BLoB: Bayesian Low-Rank Adaptation by Backpropagation for Large Language Models
Authors: Yibin Wang
Haizhou Shi
Ligong Han
Dimitris N. Metaxas
Hao Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: Bayesian Neural Network
Finetuning
Large Language Models
Primary area: probabilistic_methods
TL;DR: We introduce a principled Bayesian framework for improving large language models' generalization and uncertainty estimation.

Abstract: Large Language Models (LLMs) often suffer from overconfidence during inference, particularly when adapted to downstream domain-specific tasks with limited data. Previous work addresses this issue by employing approximate Bayesian estimation after the LLMs are trained, enabling them to quantify uncertainty. However, such post-training approaches' performance is severely limited by the parameters learned during training. In this paper, we go beyond post-training Bayesianization and propose Bayesian Low-Rank Adaptation by Backpropagation (BLoB), an algorithm that continuously and jointly adjusts both the mean and covariance of LLM parameters throughout the whole fine-tuning process. Our empirical results verify the effectiveness of BLoB in terms of generalization and uncertainty estimation, when evaluated on both in-distribution and out-of-distribution data.

BLoB: Bayesian Low-Rank Adaptation by Backpropagation for Large Language Models
Yibin Wang +11
Haizhou Shi +11
Ligong Han
Dimitris Metaxas
Hao Wang f1
Abstract
Large Language Models (LLMs) often suffer from overconfidence during inference, particularly when adapted to downstream commain-specifi tasks with limited data. Previous work addresses this issue by employing approximate Bayesian estimation after the LLMs are trained, enabling them to quantify uncertainty. However, such post-training approaches' performance severely limited by the parameters learned during training. In this paper, go beyond post-training Bayesianization and propose Bayesian Low-Rank Adaptation by Backpropagation (BLoB), an algorithm that continuously and jointly adjusts both the mean and covariance LLM parameters throughout the whole fine-tuning process. Our empirical results verify the effectiveness of BLoB in terms of generalization and uncertainty data. bayesi an-peft.
estimation, when evaluated on both in-distribution and Code is available at https: https://giiee //tpt//ggtttttttt/ cww/ Betm/Wanng-MMMaaae ML- ML--Lab/bayesieeeeeeaaaa
Introduction
Despite the recent advancements in Large Language Models (LLMM) [9, 106, 105, 66, 15, 15,4,91,99 92, 77, 12, 71], addressing the challenges of reliability and responsibility remains imperative [42, 100, 99]. LLMs often produce overconfident responses detached from factual grounding, posing potential harm users [3, 107, 44, 43, 87, 50, 118, 111, 120, 33, 68, 115, 45J. Therefore, accurately estimating response confidence (or uncertainty) is crucial to preemptively intervene before harm occurs. Current research predominantly focuses on eliciting the internal capability of uncertainty estimation of LLMs. For example, studies suggest that verbalized uncertainty yields cetter-calibrated results compared to conditional probability [87, 45].
While effective, the aforementioned methods do not offer a universal solution for expressing LLM uncertainty across all scenarios, especially when adapted [113] to domain-specific corpora, human preferences, or downstream tasks [44]. Even a well-calibrated LLM may struggle to estimate uncertainty during fine-tuning due to catastrophic forgetting of general knowledge [84]. Moreover, when applied to limited-scale downstream tasks, excessively over-parameterized LLMs can rapidly overfit, leading to overconfidence. Thus, enabling accurate uncertainty estimation of LLMs is vital for their reliable and responsible deployment.
