg/oclcc/1008143
John Blitzer, Koby Crammer, Alex Kulesza, Fernando Percira, and Jennifer Wortman. Learning bounds for domain adaptation. In Advances in Neural Information Processing Systems 20, Proceedings of
the Twenty-First Annual Conference Neural Information Processing Systems, pp. 129-136. Cur- Associates, Inc., 2007. URL https: https://pooceeingg.ee.
Laurent Dinh, Jaseha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP. In 5th International Conference Learning Representations (ICLR), Conference Track Proceedings. OpenReview.nee, 2017.
/ not. woree//ooommmmmmmm wwt.://cec....oooooooooochoo....
Sebastian Dörner. Sebastian Cammerer, Jakob Hoydis, and Stephan ten Brink. Deep learning based commu- nication over the air. IEEE Journal of Selected Topics in Signal Processing, 12(1):132-143 2018. doi: 10.1109//STSP2201...8888
Uttpttts::///oi
1109/1119912 2017 2017.278418.
Marthinus Christoffel Du Plessis and Masashi Sugiyama. class-prior change by distribution matching. Newral Networks, 50:1 10-119, 2014.
learning of class balance under
Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Doina Precup and Yee Whye Teh (eds.). Proceedings of the 34th International Conference on Machine Learning, ICMI. 2017, Sydney, NSW, Australia, 6- I1 August 2017, volume 70 of Proceedings of Machine Learning Research, pp. 1126-1132 135. PMLR, 2017.
10

---

Under review as a conference paper at ICLR 2022
Yaroslav Ganin and Victor S. Lempitsky. Unsupervised domain adaptation by backpropagation. In Proceedings of the 32nd International Conference on Machine Learning (ICML), volume 37 of JMLR Workshop and Conference Proceedings, pp. 1180-1189. JMLR.org, 2015. http:::////rccccdinggss mlr, press/
v337/annn15.
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor S. Lempitsky. Domain-adversaree training of neural networks. Journal of Machine Learning Research, 17:59: 1559:5,,, 2016. URL http::/tpp//ttt
org/ pwtp:/aaprss/eeeee paapppessvvv/111111
html
Dolores Garefa Martí, Joan Palacios Beltrán, Jesús Omar Lacruz, and Joerg Widmer. A mixture density channel model for deep learning-hased wireless physical layer design. In Proceedings of the 23rr International ACM Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems, pp. 53-62, 2020.
Andrea Goldsmith. Wireless Communications. Cambridge University Press, 2005.
Simon Haykin. Digital communications. Wiley New York, 1988.
Jakob Hoydis, Fayçal Ait Aoudia, Alvaro Valcarce, and Harish Viswanathan. Toward 6g ai ni-native air interface. IEEE Communications Magazine, 59(5):6--81, 2021.
Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterizatioo with Gumbel-Softmax, In 5th Interna- tional Conference on Learning Representatiins (ICLR), Conference Track Proceedings. OpenReview.net, 2017. URL /tt.tttss// http:////peeeeeeeeeeeeee/o/oeaaa
Jing Janng. A literature survey domain adaptation of statistical classifiers. Technical report, UIUC, 2008.
http://www.myaee
edu/ hagpaaulto/o/ooolaa/aaaaaaapaapp
Fredrik D. Johansson, David A. Sontag, and Rajesh Ranganath. Support and invertibility in domain-invariant representations. In The 22nd International Conference Artificial Intelligence and Statisties (AISTATS), volume 89 Proceedings of Machine Learning Research, pp. 527-536. PMLR, 2019. URI. http://
proceedings mlr.. prwss/ seress/v89/... v89/ /tta////aaannnnnnane 98. html
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. 3rd International Conference on Learning Representations (ICLR), Conference Track Proceedings, 2015. URL http:// arxiv orgg abs 1412. 6980
Diederik P.
Kingma and Prafulla Dhariwal. In Advances
Glow: Generative flow with invert-
ible 1x1 convolutions. (NeurIPS), pp. 10236-10245.
in Neural Information Processing Systems
2018.
URL sttp://papers.... wwappprrs.iipp...mm. www..iip..c..m.mm wttpssssiisss.pppppeeeeeese cc/paper/
8224 gww.- fooc---lowwwwte Lnvert
N Kostantinos. Gaussian mixtures and their applications to signal processing. Advanced signal processing handbook: Theory and implementation for radar, sonar, and medical imaging real time systems, pp. 3-1, 2000.
Jesus O. Lacruz, Rafael Ruiz, and Joerg Widmer. A real-time experimentation platform for sub-6 ghz and millimeter- MIMO systems. In ACM MobiSys 21, 2021. doi: 10. 10.1145/3458$644446663
Ruirui Li, Jyun- Yu Jiang, Xian Wu, Hongda Mao, Chu Chu-Cheng Hsich, and Wei Wang, Bridging mixture density networks with meta-learning for automatic speaker identification. In ICASSP 2020-2020 IEEE International Conference on Acoustics. Speech and Signal Processing (ICASSP), pp. 3522 3526. IEEE, 2020.
Jean Paul Linnartz. Ricean fading. htm, 2001.
http://wwwwiieeeeessccmmuuuu
nl wtt.o//eneeeeeeee////e/
Mingsheng Long. Yue Cao. Jianmin Wang, and Michael I. Jordan. Learning transferable features with deep adaptation networks. In Proceedings of the 32nd International Conference on Machine Learning (ICML). volume 37 JMLR Workshop and Conference Proceedings, pp. 97-105. JMLR.org, 2015. URL http aoproceedings. mlr, ptw.s//37/oogg
Mingsheng Long, Han 7hu, Jianmin Wang, and Michael I, Jordan. Unsupervised domain adaptation with residual transfer networks. In Advances in Neural Information Processing Systems 29; Annual Conference on Neural Information Processing Systems, pp. 136-144, 2016. URL ptLLstttps///prrceeeee ccc/paper/ 2016/ w/w.h//aahhaaeeee ac62 www.c6666llllooo lablecbab 62wc96e 02E0 SieeE0//642bbb-btbttee www.iiinnnnn..u.ooooo.oe.m.uu neurips html
Mingsheng Long, Zhangjie Cao, Jianmin Wang, and Michael Jordan. Conditional adversarial domain
adaptation. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural
Information Processing Systems, pp. 1647-1657, 2018. URL T.R.Ittt.... https: wwwp/cdediinnccccccccccc iineddinggg.. ngs. neur cc.
paper/201 8/wash/aahaaaaaa abwesh//ab8hbbb 1ttah//2288881133333333311333333
08dd8 Abst www.traattt.. html

---

Under review as a conference paper at ICLR 2022
Saeid Motiaan, Quinn Jones, Seyed Mehdi Iranmanesh, and Gianfranco Doretto. Few-shot adversarial domain adaptation. In Advances Neural Information Processing Systems 30: Anmual Conference on Neural Information Processing Systems, pp. 6670 6680, 2017a. URL
wtw.:eedinnnnnn...iiieeeee cc/
Sacid Motian, Marco Piccirilli, Donald A. Adjeroh, and Gianfranco Doretto. Unified deep supervised domain adaptation and generalization. In IEEE International Conference on Computer Vision ICCV, pp. 5716-5726. IEEB Computer Society, 2017b. doi: 10.1109/CCV222017.666 URL http://ss///....r//11.....
CCVV 20 609
Jorge Nocedal and Stephen Wright. Numerical Optimization, chapter 5-7. Springer Science & Business Media, 2006.
Timothy J. O'Shea and Jakob Hoydis. An introduction to deep learning for the physical layer. IEEE Transactions on Cognitive Communications and Networking, 3(4):563 575, 2017, doi: 10. 10.1109/TCCN2201..758877 URLhttps::/ddii crg/10 111.111.11111111 1109/TCCN 27583777
Timothy J. O'Shea, Tamoghna Roy, and Nathan West. Approximating the void: Learning stochastic channel models from observation with variational generative adversarial networks. In International Conference on Computing, Networking and Commmnccations (ICNC), pp. 681-686. IEEE, 2019. doi: 10.1 10.1109/CCC..2011. 8685573. URL https::/doi. org/ www.::......0.00.... 111.
2019. 8685573
Marco Saerens, Patrice Latinne, and Christine Decaestecker. Adjusting the outputs of a classifier to new priori probabilities: A simple procedure. Neural Computation, 14(1):21-41, 2002. doi: 10.1162)
089976602753284446
Kuniaki Saito, Kohei Watanabe, Yoshitaka Ushiku, and Tatsuya Harada. Maximum classifier dis- crepancy for unsupervised domain adaptation. In IEEE Conference on Computer Vision and Put- ter Recognition (CVPR), pp. 3723-3732. IEEE Computer Society, 2018. doi: 10.1109/CVPR. 2018. 00392. URL
Cheenaccessstheev.. thecvf. com/ content. cvpr 2018/html/Satt
Maximum Classifier C CVPR 2018 paper.ntm.
SIVERSIMA. EVK06002 Development Kit, 2020.
https://wwer www.ssiiiaim..mm.mm.. wtt.maaw.......eeeeeeeeeeee.ee hcom/product/
evk - 06002-007
Masashi Sugiyama and Motoaki Kawanahe. Machine Learning in Non-stationary Environments: Introduction to Covariate Shift Adaptation. MIT Press, 2012.
Masashi Sugiyama, Matthias Krauledat, and Klaus-Robert Müller. Covariate shift adaptation by importance
weighted cross validation. Journal of Machine Learning Research, 8:985-1005, 2007. URLhttp://el)
Qianru Sun, Yaoyao Liu, Tat-Seng Chua, and Bernt Schiele. Meta-transfer learning for few-shot learning.
In IEEE Conference on Computer Vision and Pattern Recognition CVPR, pp. 403-412. Computer Vision Foundation / IEEE, 2019. doi: 10.1 10..109//VP.2019....
Tianqi Wang, Chao-Kai Wen, Hanqing Wang, Feifei Gao, Tao Jiang, and Shi Jin. Deep learning for wireless physical layer: Opportunities and challenges. CoRR, abs/1710.05312, 2017. URL/attp://arxiv org/
abs abs/ 00///1110.0533
Lilian Weng. 2018.
Flow-based deep
generative
models.
lilianweng cethewewnngggthhh....e github. io/lil-log
URL
/tlt:://1llaawennn www.tttthhh... wtw.tthh...o.l1leeeee. 1o1.:uooo/o111111101111111110
flow how-bbasd-deepp www.eep-gennrattie---eeeee htmm
William Xia, Sundeep Rangan, Marco Mezzavilla, Angel Lozano, Giovanni Geraci, Vasilii Semkin, and
Giuseppe Loianno. Millimeter wave channel modeling via generative neural networks. In IEEE Globecom Workshops (GC Wkshps), pp. 1-6. IEEE, 2020. doi: 10. 100////Wsspss000.......
20005503000222096772222 URL
9367420
Hao Ye, Geoffrey Ye Li. Biing- Hwang Fred Juang, and Kathiravetpillai Sivanesan. Channel agnostic eud-to-end learning based communication systems with conditional GAN. In IEEE Globecom Workshops (GC Wkshps), pp. 1-5. IEEE, 2018. doi: 10. 100/LL.COM..200 8644250. URL URL http:::tt: /ttp://////0110000000000000
G G1.CCMM
11.COMM..... 2018 88. 8644250
An Zhao, Mingyu Ding, Zhiwu Lu, Tao Xiang, Yulei Niu, Jiechao Guan, and Ji-Rong Wen. Domain-adaptive few-shot learning. In IEEE Winter Conference on Applications of Computer Vision, WACV, pp. 1389 1398. IEFF, 2021. doi: 10.1 109/WACVV6630.0021 10201004.......... URI https: /ectps://do: .rrg 100 1199/AAV486600 2.02 00143
12

