Reformulating Strict Monotonic Probabilities with a Generative Cost Model

Authors: Yongxiang Tang, Yanhua cheng, Yanxiang Zeng, Xiaocheng Liu, Jiaochenchen, Ning Luo, Yuanpengjia, Xialong Liu, Peng Jiang
Primary area: generative models
Keywords: monotonic model, variational inference, generative model

Abstract:
In numerous machine learning contexts, the relationship between input variables and predicted outputs is not only statistically significant but also strictly monotonic. Conventional approaches to ensuring monotonicity focus primarily on construction or regularization methods. This paper establishes that the problem of strict monotonic probability can be interpreted as a comparison between an observable revenue variable and a latent cost variable. This insight allows us to reformulate the original monotonicity challenge into modeling the latent cost variable and estimating its distribution. To address this issue, we introduce a generative model for the latent cost variable, called the Generative Cost Model (\textbf{GCM}), and derive a corresponding loss function. We further enhance the estimation of latent variables using variational inference, which reformulate our loss function accordingly. Lastly, we validate our approach through a numerical simulation of quantile regression and several experiments on public datasets, demonstrating that our method significantly outperforms traditional techniques. The code of GCM is available in https://github.com/iclr-2025-4464/GCM.

Decision: reject
Reviewer scores: 3, 6, 6, 5

Reviews and discussion:
<thread>
creation date:
1730354700860

thread_type:
official_review

<thread_object>
creator: reviewer_cjwc

<content>
summary:
The paper proposes a latent variable model capturing both a monotonic part and a part that need not be monotonic. They give some background, and then propose a loss function for a variational training with neural nets.

strengths:
It is an interesting problem and their construction is quite novel and creative.

weaknesses:
The objective function and implementation leaves a lot of details missing and unexplained. The paper does not inspire confidence in the method.

questions:
Overview with some questions embedded:

Lines 054-059 could be clearer with a graphical model diagram. Edit: there is one later, nice.

Line 063 may not be quite right; p_theta(x,r) cannot be ignored in the evidence unless you drop the dependence on theta. Maybe rework the setup / explanation a bit, this shouldn’t be a big issue.

Line 101 equivalent to what?

Line 150 consider the machinery within https://arxiv.org/abs/2301.11695 and also the separate line of work on normalizing flows that all involve e.g. invertibility, monotone transformations, etc.

Line 210 (0,1) should be {0,1}

Line 189 should **y** and r have the same dimension?

Line 289 elementwise

Line 304 looks like it would suffer from high variance since pi(z) is fixed but p(z|x) depends on x.

Line 323 the combination of losses looks a little suspicious.

Line 323 Note that IWAE (Burda) is equal to ELBO (Jordan et al) for IWAE number of samples set to one; why is adding ELBO to IWAE sensible? Shouldn’t pi be something else and then no ELBO? Please expand, this is unconvincing.

Line 324 it seems like this doesn’t really match the beta vae setup; it would if you put a beta in front of the kl in line 318. What is this doing?

Line 378 how about some ablations of alpha and beta terms? Edit: there are some in appendix C.

Table 1 does not seem to match appendix C for any value of the parameters, what is missing? Please add details.

Line >= 378 how did you set all of the parameters?

Line 698 affect
</content>
</thread_object>

</thread>
<thread>
creation date:
1730658498790

thread_type:
official_review

<thread_object>
creator: reviewer_e3qf

<content>
summary:
The paper proposes the Generative Cost Model (GCM) to enforce strict monotonicity by modeling a latent cost variable, using variational inference and importance sampling. This approach avoids architectural constraints and outperforms traditional methods on synthetic and real datasets.

strengths:
The paper seems to present a novel approach to formulate the problem of strict monotonic probability, it provides convincing theoretical analysis and emprical results.

weaknesses:
The paper lacks empirical validation across diverse, high-dimensional real-world datasets, limiting the demonstrated generalizability of the Generative Cost Model (GCM). Moreover, the code was not provided in the supplementary materials to assess the reproducibility of the results.

questions:
1. Can the authors provide additional justification or empirical validation for the assumption of conditional independence between the latent variable z and revenue r given x, as this assumption may affect model flexibility in real-world applications?

2. Could the authors elaborate on the computational efficiency of the Generative Cost Model (GCM) compared to traditional monotonic models, especially when applied to high-dimensional datasets? How does the computational time compare to the benchmarks?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730677923741

thread_type:
official_review

<thread_object>
creator: reviewer_ta4m

<content>
summary:
This paper proposes a novel approach to model strict monotonic probabilities. Without loss of generality, it considers the binary classification formulation $y|x, r\sim Bernoulli(y;G(x,r))$, where $G(x,r)$ is a function that is monotonic in $r$. The target is to learn the function $G$. Instead of directly learning $G$, the paper introduces a cost variable $c$ and reformulates $G(x,r)$ as an integration $\int_{c<r} p(c|x)d c$. The paper then introduces a generative cost model to approximate the conditional distribution $p(c|x)$.

strengths:
The paper tackles an interesting problem of modeling monotonic probabilities. The problem itself is important, and the reformulation proposed by the paper is unique. Extensive experiments are conducted to support this new method.

weaknesses:
The paper lacks theoretical results on the finite sample efficiency of the proposed algorithm. For the experiment design, an important setup that requires strict monotonicity is quantile regression, where the conditional quantile should be monotonic in the quantile argument. It would be interesting to see experiments designed for it and a comparison with the existing benchmarks.