Bayesian methods emerge natural solution for learning uncertainty estimation abilities among their counterparts [88, 11, 101, 98, 29, 46, 51, 61, 97, 58, 102, 20, 110]. These methods model predictive uncertainty P(y ae, D) by marginalizing the posterior parameter distribution P(O after observing the dataset D:
P D) - Plyla ())
(I)
However, adapting the Bayesian framework L.LMs poses significant challenges. LLM architectures typically incorporate complex components, including non-linear activation functions, rendering exact
Equal Contribution. *Rutgers University. 2MIT-IBM Watson AI Lab. "Correspondence to: Yibin Wang <yibin. rrtngrrrrtrrscduu. Haizhou Shi chaizhou,she shi @ rutuers.edu>, Hao Wang chw448@@s.mtggerrccdn.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Bayesian inference of parameter posteriors intractable, i.e., unable to compute the integral precisely. Consequently, finding an accurate approximation algorithm for the true posterior distribution becomes primary challenge. Additionally, modeling parameter posterior distributions demands extra memory space, imposing a prohibitive burden on systems due the massive scale of L.MMs.
Contemporary methods leverage Arrameter-Efficiie Fine-Tuning (PEFT) to reduce the number of tunable parameters, thus alleviating computational and storage resource burdens [23, 41, 26, 121, 56, 53J. Built on this, recent research explores Bayesianizing only the PEFT module during fine-tuning to calibrate LLMs [8, 103, 116, 69], somewhat relieving the burden of introducing more parameters for posterior approximation. However, initial investigations suggest that straightforward combinations PEFT and basic Bayesian techniques like Monte-Carte Dropout (MCD, [29]) or Deep Ensemble (ENS, [51, 8, 103]) yield only marginal improvements in generalization and uncertainty estimation. The most promising results to date involve Kronecker factorized Laplace approximation, applied after maximum& posteriori (MAP) estimation provided by any optimization algorithm [1 16]. Nevertheless, we argue that such post-training procedures bifurcate posterior approximation into two stages, inevitably leading to suboptimal estimation.
To address this challenge, we propose Bayesian Low-Rank Adaptation by Backpropagation (BLoB), a Bayesian Deep Learning framework for fine-tuning LLMs with LoRA. BLoB jointly estimates the low-rank variational distributions' mean and covariance throughout the entire fine-tume tinettning stage via backpropagation. Unlike methods relying on post- test-training approximation, BLoB enables simulta- neous estimation of both the parameter mode (i.e., the mean if one assumes Gaussian distributions) and the parameter variance. Random sampling of model parameters based on variance estimation can enhance mode estimation. It thereby improves model performance in terms of accuracy and datasets, as verified by our
uncertainty estimation on both in-distribution and extensive experiments across multiple datasets. summary, our contributions are:
We propose a principled Bayesianization framework for Low-Rank Adaptation (LoRA) in Large Language Models (LLMs) by assuming that full weights' approximate posterior dis- tribution has low-rank structure containing linear combination of independent Gaussian distributions.
We show that, under mild conditions, optimization of the full-weight variational distribution can be done efficiently in the low-rank space of the weight update matrices.
We introduce BLoB, a variational Bayesian low-rank adaptation framework for LLMs that jointly learns the mean and covariance of the variational distribution during fine-tuning.
Extensive evaluations demonstrate the superiority of BLoB in terms of generalization and uncertainty estimation across different scenarios.
2 Preliminaries
In this section, we describe the notation as well as some preliminaries.
Notation. In this paper, scalars are denoted by lowercase letters, vectors by lowercase boldface letters, and matrices by uppercase boldface letters. Probability, expectation, and the datasct are denoted by P. I., and D, respectively. We use [m] 21,2.-- m to denote the set of consecutive integer numbers starting from and ending at m. For a matrix X = [Z1,-.. xx,] E m I
we use vec(X) = [ ",,"",,,"]' E [mm)x1 to denote the vectorization operation; we use
1/-
define the p-norm of matrix. We use @ and to denote the Kronecker product and the element-wise product, respectively.
|X Eis
2.1 Low-Rank Adaptation (LoRA)
Inspired by the pioneering work on identifying and leveraging the low intrinsic rank of over- parameterized models during fine-tuning [55, 2], Low-Rank Adaptation (LoRA) assumes a low rank for the network's weight updates
[41]. Typically a single linear layer, LoRA decomposes each update matrix AW B into the product of two low-rank matrices, where B E R and A E Rexn, Here, m, n, and denote the number of input neurons, output neurons, and the rank of the decomposition, respectively [41]. The forward pass of the linear layer with LoRA is formulated

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as:
z Woh + AWh Woh + BAh.
