Neural Approximation of an Auto-Regressive Process through Confidence Guided Sampling
Authors: YoungJoon Yoo
Sanghyuk Chun
Jaejun Yoo
Sangdoo Yun
Jung Woo Ha
Venue:  (2020)
Keywords: Neural approximation method
Auto-regressive model
Sequential sample generation

Abstract: We propose a generic confidence-based approximation that can be plugged in and simplify an auto-regressive generation process with a proved convergence. We first assume that the priors of future samples can be generated in an independently and identically distributed (i.i.d.) manner using an efficient predictor. Given the past samples and future priors, the mother AR model can post-process the priors while the accompanied confidence predictor decides whether the current sample needs a resampling or not. Thanks to the i.i.d. assumption, the post-processing can update each sample in a parallel way, which remarkably accelerates the mother model. Our experiments on different data domains including sequences and images show that the proposed method can successfully capture the complex structures of the data and generate the meaningful future samples with lower computational cost while preserving the sequential relationship of the data.}

Under review as a conference paper at ICLR 2020
NEURAL APPROXIMATION OF AN AUTO-REGRESSIVE PROCESS THROUGH CONFIDENCE GUIDED SAMPLING
Anonymous authors Paper under double-blind review
ABSTRACT
We propose a generic confidence- based approximation that can be plugged in and simplify the auto-regressive generation process with a proved convergence. We first assume that the priors of future samples can be generated in independently and identically distributed (i.i.d.) manner using an efficient predictor. Given the past samples and future priors, the mother AR model can post-process the priors while the accompanied confidence predictor decides whether the current sample needs resampling not. Thanks to the i.i.d. assumption, the post-processing update each sample in a parallel way, which remarkably accelerates the mother model. Our experiments on different data domains including sequences and images show that the proposed method can successfully capture the complex structures of the data and generate the meaningful future samples with lower computational cost while preserving the sequential relationship of the data.
INTRODUCTION
The auto-regressive (AR) model, which infers and predicts the causal relationship between the previous and future samples in a sequential data, has been widely studied since the beginning of machine learning research. The recent advances of the auto-regressive model brought by the neural network have achieved impressive success in handling complex data including texts (Sutskever et al.)
2011), audio signals (Vinyals et al. 2012]] Tamamori et al. 2017 van den Oord et al. (016a)). and
images (van den Oord et al. 2016b; Salimans et al.) 2017).
is well known that AR model can learn tractable data distribution p(x) and can be easily extended for both discrete and continuous data. Due to their nature, AR models have especially shown good fit with sequential data, such as voice generation (van den Oord et al. 2016a) and provide stable training while they are free from the mode collapsing problem (van den Oord et al.) (0166).
However, these models must infer each element 2; of the data
[z], "2, , TT, " .N in serial manner, requiring (NN times more than the other non-sequential estimators, which outputs x at once (Garnelo et al. 2018 Kim et al. 2019 Kingma& & Welling 2014; Goodfellow et al. 2014).
Moreover, is difficult to employ recent parallel computation because AR models always require previous time step by definition. This mostly limits the use of the AR models in practice despite their advantages.
To resolve the problem, we introduce new and generic approximation method, Neural Auto- Regressive model Approximator (NARA), which can be easily plugged into any AR model. We show that NARA can reduce the generation complexity of AR models by relaxing inevitable AR nature and enables AR models to employ the powerful parallelization techniques in the sequential data generation, which was difficult previously.
NARA consists of three modules; (1) a prior-sample predictor, (2) a confidence predictor, and (3) the original AR model. To relax the AR nature, given a set of past samples, we first assume that each sample of the future sequence can be generated in an independent and identical manner. Thanks to the i.i.d. assumption, using the first module of NARA, we can sample series of future priors and these future priors are proc-processed by the original AR model, generating a set raw predictions. The confidence predictor evaluates the credibility of these raw samples and decide whether the model needs re-sampling or not. The confidence predictor plays an important role in that the approximation errors be accumulated during the sequential AR generation process if the

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Input: X31 [x,, X2. ... x;]
Prior-sample Predictor fy
Prior values: (u M]
m ... M
p(x; Ag)
Accept
AR model
Confidence
p(X-22 (g.mper)
Predictor By
Accept
[ (Mp co Miis mpMe
Parallel Inference
p(X--j (as
Paraliel
Vay (<
Re-sample
Inference
p(X; Agg (MM)))))
Re-sample
Figure I: Conceptual description of Neural Auto-Regressive model Approximator (NARA). First, the prior-sample predictor fw predicts prior samples m. Next, we draw M samples from conditional probabilities for the given input Xşi and the prior samples m in parallel. Afterward, the confidence predictor gV estimates confidence score V of each j-th sample parallel. Finally, we accept 1 samples where k = arg min; Vj < Here, is predefined confidence threshold. More detailed figure is in Supplementary
erroneous samples with low confidence are left unchanged. Therefore, in model, the sample be drawn either by the mixture of the AR model or the proposed approximation method, and finally the selection of the generated samples are guided by the predicted confidence.
