Autoregressive Quantile Flows for Predictive Uncertainty Estimation
Authors: Phillip Si
Allan Bishop
Volodymyr Kuleshov
Venue: ICLR 2022 Spotlight (2022)

Abstract: Numerous applications of machine learning involve representing probability distributions over high-dimensional data. We propose autoregressive quantile flows, a flexible class of normalizing flow models trained using a novel objective based on proper scoring rules. Our objective does not require calculating computationally expensive determinants of Jacobians during training and supports new types of neural architectures, such as neural autoregressive flows from which it is easy to sample. 
    We leverage these models in quantile flow regression, an approach that parameterizes predictive conditional distributions with flows, resulting in improved probabilistic predictions on tasks such as time series forecasting and object detection.
    Our novel objective functions and neural flow parameterizations also yield improvements on popular generation and density estimation tasks, and represent a step beyond maximum likelihood learning of flows.

AUTOREGRESSIVE QUANTILE FLOWS FOR PREDICTIVE UNCERTAINTY ESTIMATION
Phillip Si, Allan Bishop, Volodymyr Kuleshov
Department of Computer Science, Cornell Tech and Cornell University (ps789, adb262, vk379) @coonell.edu
ABSTRACT
Numerous applications of machine learning involve representing probability distributions over high-dimensional data. We propose autoregressive quantile flows, flexible class of normalizing flow models trained using novel objective based proper scoring rules. Our objective does not require calculating computationally expensive determinants of Jacobians during training and supports new types of neural architectures, such neural autoregressive flows from which it is easy to sample.
We leverage these models in quantile flow regression, an approach that parameterizes predictive conditional distributions with flows, resulting in improved probabilistic predictions on tasks such as time series forecasting and object detection. Our novel objective functions and neural flow parameterizations also yield improvements on popular generation and density estimation tasks, and represent step beyond maximum likelihood learning of flows.
INTRODUCTION
Reasoning about uncertainty via the language of probability is important in many application domains of machine learning, including medicine (Sarial 2018). robotics (Chua al., [0188] [Buckman et al..
2018), and operations research (Van Roy et al. 1997). Especially important is the estimation of predictive uncertainties (e.g., confidence intervals around forecasts) tasks such as clinical diagnosis
(Jiang et al. 2012) decision support systems (Werling et al., 2015 Kuleshov and Liang) 2015).
Normalizing flows (Rezende and Mohamed, 2016; Papamakarios et al. 2019 Kingma et al. 2016)
a popular framework for defining probabilistic models, and can be used for density estimation (Papamakarios et al., 2017), out-of-distribution detection (Nalisnick et al., 2019). content generation (Kingma and Dhariwal 2018), and more. Flows feature tractable posterior inference and maximum likelihood estimation; however, maximum likelihood estimation of flows requires carefully designing family of bijective functions that are simultaneously expressive and whose Jacobian has tractable determinant. In practice, this limits the kinds of neural networks that can be used to parameterize normalizing flows and makes many types of networks computationally expensive to train.
This paper takes step towards addressing this limitation of normalizing flows by proposing new objectives that contribute towards alleviating the computational cost of calculating determinants of Jacobians. Specifically, we argue for training flows using an objective that is different from classical maximum likelihood and instead based on proper scoring rules (Gneiting and Raftery (007). standard tool in the statistics literature for evaluating the quality of probabilistic forecasts. We show that this objective can used to train normalizing flows and that it simplifies the computation of Jacobians in certain types of flows.
We introduce autoregressive quantile flows (AQFs), a framework that combines the above learning objective with a set of architectural choices inspired by classical autoregressive flows. Quantile flows possess characteristics that represent an improvement over existing flow supporting neural architectures that simultaneously provide fast training and sampling in addition to the usual benefits of flows (exact posterior inference and density estimation). Interestingly, quantile flows be interpreted extensions of quantile functions multiple dimensions.

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We use AQFs as the basis for quantile flow regression (QFR). approach predictive uncertainty estimation which a probabilistic model directly outputs normalizing flow as the predictive distribution. The QFR approach enables neural networks to output highly expressive probabílistic predictions that make very little assumptions on the form of the predicted variable and that improve uncertainty estimates in probabilistic and Bayesian models. In the one-dimensional case, our approach yields quantile function regression and cumulative distribution function regression, two simple, general, and principled approaches for flexible probabilistic forecasting in regression.
In addition, we demonstrate the benefits of AQF' on probabilistic modeling tasks that include density estimation and autoregressive generation. Across our sets of experiments, we observe improved performance, and we demonstrate properties of quantile flows that traditional flow models do not possess (e.g., sampling with flexible neural parameterizations).
Contributions. In summary, this work (1) introduces new objectives for flow models that simplify the computation of determinants of Jacobians, which in turn greatly simplifies the implementation of flow models and extends the class of models that can be used to parameterize flows. We also (2) define autoregressive quantile flows based on this objective, and highlight new architectures supported by this framework. Finally, (3) we deploy AQFs as part of quantile flow regression, and show that this approach improves upon existing methods for predictive uncertainty estimation.
BACKGROUND
Notation. Our goal is to learn a probabilistic model p(z) E A(R") in the space A A(R") of distributions over high-dimensional
E R4d we use Ty E to denote components of x. In some cases, we have access to features x' E X' associated with and we want to train a forecaster (IRd) that outputs a predictive probability over æ conditioned on x'.
H : X'
2.1 NORMALIZING FLOWS AND AUTOREGRESSIVE GENERATIVE MODELS
A normalizing flow defines distribution p(x)) via an invertible mapping fo RRd + Rd with
parameters 0 E θ that describes a transformation between æ and random variable E from simple prior
Rd sampled
p(=) (Rezende and Mohamed, 2016 Papamakarios et al. 2019).
We may
compute p(z) via the change of variables formula p(x)
a p(z), where Dfa(z)
8= denotes
determinant the inverse Jacobian of fa. In order to fit flow-based models using maximum
likelihood, we typically choose Se to be in family for which the Jacobian is tractable.
common way to define flows with tractable Jacobian is via autoregressive models of the form
J = hy)
h = ()(3))))
where (55;hy) is an invertible transformer, strictly monotonic function of æj. and ej is the j-th conditioner, which outputs parameters h for the transformer. As long as 7 invertible, such autoregressive models can be used to define flows (Papamakarios et al. 2019)
2.2 EVALUATING FORECASTS WITH PROPER SCORING RULES
common way to represent a probabilistic forecast in the statisties and forecasting literature is via a cumulative distribution function (CDF) F Rd - [0, 1]; any probability distribution can be represented this way, including discrete distributions. Since F is monotonically increasing in each coordinate, when y is one dimensional, may define its inverse Q [0, + called the quantile function (QF), defined as (a) = inf inf(2' E R F(x')= 200 a}.
In the statistics literature, the quality of forecasts is often evaluated using proper scoring rules (or proper scores; Gneiting and Raftery (2007)). For example, when predictions take the form of CDFs, popular scoring rule is the continuous ranked probability score (CRPS), defined for two CDFs F and G CRPS(F, G) - S,, (F(x) G(z))² dr. When only have samples II, Z' from G, we can generalize this score as 1 > Jr (F(z) Ilr= - xz))² dx. Alternatively, we can evaluate the a-th quantile Q(a) of QF Q via the check score L R R + R2 defined as L, (z, f) = a(x - f) if x 2 and (1 af x) otherwise. The check score also provides a consistent estimator for the conditional quantile of any distribution.
m
2