---

Under review as a conference paper at ICLR 2022
Han Zhao, Remi Tachet des Combes, Kun Zhang, and Geoffrey J. Gordon. On learning invariant representations for domain adaptation. Proceedings of the 36th International Conference on Machine Learning (ICML), volume 97 Proceedings of Machine Learning Research, pp. 3532. PMLR, 2019. URL tp
//proseodings. mlr, prwps /0nsss/v99/2eee 9a.htm1
13

---

Under review as a conference paper at ICLR 2022
APPENDIX
Table Commonly used notations definitions
Notution
Description
x (1.....90) m)
Inpat message or class label. Usually m 2", where is the number of bits. One-hot-cnded voctor of message with position and zeros elscwhere.
Lom or simply 1, a
xe IR' with [at]
Encoded representation or symbol vector corresponding an input message. be classified by the decoder.
ye
Chunnel output that the fealure
e (1.....
Categorical random variable denoting the mixture componeat origin. Encoder NN with purameters 0, mupping
(4.
hot-ceee coded messuge symbol vector it
(y) (1))) in (m (m)))))
Decoder NN with parameters 0g mapping the channel ouipat into probubiiities over the message set. MAP prediction of input message by decoder. Conditional density (generative) model of the channel with parameters 0.- Mixture density network that
Sty) argaas,s ()
Ja. " x)
((x) (x.
the parameters Gaussian mixture. Transfer or samplin tunction corresponding to the channel conditional density. Random vector independent Input-outgy mapping of the autoencoder with combined parameter vector 0* 10" -0" 0g)
y (x.22
captures stochusticity
channel.
fa(1.)
Da, (Fg (1)
N1 .... [.]
Affine transformation parameters per component used adapt
8a and Sai xxx
Afline and inverse affine transformations between the compoment densities of the Gaussian mixtures. Kullhack--T eibler divergence between distribations Multivariate Gaussian density with mein vector and covariance matrix E. Calegorical distribation with pv Indicator function mapping predlcatte to true and 0 if falke.
DaL g)
q.
N( |1,
E, Pr
H() [x]]
fy norm of vector
The appendices organized follows:
Appendix A provides background on end-to-end learning of a communication autoencoder, MDN-
based generative modeling, and domain adaptation.
Appendix B
provides details on the training and sampling (transfer) function of MDNs.
Appendix provides additional details on the proposed method that were omitted from the main paper. This includes:
Appendix C. discusses the feature and parameter transformations between multivariate Gaus-
sians.
Appendix derives the KL divergence between the source and target Gaussian mixtures.
Appendix discusses the validation metric used for setting the hyper-parameter A.
Appendix
discusses the computational complexity of the proposed methods.
Appendix provides details on training the MAP symbol estimation autoencoder.
Appendix D provides additional experimental results, including ablation studies of the proposed
method.
Appendix provides details on the simulated channel models that were used in our experiments.
A BACKGROUND
In this section, we provide background on the following topics: 1) components of an end-to-end autoencoder-basse communication system, 2) generative modeling using mixture density networks, 3) loss function and training algorithm of the autoencoder, and 4) a primer on domain adaptation.
A.1 AUTOENCODER BASED END-TO-END LEARNING
Milioi demse layers
Norrmal com loger
Rel
Mutiple de noce layors
(1 yr)
Input massago
One not encoetnn
Wirelsse channal
loper
Predicted
S
argmage (a (s)
Ply (x)
Pv. (m
Generalive channel model
Encoder Tranemmiter: meural network (1.)
Decoder Receiver
Ps. (yr [x) y by (x.x)
D2,(r)
Figure 7: Representation of an end-to-end autoencoder-based communication system with generative channel model.
14

---

Under review as a conference paper at ICLR 2022
Consider a single-input, single-output (SISO) wireless communication system shown in Fig. where the transmitter encodes and transmits messages from the set S (1,,2, m) to the receiver through d 2 2 discrete uses of the wireless channel. The receiver attempts to accurately decode the transmitted message from the distorted and noisy channel output y. We discuss the end-to-end learning of such system using the concept of autoencoders (O'Shea & Hoydis 2017 Dörner et al. 2018).
Transmitter / Encoder Neural Network. The transmitter or encoder part of the autoencoder is modeled as a multi-layer, feed-forward neural network (NN) that takes as input the one-hot-coded representation 1, of message E S, and produces an encoded symbol vector x = Eg. (1.) R Here, 0e is the parameter vector (weights and biases) of the encoder NN and d is the encoding dimension. Due to hardware constraints present at the transmitter, a normalization layer is used as the final layer of the encoder network in order to constrain the average power and/or the amplitude of the symbol vectors. The average power constraint is defined as E|| x o Es||| Ee. (1s) S c, where the expectation is over the prior distribution of the input messages, and is typically set to 1. The amplitude constraint is defined as |zy|| S 1, Vi E [d]. The size of the message set is usually chosen to be power of 2, L..., = 2" representing b bits of information. Following 1O'Shea& & Hoydis, (017). the communication rate of this system is the number of bits transmitted per channel use, which in this case is R b/ d. An autoencoder transmitting b bits over d uses of the channel is referred to (d, b) autoencoder. For example, a (2, 4) autoencoder uses a message set of size 16 and an encoding dimension of 2, with a communication rate R 2 bits/channel use.
Receiver / Decoder Neural Network. The receiver or decoder component is also a multilayer, feedforward NN that takes the channel output y IR its input and outputs a probability distri- bution over the m messages. The input-output mapping of the decoder NN can be expressed as De, (y) = (1 (Ily).- ... Pe. (m y))", where 0 is the parameter vector of the decoder NN. The softmax activation function is used at the final layer to ensure that the outputs are valid probabilities. The message corresponding to the highest output probability is predicted as the decoded message, ie., S(y) = argmax,es Po. (s y). The decoder NN is essentially a discriminative classifier that learns accurately categorize the received (distorted) symbol vector into one of the m message classes. This is in contrast to conventional autoencoders, where the decoder learns to accurately reconstruct high-dimensional tensor input from its low-dimensional representation learned by the encoder. The mean-squared and median-absolute error are commonly used end-to-end performance metrics for conventional autoencoders. In the case of communication autoencoders, the symbol or block error rate (BLER), defined as (S)() + S)], is used as the end-to-end performance metric.
Channel Model. As discussed in $ the wireless channel can be represented by a conditional probability density of the channel output given its input P(y x). The channel can be equivalently characterized by a stochastic transfer function y = h(x, z) that transforms the encoded symbol vec- tor into the channel output, where captures the stochastic components of the channel (e.g.. random noise, phase offsets). For example, an additive white Gaussian noise (AWGN) channel is represented by y h(x, 2) = x + z, with ~ NV 0, 0² Ia) and P(y x) = Ny x, o² Ia). For realistic wireless channels, the transfer function and conditional probability density are usually unknown and hard to approximate well with standard mathematical models. Recently, a number of works have applied generative models such as conditional generative adversarial networks (GANs) (O' Sheaa al. 20199 Ye et al.) 2018), MDNs (García Martí al. (220)), and conditional variational autoen- coders (VAEs) (Xia et al. 2020) for modeling the wireless channel. To model wireless channel, generative methods learn a parametric model Po. (y x) (possibly a neural network) that closely approximates the true conditional density of the channel from dataset of channel input, output observations. Learning a generative model of the channel comes with important advantages. 1) Once the parameters of the channel model are learned from data, the model can be used to generate any number of representative samples from the channel distribution. 2) A channel model with a differentiable transfer function makes it possible to backpropagate gradients of the autoencoder loss function through the channel and train the autoencoder using stochastic gradient descent (SGD)-based optimization. 3) It allows for continuous adaptation of the generative channel model variations in the channel conditions, and therehy maintain a low BLER of the autoencoder.
15