We evaluate NARA with various baseline AR models and data domains including simple curves, image sequences Yoo et al. 2017), CelebA (Liu al. (015a), and ImageNet (Deng al., (2099). For the sequential data (simple curves and golf), we employed the Long Short-Term Memory models (LSTM) (Hochreiter & Schmidhuber 1997) baseline AR model while PixeICNN+ (Salimans) et al. 2017) is used for the image generation (CelebA and ImageNet). Our experiments show that NARA can largely reduce the sample inference complexity even with heavy and complex model on difficult data domain such as image pixels.
The main contributions of our work can be summarized follows: (1) we introduce new and generic approximation method that can accelerate any AR generation procedure. (2) Compared to a full AR generation, the quality of approximated samples remains reliable by the accompanied confidence prediction model that measures the sample credibility. (3) Finally, show that this is possible because, under a mild condition, the approximated samples from our method can eventually converge toward the true future sample. Thus, method effectively reduce the generation complexity of the AR model by partially substituting it with the simple i.i.d. model.
PRELIMINARY: AUTO-REGRESSIVE MODELS
Auto-regressive generation model is probabilistic model to assign a probability p(x) of the data x including n samples. This method considers the data as sequence (x; ., ..n.,, and the probability p(x) is defined by an AR manner as follows:
p(x) = II ", 2;-1),
(1)
From the formulation, the AR model provides a tractable data distribution p(x). Recently, training the model parameters using the training samples Å7, the computation parallelization are actively employed for calculating the distance between the real sample 2; E Xy and the generated sample z from equation (1.. Still, for generating the future samples, it requires O(N) by definition.
PROPOSED METHOD
3.1 OVERVIEW
Figure shows the concept of the proposed approximator NARA. NARA consists of a predictor fw and confidence predictor gv. Given samples x (#1, ,,,, ) the sample predictor predicts a chunk of M number of the prior values m Afterward, using the prior samples, draw the future samples (1,++M] in parallel. We note that this is possible because the
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prior ( M] is ii.d. variable from assumption. Subsequently, for the predicted M]] the confidence predictor predicts confidence scores M]] Then, using the predicted confidence, our model decides whether the samples of interest should be redrawn by the AR model (re-sample 2;41) or they are just accepted. The detailed explanation will be descrihed in the following sections.
3.2 APPROXIMATING SAMPLE DISTRIBUTION OF AR MODEL.
Given the samples X = f vi). AR model defines the distribution of future samples ((.)) ( ;3 } follows:
po (x,))
II (e (1,, (i+11
(2)
Here, 0 denotes the parameters the AR model. The indices i, are assumed to satisfy the condition j > Vi,j E NJ. To approximate the distribution ps (x,iiiXxx)) we introduce a set of prior samples (i.-- = fw (xgx; W). where we assume that they are i.i.d. given the observation Xgi. Here, W is the model parameter of the prior-sample predictor fw(().
Based this, we define approximated distribution qo,w (X(.,-1) ma.--1) characterized by the original AR model po and the prior-sample predictor (-) as follows:
2a,w w (x(1.) Xgg (a.g-11) = pa (xy)
II
Po Po X M44 m.-1
1=++2
Compute in parallel (const time)
po(x+t1xe5)
II (m)+2
Compute in sequerntial (lineur time)
po (xsi)
Here, approximation (A) is true when m approaches X(i--- 1)- Note that it becomes possible to compute q in a constant time because we assume the prior variable m, to be i.i.d. while pa requires linear time complexity.
Then, we optimize the network parameters and W by minimizing the negative log-likelihood (NLL)
of qo.w (x [G x "ss,Jve Jw (xx:)) where ((.)) is set of samples that are sampled from the baseline AR model. We guide the prior-sample predictor fw to generate the prior samples that is likely to come from the distribution of original AR model m by minimizing the original AR model
and arior-sample predictor W jointly follows:
min log po (g)
(x+)) p go,w (X(u.) X
(xs,, / (w(xx:))
(4)
where x(9) denotes the ground truth sample value in the generated region of the training samples. Note that both (z( and its approximated distribution go, (a)) approaches to the true data distribution when (1) our prior-sample predictor generates the prior samples m close to the true samples x and (2) the NLL of the AR distribution approaches to the data distribution. Based our analysis and experiments, we later show that our model can satisfy these conditions theoretically and empirically in the following sections.
3.3 CONFIDENCE PREDICTION
Using the prior-sample predictor fw(-), model generates future samples based on the previous samples. However, accumulation of approximation errors in the AR generation may lead to an unsuccessful sample generation. To mitigate the problem, introduce auxiliary module that determines whether to accept or reject the approximated samples generated as described in the previous subsection, referred to confidence predictor.