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3 TAKING STEPS BEYOND MAXIMUM LIKELIHOOD LEARNING OF FLOWS
Maximum likelihood estimation of flows requires carefully designing a family of bijective functions that are simultaneously expressive and whose Jacobian has tractable determinant. In practice, this makes flows time-consuming to design and computationally expensive to train. In this paper, we argue for training flows using objectives based on proper scoring rules (Gneiting and Raftery 2007).
3.1 LEARNING SIMPLE FLOWS WITH PROPER SCORING RULES
We begin with the one dimensional setting, where flow fo R - R is a bijective mapping that can be interpreted as a QF. Alternatively, the reverse flow fo can be interpreted as a CDE We will use Qs. F& to denote fe and fg" respectively; our goal is to fit these models from data.
In order to fit models of the cumulative distribution and the quantile function, we propose objectives based on proper scoring rules. We propose fitting models Fg of the CDF using the CRPS:
L (Fo, x4)) = CRPS vz)
(Fo(x) I( S x))² dx.
When dealing with model Qo of the QF, we propose an objective based on the expected check score
(Qo, = / L. Ia((Qola),;)de.
where La is check score targeting quantile a. We refer to this objective as the quantile loss. This objective has been used previously to train value functions in reinforcement learning as well as conditional distributions in autoregressive models (Dabney al. 2018a b). this paper, we describe its application to modeling aleatoric predictive uncertainties.
The parametric form of Qo or Fo can be any class of strictly monotonic (hence invertible) functions. Previous works have relied affine or piecewise linear functions (Wehenkel and Louppe 202 I), sum-of-squares (Jaini et al. 2019), monotonic neural networks (Huang et al. 2018 Cao et al. 2019 and other models. Any of these choices suits our framework; we provide more details below.
So far, we have described two methods for fitting one-dimensional flow model. Their objectives are actually equivalent.
Equivalence Between the CRPS and Quantile Losses
Proposition 1. For a CDF F RR + [0, 1] and x! E IR the CRPS and quantile losses are equivalent:
(F,x') a- a- ( x'))+ a, R,
This fact appears to be part of statistics folk knowledge, and we have only ever seen it stated briefly in some works. We provide complete proof in the appendix. See (Laio and Tameal 2007 for another argument.
If the models Fe, Qo are analytically invertible (e.g., they are piecewise linear), we are free to choose fitting the CDF or its inverse. Other representations for will not lead to analytically invertible models, which require choosing training direction, we discuss below.
Practical Implementation. The quantile and the CRPS losses both involve potentially intractable integral. We approximate the integrals using Monte-Carlo: this allows us obtain gradients using backpropagation. For the quantile loss, we sample a uniformly at random in [0, 1]: for the CRPS loss, we choose reasonable range of y (usually, centered around y:) and sample uniformly in that range. This approach works well in practice and avoids the complexity of alternative methods such as quadrature (Durkan al. 2019).
3.2 LEARNING HIGH-DIMENSIONS AUTOREGRESSIVE FLOWSS
Next, extend approach nigh-dimensional autoregressive flows by fitting each conditional distribution with the quantile or the CRPS loss. We start with a general autoregressive flow defined as
xz = (23;hhg)
hg (,(3))))
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where (22,; h,) is an invertible transformer and oj is the j-th conditioner As in the previous section,
we use Ob Fh; to denote 7 (2y; h1) and (my; hy), respectively.
We train autoregressive flows using losses that decompose over each dimension as follows:
(may Fayl
(2) May Qngh
(5)
=1
CRPS loss; reverse training
quantile loss; forwards training
where T,(1) and T are, respectively, the CRPS and quantile objectives defined in the previous section and applied component-wise. We average the losses on a dataset n points {x:)==
Sampling-Base Based Training of Autoregressive Flows. Crucially, Equation 5 defines not only new objectives but also new training procedures for flows. Specifically, it defines sampling -hasid learning process that contrasts with traditional maximum likelihood training.
In order to train with objectives L(), (?), we perform Monte Carlo sampling to approximate an intractable integral. Thus, we sample a source variable (z or =) from uniform distribution and transform into the other variable, which we call the target, after which we compute the loss and its gradient. This is in contrast maximum- likellioooieee learning, where take the observed variable transform it into 2, and evaluate p(²) as well as the determinant the Jacobian.
The choice of source and target variable depends on the choice of objective function. Each choice yields a different type of training procedure; we refer to these as forward and reverse training.
Forward and Reverse Training. In forward training, we sample a = ~ ((, 1)), pass through Qh, obtaining y/- Thus, at training time, we perform generation and our loss based on the similarity between samples and the real data x. In contrast, we train standard flows by passing an input y into the inverse flow obtain latent (and sampling is involved). Our process share similarities with the training of GANs, but we don't 't rely on a discriminator. The forward training process enables us to support neural architectures that maximum likelihood training does not support, as we discuss below.
In reverse training, we sample x ~ U[Tminn Fmax)) (where the bounds in U are chosen via a heuristic), and we compute the resulting via Fg. The CRPS loss defined on seeks make the distribution uniform on average, which is consistent with the properties of inverse transform sampling. In this case 7-; hy) can be interpreted CDF function conditioned on Øzj via hj.
On Computing Determinants of Jacobians. Interestingly, since quantile flows trained using variants of the CRPS rather than maximum likelihood, they do not rely on the change of variables formula, and do not require computing the derivative 0r 0r/0= /0 for each transformer r(: This fact has a number of important benefits: (1) it greatly simplifies the implementation of flow-based models, and (2) may also support families of transformers for which computing derivatives 0r/O= would have otherwise been inconvenient.
Note however, that our approach does not fully obviate the need think about the tractability of Jacobians: we still rely on autoregressive models for whom the Jacobian is triangular. We will explore how to further generalize our approach beyond triangular Jacobians future work.
4 AUTOREGRESSIVE QUANTILE FLOWS
Next, we introduce antoregressive quantile flows (AQFs), a class of models trained with our set of novel learning objectives. Quantile flows improve over classical flow models by. among other things, supporting neural architectures that simultaneously provide fast training and sampling.
An AQF defines mapping between and that has the form æ, T hy) hy ()(2,))) in classical flow, with T(2i h2)) being an invertible transformer and cj being the j-th conditioner.
Architectures for Quantile Flows. Crucially, we are free to select a parameterization for either hy) or its inverse -- (zy; hy) and the resulting model will be trainable using of our
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objectives L(), L()). This is in sharp contrast to maximum likelihood, in which need to learn model of (Z;;h)) in order to recover the latent & from the observed data x and compute p(2).
This latter limitation has the consequence that in classical flows in which the transformer - not invertible (which is in most non-affine models), it is difficult or not possible to perform sampling (since sampling requires 7). As a consequence, most non-affine flow models can only be used for density estimation; ours is the first that be used for generation. We give examples below.
Affine, Piecewise Linear, and Neural Quantile Flows. Affine transformers have the form T(2; a, 6) = a +b, hence they are analytically invertible. A special case is the Gaussian transformer with parameters μl, o², which yields the masked autoregressive flow (MAF: Papamakarios et al. (217)) architecture. The MAF can be trained in our framework using maximum likelihood or 7, 700, 7,(2), Affine flows further extend to piecewise linear transformers, increasing expressivity while maintaining analytical invertiblity. Training affine and piecewise linear transformers provides both forwards and a reverse mapping between æ and z in closed form.
For higher expressivity, may parameterize transformers with monotonic neural networks, which can be constructed from a composition of perceptrons with strictly positive weights. Huang ett al. (2018) proposed an approach where conditioner c(Izş) directly outputs the weights of monotonic neural network which models (x, ((),_)))))) In our experiments, T(zj, c(xxz)) is simply parameterized by a network with positive weights that outputs Iz as based on 2j, T<j similarly to (Cao et al. 2019), although AQFs also support the approach of Huaag et al. (2018).
In general, any transformer family that can be used within autoregressive models can be used within the quantile flows framework, including integration- based transformers (Wehenkel and Louppe 202T) spline approximations (Müller et al. 2019 Durkan et al.
Comparison to Other Flow Families.
2019 Dolatabadi et al.) (020). piece-wiie separable models, and others.
A key feature of our framework is that it supports both forwards and reverse training while alternatives do not. For example, when training a neural autoregressive flow (Huang et al.) 2018), we need to compute from but then 7 is not analytically invertible, which precludes us from sampling from it. Our method, on the other hand, parameterizes itself using flexible neural approximator, making possible to casily perform sampling within a flexible neural autoregressive flow for the first time.
compute the log-likelihood. Thus, we parameterize 7 -- with neural network,
Additionally, modeling flexible CDFs enables to study broader classes of distributions, including discrete ones. We perform early experiments and leave the full extension to future work.
5 PREDICTIVE UNCERTAINTY ESTIMATION USING QUANTILE FLOWS
Probabilistic models typically assume a fixed parametric form for their outputs (e.g., a Gaussian), which limits their expressivity and accuracy (Kuleshov et al. 2018). Here, we use AQFs as the basis for quantile flow regression (QFR), which models highly expressive aleatoric uncertainties over high-dimensional variables by directly outputting normalizing flow.
5.1 QUANTILE FLOW REGRESSION
The idea of Quantile Flow Regression (QFR) is to train models H ::: X' (Rd [0, 1l) that directly output a normalizing flow. This approach does not impose any further parametric assumptions on the form of the output distribution (besides that it is parameterized by expressive function approximator). enabling the use of flexible probability distributions in regression, and avoiding restrictive assumptions of the kind that would be made by Gaussian regression.
From an implementation perspective, H outputs quantile flows of the form = T hz)
(6) where each conditioner C: (x g(x)) ) in function of an input x' processed by a model g(x"). We provide specific examples of this parameterization in our experiments section. The entire structure can be represented in single computational graph and is trained end-to-end using objectives T,(1), (222)
h = (y(F,))
5