(2)
where h and 2 denote the input and output of the layer. Since the rank r < min n is significantly smaller than the numbers of input and output neurons (e.g., r 8 < m n 4096 in the attention layer [41 LoRA can drastically reduce the number of trainable parameters by approximately three orders of magnitude compared to full- aallmparreeter fine-tuning, while achieving comparable performance to the full-rank fine-tuning. This also leads to similar reduction in memory consumption for storing optimizer states, thereby reducing the hardware requirements for fine-tuning LLMs to great extent.
2.2 Variational Bayesian Networks (VBNs)
Bayesian Neural Networks (BNNs) estimate the posterior distributions of network parameters rather than relying on single-point estimates [10, 102]. Due to the intractability of exact inference of the true posterior, Variational Bayesian Networks (VBNs) approximate the true posterior using variational distribution; this is done by minimizing its KL. divergence from the true posterior distribution [39, 30, 11]. Specifically, if the weights W's variational distribution q( (WWB) (0) is parameterized by 0, minimizing the divergence KL[g( W () [[P(W is equivalent to minimizing the following variational free energy with respect to 0 [67, 117, 28]:
F(D,0) A Eg の) llg P(D) W)] + KLLg(W 0) || P(W)].
The final formulation of the objective function in Eqn. offers another interpretation beyond mini- mizing the KI. divergence between the variational and true posterior distributions [11]. Specifically, the first term maximizes the likelihood of the data, while the second term regularizes the variational distribution a( g(W(), (0). We refer to the first term as the likelihood cost and the second term as the com- plexity cost. Optimizing these two terms involves balancing the expressiveness of the approximate posterior distribution and its simplicity.
Optimizing the first term of Eqn. 3 requires integrating out the parameterized variational distribution, necessitating Monte Carlo gradient estimation [52, 8 81]. Using this approach, we can incorporate the re- pe-aaaeettrrzzaiinn trick enable backpropagation of the gradient to the underlying parameter [72, 47, 78]. In Bayes By Backprop (BBB) [11], J the variational distribution is further simplified diagonal Gaussian A M(,, o )²), where log(1 + exp(p)) ensures the standard deviation is positive. Then we have the Monte-Carlo estimation of Eqn. that can pass the gradient to A:
F(D,0) K
log P(D)WA) + K
Σ k=1
llgg (Wal0) log P(We)),
(4)
where Wi = f + +og( (1+exp(p)) EE is the k-th sample of the weights yielded by parameterization and ~ (0, I). In BBB, the authors assume the prior distribution P(W) = = of) + +11 T (O, oσ) be mixture of Gaussians. Consequently, they optimize the second term based on weight sampling. In different scenarios, a simpler form of the prior, which allows for a closed-fomt solution, can also be considered. Although our proposed method largely based on the existing framework of BBB, trivially combining BBB with LoRA does not yield satisfactory results. It is important to note that our specific designs are necessary to encourage the fast convergence of the variational distribution, which will be introduced later in Sec. 3.
3 Methodology
In this section, we formally introduce our proposed method, Bayesian Low-Rank Adaptation by Backpropagation (BLoB). We begin by discussing the design choices for Bayesianizing LoRA pa- rameters in Sec. 3.1, highlighting the assumptions BLoB makes about the approximate posterior in the full- weight space. Next, in Sec. 3.2, we explore the low-rank structure of the prior distribution in the full-weight space, which in turn motivates our choice of prior distributions in the low-rank parameter space. Sec. 3.3, we introduce our parameterization method for the variational distribu- tions. In Sec. 3.4, we integrate Flipout [1 [108] into LoRA for improved sampling efficiency and faster convergence. Finally, we present the complete algorithmic description of BLoB in Sec. 3.5. Proof of the theorems and claims in this section can be found in Appendix A.