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First, we define the confidence of the generated samples follows:
(5) where ä& po (* Xg*-1) and E (1.... .3). The confidence value us provides a measure of how likely the generated samples from qo,w (-) is drawn from (o [xgk-1). Based on the confidence value Vk, our model decides whether can accept the sample Ty or not. More specifically, we choose a threshold E [0, 1] and accept samples which have the confidence score larger than the threshold e.
Uk = q0,w (2t = ã& x.s, (fw(xxx)))
When the case 1, our model always redraws the sample using the AR model no matter how our confidence is high. Note that our model becomes equivalent to the target AR model when 1. When c= 0, model always accepts the approximated samples. In practice, among approximated M samples where k(e) arg ming V4 > e. Subsequently, we re-sample =k())) from the original AR model and repeat approximation scheme until reach the maximum length.
accept k(c) samples
However, it is impractical to calculate equation (5) directly because we need the samples 2 from the original AR model. We first need to go forward using the AR model to see the next sample and come backward to calculate the confidence to decide whether we use the sample or not, which is nonsense.
To detour this problem, we introduce a network (.).) that approximates the binary decision variable hg = LPp 2 e) as follows:
h(o) = gv (x fw (x<:)),
(6)
where [ ..... h3..
The network gv)+) is implemented by auto-encoder architecture with sigmoid activation output that makes the equation (6) equivalent to the logistic regression.
3.4 TRAINING DETAILS
To train the proposed model, we randomly select the sample 2l0) [1, N] for the sequence ( in training batch. Then, we predict ((0) min( B, N - (()) sample values after ((8), where B denotes the number of samples the prediction considers. To calculate equation Q we minimize the loss of the training sample ((5), k = 1.... K. and the locations s(9) [1] as.
M
min 6,W
log (x (0004100))
1
log qa,w
(& ($(G)) (ate) 4200 Salii).
(x)
K
KM
:=1==1
Here, & ((())
(al0) 49 +201 for j E {1 {1....M. M) denotes M number of the sequences from the AR distribution
Po(x(ato, 300 +209) x (t) for i-th training data. From the experiment, we found that M sample enough to train the model. This training scheme guides the distribution drawn by NARA to fit the original AR distribution as well as to generate future samples, simultaneously. To train gv gv(-). binary
cross-entropy loss is used with h in equation with freezing the other parameters.
3.5 THEORETICAL EXPLANATION
Here, show that the proposed NARA is regularized version of the original AR model. At the extremum, the approximated sample distribution from NARA is equivalent to that the original AR model. In NARA, our approximate distribution q( ((x(t,++1] Xxi, m(.))) is reformulated follows:
j+1
20.0 0x.0 (X11441|Xxs m(ss) = po po0(1411x00 (xci)
II 1=4+2
as (xi
ml-1)
Po(Xg:41) po(X)1+22
Po (x(xx)+1).
(x(x(4+1)
4s (x (i.j).
po(xxi po(X
po po(x<;)
To (Xei m1+1)
po po(x
RAporgs.ma..)
(8)
where the parameter & denotes the network parameters of the approximated distribution g(-). There- fore, our proposed cost function can be represented as the negative log- likeliilldood of the AR model with regularizer - log R(po, 40, mia)):
min (xc.))))))))))))))))))) flog pa (x ((.+1) (xg)) + log R(po, 9o m(s)))
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1.0
0.5
800e)tence rstio
0.8
8.0
0.5
CSTM
prior sample
3.8
0.0
0.4
0.2
3.6
0.4
confidence threshold
196 200 300 390
(a) One-dimensional time series data generation results.
(h) Error, accepts over E [0.0, 1.0].
Figure (a) One-dimensional time series data generation example. Dashed lines denote the generated samples. Red lines, green, and blue lines are from the approximation model with acceptance ratio c 0.0, 0.5 and c 1.0. Black line is the generation results from the baseline LSTM. Magenta line presents the generated prior samples. (b) The graph denoting the mean & error (blue) and the acceptance ratio (red) over threshold
Note that the proposed cost function is equivalent to that of the original AR model when log (ps, qa. (i.s) O, which true under the condition of (ia)) = (Gaa and (+ = (-1-))) Here, ( fw (x . By minimizing the equation (9). R(pe, qs (i.) enforces the direction of the optimization to estimate the probability ratio of qΦ and while it minimize the gap between
(5.) so that (i.j) = fw (x&1) approaches to x(ij))-
4o (x.
4 RELATED WORK
Deep AR and regression models: After employing the deep neural network, the AR models han- dling sequential data has achieved significant improvements in handling the various sequential data including text (Sutskever et al. 2011). sound (Vinyals et al., 2012; Tamamori al. (017). and images (van den Oord et al. 2016b; Salimans et al. (017). The idea has been employed to "Flow based model" which uses auto- regressieee sample flows (Kingma Dhariwal 2018 Germain et 2015; Papamakarios et al. 2017 Kingma et al., 2016) to infer complex distribution, and reported meaningful progresses. Also, the attempts Yoo al. 2017 Garnelo et al. 2018 Kim et al. 2019) to replace the kernel function of the stochastic regression and prediction processes to neural network has been proposed to deal with supisiuper data not imposing an explicit sequential relationship.