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Estimating Probabilities from Quantile Flows. In practice, may also be interested to estimate the probability of various events under the joint distribution of æ. This may be done by sampling from the predictive distribution and computing the empirical occurrence of the events of interest. See time series experiments for details, as well as the appendix for pseudocode of the sampling algorithm.
Note that while cannot directly estimate events such as P(r1 I S 12) from variate quantile flow, we can always compute densities of the form P(z)). When fitting flows in the reverse direction, we can simply differentiate each transformer dr /dz to obtain conditional densities. When fitting flows in the forward direction, we can use the formula dt d/dd= 1/P(r(z)).
5.2 QUANTILE AND CUMULATIVE DISTRIBUTION FUNCTION REGRESSION
In the one-dimensional case, our approach yields quantile function regression and cumulative distribution function regression (CDFR), two simple, general, and principled approaches for flexible probabilistic forecasting in regression. In quantile function regression, a baseline model H outputs a quantile function Q(a); in practice, we implement this model via a neural network f(x', a) that takes input an extra [0, 1] and outputs the a-th quantile of the conditional distribution of given x'. We train the model with the quantile loss (2) while sampling
U[1,, 1J) during the optimization process. We provide several example parameterizations of f in the experiments section. Cumulative distribution function regression is defined similarly to the above and is trained with the
CRPS loss ())
Quantile vs. Cumulative Distribution Functions. The quantile function Q makes it easy to derive a-% confidence intervals around the median by querying Q(1 ax/2) and (a//2). Conversely, the CDF function makes it easy query the probability that falls in region [x,, via f(22) - f f(1) Each operation can also be performed in the alternative model, but would require bisection grid search. We suspect users will fit one model or both, depending their needs.
Extensions to Classification. A CDF can also be used to represent discrete random variables; thus our method is applicable to classification, even though we focus on regression. We may a CDF F over K numerical class labels 01 < ... < k the optimal form of F is step function. When F is not a step function, we can still extract from it approximate class membership probabilities. For example, in binary classification where classes are encoded as 0 and 1, we may use (f e) F(0)) an estimate of the probability of æ = 0. We provide several experiments in this regime.
6 EXPERIMENTS
We evaluate quantile flow regression and its extensions against several baselines, including Mixture Density Networks (MDN; Bishop 1994)). Gaussian regression, Quantile regression (QRO; in which fit separate estimator for 99 quantiles indexed by a 0.01, .., 0.99). We report calibration errors (as in Kuleshov et al. (2018)). check scores (CHK), the CRPS (as in Gneiting and Raftery (2007) and LI loss (MAE); both calibration error (defined in appendix) and check score are indexed by 0.01, ..., 0.99. Additional experiments on synthetic datasets can be found in the appendix.
6.1 EXPERIMENTS ON UCI DATASETS
Datasets We used four benchmark UCI regression datasets (Dua and Graff 2017) varying in size from 308- 1030 instances and 6-10 continuous features. We used two benchmark UCI binary classification datasets with 512 and 722 instances with 22 and features respectively. We randomly hold out 25% of data for testing.
Models. QFR and CDFR were trained with learning rates and dropout rates of 3e 3, 3e 4 and (0.2, 0.1) respectively as described in Section 5.2 All of the models were llo- neural networks with hidden layer size 64, and ReLU activation functions.
Results, QFR, on average, outperforms the baselines in each of the one-dimensional UCI tasks (Table []. Interestingly, method was also able to learn CDF for binary classification. We

