Thank you for the constructive and encouraging comments as well as the insightful questions. We are glad that you find our method "justified" by "theoretical and empirical arguments", our paper "well written"/"easy for the reader to follow", and the empirical evaluation showing our method's "consistent gains over baseline approaches". Additionally, we appreciate your recognition that our proposed necessary modifications are "a core part of the paper's contribution". Below we will address your questions in detail.

W1. The proposed method requires additional compute time and memory... by briefly mentioning it in the main paper...

This is a good suggestion. Following to your and Reviewer Adn3's suggestion, we conducted an additional study on the post-training computational cost for different methods. We calculate the inference time and maximum memory usage for LAP, standard LoRA, and BLoB. The results are presented in the table below:

These results show that compared to LAP, our BLoB can achieve comparable or better ECE and NLL with less inference time and less memory usage. Notably, our BLoB's memory overhead compared to standard LoRA is minimal, while LAP introduces significant memory overhead.

We will include the discussion above in the main body of the revised paper as suggested.

Q1. ...the KL divergence term. How important is this? ... this was not explicitly discussed in the main paper...

This is a good question. Indeed, the KL divergence reweighting is crucial to BLoB's final performance. Due to space limit, we initially relegated this discussion to the Appendix. In our revision, we will present a comprehensive ablation study covering our novel KL reweighting schedule, parameterization, and asymmetric Bayesianization of LoRA, as suggested.

Q2. Is there any particular reason why $A$ is modelled in a probabilistic way instead of $B$?...

Thank you for this insightful question. As discussed in Section 3.1 (lines 129-131), BLoB models $A$ probabilistically rather than $B$ due to the conventional LoRA initialization: $A\leftarrow \text{random init}$, $B\leftarrow 0$. The choice is indeed arbitrary; this asymmetry could be reversed (i.e., model $B$ probabilistically instead) if the LoRA initialization were flipped (i.e., $A\leftarrow 0$, $B\leftarrow \text{random init}$).

Q3. Limitations

Please refer to Q1. [Discussion of Limitations] in General Response.

Thank you for your constructive feedback and insightful questions. We are glad that you like our idea and find our method "promising" and its performance "good"/"competitive". Below we address your major questions (W2, W3, Q3, and Q5) in detail. The remaining questions will be answered in a separate Official Comment after this Rebuttal post.

W2. It seems in practice many extra efforts are needed to get the method to work, e.g., the complex KL re-weighting and rescale size of training data mentioned in Appendix B.1.

Thanks for mentioning this. Our KL re-weighting technique ensures the model fits the data well while converging to the prior distribution, akin to the BBB approach [1]. By rescaling the training data size, we maintain consistent hyperparameters across varying dataset sizes.

As Reveiwer AGd6 noted, these adjustments are not limitations but key technical contributions ("a core part of the paper's contribution") that enable BLoB to work effectively across diverse datasets.

[1] Blundell, Charles, et al. "Weight uncertainty in neural network." International conference on machine learning. PMLR, 2015.

W3.1. I don't see the need to make Theorem 3.1 and Theorem 3.2 theorems, nothing surprising is stated there... I like the idea of the paper, but I think it can be presented in a more honest way.

This is a good question. Actually our Theorems 3.1 & 3.2 are necessary because they

• shows that with a proper $\tilde{R}$, one can compute the KL divergence for the high-dimensional full weight vec($W$) simply by computing the KL divergence for $A$, which is much lower-dimension, more parameter-efficient, more memory-efficient, and faster, and
• investigate important theoretical properties of our BLoB, offering insights into its underlying assumptions and advantages.

We will include the clarification above in our revision as suggested.

W3.2. ...I don't think the results justify the "superior" claim.

For the performance of BLoB (N=10), in the ID experiments in Table 1, it achieves the best ECE and NLL performance on almost all datasets, with similar ACC compared to MLE and MAP. In the OOD experiments in Table 2, BLoB ($N=10$) achieves the best performance in 7 out of 12 metrics across the four datasets, while the second-best LAP method achieves the best performance in only 4 metrics. The results clearly demonstrate that BLoB outperforms other baseline methods.