Approximated AR methods: Reducing the complexity of the deep AR model has been explored by a number of studies, either targeting multiple domain (Seo et al. 20188; Stern et al. 2018) or specific
target such as machine translation (Wang et al. 2018 Ghazvininejad et al. 2019 Welleck et al.
2019
Wang et al. 2018 2019) and image generation (Ramachandran et al. 2017).
general approximation method for AR methods by assuming the i.i.d. condition for the "easy to predict" samples. This differentiates our approach to (Seo al. 2018) in that do not sequentially approximate the future samples by using a smaller AR model but use a chunk-wise predictor to approximate the samples at once. In addition, our confidence prediction module can seen stochastic version of the verification step in (Stern) ctt al., 20188), which helps our model to converge toward the original solution. This confidence guided approximation can be easily augmented the other domain specific AR approximation methods because our method is not limited to a domain specific selection queues such as quotation (Welleck al. 2019 Ghazvininejad al. 2019) or nearby convolutional features (Ramachandran al. 2017).
Adding step further to the previous studies, propose
EXPERIMENTS
In this section, we demonstrate the data generation results from the proposed NARA. To check the feasibility, first test method into time-series data generation problem, and second, into image generation. The detailed model structures and additional results are attached in the Supplementary material. The implementation of the methods will be available soon.
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Figure 3: Image sequence generation examples (cropped in an equivalent time-interval), From the top to the bottom, each row denotes the generation result with E (1.00, 0.75, 0.50, 0.25, 0.003.
Table I: Mean f-eerror and the acceptance ratio in image sequence generation.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Acceptance (%)
0.0
15.3
43.0
59.3
69.1
77.1
82.1
84.5
88.0
92.5
100
Mean error (&)
21.1
26.8
24.9
21.7
18.4
19.9
18.3
20.6
19.4
25.3
24.3
5.1 EXPERIMENTAL SETTING
Time-series data generation problem: In this problem, we used LSTM as the base model. First, we tested our method with simple dieeeinnniooal sinusoidal function. Second, we tested the video sequence data (golf swing) for demonstrating the more complicated case. In this case, we repeated the swing sequences 20 times make the periodic image sequences and resize each image 64 64 resolution. Also, beside the LSTM, we used autoencoder structures to embed the images into latent space. The projected points for the image sequences linked by LSTM, similar to (Yoo al.) 2017). For both cases, we used ADAM optimizer (Kingma & Ba 2015) with a default setting and learning rate 0.001.
Image generation: For the image generation task, we used PixeICNN+ (Salimans et al. 2017) as the base model. The number of channels of the network was set 160 and the number logistic mixtures was set to 10. See (Salimans et al. (017)) for the detailed explanation of the parameters. In this task, the baseline AR model (PixeICNN++) is much heavier than those used in the previous tasks. Here, we show that the proposed approximated model can significantly reduce the computational burden of the original AR model. The prior-sample predictor fw(() and the confidence estimator gv were both implemented by U-net structured autoencoder (Ronneberger et al., 2015). We optimized the models using ADAM with learning rate 0.0001. Every module was trained from scratch. We mainly used CelebA (Liu al. 2015b) resizing the samples to 64 x 64 resolution. In the experiments, randomly pick 36, 000 images for training and 3, 000 images for validation.
Training and evaluation: For the first problem, second problem, four to eight GPUs (NVIDIA Tesla P40) were used"] The training and inference code used in this section implemented by PyTorch library. For the quantitative evaluation, measure the error between the true future samples and the generated one, and also employ Fréchet Inception Distance visual quality of the generated images for the second image generation problem.
single GPU (NVIDIA Titan XP), and for the
(FID) (Heusel al., 2017) measure of the model performance and
5.2 ANALYSIS
5.2.
TIME-SERIES DATA GENERATION
Figure 2a shows the generation results the one-dimensional time-series from our approximation model with different acceptance ratios (red, green, and blue) and the baseline LSTM models (black).
The overall expreiments conducted NSML. Sung
2017 GPU system.

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Generated images of e {0.0, 0.2, 0.4, 0.6, 0.8, 1.0} on the CelebA dataset. The image in the left of the triplet denotes accepted region (white: do accept, black: do not accept), the mid is confidence map (red: high, blue: low) and the right shows the generated result.