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Table 1: Results on the regression benchmark datasets.
Check Score
CRPS
Dataset Yacht
Gaussian
QRO
MDN
QFR
CDFR
Gaussian
QRO
MDN
QFR
CDFR
0.225
0.202
0.174
0.171
0.443
0.381
0.282
0.275
0.415
Boston
1.547
1.191
1.115
0.865
2.643
1.441
1.160
0.025
2.373
Concrete
2.243
1.563
2.896
1.191
4.157
1.906
5.703
0.697
2.97
Energy
2.254
1.463
2.926
1.183
4.171
1.911
5.745
0.578
2.91
MAE
Calibration MAE
Dataset Yacht 0.627
Gaussian
QRO
MDN
QFR
CDFR
Gaussian
QRO
MDN
QFR
CDFR
0.563
0.494
0.481
0.539
0.097
0.100
0.019
0.028
0.116
Boston 3.909
3.398
2.933
2.215
2.611
0.139
0.140
0.036
0.034
0.038
Concrete 6.329
4.051
9.022
2.853
3.623
0.165
0.114
0.029
0.046
0.066
Energy 6,629
4.151
9.393
2.815
2.829
0.165
0.114
0.032
0,029
0.049
demonstrated that the CDFR method is able to compete with the neural network baseline and that our method can be potentially extended to classification, testing on the Diabetes and KC2 dataset where the Neural Network obtained metrics of 0.717 and 0.832 respectively, and our CDFR obtained 0.723 and 0. 0.796 respectively.
6.2 OBJECT DETECTION
Data. We also conducted bounding box retrieval experiments against Gaussian baselines on the data corpus VOC 2007, obtained from Everingham et al. with 20 classes of objects in 9963 images.
Models. We modified the architecture of et al. (2019) which uses
Table 2: Obj. Detection with CDFR
KL-lloss to fit a Gaussian distribution for each bounding box. The existing architecture has a MobileNet-v2 backbone (Sandler al. (2019). Keeping the backbone unchanged, the quantile regressor method adds an alternate head with separate layers, which adds uncertainty bounds to the original output. The original Gaussian method was trained for total of 60 epochs, at
Method
CRPS
Gaussian
6.85
with a learning rate of 2c-4 which is decremented by a factor epochs 30 and 45. The quantile flow layer is trained for additional 20 epochs, with the same learning rate.
CDFR
6.15
Table 3: Results on the object detection task
Results. We implement QFR in conjunction in standard model, as well as one that uses variational dropout (VD; Gal and Ghahramani (2016)) In
Method
Cal. MAE
MAP@50
CRPS
each case, the QFR layer improves uncertainty estimation (as measured by CRPS; Table 3). with the best results coming from model trained with QFR and VD. MAP@ 50 is defined in the appendix. This suggests our methods improve both aleatoric and epistemic uncertainties (the latter being estimated by VD). We illustrate examples of predicted
Gaussian
0.133 0.103
0.796
8.89
QFR
0.796
8.59
Gauss. VD
0.771
8.73
QFR VD
0.772
7.99
uncertainties real images in Figure [ We also swapped QFR for CDFR, while keeping approximately the same architecture, and compared to the Gaussian baseline using the CRPS. CDFR also outperforms the Gaussian baseline (Table 2).
6.3 TIME SERIES FORECASTING
Data. Our time series experiment uses quantile flows to forecast hourly electricity usage. We follow the experimental setup of Mashlakov et al. (202T) using their publicly available code. We use the 2011-200 1-2014 Electricity Load dataset, popular dataset for benchmarking time series models. contains 70,299 time series derived from hourly usage by 321 clients across three years. There exist different ways pre-process this dataset; follow the setup Mashlakov et al. (2021). Our models predict hourly electricity usage 24 hours into the future for each time series, for a total of 1,687, 176 predictions. There four covariates.
Models. We benchmark two state-of-ft oftte-of-ttheart time series forecasting models: DeepAR (Salinas al. 2019) and MQ-RNN (Wen et al.] 2018). Both autoregressive LSTM architecture in
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RMAIL ROBERT
HM EETT
LDEW
LOEWS
DEW LOEWS
person: 54%
51%
car. 94%
Figure 1: Examples of bounding box uncertainties the object detection task. The thicker boundary represents the original model prediction, and the thinner lines represent ten quantiles (hetween 0.1 and 0.9) predicted via QFR. High-contidence boundaries (such as those of the car) are closer together, while low-confidence boundaries (e.g., the bus which is blurred and partially occluded) are asymmetrically spread out, reflecting the advantages of Gaussinn uncertainty estimation.
Mashlakov et al. 2021); DeepAR outputs Gaussians each time step, while MQ-RNN estimates five conditional quantiles, which are followed by linear interpolation.
We train recurrent model with same LSTM architecture and replace the predictive component with a quantile flow. Each transformer 7zz;; hx) is a monotone neural network with one hidden layer of 20 neurons, conditioned on the output h2 of the LSTM module, acting as the conditioner (y