Q3. What is the difference between BBB and BLoB?... How many samples are used to evaluate BBB

The key distinctions between BLoB and BBB are:

1. Asymmetric Bayesianization (AB): BLoB models the approximate variational distributions for only one LoRA component $A$. This technique is crucial, as classic BBB consistently fails on various datasets without it. In fact, to produce meaningful results for BBB in Tables 1 and 2, BBB has to incorporate our proposed AB, giving it the baseline BBB an unfair advantage.
2. Novel Parameterization and Training: BLoB introduces a new method to parameterize the standard deviation of matrix $A$ and employs a different KL-reweighting strategy during training. Table 1 demonstrates that these innovations are essential for successful uncertainty estimation.

As mentioned by Reviewer AGd6, these certain distinctions "are necessary for the algorithm to work in practice" and "are a core part of the paper's contribution, and they are discussed thoroughly and justified with a mix of theoretical and empirical arguments."

Number of Samples. For fair comparison, both BLoB and BBB use $N=10$ samples in Table 1. We will clarify this in the revision.

Q5. I would be interested to see Laplace added to Table 8 to show a fair comparison of the computation cost compared with Blob.

This is a good question. We divide the computational cost into two phases: training and post-training (inference on test data). LAP's training cost is equivalent to LoRA's, as shown in Table 8, since it only involves MAP training. For a comprehensive comparison of post-training computational costs, we calculate the inference time and maximum memory usage for LAP, standard LoRA, and BLoB. The results are presented in the table below. The complete table, including the corresponding ECE and NLL for reference, is provided as Table 1 in the PDF attached to the General Response.

These results show that compared to LAP, our BLoB can achieve comparable or better performance with less inference time and less memory usage. Notably, our BLoB's memory overhead compared to standard LoRA is minimal, while LAP introduces significant memory overhead.

The remaining questions are answered in a separate Official Comment titled "Remaining Questions for Reviewer Adn3" below.

W1. Not clear about the method's limitations.

Please refer to Q1. [Discussion of Limitations] in General Response.

Q1. What it mean for the mean matrix of q(A) to be an output of a neural network (Line 186-187)?

We are sorry for the typo. $q(A)$ is not the output of a neural network. It is directly modeled as a Gaussian distribution $\mathcal{N}(M, \Omega)$, with $M$ and $\Omega$ as learnable parameters.

Q2. In Eq. (70), it seems different mini-batch has different KL re-weighting hyperparamers lambda? How sensitive is the training to the value of lambda?

Yes, different mini-batches has different $\lambda$ values. When we use the typical settings of $\lambda_i = \frac{2^{M-i}}{2^M-1}$ [1] or $\lambda_i = \frac{1}{M}$ [2], the model struggles to fit the data distribution while converging to the prior distribution, resulting in an "NaN" NLL loss. Our setting of $\lambda_i = \frac{2^i}{2^M-1}$ allows the model to fit to the data in the early stages of training and achieve a reasonable level of complexity cost in the later stages. This re-balance scheme of data likelihood and complexity cost has proven effective across multiple datasets and can be potentially applied to other VI methods.

We will include the discussion above in the revision as suggested.

[1] Blundell, Charles, et al. "Weight uncertainty in neural network." International conference on machine learning. PMLR 2015.

[2] Graves, Alex. "Practical variational inference for neural networks." NeurIPS 2011.

Q4. I would be interested to see a figure plot of model performance versus the number of MC samples. ...

This is a good suggestion. Following your suggestion, we have run additional experiments for model performance versus the number of MC samples on WG-S dataset. The results are presented in the table below. The corresponding figure is provided as Figure 1 in the PDF attached to the General Response.

We can see that in general, the performance improves with more MC samples, but saturates after around $N=10$.

Q6. Why is Laplace performing so poorly on ARC-C dataset? The original Bayesian LoRA paper it seems to perform much better on that dataset.

Please refer to Q2. [Reproducing LAP's Results on the ARC-C Dataset] in General Response.

Q7. Line 236, MLE and MAP are not uncertainty estimation methods.

We apologize for the confusion. We will remove "uncertainty estimation" when describing MLE and MAP in the revision as suggested.

Q8. I don't entirely agree with the claim made in Lines 46-47. The possible suboptimal estimation of Laplace... never know if the complex NN has really converged to MAP; (2) setting prior precision...