Figure 4:
From the figure, from the prior sample generation result (magenta), the prior samples m converged to the true samples as claimed in Section 3.5
that both models correctly generates the future samples. Please note that,
The graph in Figure 2b shows the acceptance ratio and the -errrr over the confidence threshold (0, 1]. The error denotes the distance between the ground truth samples and the generated ones. As expected, our model accepted more samples as the threshold decreases, However, contrary to our initial expectations, the thres-h graph shows that the less acceptance samples does not always bring the more accurate generation results. From the graph, the generation with an intermediate acceptance ratio achieved the best result. Interestingly, we report that this tendency between the acceptance ratio and the generation quality was repeatedly observed in the other datasets well.
Figure 3 shows the image sequence generation results from NARA. From the result, we can see that the proposed approximation method is still effective when the input data dimension becomes much larger and the AR model becomes more complicated. the golf swing dataset, the proposed approximation model also succeeded to capture the periodic change of the image sequence. The table 3 shows that the proper amount of approximation can obtain better accuracy than the none, similar to the other previous experiments. One notable observation regarding the phenomenon is that the period of image sequence was slightly changed among different ratio of approximated sample acceptance (Figure One possible explanation would be that the approximation module suppress the rapid change of the samples, and this affects the interval of single cycle.
5.2.2 IMAGE GENERATION
Figure 4 shows that method can be integrated into PixelCNN+ and generates images with the significant amount of the predicted sample acceptance (white region). We observed that the confidence mostly low (blue) in the eyes, mouth, and boundary regions of the face, and the PixeICNN is used to generate those regions. This shows that compared to the other homogeneous regions of the image, the model finds relatively hard describe the details, which matches with our intuition.
The graphs in Figure present the quantitative analysis regarding the inference time and the NLL in generating images. In Figure 5a the relation between inference time and the skimming ratio is reported. The results show that the inference speed is significantly improved as more pixels are accepted. Table 2 further supports this that our approximation method generates fair quality of images while speeds up the generation procedure 5 10 times faster than the base model.
In the image generation example also, we found that the fair amount of acceptance can improve the perceptual visual quality of the generated images compared to the vanilla PixeICNN+ (Table 2). Our method benefits from increasing the acceptance ratio to some extent in terms of FID showing a U-shaped trend over the variation, similar to those in Figure 2b Note that a lower FID score identifies better model. Consistent with previous results, approximation scheme leams the peian-pror& of the images and guides the AR model to prevent generating erroneous images. The confidence maps and the graph illustrated in Figure 5b, and 5
conjecture that the proposed
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Goncration specd
Accept stoceeeeeeeeeoooo mile
No skin
Tull skim
Test //mm
39.b
1.0
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10"
nixels 158
480
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spochs
epochs
(a) Generation speed (pixels/sec) over e [1.0,0.0]
(b) NLL Curve.
(c) & error of the prior mi-
Figure 5: Quantitative analysis of the proposed method on CelebA dataset. (a) shows the generation speed over the threshold (b) shows the NLL of the inference with and without any acceptance. (c)
presents the -loss between the prior m and the pixel value for every epoch.
Table 2: The speed up and visual quality of the results on CelebA dataset. The speed-up of the inference time with (SU) and without anchors (SU AA) compared to that of PixeICNN+ is measured. acceptance ratio (AR) without anchors and FID (lower is better) are also displayed.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
SU
1.0x
1.2x
1.7x
2.2x
2.8x
3.5x
4.1x
4.5x
5.3x
6.8x
12.9x
SU AA
1.0x
1.2x
1.7x
2.4x
3.1x
4.1x
5.2x
5.9x
7.4x
11.0x
59.4x
AR (%)
0.0
15.3
43.0
59.3
69.1
77.1
82.1
84.5
88.0
92.5
100
FID
56.8
50.1
45.1
42.5
41.6
42.3
43.4
45.9
48.7
53.9
85.0
support this conjecture. Complex details such as eyes, mouths and contours have largely harder than the backgrounds and remaining faces.
In Figure 5b and Figure Sc. the graphs show the results supporting the convergence of the proposed method. The graph in Figure shows the NLL of the base PixeICNN+ and that of our proposed method under the full-accept case, i.e. fully believe the approximation results. Note that the NLL of both cases converged and the PixeICNN achieved noticeably lower NLL compared to the fully accepting the pixels at every epoch. This is already expected in Section 3.2 that the baseline AR model approaches more closely to the data distribution than our module. This supports the necessity of re-generating procedure by using the PixeICNN++, especially when the approximation module finds the pixel has low confidence.
The graph in Figure pressents the & distance between the generated prior pixel and the corre- sponding ground-truth pixel x in the test data reconstruction. Again, similar to the previous time-series experiments, the model successfully converged the original value (m approaches to x). Combined with the result in Figure 5b) this result supports the convergence conditions claimed in section 3.2 Regarding the convergence, we compared the NLL of the converged PixelCNN+ distribution from the proposed scheme and that of PixeICN++ with CelebA dataset from the original paper (Salimans) al. 2017).