---

Under review as a conference paper at ICLR 2022
A.2 GENERATIVE CHANNEL MODEL USING A MIXTURE DENSITY NETWORK
In this work, we use an MDN Bishop 1994 2007) with Gaussian components to model the conditional density of the channel output given its input. MDNs can model complex conditional densities by combining a (egg. mixture of Gaussians). The MDN learns to predict the parameters of the mixture model o(x) as a function of the channel input This can be expressed as Φ(x) Ma,(x), where 0g is the parameter vector (weights and biases) of the neural network. The parameters of the mixture model defined by the MDN are a concatenation of the parameters from the & density components,
neural network with standard parametric mixture model
ie., "(x))" = [x)", Φx(x)"), where (x(x) is the parameter vector of component i. Fo- cusing a Gaussian mixture, the channel conditional density modeled by the MDN is given by
k = = Σ
Po. (x)
Σ
Po. (K =
(x) N (4) of
(13)
where [xx E IRd is the mean vector, (x) E Ied is the variance vector, and (x) E (0, 1] is the weight (prior probability) of component i. Also, K is the latent random variable denoting the mixture component of origin. We have assumed that the Gaussian components have a diagonal covariance matrix, with of (x) being the diagonal elements The weights of the mixture are
parameterized using the softmax function as TT(x) = (x(()) Z==1 E05lx) Vi in order to satisfy the probability constraint. The MDN simply predicts the un un-normalized weights a; (;(x) E IK (also known as the prior logits). For Gaussian MDN, the parameter vector of component i defined as (x(x)² [a,(x), [xxx (x))),, and its output parameter vector Φ(xx has dimension k &(2d+ 1). Details on the conditional log- oolliielihood (CLL) training objective and the transfer function of the MDN, including differentiable approximation of the transfer function, are discussed in Appendix B
A.3 LOSS FUNCTION AND TRAINING OF THE AUTOENCODER
In this section, we provide formal discussion of the end-to-end training of the autoencoder. First. let us define the input-output mapping, of the autoencoder as (1.) = Do, (he, (Eo, (1.), = (Do, he. (-. ) o Eg. (1.), where g² ( 0 07] is the combined vector of parameters from the encoder, channel, and decoder. Given an input message E S, the autoencoder maps the one-hot- coded representation of s into an output probability vector over the message set. Note that, while the encoder and decoder neural networks are deterministic, forward pass through the autoencoder is stochastic due to the channel transfer function he. The learning objective of the autoencoder is to accurately recover the input message at the decoder with high probability. The cross-entropy (CE) loss, which is commonly used for training classifiers, is also suitable for end-to-end training of the autoencoder. For an input & with encoded representation
Eg, (1,), channel output y ho, (x, z),
and decoded output De, (y) = [Pe (1 [y), Pe. (m |')'" the CE loss is given by
(1,,
1 log fl.) = - 1. log Do, (Eg, (1.), z)) log Pa (s ho. (Eo, 21)),
(14)
which is always non-negative and takes the minimum value 0 when the correct message is decoded with probability 1. The autoencoder aims to minimize the following expected CE loss over the input message set and the channel output:
Elea fo(ss)) - Σ p(s) / Po. (y Ee. (1.)) log Po. (s Iy) dy.
m
--E=--
(15)
Here p(s), Vs S is the prior probability of the input messages, which is usually assumed to be uniform in the absence of prior knowledge. In practice, the autoencoder minimizes an empirical estimate of the expected CE loss function by generating large set of samples from the channel
conditional density given each message. Let yr(+) = {y! = he. (Eg, (1.), Zn), n = 1,...,,.. N} denote a set of independent and identically distributed (iid) samples from Pe. (y Ee. (1.)). the
'The diagonal covariance assumption does imply conditional independence of y as long as k
16

---

Under review as a conference paper at ICLR 2022
channel conditional density given message s. Also, let } U,)%+ denote the combined set of samples. The empirical expectation of the autoencoder CE loss (15) is then given by
N
Cauto Ca (0:3) = Σ p(s)
log Pa. (s he. (Ee, (1.), zn)).
(16)
12=1
It is clear from the above equation that the channel transfer function heg should be differentiable in order to be able to backpropagate gradients through the channel to the encoder network. The transfer function defining sample generation for a Gaussian MDN channel is discussed Appendix B
The training algorithm for jointly learning the autoencoder and channel model (based on (Garcia Marti
ct al. 20200) is given in Algorithm [T It is an alternating (cyclic) optimization of the channel
parameters and the autoencoder (encoder and decoder) parameters. The reason this type of alternating optimization is required is because the empirical expectation of the CE loss Eq. (T6) valid only when the channel conditional density (i.e., 0.) is fixed. The training algorithm can be summarized as follows. First, the channel model is trained for Noe epochs using data sampled from the channel with initial encoder constellation (e.g.. M-QAM). With the channel model parameters fixed, the parameters of the encoder and decoder networks are optimized for one epoch of mini-batch SGD updates (using any adaptive learning rate algorithm e.g., Adam (Kingma & Bal 2015)). Since the channel model is no longer optimal for the updated encoder constellation, it is retrained for No epochs using data sampled from the channel with the updated constellation. This alternate training of the encoder/decoder and the channel networks is repeated for Noe epochs or until convergence.
Algorithm 1 End-to-end training of the autoencoder with a generative channel model
1: Inputs: Message size m; Encoding dimension d; Initial constellation (Eo(1.), Vs E S); Number of optimization epochs for the autoencoder Nue and channel Nee.
2: Output: Trained network parameters 0.. θ.. 0
3: Initialize the encoder, channel, and decoder network parameters.
4: Sample training data ! from the channel using the initial constellation.
5: Train the channel model for No. epochs to minimize (0.; )
6: for epoch = 1,.... Naa:
7: Freeze the channel model parameters 0..
8: Perform a round of mini-batch SGD updates O. and 0₂ with respect to Lanto(0;
9: Sample training data ( from the channel with the updated constellation (Eg, (1,), Vs E S).
10: Train the channel model for Noe epochs to minimize (o D(
11: Return te, 0c, O2.
Finally, we observe some interesting nuances of the communication autoencoder learning task that is not common to other domains such as images. 1) The size of the input space is finite, equal to the number of distinct messages m. Because of the stochastic nature of the channel transfer function, the same input message results in a different autoencoder output each time. 2) There is theoretically no limit on the number of samples that can be generated for training and validating the autoencoder. These two factors make the autoencoder learning less susceptible to overfitting, that is a common pitfall with neural network training.
A.4 A PRIMER ON DOMAIN ADAPTATION
We provide brief review of domain adaptation (DA) and discuss the key differences of our problem setting from that of standard DA. In the traditional learning setting, training and test data are assumed to be sampled independently from the same distribution P(x, y), where x and y are the input vector and target respectively 0 In many real world settings, it can be hard or impractical to collect a large labeled dataset Dÿ for target domain where the machine learning model (e. g.g DNN classifier) is to be deployed. On the other hand, it is common to have access to a large unlabeled dataset DV
TThe notation used in this section is different from the rest of the paper, but consistent with the statistical
learning literature.
17