This is a good point. We agree that these are also major reasons that lead to Laplace approximation's poor performance. We will include these two additional reasons in our revision as suggested.

Q9. Notation issue and missing Bolding

We are sorry for these typos and will correct them in the revision as suggested.

Thank you for your constructive feedback and insightful questions. We are glad that you find our proposed methodology "competitive" with "simplicity", "efficiency", and "good contribution". Below we will address your questions in detail.

W1. The flow of the paper is rather poor, and the presentation of the method overly complicated. I prefer the authors to simply start from (5) and...

We sincerely appreciate your valuable feedback on the organization of our paper. We will make every effort to incorporate your suggestions to ensure that our model presentation is clearer and more understandable.

W2. ...the problem of having to involve imprior priors... In short, the theorems are unneccessary and need fixing because of improper distributions.

This is a good question.

Degenerate Gaussians. Note that in our context, degenerate Gaussian distributions are still valid for probabilistic inference, as demonstrated by Schoeman et al. [1] (Eq. 12, 16-18), because

• their probability density is well-defined, and
• the KL divergence between two degenerate Gaussians is computable.

Due to space constraint, we provide detailed discussion in a separate Official Comment titled Degenerate Gaussians are Proper for Probabilistic Inference below this Rebuttal.

Necessity of Theorems 3.1 & 3.2. Our Theorems 3.1 & 3.2 are necessary because they

• shows that with a proper $\tilde{R}$, one can compute the KL divergence for the high-dimensional full weight vec($W$) simply by computing the KL divergence for $A$, which is much lower-dimension, more parameter-efficient, more memory-efficient, and faster, and
• investigate important theoretical properties of our BLoB, offering insights into its underlying assumptions and advantages.

We will include the clarification above in our revision as suggested.

[1] Schoeman, Johannes Cornelius, Corné E. van Daalen, and Johan A. du Preez. "Degenerate Gaussian factors for probabilistic inference." IJAR 2022.

W3. Eq (3) to remove the $\min_\theta$ prefix, and to introduce the prior and likelihood.

This is a good suggestion. We will remove the prefix in the revision.

W4. Section 4: Shouldn't ECE and NLL be the metrics in focus because this is about "Bayesian"?

Yes, ECE and NLL are indeed the primary metrics for evaluating LLMs' uncertainty estimation. However, we must also consider the model's accuracy to ensure a balance between uncertainty estimation and predictive performance. For example, a random classifier could achieve perfect calibration (zero/optimal ECE) and potentially lower NLL than a poorly calibrated model. Thus, while prioritizing NLL and ECE, it is also crucial to maintain ACC comparable to MLE and MAP baselines.

W5. Section 4: The numbers in the table does not justify saying BLoB is the best.

For the performance of BLoB ($N=10$), in the ID experiments in Table 1, it achieves the best ECE and NLL performance on almost all datasets, with similar ACC compared to MLE and MAP. In the OOD experiments in Table 2, BLoB ($N=10$) achieves the best performance in 7 out of 12 metrics across the four datasets, while the second-best LAP method achieves the best performance in only 4 metrics. The results clearly demonstrate that BLoB outperforms other baseline methods.

W6. "Bayesianization" is a mouthful --- can we think of a better derived word?

Thanks for mentioning this. We follow prior work to use "Bayesianization" as the noun form of "Bayesianize" [1]. This term accurately captures the process of transforming a deterministic component into a Bayesian one.

[1] Bacharach, Michael, and Susan Hurley. "Issues and advances in the foundations of decision theory." Foundations of decision theory (1991): 1-38.

Q1. Line 137: What is it that requires accurate estimation?

The term "accurate estimation" refers to estimating $E_{A, B}[BAx]$. When both $A$ and $B$ are modeled as Gaussian posteriors, $E_{A, B}[BAx]=0$ holds before fine-tuning. However, accurately estimating this expectation requires an impractically large number of weight samples. In our BLoB approach, we model $B$ as deterministic and initialize it to 0 (similar to LoRA), avoiding this issue and leading to faster convergence with fewer samples. We will include the discussion above in the revision.

Q2. How does more than one sample (for N>0) used in generating a single output?