Results. Figures 2a and 2b compare predictions from DeepAR and AQF (in blue) on illustrative time series (red). We take samples from both models and visualize the (0.1, 0.9) quantiles in shaded blue. While both time series underpredict, the Gaussian is overconfident, while the samples from AQF contain the true (red) curve. Moreover, AQF quantiles expand in uncertain regions and contract with certainty. We also compare the three methods quantitatively in Table 4 Both achieve similar levels of accuracy (ND and RMSE are as defined in [Salinas al. (2019), but we have included them in the appendix as well), and AQF produces better uncertainties. We also found that for 37.2% of time series, DeepAR and AQF performed comparably for CRPS (CRPS within 5% of each other); 30.3% of time series DecpAR fared better, and on 32.1% 1% AQF did best.
Table 4: Time series experiments
Method
ND
RMSE
CHK
DeepAR
9.9%
0.692
89.15
MQ-RNN
9.9%
0.712
88.91
AQF (ours)
9.8%
0.723
88.02
6.4 GENERATIVE MODELING
A key feature of AQFs is that they can be used for generative modeling with expressive neural parameterizations since they can be trained in the forward direction, unlike Huang et al. (2018). We benchmark three different methods for generative flows, MAF trained via maximum likelihood (MAF-LL) and with our quantile loss (MAF-QL), and our proposed method, Neural Quantile Flow (NAQF) with the same number of parameters on various UCI datasets. The models consist of an LSTM layer followed by a FC layer, with 40 hidden states in each. The MAFs generate the parameters for Gaussian distribution from which the results are sampled.
Results. Figure shows the error curves of the MAF models trained with log-likelihood and with our loss. Interestingly, there is little correlation between the two, which is reminiscent of other model classes trained with different objectives (c.g., LL-Flows and
1256
1250
2000
2000
300
250
150
(a) DeepAR Forecaster
(b) Autoregressive Quantile Flows
Figure 2: Examples of Predictive Uncertainties on the Times Series Forecasting Task
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Gas
Power
BOSI 500
déjectie.
déjective
Tranned
Yrined with Lt dideetivee ML
Trirned with LL elpestive NLL
022
Ttanned with LI déjective
Yained with atjective ML
Tranned with OL otjective NL.
Thaned with OL otgectiv NLL
-200
2300 5000 7500 10000 12500 15000
5000 10000 15000 20000 25000 20000
5000
10000
19520
30500
Figure 3: Log likelihood and CRPS
for the MAF three UCI Datasets.
GANs); in some cases, likelihood can even diverge as we train with our loss. Table evaluates CRPS trained with each model. The best results achieved with the NAQF, even though it uses the same number of parameters, indicating the value of neural transformers.
Table 5: CRPS on UCI datasets.
6.5 IMAGE GENERATION
Dataset
MAF-LL
MAF-QL
NAQF
We demonstrate the ability of our model to generate samples using a neural parameterization by generating images of digits from sklearn (Figure(4). We use 5 LSTM layers with 80 neurons; all models trained to convergence with learning rate le-3, with 500 epochs for the NAQF and MAF-QL, while the MAF-LL given extra 500 epochs to reach convergence. Additionally, we created SVM discriminator to differentiate the real data versus the sampled data from the model (D-Loss Table 6J defined formally in appendix). The better the sampled data is, the lower the accuracy achieved by the discriminator.
BSDS 300
.044
.036
.033
Miniboone
.567
.561
.525
Gas
.645
.565
.513
Power
.542
.506
.502
Hepmass
.617
.614
.523
Results. The NAQF generates signifincantly better samples that are the most challenging to discriminate from real samples, and also echieves the best CRPS values. The MAF method trained with LL achieves worst results and visibly worse samples, despite having the same number of parameters and being trained for longer. Note that despite the MAF-QL
Table 6: Image Generation Experiments
Method
CRPS
LL
Q-Loss
D-Loss
MAF-LL
2.31
-45
0.625
0.978
MAF-QL
2.16
-6187
0.281
0.779
NAQF
2.14
n/a
0.253
0.723
having a considerably worse NI.I,, the samples generated are still more representative the data, according to the discriminative model, and visually the sample quality is much higher.
(a) Real Digits
(b) MAF-LL
(c) OL
(d) NAQF
Figure 4: Images of digits sampled from four autoregressive flow methods.
DISCUSSION AND CONCLUSION
Our work extends the framework flows Pwwssaaaaaaaiosss et al. (2017: 2019) to use a novel learning
objectives. Similarly, it generalizes implicit quantile networks Dabney al.) (20T8b) into
normalizing flow framework. Our work is an alternative to variational (Kingma and Welling. 2014).
adversarial (Goodfellow al. 2014). or hybrid (Larsen et al. 2016 Kuleshov and Ermon 2017)
generative modeling and extend structured prediciton models (Lafferty et al. 2001] Ren et al. 2018)
Interestingly, quantile flows can be seen generalization of quantile functions to higher dimensions.
While several authors previously proposed multi-variate generalizations of quantiles (Serfling 2002
Liu and Wu 2009). neither of them possess all the properties of univariate quantile function, and ours is an additional proposal.