---

Under review as a conference paper at ICLR 2022
from the target domain, and large labeled dataset D from different but related source domain Both D', and Dj are much larger than Di, and in most cases there is no labeled data from the target domain (referred to unsupervised DA). For the target domain, the unlabeled dataset (and labeled dataset any) are sampled from an unknown target distribution, i.e., x E D ~ Px(x) and (x E Di (x(x. y)- For the source domain, the labeled dataset is sampled from an unknown source distribution, i.e., (x, y) E D ~ (x, y) The goal of DA is to leverage the available labeled and unlabeled datasets from the two domains to learn predictor, denoted by the parametric function ÿ (x), such that the following risk function w.n.t the target distribution is minimized:
[e] = B(x,)) (([oxx)) = Σ / Pi(x, ((fo(x),) dx,
where E(j. 3) is a loss function that penalizes the prediction ÿ for deviating from the true value y (e.g., cross-entropy or hinge loss). In a similar way, we can define the risk function w.r.t the source
distribution [o) A number of seminal works in DA theory (Ben-David et al., 2006: Bliizer et al. 2007 Ben-David et al. 2010) have studied this learning setting and provide bounds on Re in
terms of [o] and the divergence between source and target domain distributions. Motivated by
this foundational theory, a number of recent works (Ganin & Lempitsky [2015] Ganin et al. 2016
Long et al.] 2018 Saato et al. [0188] Zhao et al.] 10119Johansson et al. (019)) have proposed using DNNs for adversarially learning a shared representation across the source and target domains such
that a predictor using this representation and trained using labeled data from only the source domain also generalizes well the target domain. An influential work in this line of DA is the domain adversarial neural network (DANN) proposed by (Ganin & Lempitsky (015)) and later by (Ganin al. (016). The key idea behind the DANN approach is to adversarially train a label predictor NN and domain discriminator NN in order to learn a feature representation for which i) the source and target inputs are nearly indistinguishable to the domain discriminator, and ii) the label predictor has good generalization performance on the source domain inputs.
Special Cases of DA. While the general DA problem addresses the scenario where (x y) and (x(x.u) y) are different, certain special cases of DA have also been explored. One such special case covariate shift (Sugiyama et al., 2007]] Sugiyama & Kawanabe, 2012). where only the marginal
distribution of the inputs changes (i.e., (x (x) P. (x)), but the conditional distribution of the target
given the input does not change (i.e., ( x) Re P. (y x)). Another special case is the so-called
label shift class-prior mismatch (Saerens et al. 2002 Du Plessis & Sugiyama 2014), where only
the marginal distribution of the label changes (i.e., Pr( f Ps(y)), but the conditional distribution
of the input given the target does not change (i.e., (x yy) Rs Px(x (1)). Prior works have proposed
targeted theory and methods for these special cases of DA.
B MDN TRAINING AND SAMPLE GENERATION
In this section, we provide details on the MDN training, followed by discussion on the sampling function of an MDN, and how to make the sampling function differentiable to enable backpropagation- based training of the autoencoder. Given a dataset of input-output pairs sampled from the channel D. = {xx,, Yn), n = I.... Ne). the MDN is trained to minimize the negative conditional log- likelihood (CLL) of the data given by
N
(o D.) = No
1
Σ
log Po. (yn (x))
(17)
n=1
With a large No. the MDN can learn a sufficientl-ccomplex parametric density model of the channel. The negative CLL objective can be interpreted as the sample estimate of the Kullback- divergence between the true (unknown) conditional density P(y x) and the conditional density modeled by the MDN Po. (y x). Therefore, minimizing the negative CL. finds the MDN parameters 0. that lead to a close approximation of the true conditional density. Standard SGD--based optimization methods such as Adam (Kingma & Ba 2015) can be applied to find the MDN parameters O. that (locally) minimize the negative CLL.
After the MDN is trained, new simulated samples from the channel distribution can be generated from the Gaussian mixture using the following stochastic sampling method:
One could have multiple source domains in practice; we consider the single source domain setting.
18

---

Under review as a conference paper at ICLR 2022
1. Randomly select channel input from the categorical prior distribution {p(x), x 2).
Randomly select a component K' = according to the mixture weights { (x). Tx (x)).
3. Randomly sample from the standard d-dimensional Gaussian density z ~ N:: (.. Ia).
4. Generate the channel output as = of @ z + [x(x).
Recall that refers to the element-wise product of two vectors. The channel transfer or sampling function for a Gaussian MDN can thus be expressed as
= he, (x, =
Σ 1(K = (x) O2 (4((x),
(18)
where K Cat(TT (x),..., (x)) and
(- O, Note that this transfer function is
differentiable w.r.t parameters T. (x) and the MDN weights predicting it, because of the indicator function. As such, it is not directly suitable for SGD (backpropagation) based end-to-end training of the autoencoder. We next propose a differentiable approximation of the MDN transfer function based on the Gumbel softmax reparametrization (Jang et al. 2017), which is used in our autoencoder implementation.
B.1 DIFFERENTIABLE MDN TRANSFER FUNCTION
Consider the transfer function of the MDN in Eq. 18 We would like replace sampling from the categorical mixture prior Ca)(m) ... (x]) with differentiable function that closely approximates it. We apply the Coumell-Softmal reparametrization (Jang et al., 2017) which solves this exact problem. First, recall that the component prior probabilities can be expressed in terms of the prior logits as:
(x:
(x)
eas(x)) E [A]].
Dj=1
Consider & iid standard Gumbel random variables GI, , id Gumbel(O, 1). It can be shown that, for any x E X, the random variable
S(x) arginax G3 + a (x) iE[k]
(19)
follows the categorical distribution Cat( (T (x).... & (x)). This standard result is known as the Gumbel-max transformation. While Eq. (19) can be directly used inside the indicator function in Eq. (18), the argmax will still result in the transfer function being non-differentiable. Therefore, we the following temperature-scaled softmax function as smooth approximation of the argmax
expl(G, + (x)))//]]
(x(x)) Σ exp + (=( Vi Vi E [k],
(20)
where r >> O is temperature constant. For small values of r, the temperature-scat softmax will closely approximate the argmax, and the vector [S,(x (x; (x will closely approximate
the one-hot vector [I()(x) = 1), 1(S(x) k)].
Applying this Gumbel softmax reparametrization in Eq. 18 we define a modified differentiable transfer function for the Gaussian MDN as
y he. (x,z)
S((xT) oz + (x))...
(21)
With this transfer function, it straightforward to compute gradients with respect to the prior logits (x),, Vi. Another neat outcome of this approach is that the stochastic components (Gumbel random variables G.) are fully decoupled from the deterministic parameters a; (x(x) in the gradient calculations with respect to (; r). In experiments, we used this Gumbel-softma based smooth transfer function while training the autoencoder, but during prediction (inference), we use the exact argmax based transfer function. We found 0.01 to be good choice for all the experiments.
C ADDITIONAL DETAILS ON THE PROPOSED METHOD
In this section we provide additional details on the proposed method that were omitted from the main paper (sections 3 and 4).
19

---

Under review as a conference paper at ICLR 2022
C.I TRANSFORMATION BETWEEN MULTIVARIATE GAUSSIANS
We discuss the feature and parameter transformations between any two multivariate Gaussians. This result was applied in $ to formulate the MDN adaptation. Consider first the standard transformation from y ~ N- μ4 Σ) to y ~ N(: p. Σ) given by:
Apply a whitening transformation z = D 1/2 UT (y (1) such that z ~ N(.10,I).
Transform z into the new Gaussian density using ý = UD!! 1/2 + j..
We have denoted the eigen-decomposition of the covariance matrices by E = UDU and Σ ÜDÜT, where U and Ú are the orthogonal eigenvector matrices, and D and D the diagonal eigenvalue matrices. Combining the two steps, the overall transformation from to y is given by
y = 1D2 D 1/2 U (y - 11) + p.
then is easily verified that the covariance matrices are related by Σ = CECT. In general, the mean vector and covariance matrix of any two Gaussians can be related by the following parameter transformations:
Suppose we define the matrix C = -
n + b and Σ CEC
with parameters A E RRddd b E Rd, and C E Rddd, Substituting the above parameter transforma- tions into the feature transformation, we get
= C - n) + Ap+b.
C.2 DIVERGENCE BETWEEN THE SOURCE AND TARGET GAUSSIAN MIXTURES
Referring provide detailed derivation of the KLD between the source and target
3.2
Gaussian mixtures under the assumption of one-to-one association between the components.
Ps. (y, K |x) P (y, x
D. Po.) EPor
log
k
Po. (y, K = =i(x)
= Σ p(x) Σ
Pa. ky,
= == i log
dy
Pa. (y, K = i i
XEX
1= Rd
Pa. (K ilxx) / (xl, log
Po. (K = il x) P 0.
(y
x, K ii
Σ p(x)
Σ
x, =) dy
P (K
x) P (y
xEX
(x( [oog (x)
N (y 11 (x) ou (x))
Σ XEX
p(x)
Σ
(x)
N (y)lm(()), ox())
+ log
dy
N (y pr, (x). /3 (x))
1=1
Ra
log Ti(x)
Σ
p(x)
(x)
(x)
XEX
Σ XEX
p(x)
(x) DEL. (N (xx) N (1
(22)
1=1
The second term in the final expression involves the KLD between two Gaussian densities, which (for general covariances) is given by
1 det E
DKL. (N(122))N((1277
log
- Σ)
2
det E)
d
1) E - (īa 4)) 2
20