questions:
No additional questions.
</content>
</thread_object>

</thread>
<thread>
creation date:
1731026157323

thread_type:
official_review

<thread_object>
creator: reviewer_njcz

<content>
summary:
The paper models a conditional monotonic distribution as a marginalization over latent variables. The key idea is to modify the monotonic modeling problem into modeling an element-wise cumulative distribution function (over a cost variable C). To actually model the latter, the authors introduce a latent variable modeling problem and solve it via a standard importance-weighted likelihood estimate with or without an additional ELBO term. The paper presents improved results in two experiments over a variety of baselines.

strengths:
- Clean reformulation of the monotonic problem into a classification over a latent variable.
- Well-motivated techniques to solve the problem.
- Comparison against a variety of baselines.

weaknesses:
No confidence intervals in the results tables. It's not clear that the improvement achieved by GCM is large enough (even over the simple MLP) to justify the much more complicated modeling procedure.

questions:
- How wide are the confidence intervals for each of the metrics? Just do standard bootstrap if possible and report please.
- Why introduce the extra prior $p_\theta(Z)?$ Also, why is the prior different from $\pi(Z)$?
</content>
</thread_object>

</thread>
<thread>
creation date:
1732870880297

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Key points of the final revision (part I)

comment:
Dear all reviewers, thank you for all the comprehensive reviews and valuable suggestions, we have submitted the final revision, and here are the modifications we made comparing to the original paper.

## 1. We revised the latent distribution and simplified the loss functions of GCM and GCM-VI.

The latent distribution we used in the original paper is categorical, and to deal with it, we have to use importance sampling to estimate evidence. However, if we traverse all possible values of $z$, it is equivalent to the law of total probability, which gives the exact estimate of the evidence by:

$p_\theta(y|z,r)= \frac{1}{K}\sum_{j=1}^{K}p_{\theta_1}(z=k|x)p_{\theta_2}(y|z=k,r).$

In this case, we can directly optimize the evidence and we have the loss function:

$\mathcal L_{GCMcate}(\theta; x,  r, y)= -\log \left[\frac{1}{K}\sum_{j=1}^{K}p_{\theta_1}(z=k|x)p_{\theta_2}(y|z=k,r)\right].$

For complex latent variables, for example, $z$ form the Gaussian distribution, we cannot use the law of total probability (impossible to integrate on $z$) or importance sampling (high variance). We follow the classical reparameterization trick and reformulate the loss of GCM as:

$\mathcal L_{GCM}(\theta; x,  r, y)= -\log \left[\frac{1}{K}\sum_{k=1}^{K}p_{\theta_2}\left(y|{x},{r},{z}_k\right)\right],$

where $z_k \sim p_{\theta_1}(z|x)$. To make a better estimate of $z$, we adopt a recognition model $q_\phi(z|x,r,y)$, similar to the IWAE, the variational version of GCM (GCM-VI) has a loss function formulated as:

$\mathcal L_{GCM-VI}(\theta, \phi; x,  r, y)=  -\log \left[\frac{1}{K}\sum_{k=1}^K\frac{p_{\theta_2}(y| z_k,  r)p_{\theta_1}(z_k| x)}{q_{\phi}(z_k| x,  r, y)}\right],$

where $z_k \sim q_{\phi}(z|x,r,y)$. The three revised loss functions are simplified compared to the original paper. We remove the hyperparameters $\alpha$ and $\beta$ in the original version formulated as $\mathcal L=-LL-\alpha ELB+\beta D_{KL}(p_\theta(z|x)\|p(z))$. So we now have fewer hyperparameters and the loss function is cleared without a redundant linear combination of $LL$ and $ELB$.

We performed an ablation study on these three loss functions in Appendix C.2, and the results are as follows. 

|Method|	Adult ACC|	COMPAS ACC|	Diabetes ACC|	BlogFeedback RMSE|
|---------------------|----------------------------|----------------------------|----------------------------|----------------------------|
|GCM-Categorical|0.8850$\ \pm$0.0013| 0.6983$\ \pm$0.0010| 0.8443$\ \pm$0.0003| 0.0988$\ \pm$0.0010|
|GCM-Gaussian|0.8854$\ \pm$0.0013| 0.6991$\ \pm$0.0011| 0.8441$\ \pm$0.0001| 0.0994$\ \pm$0.0003|
|GCM-Gaussian-VI|0.8858$\ \pm$0.0014| 0.7011$\ \pm$0.0011| 0.8442$\ \pm$0.0002| 0.1005$\ \pm$0.0004|

We can see that GCM-VI and GCM-categorical perform the best, this is consistent with their objectives, since GCM-categorical is trained by the exact $LL$ and GCM-VI provides a better estimation of the latent $ z$ than the original GCM.
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Metareview:
metareview:
The paper addresses the problem of conditional monotonic distribution modeling by representing it as a marginalization over latent variables. The central idea is to reformulate the monotonic modeling challenge into modeling an element-wise cumulative distribution function (CDF) with respect to a cost variable CCC. To achieve this, the authors introduce a latent variable modeling framework, which is solved using a standard importance-weighted likelihood estimation, optionally incorporating an ELBO term. The original paper has many issues and errors. For example, the basic conditional independence assumption was found unnecessary in the rebuttal period. These major changes are not fully verified in the rebuttal phase and it was hard for the reviewers to fully review the new version again. I do not think the current version is ready for acceptance.