Revisiting Unbiased Implicit Variational Inference

Thank you very much for your constructive feedback. We have made the following revisions in response to your comments:

1. Computational Cost Comparison: Following your suggestion to investigate computational costs, we have now performed comparisons based on the Conditioned Diffusion Process task for all methods. To facilitate a fair comparison, we use the total iterations of KSIVI (outer loop iterations × inner loop iterations) instead of just the outer iterations as done in Sec 5.3. This comparison highlights that AISIVI learns much faster per iteration than KSIVI (as shown in the table), but also emphasizes the extraordinarily fast implementation of KSIVI when comparing run times. Overall, the comparison confirms that only the gold standard KSIVI and our AISIVI reach log marginal likelihood values > 70,000, while all other methods notably lag behind. This result aligns with the visual comparison in Fig. 4 and demonstrates that it is possible to achieve such performance without relying on a kernel-based approach.

2. Explanation for Consistency: Regarding your question

On line 250 in column 2: "This alternating training is possible since sIS,k(z) is a consistent estimator of the score gradient for any τε|z with supp(qε|z) ⊂ supp(τε|z)."

about the consistency of alternating training: This statement means that as long as K (the number of importance samples) is large enough, we can draw from any proposal distribution that satisfies the support assumption. Since the score gradient estimator ($s_{\text{IS}}$) is consistent, it converges almost surely. This allows us to keep the proposal distribution fixed while updating the SIVI model, and update the flow afterwards with our unbiased forward KL gradient, which moves the current CNF closer to the optimal proposal distribution (the target) for the new SIVI model. This enables convergence to the global solution. We will make this more clear in an updated paper version and thank the reviewer for this question.

3. Typo and Reference Update: We have corrected the typo and greatly appreciate the reference to Andrade (2024). We will replace the current citation with Andrade (2024), as it provides a more relevant theorem on variance underestimation.

4. Algorithm Clarification: We updated the algorithm description to make it clear that θ refers to the parameters of τ (the normalizing flow). This change should clarify the notation and its role in the model.

We hope these revisions address your concerns. Thank you again for your detailed feedback, which has been incredibly helpful in improving the clarity and rigor of our work.

Thank you for your thoughtful feedback and valuable suggestions. We have made several improvements to address your concerns:

1. Improved Figures and Tables: We have revised Figures 2, 3, and 4 for better clarity and ensured they are color-blind-friendly. Specifically, we have revised the figure of Experiment 3 to better illustrate the performance differences. Additionally, we have added a table for clearer comparisons. The updated results now show that in Experiment 3, AISIVI and KSIVI outperform all other methods in terms of log marginal likelihood.

2. Additional Baselines: We have expanded our comparisons to include UIVI and IWHVI. Furthermore, for the multimodal example, we added a normalizing flow baseline. The results demonstrate that normalizing flows either smooth out low-density regions or introduce additional fragments (as observed with the neural spline flow).

3. Convergence Speed and Stability: KSIVI remains the fastest in terms of convergence speed, but our method is the only one that matches its performance. Interestingly, AISIVI does not require very large inner batches if the proposal distribution (e.g., a normalizing flow) is sufficiently flexible and learns faster per iteration than KSIVI. Our framework efficiently supports computationally expensive proposal distributions, such as CNFs, since we do not need to backpropagate through the proposal distribution when computing path gradients. This property is one of our main novel contributions. However, without a proposal distribution, large batch sizes are indeed necessary, as demonstrated by BSIVI.

4. Computational Cost Comparison: Following the suggestion to investigate computational costs, we have now performed comparisons based on the Conditioned Diffusion Process task for all methods. To facilitate a fair comparison, we now use the total iterations of KSIVI (outer loop iterations × inner loop iterations) instead of the outer iterations as done in Sec 5.3. While this shows that AISIVI learns much faster per iteration than KSIVI (as seen in the table), it also highlights the extraordinarily fast implementation of KSIVI when comparing run times. Overall, the comparison clearly shows that only the gold standard KSIVI and our AISIVI are able to reach log marginal likelihood values >70,000, while all other methods lag behind significantly. This confirms the visual comparison of Fig. 4 in the paper and demonstrates that such performance levels can be achieved without relying on a kernel-based approach.

5. Citations: We will incorporate all the suggested references and are very grateful for your recommendations, which further enhance our paper’s positioning within the broader literature.

Thank you again for your constructive feedback, which has helped us refine and strengthen our work. We hope these improvements address your concerns.