---

Under review as a conference paper at ICLR 2022
Applying this result to the KLD term in Eq. (22
which has diagonal covariances, get
DEL (N
(-1,
(x)) N
7. (x), (x()))
Σ
log Gij
+
z +
1
(ags Mey (x) + baj Mr)(x))
d
(23)
oz)(
2
j=1
The other term in Eq.
(22) involving the KLD between the component prior probabilties can be expressed as a function of the adaptation parameters [81, 71, ... 18,, 7k] as follows:
(x)
k
(x))
(x)
(xx)
(x)
log
Σ
log
log
(x)
x(x)
2(x)
2(x)
1=1
a=1
(x))
log(
3 ax(x)) +-
log( (x(()))
(a:(x) Bx)s()) 76); (24)
x(x)
where =(x) شソー (x)) and 3(x)
+ are the normalization terms in the softmax function. Substituting Eqs. (23) and 24 into the last step of Eg. (22) gives the KLD between
the source and target distributions as a function of the adaptation parameters 1/.
C.3 VALIDATION METRIC AND SELECTION OF
The choice of À in the adaptation objective (Eqs. and (8)) is crucial it sets the amount of regularization most suitable for the target domain distribution. We propose a validation metric for selecting À based on the CLL of the inverse-affine cuffine-transformet target dataset with respect to the source mixture density. The reasoning is that, if the adaptation finds a solution zó that is a good fit for the target dataset, then the inverse feature transformations based that solution should produce a transformed target dataset that has a high CLL with respect to the source mixture density. The validation metric is the negative CLL of the inverse-transformed target dataset, given by
va ;b;
log Po. ( (g (y((9) (()..
(25)
N
Here (00) is the best component assignment for the sample (x/ (t) yy!"), given by
argmax kE(k]
Pa. (K i (t) (!)).
(26)
The above equation is simply the maximum-a- posterrioi (MAP) rule applied to the component posterior of the target Gaussian mixture defined as
(x) ( (.) (x)) Pa. (K (K= i|x,)) Σラ (x) P (x), (x))
Vi [A).
Note that the validation metric (25) is based on the source Gaussian mixture (with parameters 9z). but the MAP component assignment for each target domain sample Eq. (26) is based on the target Gaussian mixture (with parameters O.). The adaptation objective is minimized with À varied over range of values, and in each case the adapted solution z evaluated using the validation metric. The pair of X and d resulting in the smallest validation metric is chosen as the final adapted solution.
C.4 COMPLEXITY ANALYSIS
We provide an analysis of the computational complexity of the proposed adaptation methods.
MDN Adaptation.
The number of free parameters being optimized the MDN adaptation objective (Eqs. 7 or is given by لقا A (d2 + d + 2). This is much smaller than the number of parameters in typical MDN, even considering only the final fully fully-connected layer (see Table a comparison). Each step of the BFGS optimization involves computing the objective function, its gradient, and an estimate
21

---

Under review as a conference paper at ICLR 2022
of its inverse Hessian. The cost of one step of BFGS can thus be expressed as O(N dd | (bb)))) Suppose BFGS runs for a maximum of T iterations and the optimization is repeated for L values of X, then the overall cost of adaptation is given by O(L No d |b))). Note that the optimization for different À values can be easily solved in parallel.
Test-time adaptation at the decoder.
The computational cost of the proposed adaptation methods (for processing a single input) at the decoder are only slightly different from each other. The affine transformation method of $4.1 computes the posterior distribution Pg. (x, i Iy) over the set of symbols and components (of size m k). Computation of each exponent factor in the posterior distribution requires ((d*) operations for the full-covariance case, and ((d) operations for the diagonal covariance case. This corresponds to calculation of the log of the Gaussian density. Therefore, computation of the maximum of the posterior distribution requires O (I: m (k m dd)) = O (k² m² d³) operations for the full-covariance case (and similarly for the diagonal case).
The MAP symbol estimation method of $ 4.2 computes the posterior distribution Pg. (x y) over the set of symbols (of size m). Computation of each exponent factor in the posterior distribution requires O d operations for the full covariance case, and O d) operations for the diagonal covariance case. This corresponds to calculation of the log of the Gaussian mixture density with k components. Therefore, computation of the maximum of this posterior distribution requires O (m (m k d²)) = O (k m² d²) operations for the full-covariance case.
Therefore, the computational complexity of the affine transformation method larger than that of the MAP SE method by factor of k. Usually k is not large. However, the computational cost of both methods can be relatively high for large modulation orders (e.g., m 256 or 1024 symbols).
A key advantage of the proposed adaptation is that it is very computationally efficient to implement at the receiver end of communication system. Typically, d is very small for communication problems, and d = 2 coincides with standard modulation techniques (e.g., QPSK). The number of symbols or messages m can vary from to 1024 in powers of The number mixture components k can be any positive integer, but is usually not more than a few tens to keep the size of the MDN practical.
C.5 MAP SYMBOL ESTIMATION AUTOENCODER
In this section, we discuss some details of the adapted decoder with MAP symbol estimation that were not addressed in 4.2 The MAP SE method for adapting the decoder introduced the following transformation layer prior to the decoder in order to estimate the channel input x from the channel output
map (y) argmax Pa. (x XEX
argmax log Pe. (y x) + log p(x).
xEX
As discussed in $ the presence of the non-differentiable argmax poses a problem for mackpropagation-boe hased training of the autoencoder. We propose address this using a temperature- scaled softmax approximation to the argmax, similar to the method discussed in Appendix B.T
Consider the posterior distribution of the channel input given the channel output
p(x) Po. (y x)
exp(9e. (x,y) Exex exp(ge. (x', yy)"
Po. (x = Σx Ex'ex p(x') Po. (y x')
(27)
where 9a. (x,y) log Po. (y x) + log is defined for convenience. Let us introduce temperature constant T > in the softmax function, and define the temperature-scaled posterior
distribution
exp(yo, (x, x),
(x Exex explyo. (x)
(28)
For large t, the above posterior approaches uniform distribution. For small - it approaches distribution with probability for x corresponding to the maximum exponent, and Os elsewhere. Based on this observation, we define the following smooth approximation of the MAP SE layer
X xx)
Σ P(7) (xly)x
(29)
X
22

---

Under review as a conference paper at ICLR 2022
This can be interpreted the conditional expectation of x given y with respect the temperature- scaled posterior distribution (28). We can show that this smooth MAP estimate approaches the true MAP estimate in the limit as T approaches 0, i.e.,
lim Σ Pe (xly)x=
argmax Po. (xlx)
XEX
XEX
Training Based on Temperature Annealing.
The MAP symbol estimation autoencoder uses the smooth MAP estimate (29) during training, and the exact MAP estimate during inference. In order to have good convergence and prevent the training from getting stuck at poor solutions, we do not fix the temperature T to a small value throughout. temperature annealing schedule during training. Specifically, is initialized to reasonably large value (e.g., T 11,, and it is decreased by an exponential factor r E (0, 1) at the end of every r 2 1 epochs. The solution (autoencoder parameters) at the end of epochs for the current temperature is used to initialize the training at the next lower temperature. This process is continued until a small final temperature T, is reached. In our experiments, we set the constants related to temperature annealing follows: T; 1, T 0.05, 10, 17 = 0.7169. This choice of 1 ensures that there are 10 temperature steps including the initial and final values.
Instead, we decrease т according
D ADDITIONAL EXPERIMENTS
D.1 EXPERIMENTAL SETUP
We implemented the mixture density network and communication autoencoder models using TensorFlow (Abadi et al.) (0155) and TensorFlow Probability. We used the BFGS optimizer implementation available in TensorFlow Probability. The code base for our work can be found at https:
Table 4: Architecture of the Encoder, MDN channel, and Decoder neural networks. FC - fully connected (dense) layer; @ denotes layer concatenation; ELU exponential linear unit; number of messages; d encoding dimension; k number of mixture components; Tu size of a hidden layer.
Network
Layer
Activation
All the experiments were run on a Macbook Pro laptop with GB memory and 8 CPU cores. Table summarizes the architecture of the encoder. MDN (channel model), and decoder neural networks. Note that the output layer of the MDN a concatenation (denoted by +) of three fully-connected layers predicting the means, variances, and mixing prior logit parameters of the Gaussian mixture. The size of the hidden layers 74 was set to 100.
domain haw.iiaaadppaatoo--o000oa
FC, m x 12
ReLU
Encoder
FC, nh x d
Linear
Normalization (avg. power)
None
FC, ni.
ReLU
MDN
FC, ma X vig.
ReLU
FC, my x (means)
Linear
@ FC, ny, x kdd (variances)
ELU + 1 1++
@ FC, R2 x & (prior logits)
Linear
FC, d 74.
ReLU
Decoder
FC, na x m
Softmax
The parameters ab of the proposed adaptation method are initialized as follows for each component d:
A, La, b, 0, C; Ia. 8 1, 7 O,
where La is the d x d identity matrix. This initialization ensures that the target Gaussian mixture is always initialized with the source Gaussian mixture. The regularization constant in the adaptation objective was varied over 16 equally-spaces values on the log-scale (base 10) with range 10 to 100. The À value and zb corresponding to the smallest validation metric are selected as the final solution.
We used the Adam optimizer (Kingma & Ba 2015) with fixed learning rate of 0.001, batch size of 128, and 100 epochs for training the MDN. For adaptation of the MDN using the baseline methods transfer and transfer-last-layer Adam is used with the same learning rate for 200 epochs. The batch
size is set as = max(10, 10, 0.1 !N!") where N:!") is number of adaptation samples in the target domain. For training the autoencoder using Algorithm[] we found that stochastic gradient descent (SGD) with Nesterov momentum (constant 0.9), and an exponential learning rate schedule between 0.1 and 0.005 works better than Adam.
23