CONCLUSION
In this paper, we proposed the efficient neural auto-rogressive model approximation method, NARA, which can be used various auto-regressive (AR) models. By introducing the prior-sampling and confidence prediction modules, we showed that NARA can theoretically and empirically approximate the future samples under relaxed causal relationships. This approximation simplifies the generation process and enables our model to use powerful parallelization techniques for the sample generation procedure. In the experiments, we showed that NARA can be successfully applied with different AR models in the various tasks from simple complex time-series data and image pixel generation. These results support that the proposed method can introduce a way to use AR models in a more efficient manner.
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Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer- assisted intervention, pp. 234-241 Springer, 2015.

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Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. PixelCNN++: Improving the PixeICNN with discretized logistic mixture likelihood and other modifications. In International Conference Learning Representations, 2017. URL
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Under review as a conference paper at ICLR 2020
A IMPLEMENTATION DETAILS
Time serious sample generation: For the 1-dimensional samples case, the LSTM consists of two hidden layers, and the dimension of each layer set to 51. We observed predicted p = 20 future samples. The chunk-wise predictor (.( and the confidence predictor gv(.) were defined by single fully-connected layer with input size 200 and the output size 20. In this task, our model predicted every 20 samples by seeing the 200 previous samples.
200 steps and
For the visual sequence case, used autoencoder structures to embed the images into latent space. The encoder consists of four contvvactivation plus one final Conv filters. Here, the term "Conv" denotes the convolutional filter. Similarly, the decoder consists of four "Conv transpose-activation" with one final Conv transpose filters. The channel sizes of the encoder and decoder were set to {32,32, 32, 32, 64) and (64, 32, 32, 32). All the Conv and Conv Transpose filters were defined by 3 The activation function was defined as Leaky ReLU function.
The embedding space was defined to have 10-dimensional space, and the model predicts every future samples given 50 previous samples. The prior sample predictor and the confidence predictor are each defined by a fully-connected layers, and predict future samples in conjunction to the decoder. We note that the encoder and decoder were pre-trained by following (Kingma & Welling 2014
Image generation: The prior sample and confidence predictors consist of an identical backbone blocks,
network except the last block. The network consists of four number of and the four number of "Conv transpose- granspse-BN-Acttvve BN-Activation" block. All the Conv and Conv Transpose filters were defined by 4 4. The last activation of the prior sample predictor is tangent hyperbolic function and that of confidence predictor is defined as sigmoid function. The other activation functions defined Leaky-Rel.U. Also, the output channel number of the last block are set to 3 and 1, respectively. The channel size of each convolution filter and convolution- convolution-transpoe transpoos filter was set to be (64, 128, 256, 512] and {256, 128, 64,
We used batch normalization (Toffe & Szegedy 2015) for both networks and stride is set to two for all the filters. The decision threshold is set to the running mean of the log((9.w (-)). We set x. 2.5 for every test and confirmed that it properly works for all the cases.
B SUPPLEMENTARY EXPLAIN ON PROPOSED SAMPLE GENERATION
Figure 6 shows the detailed process of sampling when the proposed NARA module is attached to the mother AR model. The diagram describes the Sinusoidal function generation example, which is the simplest example in our paper. In each auto-regressive step, we decide if we use the approximated value predicted by our sample predictor (predict the value in chunkwise manner assuming i.i.d condition) or sample the value with the mother auto-regressive model. This decision is conducted based on the confidence estimation by the Confidence predictor of the paper. The chunk- estimation step can be boosted by recent parallel computing methods different from the mother auto- regressive model, and this can possibly relax the computation burden of the original regressveee model. More importantly, this scheme can be applied to various auto-regressive models.
C SUPPLEMENTARY EXPERIMENTS
C.I SUPPLEMENTARY GENERATION RESULTS
In addition the results presented in the paper, we show supplement generation examples in below figures. Figure 3 and Table 3 present the image sequence generation result from the other golf swing sequence. In this case also, we can observe the similar swing cycle period changes and acceptance ratio-error tendencies reported in the paper. Our approximation slightly affects the cycle of the time-serious data, and the result also shows that the approximation can achieve even better prediction results than "ncne-acceptance" case.
Figure I1 and 12 shows the additional facial image generation results among E [0.0, 1.0). We can that pixels from boundary region more frequently re-sampled compared to other relatively simple face parts such as check or forehead. Also, we tested our model with ImageNet classes, and the results presented in Figure 10)
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[xp Mapp Misse
Original AR model (parallel inference)
Input price
Ngi- [x,, Xas ooo x,]
Input
Prior-sample Predictor fw
[ Xy Ap Xiep X
(parallel inference)
Input sampled values
Confidence Predictor Bv (parallel inference)
MJ ma m
yomuter
Prior values
G/p .... 9-M k argming Guy S
Confidence scores
Accept: [Xi..ro Nx b.... Re-sample: [ [x x'mxa]
Approximated samples
Figure 6: Detailed description of the sample generation process.