3

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Pretrained Weights
B
popy Prefeiined
M
Pretrained Weights W3 E 3 gmionn
M
W, 2
We
A
*
R
(6) LoRA
Laplace LoRA
BLeB (Our)
Figure I: Overview of our Bayesian Low-Rank Adaptation by Backpropagation, i.e., BLoB (right) as well comparison with existing methods such as LoRA (left) and Laplace LoRA (middle).
3.1 Low-Rank Variational Approximate Posterior Distribution: LoRA Bayesianization
Asymmetric LoRA Bayesianization. In LoRA [41], the weights treated asymmetrically. A is randomly initialized, usually from the standard normal distribution or using Kaiming initialization, while B 0 is initialized ties of the pre-trained weights at the start of fine-tuning. The trivial solution of estimating the variational approximate posterior for the entire set of LoRA parameters training convergence. For example, consider the Gaussian posteriors q( (A - (AAMM A. 52%) and q( (B(0) NV o, 222)), where Q2 and f2 are variance estimates added A and B, respectively. Although the expectation EA, (Wo + Ba)=] functionality of the pre-trained model, accurate estimation requires an impractically large number of weight samples. Such variational distributions lead to significant fluctuations during the early stages of fine-tuning, unless the initial variance of B. S2g + 0 is intentionally minimized towards zero. Therefore, we take an asymmetric approach to initialize $2 0 and keep it fixed throughout the fine-tuning process. This, in effect, gives up Bayesian modeling of the B component and focuses only on the posterior of A in LoRA, as shown in Fig. 1.
matrix ensure that the model fully retains the capabili-
significantly hinder
Wn + EA, B Ax] Wn preserves the
Additional Advantages. In addition reducing sampling noise and improving convergence speed, our Bayesianization design has two further advantages. First, compared to modeling the variational distributions of both A and B, our approach significantly reduces additional memory cost by approximately 50% per layer. Second, our design is equivalent to finding a posterior estimate for the full- full-weight matrix with low-rank structure. For instance, by assuming deterministic and Bayesianizing P(A = ([AMM, s2²), each element of the full weight matrix Wij is calculated as
WG Wo,is
Σ Bu Auji k=1
where Auj N(Mkj. (k is drawn independently Vk E [r]. It is noteworthy that due to the low-rank structure defined in Eqn. 5, the full-weight parameters of W are no longer independent from each other. The correlation among them can be reflected by the following theorem:
Theorem 3.1 (Variational Distribution of the Full-Weight Matrix in BLoB). With the pre-trained weight matrix Wo E Rmxn and the low-rank weight update matrix B E RRxr, suppose that the variational distribution of the other low-rank update matrix A E R is Gaussian with {(A 0 = [M, (t}) = II,,,NN (Asgg Mag S2, where M = [My] E Tgrx and f = [2az] E TR are its mean and standard deviation, respectively. The equivalent variational distribution defined on the full weight matrix W as in Eqn. 3 is given by
g(vec(W)) 0) (vee(W)))u, Eg), where
(6)
[bg = vec Wo + BM).
(7)
Σg I & B [diag(vec(12)3)] [I, & B...
(8)
Theorem 3.1 shows that our asymmetric LoRA Bayesianization is equivalent to using a Gaussian variational distribution for the full weight W (i.e., Eqn. 6), with a flexible covariance matrix (i.e., Eqn. 8), to approximate the postetior distribution of the full weight W.

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Remark. The covariance matrix Ey is strictly singular. which consequently inspires us to design prior P(W) with such low-rank structure in Sec. 3.2. Previous work on low-rank Gaussians rypically considers covariance with similar structure D + Eq. where D is diagonal [89, 70, 86, 90, 73]. However, sampling from a Gaussian with this structure requires sampling noise of the same shape as the full-weight matrix, which is not parameter-eftcient we therefore do not adopt this in our work.
3.2 Low-Rank Prior Distribution
In Eqn. 3, optimizing the KL divergence between the variational and prior distributions in the space of full weights can be burdensome. Therefore, we assume the prior distribution of the full weights to be a low-rank Gaussian, with its mean centered the pre-trained weights vee (o) and its covariance matrix parameterized by rank-r' matrix R E ((mm)xrr;
P(vec( W)) (V(eoc W) Ep);
where
MM vec( (W))
(9)
Σp RR
Assuming low-rank prior distribution and designing appropriate R. allows optimize the KI. divergence in the decomposed low-rank weight space, as suggested by the following theorem.