---

Under review as a conference paper at ICLR 2022
D.2
DETAILS ON THE FPGA EXPERIMENT
Referring to the experiment in $5.3 for the real and over-the-air traces we used the platform from Lacruz al. (2021). This illra-wide-band mm-wave transceiver baseband memory-based design is developed on top of an ZCU111 RFSoC FPGA. This evaluation board features a Zynq Ultrascale + ZCU28DR, This FPGA is equipped with 8 AD/DA converters with giga-sampling capabilities. which make it ideal for RF system development; the 4 GB DDR4 memories contain RF-ADCs with up to 4 GSPS of sampling rate, and RF-DACs with up to 6.544 GSPS. This board also includes quad-core ARM Cortex-A53 and dual-core ARM Cortex-RS R5 real-time processor.
For the radio frequency, we used 60 GHz RF front-end antennas. These kits include 16 + 16 TRX patch array antenna plus the RF module with up/down conversion from baseband to I/Q channels, and TX/RX local oscillator (LO) frequency control. The antennas use 57 71 GHz, range of frequencies that cover the unlicensed 60 GHz band for mm-wave channels, and are managed from PC Host via USB.
15-
X
3.0
0.3
0.
00-
-05
-05-
-05
-10
-15
-15
-0.5
10
15
1.0
05
05
10
25
2.5
-1.0
0.3
(a) AWGN channel.
(b) Uniform fading channel.
(c) Riccan fading channel.
Figure 8: Plot of different simulated channel models with a 16- 16-QAM constellation as the channel input for an SNR of 14 dB.
Pua
ctegle.
1.02
1.00
A4
097
0.00
510
-135
-0,2
-575
-052
-0.00
-150
-3.00 -0.55 -0.0
+0.35 0.0
0.5
050 0.5
3.00
-0.55 -059
-0.5
0.25 035
133
-1.0 -2.55
10.90
-229
0.00
0.30
3.55
1.00
(a) AWGN channel.
(b) Uniform fading channel.
(c) Ricean fading channel.
Figure 9: The symbol constellation learned by the autoencoder on different channel distributions. Starting from the 16-QAM constellation (red circles), the autoencoder learns custom constellation (black triangles).
We implemented hardware on the loop training. For the experimentation on real traces, we use Matlab as a central axis. The PC host running Matlab is connected to the platform via Ethernet. The FPGA can transmit different custom waveforms like 16-QAM frames from the 802.1 aa and 802.11ay standards, with 2 GHz of bandwidth. The frames are sent over-the-air via 60 GHz radio frequency kits, and the samples are stored at the FPGA DDR memory. We decode the received data from the transmission, removing the preamble and header fields and extracting the symbols to train the MDN. We add preamble to the generated constellation from the MDN for packet detection purposes, and transmit again the new waveforms ever-the-air. Finally, the adaptation is performed offline with the decoded symbols from the custom autoencoder-lcamme constellation.
Source and Target Domains.
For the experiment in $5.3 we introduced distribution changes via IQ imbalance based distortions to the symbol constellation, and evaluated the adaptation performance as a function of the level of imbalance. The source domain would be the original channel, the over-ther the-air link between the transmitter and receiver which the training data collected. This source domain data is used
24

---

Under review as a conference paper at ICLR 2022
for training the channel and the autoencoder. The target domain would be modification of the source domain where the symbols used by the transmitter (c.g.. 16-QAM) are distorted by modifying the in ih-phase and pnndraturephhase (IQ) components of the RF signal. This causes change in the distribution observed by the receiver (decoder), leading to drop in performance without any adaptation.
D.3 COMPARISON WITH BENCHMARK RETRAINED AUTOENCODER
For the experiment setting of $5.2 we include an additional benchmark comparison with a fully- retrained autoencoder. The retrained autoencoder's performance would be the best-case achievable performance by any adaptation method, given that retraining uses plenty of labeled training data and trains for much longer. Note that when we say the autoencoder is retrained using Algorithm T implied that the MDN channel model is also retrained.
Recall that for the adaptation methods, the autoencoder is trained once using channel data from a source distribution A, at an SNR of 14 dB. The adaptation methods use small target dataset from a target distribution B, whose SNR is varied from 8 dB to 20 dB. The following protocol used for retraining the autoencoder. Suppose we are adapting from source distribution A to target distribution B, the autoencoder is retrained only on data from the target distribution B for each SNR from 8 dB to 20 dB. We used 25, 000 samples for training the MDN channel and 300, 000 samples for training the autoencoder. The MDN channel is optimized for 100 epochs during a single outer epoch of optimizing the autoencoder (encoder and decoder networks). The iterative training of the autoencoder and MDN channel is repeated for 20 outer epochs (which we found to be sufficient for convergence).
The performance comparison with a retrained autoencoder for the proposed adaptation methods i) affine transformation and ii) MAP symbol estimation are given in Fig. 10 and Fig. respectively. Note that the plots for the proposed and baseline methods are exactly the same as in Fig. and Fig. 5 Only the retrained autoencoder (labeled "Retrained autoenc") has been added to the plots. As one might expect, the retrained autoencoder has significantly lower BLER than the other methods. In scenarios such as adaptation from AWGN to Uniform fading, the distribution change can be so large (see Fig. (8) that it may not be sufficient to just adapt the channel model in order to close the gap. It may be necessary to also optimize the encoder's constellation and the decoder for the target channel distribution.
Figure 9 shows the optimal constellation learned by the autoencoder (in black) for different channel distributions at 14 dB SNR. The constellation for Uniform fading resembles M-PSK (phase-shift keying) modulation, while the constellation for AWGN is very different. This implies that the optimal constellation for an AWGN channel is very different from that of Uniform fading channel.
Aftin altra dey
6.11
Elch Dre Aae
1200
150022
L065
Homie 2000
(a) AWGN to Ricean fading.
(b) AWGN to Uniform fading.
(c) Ricean fading to Uniform fading.
Figure 10: Comparison of the affine transformation based adaptation to a fully-retrained autoencoder.
D.4 COMPARISON BETWEEN THE PROPOSED DECODER ADAPTATION METHODS
In this experiment we compare the performance of the proposed decoder adaptation methods in
4.1 and $4. on simulated channel variations. Figures 12 13 and 14 compare the performance
of the two methods different source/target channels using 20, 40, and 60 samples per symbol for adaptation. We observe that in most cases, the MAP SE method outperforms the affine transformation based method. This could be due to the fact that MAP SE has an optimal transformation from the
25

---

Under review as a conference paper at ICLR 2022
0.42
00:08
14645
33642
18:24
3340M
(a) AWGN Ricean fading.
(b) AWGN Uniform fading.
(c) Ricean fading to Uniform fading.
Figure 11: Comparison of the MAP SE based adaptation to fulla-retrained autoencoder.
channel output y to the most probable input x, which better compensates for changes in the channel distribution.
St, M000.1
PAP 2e, shi2440
(a) samples per symbol.
(b) 40 samples per symbol.
(c) 60 samples per symbol.
Figure 12: Comparison of the affine transformation and MAP SE adaptation methods when the source channel is AWGN and the target channel is Uniform fading. 16-QAM is included simply as reference.
16-0MM
345
8:00
4009
0.00
2008
02090
30000
03.05
04
24
12
(a) 20 samples per symbol.
(b) 40 samples per symbol.
(c) 60 samples per symbol.
Figure 13: Comparison of the affine transformation and MAP SE adaptation methods when the source channel is AWGN and the target channel is Ricean fading.
D.5 PERFORMANCE UNDER No DISTRIBUTION CHANGE
We studied the performance of the MDN adaptation method when the source and target domains are the same, and compared with the two baselines using 50 randomly generated Gaussian mixture datasets. Figure 15 illustrates one such randomly generated training, adaptation, and test data set (the source and target distribution are not the same in this figure). The results in terms of conditional ligeliikl are shown in Table S (second line). Compared to the results when the source and target domains are different, baselines when the source and target domains. This demonstrates that the proposed MDN adaptation is capable of handling well scenarios where the distribution does not change, changes minimally.
find that method has significant better performance compared to the
26

---

Under review as a conference paper at ICLR 2022
34
0 Cree Phae
Sgraltz--diee rato
Signalte--
(a) samples per symbol.
(b) 40 samples per symbol.
(c) samples per symbol.
Comparison of the affine transformation and MAP SE adaptation methods when the source channel is Ricean fading and the target channel is Uniform fading.
Figure 14:
Training Data
Adaptation Data
Test Data
1.0
0.5
0.0
-0.5
-1.0
Figure 15:
Training, adaptation, and test dataset generated from different random Gaussian mixtures for the source and target domains.
D.6 PERFORMANCE WHEN THE NUMBER OF COMPONENTS ARE MISMATCHED
We also studied how much impact is introduced if the number of components is mismatched in the ground truth source and target Gaussian mixtures. Using 50 datasets, these results are summarized in Table (third line). Comparing the results with that of the case when the number components are matched, we find that although the performance is slightly worse than the case when the number of components are matched, the performance drop is not significant.
Gaussian mixture
D.7 SELECTION OF THE HYPER-PARAMEN PARAMETER
We investigated the sensitivity of the proposed automatic selection of À (using the validation metric) by comparing the performance of automatic selection with that of using different fixed values. These results are summarized in Table 6 Note that the automatic selection searches over 16 log-spaced values ranging from le-5 to 100. From the table, we observe that the conditional log- ligeeliloood corresponding automatic selection of À is sometimes worse than that of other fixed values of X. However, the results are better than those when is 0 (except for Ricean fading to uniform fading). Therefore, by setting non-zero X, the proposed adaptation method increases the final conditional
27