C.2 ANALYSIS ON WHEN CONFIDENCE MODULE FAILS.
If the prior-sample predictor fw performs worse than expectation, the confidence module gv will reject all prior samples; hence, in this case, the model repeats to draw samples using the original AR model sample-by-sampl manner. However, in an even worse situation, the confidence module could always accept prior-samples. including low-quality prior samples. To simulate the failure or both fw and gv, we manually fix the "accept region" regardless of the predicted confidence score. We select ten samples with the lowest confidence scores and report the results in Figure X In the figure, we can observe that the drastic error can occur when both the pripr-sample predictor fw and confidence module gv fail. However, in practice, confidence module gv can detect the drastic error during the sampling stage as shown Figure 8b hence, such extreme error will occur rarely.
C.3 APPLYING NARA WITH FASTER PIXELCNN+
To show that method can be added diverse AR mode, we also combined skimming method to the fast version of PixeICNN++P Figure 9 and Table 4 show the generated samples and the generation time using the incorporated model (skim+Fast PixeICNN++). Due the lack of time to fully investigate all datasets with the new implementation, we conducted the experiment on CIFAR-10 10, which the most widely used dataset PixeICNN works and more diverse than the CelebA dataset. From the results, we show that our method, augmented with Fast PixeICNN++, can make the baseline algorithm much faster as we suggested in the paper.
Table 3: Mean -error and the acceptance ratio over image generation sequence.
6
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Acceptance (%)
0.0
18.0
35.0
44.5
56.1
61.0
70.5
85.5
92.5
99.0
100
Mean Error
2.20
2.61
2.21
1.79
2.28
2.10
1.91
1.87
1.94
2.38
2.45
Raamccanndann, Prajit, et al. "Fast generation for convolutional autoregressive models." arXiv preprint arXiv: (2017).
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Figure 7: Image sequence generation examples (cropped in an equivalent time-interval). From the top to the bottom rows, each denotes the generation result with {1.00, 0.75, 0.50, 0.25, 0.00}.
(a) Simulated failure samples.
(b) Actual confidence
Accept region.
Figure Simulated failure samples in generating face images by fixing the "accept region" regardless of actual confidence scores. We report ten samples with lowest actual confidence scores.
0.0
0.3
0.8c
0.9c
Figure 9: Generated CIFAR-I0 images when combined with Fast PixeICNN++. Left (NARA Fast PixcICNN++), center (Fast PixeICNN++), right (PixcICN+++). The numbers on top denote 1 - e.
Table Generation time per an image and skimming ratio over on CIFAR-10 Dataset. Here, [F] denotes the original Fast PixeICNN++. Note that the maximum acceptance ratio is 28 28/32 = 87.5% since our method starts from the four lines which is already drawn by the original Fast PixeINNN++.
(CIFAR-10)
1.0
0.8
0.6
0.4
0.2
0.0
[F]
Time (sec)
24.2
23.2
19.6
16.1
12.5
4.87
23.8
AR (%)
0.00
11.0
25.4
40.7
55.5
87.5
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(a) Balloon Flower
(b) Ladybug
(c) Tree-martin
(d) Sandbar
Figure 10: ImageNet generation examples.
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+
Figure II: CelebA generation result in 64 x 64. {0.0, 0.1, 0.2, 0.2,0.3,00.. 0.4,0.5..
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Figure 12: More selected CelebA generation result 64 x 64. according to E (0.6, 0.7, 0.8, 0.9, 1.0].