Theorem 3.2 (Efficient Computation of Weight KL Divergence). Suppose the pre-trot pre-trained weights Wo, update matrix B, and the variational distribution qf A() are defined as in Theorem 3. 3.1, and the prior distribution of the full-weight matrix P vco W)) is defined as Eqn. 9. Consider the Gaussian prior distribution P(A) II. (A,5l0, of)): then have:
KL ((vec W) B, P W))] = KL (L(((A)))P(A))..
(10)
R R & R). where R satisfies RR BB".
Theorem 3.2 shows that with proper R. one can compute the KL divergence for the high-dimensionn full weight vec( W) simply by computing the KL divergence for A. which is much lower dimee-dimensoon. more parameter-efficient more mnemory-efficien, and faster. Note that the Gaussian distributions we define for both the prior and the posterior are degenerate. However, they are valid for proba- bilistic inference [83], as (i) their probability density is well-defined, and (ii) their KI. divergence is computable under the assumptions of Theorem 3.2. See Appendix A. for detailed discussion.
Concretely, we assume that the prior distribution in BLoB follows the low-rank structure described in Theorem 3.2 and minimize the KL divergence term for the low-rank component A using its analytical solution in Eqn. 3:
KL (M, (M)))))P( = 2 - (I ||0)
Σ
log Pay-
(11)
3.3 Parameterization of the Low-Rank Variational Distribution
The parameterization of the Gaussian variational distribution q(A(0) significantly affects the conver- gence speed of the KL term in Eqn. 11. The mean matrix M of g (() has no additional constraints, we therefore parameterize it directly as the output a neural network. Each entry of (AA())ss diago- nal covariance matrix Ω (i.e., standard deviation) is non-negative; we therefore use element-wise parameterization Qz G. where G = [G,y] Rrxx the real parameter matrix that determines the standard deviation Ω. Since Ω is usually initialized with small positive values close to zero, our parameterization method provides large gradients initially, contributing to the rapid decrease of the KL term. We further show, both theoretically and empirically, that our parameterization method, unlike BBB's softplus function log + exp(-)) is crucial for the fast convergence of when ((A close to the prior distribution P(A) (see more analysis in Appendix A.2).
3.4 On Improving the Sample Efficiency of BLoB
Improving Sample Efficiency with Flipout. One main challenge in estimating the variational distribution (i.c., the approximate posterior) during fine-tuning lies in the sample efficiency of the
5

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Algorithm Bayesian Low-Rank Adaptation by Backpropagation (BLoB)
Require: dataset D, pre-trained weight Wo, low-rank component B, 0 {M, G} for parameteriz- ing the mean and variance of A; Require:
prior standard deviation op: initialization hyperparameter number of input features Require: number of samples during training K, number of iterations T,, learning rate fy;
M ~ u (-V
1: G U( V2 E),
Initialization of A's parameters.
2:
8+0
Di Initialization of B.
3: fort=
do Sample a matc- of data D; ~ D containing b samples.
4:
5: for I,... K do
6:
Sample batched noise Eg ~ N (N, I). 7: Let [Ei} + BLoBFlipout (kk)
Sample the noise. Eqn. 12
8:
Let (Ak = M G Eaj 1ーー
9:
end for
E Ej=1 log P(D:) B) + 207 (M + || - - Eig log Gaj- 1 Let F: -T KT kk=
10:
11:
D Eqn. 13 and 11.
12:
Calculate the gradient w.r.t. the parameters:
AM = 8F./aM, Ac = 8F,/aG, AB = 8F./aB.