---

Under review as a conference paper at ICLR 2022
Table 5: Conditional log- log-likelihood of the MDN adaptation methods on test data from the target domain. The source and target domain datasets are generated from random, class-conditional Gaussian mixtures. Same G.M refers the special case where there is distribution change. For the case #components mismatched, the number of components in the source and target Gaussian mixtures different.
Source / Target
#Target samples
Proposed
Transfer
Transfer-last-layer
distribution
median
95% CI
median
median
95% CI
80
0.78
(-0.44, 1.07)
-2.45
(-5.83, -0.49)
-2.87
(-7.14, -0.52)
160
1.16
(0.86. 1.37)
0.80
(-0.01, 1.34)
0.66
(-0.28, 1.06)
Different GMMs
240
1.30
(1.05, 1.51)
1,41
(0.98, 1.63)
1.32
(1.01, 1.58)
320
1.44
(L.15, 1.68)
1.63
(1.31, 1.79)
1.57
(1.33, 1.76)
Same GMM
160
2.11
(1.94, 2.31)
0.91
(0.06, 1.39)
0.54
(-0.53, 1.26)
Different GMMs,
#component mismatched
160
0.99
(-0.83, 1.36)
0.61
(-0.58, 1.18)
0.40
(-0.83, 1.04)
Table 6: Conditional log- lig-likelihodd of the proposed MDN adaptation for different fixed values of the asper-parameter X. This is compared with the automatic selection of À based on the validation metric.
Source channel
Target channel
À values
Proposed
median
95% CI
0
-0.47
(-2.21, 0.48)
le-5
-0.87
(-2.26, -0.47)
le-4
-0.06
(-1.30, -0.56)
le-3
-0.29
(-1.90, 0.64)
AWGN
Uniform fading
le-2
0.17
(0.79,0.611
le-1
-0.12
(-1.37, 0.59)
1
-0.74
(-1.97,0.03
auto
-0.29
(-1.97, 0.588
0
1.21
(1.02, 1.38)
le-5
1.23
(1.03, 1.37)
le-4
1.21
(1.02, 1.37)
le-3
1.26
(1.01,
AWGN
Ricean fading
le-2
1.25
(1.10, 1.35)
le-1
1.30
(-1.06, 1.38)
1
0.87
(-0.18, 1.29)
auto
1.24
(0.80, 1.38)
0
-0.14
(-1,89, 0.64)
le-5
-0.14
(-1.66, 0.61)
le-4
-0.30
(-1,52, 0.66)
le-3
-0.12
(-1.36, 0.62)
Ricean fading
Uniform fading
le-2
0.12
(-1.08, 0.65)
le-1
-0.11
(-0.87, 0.70)
1
-0.54
(-1.43, 0.53)
auto
-0.41
(-4.24, 0.70)
log-likelihnn likelihood most cases. A more extensive study exploring some alternate methods for selecting À would be beneficial.
E SIMULATED CHANNEL VARIATION MODELS
We provide details of the mathematical models used to create simulated channel variations in our experiments. These models frequently used in the study of wireless channels (Goldsmith) 2005).
28

---

Under review as a conference paper at ICLR 2022
E.1 UNIFORM FADING MODEL
The channel output y E R' for this model as a function of the channel input (modulated symbol vector) Rd is given by
Ax + n,
where A ~ Unif(O, a) is uniformly-distribun Gaussian noise vector. Both A and n are assumed to be independent of each other and x. The average power in the signal component of y is given by
scale factor, and ~ (- 0, og Ia) is an additive
Pang
A B[I
2
Pang
where Pang denotes the average power in the channel input The noise power in this case is given by ||||||| o- The signal-to-noise ratio (SNR) for this model is therefore given by
Es
E A |
a² Pavg
No
R
6 R
where the communication rate of the system in bits/channel use. We select the fading factor such that the channel output has a target SNR value using the following equation:
6 R (Eg/No)
(30)
Pavg
E.2 RICEAN AND RAYLEIGH FADING MODELS
The channel output for the Ricean fading model is given by
Ax+n,
where A is diagonal matrix with the diagonal elements ai, ~ Rice(- v, ).) following
sid
Rice distribution, and ~ (- 0, o Lạ) is an additive Gaussian noise vector. is assumed that and A are independent of each other and of x. Note that Rayleigh fading is special case of Ricean fading when the parameter / For this model, the average power in the signal component y given by
Pang = Bl| A x
Σ
Ela ag)
Σ
Bla B[A
(2 12) 1 (20 + +1²) Pang:
where Pavg denotes the average power in the channel input x. We used the fact that the second moment of the Rice distribution is given by E[a²] 2 + v². is useful to consider the derived parameters K 22 22 which corresponds to the ratio of power along the line-of-sight (LoS) path to the power along the remaining paths, and Ω 2 + y/² which corresponds the total power received along all the paths. The SNR for this model is given by
Es =
A
(203 + +2)
No 2 RE
2 2Ro
For given input average power and target SNR, the parameters of the Rice distribution can be set using the equation
2 2Ro (Eo/No)
2 20
2
Pavg
To create channel variations of different SNR, we fix the variance o² and vary the power of the LoS component v², Suppose the smallest SNR value considered is Smin- we set of using
2 Rog Smin
2 202
(31)
Pang
29

---

Under review as a conference paper at ICLR 2022
and set to achieve target SNR Es/No using
v² 2 Roo (Es/No Smin)
(32)
Pivg
For this choice of parameters, the power ratio of LoS non-LoS components is given by
/ Nu
K
1.
Smin
The K-factor for Rician fading in indoor channel environments with an unobstructed line-of-sight is typically in the range dB to 12 dB (Linnartz, 200T). Rayleigh fading is obtained for K = 0 (or 0).
Finally, note that the vector x is composed of one more pairs of in-phase and quadrature (IQ) components of the encoded signal (dimension can be expressed as 2m). Since each IQ component is transmitted
single RF signal, the Ricean amplitude scale is kept the same for
successive pairs of IQ components in x. In other words, the amplitude scales are chosen to be
a1, "1 am, am- This does not change any of the above results.
30


