reviews: 
Comments: In this paper, the authors propose an approach that reformulates the inference problem as a classification task using contrastive learning and the generation of soft labels. 

These soft labels, derived from an unnormalized target distribution, are employed in a cross-entropy loss to fit the variational distribution. The authors demonstrate the efficacy of their methodology through a series of experiments on Bayesian inference.
Strengths: The paper presents an elegant method for fitting the variational distribution by training the model to solve a classification problem. 

I identify two key benefits of the proposed approach:

1. There is no need for samples from either the negative or the target distribution.
2. The densities of the target and negative distributions need only be known up to a normalizing constant.

Overall, the paper is well-written and easy to follow.
Weaknesses: I note the following weaknesses in the paper:

### Major:

1. Deep Learning examples:

The experimental evaluation feels modest and misses an opportunity to show the benefits of the proposed methodology in deep learning scenarios. The authors focus solely on "classical" examples of Bayesian inference. I think it would be very illustrative to consider deep learning applications.

For example, in the related work section, the authors mention papers on Variational Auto-Encoders (VAE). It would be interesting to see how the proposed methodology performs in this context.

I see several ways your approach could be applied:

- End-to-End VAE training: Instead of optimizing the standard ELBO (or alternatives like IWAE [1]), incorporate the SoftCVI training procedure and evaluate the negative log-likelihood.
- Fitting variational distribution for pretrained VAE: For an already pretrained VAE (decoder) with a complex, multimodal posterior, apply your training procedure to fit a variational distribution to this posterior. Then demonstrate that the variational distribution fitted with SoftCVI covers more modes and provides more diverse samples (see e.g. [2]).

I appreciate the example with the SLCP task. Similarly, an example with a "circular" posterior in [3] could be considered for demonstration.

2. Training curves with different $K$:

Since $K$ is a hyperparameter we can choose as we wish, it's important to understand how it influences the results. In Figures 1 and 6, two different choices of $K$ are considered, but there is no discussion on which is better. It would be beneficial to see not only the final evaluations but also the training curves for different $K$ values. Given that $K$ can range widely (from 2 to infinity), practical guidance on selecting $K$ would be valuable.

Additionally, training curves are interesting in the context of comparison with SNIS-fKL, as the variance of estimation will be greater there, and it is interesting to see it.

### Minor:

1. Motivation in Section 2.1:

Initially, I found the beginning of Section 2.1 a bit confusing. It mentions a scenario with $K−1$ negative samples and only 1 positive, which doesn't seem to match the scenario in the paper. As I understand it, we have samples only from the proposal distribution and assign probabilities to them as if they come from the true distribution to generate soft labels. We don't assume that only one sample comes from the true distribution. If my understanding is incorrect, please clarify. Otherwise, consider rephrasing this section for clarity.


2. Typo:

Line 329: "in contrast **to to**".

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I am positive about the methodological contributions. With the incorporation of extensions to the experimental section, I am willing to significantly increase my evaluation score.


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References:

[1] Burda, Y., Grosse, R., & Salakhutdinov, R. (2015). Importance Weighted Autoencoders. arXiv preprint arXiv:1509.00519.

[2] Thin, A., Kotelevskii, N., Denain, J. S., Grinsztajn, L., Durmus, A., Panov, M., & Moulines, E. (2020). MetFlow: A New Efficient Method for Bridging the Gap Between Markov Chain Monte Carlo and Variational Inference. arXiv preprint arXiv:2002.12253.

[3] Thin, A., Kotelevskii, N., Doucet, A., Durmus, A., Moulines, E., & Panov, M. (2021). Monte Carlo Variational Auto-Encoders. In International Conference on Machine Learning (pp. 10247-10257). PMLR.
Recommendation: 6
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 3