---

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12

---

A UCI DATASETS EXPANDED
Yacht. The yacht dataset consists of 308 instances, features and continuous target.
Boston. The boston house prices dataset consists of 506 instances, 14 features and 1 continuous target.
Concrete. The concrete dataset consists of 1030 instances, 8 features and continuous target.
Energy. The energy efficiency dataset consists of 768 instances, 10 features and I continuous target.
Classification Datasets
KC2. The kc2 dataset consists of 522 instances, 22 features and binary target.
Diabetes. The diabetes dataset consists of 769 instances, features and binary target.
Multivariate Dalasets
ENB. The enb dataset consists of 768 instances, 10 features and continous targets.
B PROOF OF PROPOSITION
Proposition 2. For a CDF F: : + [0,1] and y E the CRPS and quantile losses are equivalent:
( (F. 3²) L (2) (F 3/) + a, L.beR,ax R
(7)
Proof. Recall that the pinball loss is defined as
Ly f) = a(y f) if 2 f and (1 a1- y) otherwise. Observe that can equivalently write
2La (a), v') |P- (a) - s/1 2( 0.5)(F- y/)
We take the expected pinball loss and perform the change of variables y = 1' (a):
2 S La(F s')da = / [aF s'l 2(a 0.5) -
y²)] da
1ly v1 (2 1)(y f()ldd
where (yy is the probability density function associated with F. Using integration by parts and the fact that F(x') T(xx' S x) obtain
and 0c,
1
fz') = 00 (F((') - I(xx'1=''0 s x))² dx.
C EXTRA FIGURES
Sepezsooe Ttrt
60
022
66
69
(a) OpenML. KC2 ROC
(b) OpenML. Diahetes ROC
D SYNTHETIC EXPERIMENTS
In addition to the real-world dataset experiments, we also conducted experiments with synthetic datasets, which provides additional insights into the properties of autoregressive quantile flows.
13

---

Table 8: Calibration Metrics and Accuracy
Metric
AQF
Gaussian.
RMSE
1.017
1,016
Calibration 0.25
0.236
0.342
Calibration 0.5
0.493
0,510
Calibration 0.75
0.746
0.684
Table 9: Results the 1D synthetic data benchmarks.
Check Score
CRPS
Dataset
AQF
Gaussian
MDN
Poisson
True
AQI
Gaussian
MDN
Poisson
True
Gaussian
0.302
0.330
0.308
0.294
0.586
0.600
0.610
0.58T
Beta
0.233
0.251
240
0.224
0.460
0.547
0.493
0.443
Poisson
0.487
0.517
0.537
0.502
0.452
0.965
1.027
1.059
0.993
0.898
D.1 ONE-DIMENSIONAL EXPERIMENTS
We start with a series of nneedimennional experiments involving the simple 1D function which we add additive Gaussian, Beta, and Poisson noise.
Doissonnoiss
The Gaussian and Beta distribution experiments were generated from base function of f(x') sin(2' (a'/// /2) + x' /11 such that x = f(x') + +N(0,1) and f(x') + B(2,5) 5) respectively. The Poisson distribution was generated from P(f(x') + C),, where C is a constant term to make all terms non-negative.
We these functions using autoregressive quantile flows (AQFs) as well as the following Baselines: Gaussian regression, Mixture Density Networks (MDNs), and Poisson regression. All models implement the same neural networks and differ only how the final layer represents aleatoric uncertainty.
We calculated check score and CRPS metrics for each model. addition, we calculated the metrics for the true distribution, AQF is much closer to the true distribution compared to the other models with Gaussian assumptions, especially when performed non-Gausee Gaussian distributions.
upper bound on the performance of the models. As shown in Table g
This shows that fitting an expressive AQF model better captures predictive uncertainty than alternative models that make parametric assumptions parametric distribution (e.g., Poisson or Gaussian) works well if the underlying data actually follows that distribution. When using an AQF, we don't 't need select predictive distribution type, it just works out of the box as if we knew it.
the output distribution. In other words, using
We also report calibration error, defined as the absolute difference between the percentage of points below the predicted quantile curves and the actual quantile value (Table For a perfectly Gaussian noise distribution, the QRF (analogous to AQF 1-dimension) and QRO methods (QRO consists of fitting a separate estimator for 99 quantiles indexed by a 0.01,.... 0.99) comparable to Gaussian regressor in terms of calibration. When the distributions become non non-Gaussian, QRF is adaptable whereas the Gaussian fails to capture the distribution. Thus, the Gaussian maximum likelihood loss tends to be overconfident, a fact also demonstrated in Kuleshov et al. (2018); AQF addresses the overconfidence.
Table 7: Calibration Error on ID datasets
Dataset
QRF
QRO
Gaussian
Gaussian
1.01
1.03
0.952
Beta
1.39
1.75
4.63
Poisson
2.76
2.57
17.64
Supervised Accuracy and Calibration at Key Quantiles We also report calibration values at key quantiles (25%, 50%, 75%) in Table We also measure predictive accuracy in terms of RMSE and demonstrate that our method obtains good estimates of uncertainty without sacrificing supervised performance in terms of regression quality.
14