Reviews:
[{"review_id": "ixACQoF8Hm9", "reviewer": "Reviewer_9VpG", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "1. The authors only mentioned adapting the MDN model in the abstract and introduction. Readers will be confused why it can be faster than retraining the autoencoder. They revealed that they only train a very small adaptation layer rather than the MDN model on page 3. In my opinion, they should put figure 1 on page 1 to avoid confusion about the speed.\n\n2. When discussing the shortcoming of existing solutions, the author mentioned that the rapidly changing channel leaves not enough time to collect the data to train the autoencoder. But they have the same problem here, they only have a tiny dataset to train the adaptation layer, why overfitting will not happen if they only have such a small dataset?\n\n3. Where does the small labeled dataset come from, and how often to recollect such a dataset and re-run the adaptation?  As the authors mentioned, the channel is changing rapidly, if the channel has changed, the adaptation layer needs to be updated. But collecting the dataset too frequently will leave not enough time to transmit actual data. How to solve this problem?\n\n4. The regularization constant lambda needs to be set carefully for a given pair of source and target distributions. The dependency of lambda choice on the source-target distribution pair makes the proposed solution very impractical. Why not use a \"dynamic'' lambda like in ADMM? Is it because a fixed lambda converges faster?\n\n5. No benchmark comparison. There are existing solutions that retrain the entire MDN. The authors should compare with them to show that retraining a small adaption layer does not cause too much accuracy loss compared to retraining the entire MDN.", "summary_of_the_paper": "This paper proposes an end-to-end learning framework for communication systems using autoencoders. The basic idea is to only retrain a small adaptation layer between the MDN channel model and the decoder. Compared to retraining the autoencoder or the entire MDN channel model, the proposed solution runs faster and fits the dynamic communication channel environment better.", "summary_of_the_review": "Overall, the proposed idea looks OK. But it is unclear why training with such a small dataset will not lead to overfitting. Also, the overhead of recollecting the labeled dataset and retraining under a rapidly changing channel is not discussed.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "WDyK2BYVYBn", "reviewer": "Reviewer_S81w", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The addressed problem is of practical relevance and the authors carry out experiments on both real and simulated data, with good results.\nHowever, I have some reservations.\n\nWhat is the focus of the paper? Is it the domain adaptation approach or the specific problem of domain adaptation (DA) of autoencoders using an MDN as the channel model. I am asking because from the abstract it seems that the main focus is on the adaptation technique and the autoencoder case is just an application examples, while from the introduction section it seems that the main focus is on the autoencoder. \n\nThe authors assume that the source and target distribution have the same number of components with a one-to-one correspondence. How realistic is the working hypothesis? In the appendix the authors report the results if the number of components is mis-matched. I think it should be included in the main body of the paper instead.\nAlso, how realistic is the working hypothesis that labelled samples from the target domain an be collected before the channel changes again? Wireless channels can vary very rapidly, so I presume that the proposed approach applies only relatively stable channels.\n\nIt would be interesting to include cases in which the domain adaptation fails. I presume that if the channel changes significantly then the proposed adaptation might fail.\n\nIn (section 4.1), the authors adopt the MAP rule to determine the channel input and the Gaussian component. How complex is the MAP rule? I presume that if the number of Gaussian components and the constellation size become large, it might be an issue. (The same question applies to section 4.2 in which the MAP rule is applied as well). I am not sure if in Appendix C.4 the authors included the complexity of MAP rule. I think they should as it is part of the overall approach.\n\nI am confused about the way results are reported in Table 1. The authors report the estimated conditional likelihood p(x/y) estimate by the various methods, but what is the true p(x/y)? How to evaluate if the estimated conditional likelihood is close to the true one? Also, it seems that for large number of samples the “Transfer-last-layer” outperforms the proposed approach. Why is this the case?\n\nWhy in Fig 4 and 5 the 16 QAM performs less than the no adaptation?\n\nUnder “observations and takeaways” the authors state that “Between the two adaptation methods, MAP SE performs marginally better than the Affine method.” Which one of the two would the author recommend? Are they equivalent in terms of complexity?\n\nSince the authors claim that the proposed approach is fast and light-weight, the authors should include a study of the complexity analysis in the main body of the paper (I suggest move it there from the appendix), including also the training and running times of the proposed approach. I see that they include the number of trainable parameters, which is very important, but it’s only partial information.\n\nThe baselines included in the numerical results are very similar to the proposed approach. It would be interesting if the authors could include other methods as well, maybe even more complex, to analyze the trade-off between performance and complexity offered by  the proposed approach.\n\nMaybe the authors could include a future work section in the conclusion (if they assume to continue research on this topic) and discuss the generality of this approach. What are other possible application areas?\n", "summary_of_the_paper": "This paper addressed the problem of domain adaptation when the target domain has only few labeled samples. It proposes to use an MDN with Gaussian components to represent the source distribution and then adapt the MDN parameters to match the target distribution to the source distribution.\nThis approach is adopted to address the problem of adapting a communication autoencoder (trained on the source distribution) to samples drawn from the target distribution without worsening the error rate of the autoencoder. The paper includes results on simulated and real data.\n", "summary_of_the_review": "The addressed problem is of interest and the authors test the proposed technique both on simulate and real data. However, the main technical contribution of the paper relies on the well-known property of Gaussian distributions (hence the paper provides only marginal novelty) and the two working hypothesis (same number of components between the two Gaussian distributions and a one-to-one\ncorrespondence between the components) are not properly discussed from a practical viewpoint. Finally, considering the fact that the authors promote their approach because it is fast and light-weight, a proper complexity analysis is missing in the paper (the one in the appendix is partial and does not seem to include the cost of the MAP rule).", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "ThRVxo8VOf", "reviewer": "Reviewer_wact", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths\n\nThe need for domain adaptation in communication systems is practically important and the paper does a good job motivating it. The paper is reasonably well written and provides a good background on applications to wireless communications.\n\nWeaknesses\n- On the theory side: the proposed domain adaptation method involves maximizing a regularized log-likelihood and is optimized using known methods such as BFGS quasi-newton methods. \n- On the application side:  the block error rates of 0.3 etc in Fig. 5 are too high to be of practical interest in communications applications. I would argue that none of the schemes presented are practical for the models of interest.\n- The authors notes that for AWGN to Ricean fading, the performance of the adapted network is sometimes worse than the baseline. The authors’ explanation for this phenomenon is that the “distribution of the two domains are not very different. ”. However, if the distribution of the two domains are close, we should expect the performance of the adapted system stays the same as or slightly better than the baseline. Furthermore it should be noted that the AWGN to Ricean fading is of practical interest as both models involve a dominant line-of-sight component. \n", "summary_of_the_paper": "The paper considers an application of deep-learning to communication systems. Due to the nature of deep neural networks that always learn to work on training distribution, the current deep-learning based methods often perform badly when there is a domain shift between training and testing data. And there is no surprise that this problem occurs in communication systems using neural networks. The authors propose  a fast and light-weight method for adapting a Gaussian mixture density network (MDN) using only a small set of target domain samples. This method is well-suited for the communication setting, where the distribution of target data changes rapidly (e.g., a wireless channel), making it challenging to collect a large number of samples and retrain. The authors propose a method for adapting the auto-encoder without modifying the encoder and decoder neural networks, and adapting only the MDN model of the channel. The method utilizes feature transformations at the decoder to compensate for changes in the channel distribution, and effectively present to the decoder samples close to the source distribution. Some experimental results are provided to evaluate the proposed approach.", "summary_of_the_review": "The paper has significant weaknesses in the proposed applications side and has limited novelty on the algorithmic side. I therefore do not recommend acceptance. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "LxJWdo6ME7Q", "reviewer": "Reviewer_8ana", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strength:\n(1)\tThe paper proposes a new MDN based adaptation model for e2e communication system. \n\n(2)\tThe proposed method is verified by both synthetic and real-world datasets.\n\nHere are some detailed comments:\n\n(1)\tIs the adaptation method applicable to other e2e autoencoder models, e.g., GAN or VAE. In another words, is there any specific advantage of selecting MDN?\n\n(2)\tRelated works on transfer learning miss important references on semi-supervised domain adaptation, e.g., [ref1] and [ref2]. Please also find more semi-supervised transfer methods online. \n\n[ref1] Adaptive Knowledge Transfer based on Transfer Neural Kernel Network\n\n[ref2] Adaptive Consistency Regularization for Semi-Supervised Transfer Learning\n\n(3)\tDo you also assume the marginal distribution of x, (or s) is also different between the source and target domains?\n\n(4)\tDiagonal covariance implies the inputs are independent, which is a quite strong assumption, it is necessary to work on the general covariance case.\n\n(5)\tIt is unclear how to ensure $\\mathbf{C}_i$ to be diagonal. In fact, I think it is infeasible to make $\\mathbf{C}_i$ diagonal according to the derivation in appendix C.1. \n\n(6)\tWhat’s difference between $\\hat{\\pi_i}$ and $\\hat{\\alpha_i}$.\n\n(7)\tThe objective eq.(6) calculates the KL-divergence of two gaussian mixtures. This requires the data size of the source and target to be the same. Is the source data size reduced to match the small number of target data points. In this way, it is not very convincing to estimate the true conditional probability using limited data points.\n\n(8)\tThe comparison methods lack the state-of-the-art semi-supervised domain adaptation methods, which is less convincing. \n\n(9)\tExperiments on simulated and real-world datasets are imbalanced. More real-world experiments are expected.  Moreover, the current setting of section 5.3 is not very clear. How is the transfer setting formulated, i.e., please give the source and target clearly. \n\n", "summary_of_the_paper": "In this paper, the authors propose an MDN based adaptation model to e2e communication system. Specifically, it transforms the inputs to the decoder such that their class-conditional distributions are close to that of the source domain. Experiments are done on both synthetic and real-world datasets. ", "summary_of_the_review": "Overall, the paper needs further improvements to reach ICLR standard. See detailed comments above.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "D89ZYR4X-Ur", "reviewer": "Reviewer_brRc", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "*Strong Points*\n1. Their proposed method is simple, but shows solid gain over baselines when target samples are limited. Applying affine transformation to get the parameters of the Gaussian mixture model in the target domain is a convincing approach. And, their simple approach is showing good gains in several cases. \n2. Judging from their description of the related work, this problem is not much studied in the wireless communication area. Exploring a new and practical scenario is a strong point in this paper. \n\n*Weak Points*\n1. The method is basically specialized for wireless communication with an autoencoder. Although the first several sentences of the abstract seem to describe more general scenarios, their methods are tailored for the wireless communication experiments, and unclear whether the proposed method is widely applicable in other recognition problems, e.g., image classification and sentiment analysis. \n2. It is hard to judge the novelty of the proposed method since their baseline is very naive. There is no baseline borrowed from the literature of few-shot classification or domain adaptation work for limited target samples, e.g., \"Few-Shot Adversarial Domain Adaptation, Neurips 2017\". I understand that it may be hard to directly borrow such baselines since they are not designed for wireless communication. \n3. Related to 2, they are not covering several few-shot domain adaptation works, such as  \"Few-Shot Adversarial Domain Adaptation, Neurips 2017\". Their work is not the first to study the problem. It was confusing. \n", "summary_of_the_paper": "This paper presents a method for domain adaptation, where we have access to only a small number of labeled target samples and aim to adapt a model with a small training cost. They propose a fast and light-weight method for adapting a Gaussian mixture density network\n(MDN). Specifically, they transform the parameters of source Gaussian mixture models to those of the target domain by applying affine transformations.  They make the assumption of one-to-one pairing between the components of the mixture models. To optimize a model, they regularize training by the Kullback-Leibler divergence between two paired mixture models. This method is specialized for e2e learning for a communication system using an auto-encoder.  They evaluate the performance of the model on several adaptation scenarios in wireless communication and show a good advantage over baselines when enough target samples are not available. ", "summary_of_the_review": "From the weakness I described above, I recommend this paper below the acceptance threshold. The work may have high importance in the application to wireless communication, but lacks discussions related to existing domain adaptation work and comparison to them. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a domain adaptation method that is specific to auto-encoder and wireless domain. The proposed method shows solid gain over baseline and is simple. There were multiple complaints on reviewer's side regarding clarity of the abstract and application of the work and related work that was responded to by the authors during the rebuttal period. However, the modifications were not enough to address all concerns. Mainly, the assumptions for the method to work such as source and target having the same number of components and diagonal covariance. It would help the paper to discuss the cases where the model fails. In addition, the paper will benefit from a stronger baseline.
Decision: Reject