16


Reviews:
[{"review_id": "ryxp5HFpFr", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper addresses the sequential limitation of autoregressive model when doing sampling. Specifically, instead of sampling next observations in a sequential fashion, this paper generates future observations in parallel, with the help of i.i.d. future priors. With the help of learned confidence model, the model is able to get trade-off between speed and approximation accuracy. Experiments on synthetic data and image generation with PixelCNN++ show the comparable results while being significantly faster than baseline.\n\nOverall the paper is well motivated, with an interesting design of the variational distribution to approximate the true autoregressive distribution. The design of the confidence model looks a bit heuristic, but the trade-off ability between efficiency and quality it brings is also quite interesting. \n\nBelow are some minor comments:\n\n1. The theoretical analysis is basically comment about the objective which is less interesting. However more interesting guarantees would be: 1) with the additional correction term added, how would it help with reducing the variance; 2) As the q_{\\theta, \\phi} is always in a limited form due to the parallelism requirement, how bad the approximation could be in the worst case ---- I’m not asking for these results, but any form of discussion would be helpful.\n\n2. The author only compared with the raw PixelCNN++. Would any of the existing AR-speedup method be applicable for a comparison? \n", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "N/A", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "H1l8W_ndtS", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper presents a technique for approximately sampling from autoregressive models using something like a a proposal distribution and a critic. The idea is to chunk the output into blocks and, for each block, predict each element in the block independently from a proposal network, ask a critic network whether the block looks sensible and, if not, resampling the block using the autoregressive model itself.\n\nIn broad strokes the approach makes sense. It assumes, essentially, that parts of the sequence are hard to predict and parts are easy and, if there are enough easy parts, this procedure should lead to faster inference.\n\nThe paper's writing is not ideal. There are some grammatical mistakes that harm reading (for example, the second paragraph of the introduction says \"However, these models must infer each element of the data x ∈ RN step by step in a serial manner, requiring O(N) times more than other non-sequential estimators\", where it is unclear what is O(N) more than what, how is this measured, etc). That said I was mostly able to follow all key points.\n\nThe authors do not point out the obvious connection to GANs, which also rely on a critic network to decide whether a sample looks like it comes from the correct distribution, except in GANs the critic is jointly trained with the generator (as opposed to here where it's trained after) and in GANs the critic is only used at training time, while here the critic is used to accelerate sampling (the better the critic the faster this method can sample).\n\nI wish the experimental results were a little more explicit about the time vs quality tradeoff; I expected to see more plots with pareto curves, since as-is it's hard to judge the magnitude of the tradeoffs involved. I'd also like a more thorough analysis on why there is a non-monotonic tradeoff in some experiments (table 1, figure 2(b)) between the amount of approximation and the sample quality; this makes me think something else is going on here as this approximate inference method should just decrease quality, never increase it.\n\nOverall I lean towards accepting the paper, but I encourage the authors to revise the writing and to add a few plots explicitly showing the time vs quality tradeoff both in likelihood (wrt the full model) and in downstream metrics like FID.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}, {"review_id": "ByxpVK-jtB", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors consider the problem of sampling time series.\nTo solve the problem they propose a method that is based on the autoregression model. The novelty here lies in the proposed sampling methods: we start with a sampling of a prior and then try to generate data according to the restored distribution. We learn two functions: signal recovery and confidence prediction.\nThe main hyperparameter of the algorithm $\\varepsilon$ identifies how much samples we accept.\nThe distinctive feature of the algorithm is speed-up for the sample generation process.\n\nWeak reject\n\nThere are a significant number of works on video generation, see e.g. [1, 2], references therein and articles that cite these two articles. The problem setting seems pretty similar. It seems like a good idea to compare to these methods (and it seems that video generation is a very resource-demanding procedure, and they don't use parallel applications similar to proposed in the paper. What is the reason?) Most of the approaches use only one frame to generate video, but it seems that LSTM in these methods will benefit from using of multiple frames as input (and will be able to transfer information in autoregression manner by transferring all they need in a hidden state).\nThe article, in my opinion, will benefit from comparison to these approaches or at least by using some benchmarks from these works to demonstrate feasibility of the considered approach, also it seems that these works are good for demonstration of parallelization capabilities (as in many cases the same idea applies).\n\nNot minor comments:\n1. In Figure 2 (a) it is not clear how the data and prediction were generated. According to the procedure in Figure 1 and text we use the same input for all approaches. However solid lines for different epsilons are different.\n2. The effect of the dependence of recovery of quality for Figure 2 (b) is not explained and is controversial: we get the smallest error for intermediate acceptance ratio, however, there is also a decrease of error if we further increase the gauge threshold (btw the term gauge threshold is new to machine learning community, consider replacement of it)\n3. More simpler examples will benefit the paper, as we'll be able to know more fundamental properties of the proposed approach. \n\n\nMinor technical comments:\n1. s. 3.1. predictor predicts\ncommas in equation (8)\n2. Figure 2: no axis labels for the left plot, use for label \"acceptance ratio\" red color font & for label \"L1 error\" blue color font\n3. Table 1 bracket after l_1 is missing\n4. Maybe $\\sigma$ is not the best designation of confidence, as it can be confused with the variance\n5. Figure 1: some indexes should be not $x_{i + 2}$, but $x_{i + j}$, $x_{i + M}$. Also, some \">\" before \\epsilon should be \"<\"\n6. \"a auto-encoder architecture\" -> \n\"an auto-encoder architecture\"\n\n[1] J. He et al. Probabilistic Video Generation using Holistic Attribute Control. 2018. ECCV\n[2] E. Denton, R.Fergus. Stochastic Video Generation with a Learned Prior. 2018.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper presents a technique for approximately sampling from autoregressive models using something like a a proposal distribution and a critic. The idea is to chunk the output into blocks and, for each block, predict each element in the block independently from a proposal network, ask a critic network whether the block looks sensible and, if not, resampling the block using the autoregressive model itself. 
The idea in the paper is interesting, but the paper would benefit from
- a better relation to existing methods
- a better experimental section, which details the hyper-parameters of the algorithm (and how they were chosen) and which provides error bars on all plots (and tables)
Decision: Reject