Update the parameters:
M -- M- nAm;
G G A
B B B
14: end for
weights [109, 25, 58]. During mini-batch stochastic gradient descent, batch of examples typically share the same weights drawn from the variational distribution. This can lead to slow convergence of the likelihood cost in Eqn. 3. Drawing inspiration from [108], introduce the technique of flipout to speed the sampling procedure of our low-rank variational distributions (A)
LoRA Flipout. Unlike the original approach, which applies rank-1 random flipping to the full weights, we apply flipout exclusively to the low-rank component A. Specifically, suppose we have mini-batch of input vectors H Raxb where represents the batch size. We randomly sample two low-rank flipping matrices S E
and T { 1-1,+ +1] --1,+1 Denoting as E E gixn the weight noise sampled for this mini-batch. the batched output Z after applying flipout is then
= W.H B(MH + [EE 22) - T)).
(12)
crucial that the independent noises added to the low-rank weight noise AA é E S ensure sampling independence across examples within mini-batch, thereby enhancing the sampling efficiency of the algorithm. This is done without violating the assumptions outlined in Theorem 3.1 and 3.2. As illustrated algorithm 1, we use Egy to represent the equivalent noise applied to parameter A for the j-th example in the k-th batch after BLoBFlipout. Due to the low-rank structure of our Bayesianization method, the computational overhead of employing flipout in BLoB is also minimal.
3.5 BLoB: Final Algorithm
We are now ready to present our full BLoB algorithm.
During training, under the assumptions outlined in Theorem 3.1 and 3.2, optimizing the evidence lower bound on the full weight W be efficiently done in the low-rank space, using the following final objective function:
F(D, B, - EEq( B. log P(D)W) KL(G(W (B,,) P(W))
EE,AA P(D)A, B)] + KL[g(A|0) || P(A)),
(13)
where = (M denotes the set of the parameters underlying the variational distribution of the low-rank matrix A. Additionally, to trade off between data fitting and posterior approximation,

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Table I: Performance of different methods applied to LoRA on Llama2-7B pre-trained weights, where Accuracy (ACC) and Expected Calibration Error (ECE) are reported in percentages. The evaluation is done across six common-sense reasoning tasks with shared hyper-parameter setting after 5,000 gradient steps. We use to represent the number of samples during inference in BLoB. "f" and " indicate that higher and lower values preferred, respectively. Boldface and underlining denote the best and the second-best performance, respectively.
Datasets
Metric
Method
WG-S [82]
ARC-C [18]
ARC-E [18]
WG-M [82]
OBQA [65]
BoolQ [17]
MLE
68. 10.99
29.10
85.65
74.53-a0
81.52=0.25
86.53
MAP
68.621971
67. 2045
86.55=055
75.61 :an
81.38_0As
86.50gm41
MCD [29]
69 69.4620.62
68. 1.30
86.21
76.45=004
81.72-6.10
87.29
ENS [51.8, 103]
69.57:40m
66.20 2.000
84,40=081
75.32
81.38=0091
87.09=n.1
ACC (1)
BBB [11]
56.54
68. 13217
85.86 05.8
73.63=2244
82.06=0.59
87.21=002
LAP [166]
10.20
66.78_0.9
80.05
15.55
02.12:2067
86.95
(N=0)
70.89
1..83
86.682040
1..55
82.73
86.80
BLoB (N=5)
66.30
1.15
84.74
12.89
81.79=0:94
86,47=015
BLoB (N=10)
69.07
68.81
85,56
73.69
81.52200.44
86,99
MLE
29.83-ao.5
29. 1.9
13. 1.99
20.62.a074
12.55
3.18
MAP
29.7614987
29.4
12.07:a55
23.07:0044
13,26-0.2
3.16:1023
MCD [29]
27. 27.98
27.53
12.20
19.55
13.10=0.11
3.46
ENS [51,8, 103]
28.52
19:16722
12.57=0058
20.86
15.34=6227
ECE (4)
BBB [1]]
21.81
26.23-1147
=2.28
15.76
11.38=1007
3.74
LAP [116]
4.15
16.25
33.29
7.40
8.70
1.30
BLoB (N=0)
20.62.1083
20.61
9,4321
11.23
8.36
2.46
BLoB (N=5)
10.89
11,22+aa5
6. 16:62222
4.51-0035
3.40
1.63
BLoB (N=10)
9.35
9,59-141
3.64
3.01
3.77
MLE
3.17
2.85-027
1.17:10
0,95
0.73
0.32-10000
MAP
2.462034
2.66-6.11
0.90
1.6220.99
0.75
0.33:2000
MCD [29]
2.79
2.67.2015
1.000.14
1.02
0.77
0.31 1:0000
ENS 8, 103]
2.71
2.46-0.22
0.82
1,25=0083
1 1.622004
0.57::0.02
NLL (L)
BBB [11]
1.40a055
2.23-0004
0,91
0.84=0.15
0.66:ans
0.31:
LAP [116]
0.6020.0
1,03
0.88
0.57
0.52:um
0.31
BLoB (N=0)
2.91
1.19
0.56
0.60
0.56
0.32:1000
BLoB (N=5)
0.68
0.90-0.01
0.46:8.0
0.56=000
0.53
0.32+4000
BLoB (N=10)
0.63
0.78.