This is a good question. Ideally, a Bayesian neural network's output is the expected output $E_{q(W|\theta)}[P(Y|W,X)]$, where $X$ is the input, $Y$ is the output, and $q(W|\theta)$ is the approximate posterior of the parameters. In practice, we approximate this expectation through sampling: $E_{q(W|\theta)}[P(Y|W,X)]\approx \frac{1}{N}\sum_{n=1}^{N} P(Y|W_n, X)$, where $W_n\sim q(W|\theta)$ is the n-th sample drawn from the approximate posterior, and $N$ is the total number of samples.

Q3. Limitations

Please refer to Q1. [Discussion of Limitations] in General Response.

In this separate Official Comment, we will elaborate on two key reasons why degenerate Gaussian distributions are valid for probabilistic inference:

1. Their probability density is well-defined.

According to Schoeman et al. [1], the probability density of a degenerate Gaussian distribution can be factorized into the product of the density of a non-degenerate Gaussian distribution in a lower-dimensional linear subspace and a Dirac delta function (Eq. 12 in [1]). This density function has an alternative expression in the form of limit (Eq. 16-18 in [1]), which aligns with our main theorem 3.1 and 3.2:

$$p(x) = \lim_{a\rightarrow 0}\mathcal{N}(x | \mu, Q(\Lambda^{-1} - aI)Q^T + a I) = \mathcal{N}(x | \mu, \Sigma=\lim_{a\rightarrow 0} Q(\Lambda^{-1} - aI)Q^T + a I).$$

Here $\Lambda^{-1}$ is the precision matrix of the lower-dimensional non-degenerate Gaussian distribution, and $Q$ is the linear transformation matrix that expands this to high dimensions. In our main paper, we use the simplified notation $\mathcal{N}(x | \mu, \Sigma)$ for readability, omitting the limit. The full limit notation and proof appear in Appendix A.1. We will clarify this simplification in our revision.

2. The KL divergence between two degenerate Gaussians is computable under certain conditions.

Secondly, the KL divergence between two degenerate Gaussians can be computed under the condition that they "have support on the same lower-dimensional manifold" (Eq. 44 in Schoeman et al. [1]). This condition has also been clearly stated as satisfied in our Theorem 3.2 ($RR^{T} = BB^{T}$, line 177), and we also include a detailed derivation of how we reach this condition in Appendix A.1 (line 676-678).

[1] Schoeman, Johannes Cornelius, Corné E. van Daalen, and Johan A. du Preez. "Degenerate Gaussian factors for probabilistic inference." IJAR 2022.

W1 + W2. I understand that the method is sound because your degenerate Gaussians are on the same subspace. It is now clear to me that you also understand this constraint. I'll increase the soundness score, and the overall score.

However, I still believe that the presentation can be vastly simplified (and hence the paper more accessible) by simply starting from (5). The relation to full Bayesianization can be deferred to discussion.

W5.  It is "generally better" than the other methods. You can also say it is "the best on average" if you do some averaging either over the score or by counting wins. You cannot just say it is "the best".

Q2. I see that the "experimental setup strictly adheres to the original LoRA [2] framework". It is good to say this explicitly in the paper. For ACC, do you generate, say N=10, and then count if any is correct? Is there some discount because you have N chances?

Thank you for your valuable suggestions on enhancing the presentation quality of our work. Following your suggestions, we have refined Sections 3.1 and 3.2. Below is the outline of our updated version:

Section 3.1: Low-Rank Variational Approximate Posterior Distribution

1. Main Method: Introduction of Asymmetric Bayesianization Scheme

   • Calculating each entry of the full weight matrix: $W\_{ij}=W\_{0,ij} + \sum\_{k=1}^r B\_{ik}A\_{kj}$ (i.e., Eqn. 5 of the paper)
   • $A$ modeled by independent Gaussians: $P(A|\theta)=\mathcal{N}(A|M,\Omega^2)$
   • $B$ modeled as deterministic values

2. Presentation of Theorem 3.1

   • Statement: Asymmetric Bayesianization corresponds to assuming a low-rank Gaussian variational distribution for the full weight $W$