Thank you for your thorough review and insightful feedback. We greatly appreciate your suggestions, and we have addressed the following points:

1. High-dimensional Applicability: We argue that the definition of "high-dimensional" depends on the specific goal. Finding an adequate approximation is achievable in higher dimensions, but our target for Experiment 3 was the true distribution. As shown in the newly added table and improved figure, only our method and KSIVI from all SIVI variants come close to matching this distribution. Thus, for this scenario, 100 dimensions can be considered sufficiently high, as no other SIVI variant can achieve such good results. We will clarify in the final manuscript what we mean by 'high-dimensional' in our setting to avoid any confusion.

2. Computational Cost: We have now included UIVI in Experiment 3, and as you suggested, we’ve focused on the log marginal likelihood. It is clear that UIVI’s performance does not come close to AISIVI, even for a similar simulation budget. Note that we can only afford an inner batch size of 2 because of the expensive inner MCMC loop, which further emphasizes our point about drastically improved computational costs. Similar findings are reported in Appendix D of SEMI-IMPLICIT VARIATIONAL INFERENCE VIA SCORE MATCHING by Longlin Yu et al., which further supports our argument. Thank you for strengthening our paper with this suggestion.

3. Reparameterization Trick and IWHVI: We have corrected our previous claim, and indeed, the reparameterization trick can be applied to IWHVI. We appreciate your clarification on this point.

4. Optimization Strategy: The alternate optimization scheme is a key aspect of our method and one of the main contributions. By using the path gradient and alternating optimization, we avoid backpropagating through the proposal distribution. This allows us to process arbitrarily large inner batches, whereas a simultaneous optimization approach is limited by memory constraints. This design enables us to use a more computationally expensive proposal distribution (conditional normalizing flow), and as shown in our updated experiments, AISIVI outperforms IWHVI in terms of performance within a similar computational budget. This now serves as an ablation study comparing simultaneous and alternating training approaches. We have also included IWHVI as a strong baseline, as suggested.

5. Typo: We have corrected the typo in Eq. 15, as you pointed out.

Thank you again for your valuable feedback and for helping to improve the clarity and strength of our paper. We hope these changes address your concerns effectively. Should the reviewer find the response satisfactory, we would appreciate reconsidering the initial score. Otherwise, we remain fully committed to addressing any remaining concerns during the second author response phase.

1. Experiment 1 Comparison: We agree that no comparison is needed for Experiment 1, as it primarily serves as a sanity check rather than a competitive benchmark.

2. Correlation in Experiment 2: We computed all possible correlations separately for the true $\beta$ and the estimated $\beta$, considering each coordinate individually. A negative correlation, $\rho_{ij}$, means that $\beta_i$ and $\beta_j$ (where $\beta$ represents here for example the true $\beta$ in this case) are negatively correlated. After obtaining these correlations for both the true and estimated $\beta$, we matched them so that the best possible outcome would form a diagonal line, ensuring alignment between the learned and true $\beta$. In Eq. 36, this can also be seen since the correlations are always computed with respect to a fixed parameter vector. We will adapt the manuscript to make this process clearer for the reader. Our results demonstrate the state-of-the-art performance among SIVI variants. We will clarify this point again in the paper and thank the reviewer for the comment.

3. Experiment 3 Evaluation:

“From the results and the figures, it seems like all methods are doing equally well.”

In Experiment 3, we included a table demonstrating that only our proposed method (AISIVI) performs on par with the state-of-the-art KSIVI method. Additionally, we have improved the figure to better highlight that AISIVI is the only method capturing variance as effectively as KSIVI.

4. Code Availability: We have inquired with the area chair about the permissibility of providing the code currently hosted on (Anonymous) Github (but the rebuttal rules only allow figures and tables to be linked).

5. Notation and Equation Clarifications: The use of subset notation (⊆) is indeed correct, given our definition of $\mathcal{R}$. However, we recognize that this notation might lead to misunderstandings and will adapt it to a more standard form. We have also incorporated the suggested clarification for Eq. (2).

6. KL Divergence Normalization: As far as we know, there is no standard way to normalize KL divergence. However, to provide better intuition, we have added a figure depicting different training states alongside their KL values, helping to contextualize the reported numbers.

We hope these clarifications address your concerns and demonstrate our commitment to improving the manuscript. Thank you again for your valuable feedback. Should the reviewer find the response satisfactory, we would appreciate reconsidering the initial score. Otherwise, we remain fully committed to addressing any remaining concerns during the second author response phase.