0.40
0.54:0.00
0.50=a001
0.31:4.00
we employ a KI. re-weighting scheme, which is detailed in Appendix B.1. The full algorithmic description of BLoB training shown in Algorithm
During inference, for an input æ, we approximate the expected output distribution P(y|æ) BLoB by drawing N samples from the variational distribution q (W)))). Empirically, N 10 provides good balance between estimation quality and computational efficiency:
E Eo W]]
Plyl. W.), W. ~ (W
(14)
Experiments
In this section, we compare BLoB with existing methods on real-world datasets. Sec. 4.1 introduces the experimental settings, including baselines, fine-tuning, and evaluation protocols. We then evaluate BLoB's generalization and uncertainty estimation abilities in both in-distribution (Sec. 4.2) and out-of-distributtin scenarios (Sec. 4.3).
4.1 Settings
Fine-tuning and Evaluation. We implement BLoB in the PEFT library [63] and fine-tune the LlaMA2-7B [92] model on common- sense reasoning tasks. Following Laplace-LoRA [116]. we apply LoRA to output layer as well as the queries and values of the attention layers. For hyperparameters, we strictly adhere to the default settings in the PEFT library and the original LoRA paper [63, 41] to ensure maximal reproducibility. This includes the number of training steps, learning rate, and LoRA rank (see Appendix B. for details). For common sense reasoning tasks, select
7

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the next token logits corresponding to possible answers from each dataset and fine-tune the LLM to maximize the likelihood of the correct token. For evaluation, in addition to Accuracy (ACC), we use Expected Calibration Error (ECE [31]) and Negative Log-Likelihood (NLL) to assess the models' uncertainty estimation ability (see Appendix B.2 for details).
Baselines and Implementation Details. We compare BLoB with state-of-the-ee uncertainty estima- tion methods applied the LoRA adapters of LLMs, including Monte-Carlo Dropout (MCD) [29], Bayes By Backprop (BBB) [1]], Deep Ensemble (ENS) [51, 8, 103], and the latest Laplace- LoRA (LAP) [116]. We also report the performance of two standard PEFT baseline methods for reference: Maximum Likelihood Estimation (MLE) [41] and Maximum A Posteriori (MAP).
For MLE, we use the LoRA implementation. For MAP, we use weight decay rate of 1c - 5. For MCD, we use an ensemble of 10 LoRAs with a dropout rate of p = 0.1. For ENS, we independently fine-tune 3 LoRAs and average their legits during evaluation. For BBB, we adopt the default settings from the Bayesian-Torre Torch library [48] and only Bayesianize the A matrix, similar to BLoB. We sample N 10 times for BBB during test. We re-implement LAP and apply it to the MAP checkpoints. We keep all BLoB-spccific hyperparameters consistent across all datasets. Typically, we set the number of samples K I during training for all our BLoB experiments, which highlights BLoB's sampling efficiency. As shown in Table I, we also report BLoB's performance with different numbers of samples during Bayesian inference, where N 0 indicates directly using the mean of the weight distribution for prediction.