---

Free pridd
points uasodian sagrasser
ADP
Tiee
(a) AQF
(b) Gaussian
Figure 6: 80% Confidence Intervals obtained via AQF (left) and Gaussian regression (right). Note that confidence intervals obtained from Gaussian regression (right, blue) differ significantly from the true confidence intervals (right, red). Confidence intervals obtained from AQF (left, blue) perfectly overlap with the true intervals (left, red).
In Figure and 6bb report 80% confidence intervals for AQF and Gaussian regression methods on the synthetic beta noise data for visualization of the quantile predictions.
D.2 MULTI-DIMENSIONAL EXPERIMENTS
We also conducted multi-dimeenional experiments with a d-dimensional target æ; we model the first dimension by = sin /3) + x' + NO0,, 1); then, further dimensions are modeled by ya = 5 + Fd-1 + (0, 1). use both the marginal CRPS and the joint CRPS metrics, as described in Jordan et al. (018), for both simple 2d-experiment well as more complex 50-dimensional experiment.
The methods we evaluate are our proposed AQF, Gaussian Markov Random Field (a type of structured prediction algorithm that assumes a multi-variate Gaussian distribution on the output), and model in which we train multi-variate Gaussian distribution using quantile loss instead of log likelihood (denoted as "Quantile" " in the table). As in the ID synthetic experiments, we also compare metrics against samples drawn from the true distribution to obtain upper bound.
Table TO shows that AQF does a better job at capturing the high-dimensional distribution than models that assume more restricted parametric form, such the Gaussian MRF. Simply replacing the objective function with quantile loss instead standard Gaussian noise does not provide as significant of gain as AQF does. In other words, existing models (e.g., a Gaussian MRF) are not sufficiently expressive to capture the multi-variate distribution that he we have defined. In contrast the AQF layer is able to achieve close to approach the theoretical upper bound without making any additional modeling assumptions.
Table 10: Results on Multidimenssonal Synthetic Datasets
CRPS (Marginal)
CRPS (Joint)
Dataset
AQF
Gaussian
Quantile
True
AQF
Gaussian
Quantile
True
2D
0.624
0.669
0.634
0.585
0.686
0.701
0.695
0.646
50D
1.045
1.133
1.058
0.970
1.238
1.345
1.281
1.163
15

---

E ALGORITHM
Algorithm Training Neural Network with the Quantile Loss
Sample (((0, 1)
Sample training instance (x', (d'x)) 2) from the dataset (or batch of instances) Compute prediction Bpred f(z',a)
Compute quantile loss La (x, p(rea).
Take a gradient step quantile loss Tea (2r, F-red)-
F ADDITIONAL DEFINITIONS FOR USED METRICS
F.1 CALIBRATION MAE
If we have 1-dimensional model which predicts curve F(g) for quantile q, then have that Calibration MAE on test set given by
P(X < F(9)) gl
99 9=1
F.2 MAP@50
MAP@ 50 denotes the mean average precision for an IoU value of 0.5. IoU is computed as, given ground truth boxes G and predicted box B, GnB precision-recet recall curve given the IoU.
GUB 2 0.5. MAP is then calculated as the AUC of the
F.3 RMSE AND ND (TIME SERIES)
We will define data point ži,t to be the t-th time in the i-th context window, and E to be the prediction for point z.
Then RMSE is defined as
RMSE
Σ 2
N(T to)
ND (Normalized Deviation) is calculated
Ee, |2,,1 2i,l
ND
Ex,t [iiel
F.4 LL. Q-Loss, AND D-LOSS (IMAGE GENERATION)
Log Likelihood and Q-Loss (quantile loss as defined in our paper) with respect to the digits generation task are optimization objectives and not metrics. We measure these with depict how one affected in the same model trained on the other loss function.
D-Loss (D R {0, 1}}) is measured as the accuracy with which a SVM can discern the difference between generated samples S and true samples Y, which are labeled 0 and I respectively.
Given validation set with m data points and labels X (0, 1}. we have that
D-Loss = m Σ D(x xz + (1 ((x(())) - (()))
1
16

---

F.5 MNIST EXPERIMENTS
For MNIST, we trained autoregressive models with the PixelCNN architecture, Each model was trained for 300 epochs with learning rate of 1e-3, and has hidden layer size 784.
7
9
3
9
3
0
2
0
7
7
P
8
5
5
7
5
4
7
3
/
2
3
PixeIMAF-LL
(b)
PixelMAF-QL.
Figure 7: MNIST samples
Table 11: MNIST Generation Experiments
Method
U-CRPS
CRPS
NLL
PixeIMAF-LL
.128
.279
-6011
PixeIMAF-QL
.099
.215
-5888
Results. From the samples in Figure 7 the results generated from the MAF-QL seems to be much more visually accurate than MAF-LL, which is trained on LL-based loss, even though they have the same architecture.
17