3. Discussion on the Choice of Asymmetric Bayesianization

   • Advantages over Bayesianizing both $A$ and $B$:
     • Stable training at the early stage
     • Faster convergence of parameters during training
     • Lower memory cost

Section 3.2: Low-Rank Prior Distribution

1. Presentation of the Prior Distribution on the Low-Rank Component $A$

2. Corresponding Prior Distribution on the High-Dimensional Space of the Full Weight Matrix $W$

3. Presentation of Theorem 3.2

   • Statement: Full-weight KL divergence can be efficiently computed on the low-rank component

4. Dedicated Remark on the Legitimacy of using Degenerate Gaussians for Probabilistic Inference (as in Our Previous Discussion)

Thank you once again for your suggestions. We believe this revision will make the paper more accessible for our audience and help address any potential areas of confusion.

Thank you for your constructive feedback and insightful questions. We are glad that you find the problem we address "important", our empirical evaluation of BDL on LoRA "make sense", and our paper "clear". Below we will address your questions in detail.

W1. Novelty is limited compared to Yang et al. [1], which implements a very similar idea via Laplace approximations instead of the Bayes-by-Backprop variational posterior.

Please refer to Q3. [Novelty of BLoB] in General Response.

W2. The method by Yang et al. ("LAP") is not replicated successfully in Table 1.

Please refer to Q2. [Reproducing LAP's Results on the ARC-C Dataset] in General Response.

Q1. As $N\rightarrow \infty$, would we expect the accuracy to converge to those obtained with $N=0$ (mean prediction)? Is BLoB an unbiased estimator of the mean prediction?

This is a good question. Ideally, a Bayesian neural network's output is the expected output $E_{q(W|\theta)}[P(Y|W,X)]$, where $X$ is the input, $Y$ is the output, and $q(W|\theta)$ is the approximate posterior of the parameters. In practice, we approximate this expectation through sampling: $E_{q(W|\theta)}[P(Y|W,X)]\approx \frac{1}{N}\sum_{n=1}^{N} P(Y|W_n, X)$, where $W_n\sim q(W|\theta)$ is the n-th sample drawn from the approximate posterior, and $N$ is the total number of samples. When $N=0$, we use the mean of the weights to make predictions $P(Y|E_{q(W|\theta)}[W], X)$. Due to the nonlinearity of neural networks $P(Y|W,X)$, as $N \rightarrow \inf$, the average output converges to the expectation, but typically differs from the mean weight prediction. Thus,

• the sample mean BLoB uses, $E_{q(W|\theta)}[P(Y|W,X)]\approx \frac{1}{N}\sum_{n=1}^{N} P(Y|W_n, X)$, is an unbiased estimator of $E_{q(W|\theta)}[P(Y|W,X)]$, but
• BLoB is not an unbiased estimator of the mean prediction $P(Y|E_{q(W|\theta)}[W], X)$, and the mean prediction is not an unbiased estimator for the Bayesian prediction $E_{q(W|\theta)}[P(Y|W,X)]$ either. To summarize, when $N \rightarrow \infty$, $\frac{1}{N}\sum_{n=1}^{N} P(Y|W_n, X) \rightarrow E_{q(W|\theta)}[P(Y|W,X)] \neq P(Y|E_{q(W|\theta)}[W], X)$.

To provide more context, we report the results for $N=0$ to $N=160$ on the WG-S dataset in the table below, demonstrating improved uncertainty estimation with increased number of samples but not converge to that of $N=0$.

Q2. Why is an ensemble used here? Shouldn’t the uncertainties come from the dropout variational posterior? Why are multiple models necessary?

We apologize for the confusion. By “ensemble of 10 LoRAs”, we meant to say “For MCD, we sample 10 times from the variational posterior distribution of LoRA with a dropout rate of $p = 0.1$ during inference." We will clarify this in the revision.

Q3. Will you open source the code on publication?

Thank you for your interest. The implementation of our method is straightforward and compatible with different LLMs. In fact, we have cleaned up the code, and will release the code upon acceptance of the paper.

Q4. Suggestions & Typos

We sincerely appreciate the reviewer's careful reading and for pointing out the typos. We will fix them in the revision as suggested.

Q5. Limitations

Please refer to Q1. [Discussion of Limitations] in General Response.