Reviews:
[{"review_id": "zN_rR3sSI28", "reviewer": "Reviewer_UtzJ", "summary": "", "questions": "", "limitations": "", "rating": 8, "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": "Strong points:\n- The proposed objective functions (quantile loss and continuous ranked probability score) appear well-motivated and have nice theoretical properties.\n- The generalization to quantile flows is well-motivated and well-described.\n- The implementation of all methods is clearly described.\n- The paper is well-structured and introduces complex topics well.\n- The analyses of the UCI datasets are comprehensive.\n\nMajor concerns:\n- Since the manuscript proposes several new objective functions and architectures, it would be nice to have comprehensive experiments on synthetic data. In particular,\n    - how well does the method for quantile function regression capture the true quantile function in a univariate case?\n    - do the prediction intervals behave as expected from the theory?\n    - how well does the method estimate the cdf at various points (e.g., the median, the 25th and 75th percentiles)?\n    - compare this method vs maximum likelihood in terms of timing and accuracy\n- Some of the terms aren't defined (e.g., the column headings in Table 3, Table 4)\n\nMinor concerns:\n- check throughout for typos", "summary_of_the_paper": "The paper proposes an interesting approach for estimating densities and providing uncertainty estimates for these densities. The paper extends normalizing flows in several ways: (1) by using objective functions based on proper scoring; (2) by using autoregressive quantile flows; and (3) by defining quantile flow regression.", "summary_of_the_review": "I vote for accepting the paper subject to some additional experiments.", "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": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "uxsbarcW21N", "reviewer": "Reviewer_AxZh", "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": "Overall, the proposed autoregressive quantile flow is a reasonable application of normalizing flow in quantile regression. It is also extendable to various different flow transformations and applicable to a wide variety of regression problems. However, this paper has not provided a very clear and strong motivation of why using normalizing flow (or autoregressive flow in particular) for uncertainty estimation, given there are many models that can be trained by gradient-based methods, one can simply replace their objective functions by quantile/CRPS loss. By choosing an appropriate model according to data property, the performance should also be no worse than normalizing flow. Moreover, many other approaches (e.g., diffusion model) also support forward and reverse training, which is not a unique feature of this method. In addition, the authors may also want to explain the procedure more clearly by adding an algorithm box into the main paper.", "summary_of_the_paper": "This paper proposed a quantile regression method for uncertainty estimation based on autoregressive quantile flow. The flow model can be trained in both forward and reverse setting using different loss functions, and the quantile flow framework can be combined with other linear or non-linear transformations. The authors have conducted diverse empirical evaluations on objective detection (bounding box regression), time series forecasting, and generative models to demonstrate the advantage of the proposed method.", "summary_of_the_review": "This paper has proposed a normalizing flow-based method to address the predictive uncertainty in regression problems. The significance of novelty can be further improved before it can be accepted.\n\n## Update after Rebuttal\n\nThe authors have addressed most of the concerns. It would be better to clarify the difference with diffusion models in the main paper. Good luck!", "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": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "qcbJQh4-NcP", "reviewer": "Reviewer_85oZ", "summary": "", "questions": "", "limitations": "", "rating": 8, "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 strengths of this article are as follows:\n1. The proposed objective is novel, which does not need to compute the inverse of the Jacobian, which is intractable in most cases. Moreover, it enables complicated transformers like deep neural networks to be applied in flow models and simplifies the implementation, thus having good potential for future applications.\n2. Benefit from the design of objective functions, the proposed framework supports a sampling-based training paradigm. Such a mechanism embraces a broader range of loss functions to the flow framework.\n3. It is also interesting to see that the proposed quantile flows could also provide the distribution estimation of model outputs, which can be applied to provide uncertainty of predictions. \n\nDespite the main contributions of the article, it could still be improved from the following aspects: \n1. Providing more detailed descriptions of the evaluation metrics for better understandability.\n2. Some typos are found:\n  - In the second paragraph of subsection 2.2, “the quality of forecasts if often...” -> “the quality of forecasts is often...”.\n  - In eq. (1), “F_\\theta (y_i)” -> “F_\\theta (y)”.\n  - In the third paragraph of subsection 6.1, a period is missing at the end of the first sentence.\n3. Please cite the source of datasets or provide links. Also, please unify the reference format and cite the following paper: \nSandler, Mark, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. \"Mobilenetv2: Inverted residuals and linear bottlenecks.\" The IEEE conference on computer vision and pattern recognition, pp. 4510-4520. 2018.\n", "summary_of_the_paper": "This paper proposes a novel framework for training flow models named Autoregressive Quantile Flows (AQF). The proposed method utilizes a new objective by evaluating forecasts with proper scoring rules, including the continuous ranked probability score and the check score. The advantages of the proposed objective are 1) it could avoid the explicit calculation of the determinant of the Jacobian matrix and 2) it could also provide uncertainty estimation for predictions. Experiments on multiple tasks including regression, object detection, time series forecasting, and generation validate the effectiveness of this framework.", "summary_of_the_review": "This article provides a novel and interesting idea to improve flow models in multiple aspects. I think this work can inspire other researchers in the community, so I tend to accept the paper after some minor modifications.\n\nPost-Rebuttal\n----\nThe paper has its merits and the authors have revised the manuscript according to the comments. \n\nThus, I keep the score and recommand 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": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a framework for training autoregressive flows based on proper scoring rules. The proposed framework is shown to be a computationally appealing alternative to maximum-likelihood training, and is empirically validated in a wide variety of applications.

All three reviewers are positive about the paper and recommend acceptance (one weak, two strong). The reviewers describe the paper as well written and well motivated, and recognize the paper's contribution as significant.

Overall, this is a nice and promising methodological exploration of flow-model training that is worth communicating to the ICLR community.
Decision: Accept (Spotlight)