4.2 Results on In-distribution Datasets
We fine-tune Llama2-7B six common- reasoning tasks: Winogrande-small (WG-S), imnogrande-medium (WG-M) [82], ARC-Challenge (ARC-C) [18], ARC-Easy (ARC-E) [18], Open- BookQA (OBQA) [65], and BoolQ [17]. For all baseline methods, using the same pre-trained L.M backbone, we maintain consistent hyperparameters across all datasets and do not use additional validation sets to achieve higher performance (See Appendix B.3 for detailed settings).
Table I shows the performance of BLoB compared to the baselines, including ACC, ECE, and NLL, on the in-distribution test set with the pre-trained Llama2-7B model. The high ECE and NLL for MLE indicate overconfidence in LLMs during conventional fine-tuning, except for BoolQ due to its large dataset size. Simple but popular baselines like MAP, MCD, and ENS show mixed results in terms of NLL and/or ECE, highlighting the challenge of uncertainty estimation during LLM fine-tuning. LAP, the most competitive post-training baseline for uncertainty estimation, significantly reduces NI.L and ECE on some datasets but lacks consistent performance, as indicated by its failures on ARC-C and ARC-E. BBB mitigates the overconfidence issue in LLMS across almost all datasets, showcasing the advantage of jointly optimizing the mean and covariance of the variational weight distributions during fine-tuning. However, there remains considerable room for improvement.
BLoB consistently achieves better or comparable performance across all datasets. With the number of samples during inference set to N = 10, the same as MCD, BLoB provides the best uncertainty estimation performance, significantly reducing NLL and ECE, and greatly mitigating overconfidence while maintaining comparable or better ACC than MLE. Even with half the number of samples, N = 5, BLoB still delivers performance comparable to that of N = 10 and outperforms other baselines on most datasets. By abandoning the modeling of the posterior distribution, prediction using the mean of the weight distribution, i.e., BLoB (N=0) sacrifices some degree of calibration in exchange for improved accuracy. Appendix C.5 presents the trade-off between accuracy and calibration, which is controlled by the standard deviation of the prior Gaussian distribution).
Besides Llama2-7B, we also include additional results for RoBERTa-base [60] on text classification tasks in Appendix C. Our method consistently achieved either the best or runner-up performance across nearly all datasets, demonstrating its versatility across different architectures.
'AP encounters training failures on the ARC-C dataset under unified setting, where the same hyper- parameter configuration applied across all datasets. To achieve competitive performance with LAP on the ARC-C C dataset, we deviate from this unitied setting, allowing dataset-speciffc hyperparameters for LAP. Note that this introduces an unfair advantage for LAP. Specifically, for the LAP on ARC-C dataset, we set the dropout rate to 0.I, the learning rate to 5e-5, and applied early stopping at the 5, 5.000-th iteration (out of 10,000).
8

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Table 2: Performance in-distribution and out-of-distribution datasets. All the uncertainty estimation methods are applied to the LoRA adapter added upon the pre-trained Llama2-78 weights.
Datasets
Metric
Method
In-Dist.
Smaller Dist. Shift
Larger Dist. Shift
OBQA [65]
ARC-C (18)
ARC-E [18]
Chem [38, 37]
Phy [38, 37]
MLE
81.52:1025
66.20
75. 15.12
40,62:225
28. 1.20
MAP
81.38=0091
69.59+aas
75.47=073
44.79:0000
28.47=11.0
MCD [29]
81.72=0010
69.03
76.00=1.5s
=
29.17+454
ENS [S1, 103]
81.38-2065
67
75. 18
43.75==1004
30.5662222
ACC (t)
BBB [1]]
82.06
67.25
75
42.36:1049
30.21
LAP [116]
82.12
69.14.1115
74.94.006
44.101130
31.6010.49
BloB (N=0)
82.73:0041
69.93-1120
76.88:20.41
41.67=225
31.94:1/22
BLoB (N=S)
81.79
68.36.11.9
75.82.21.15
40.621..07
32.64
BLoB (N=10)
81.52.10.4
67.71
76.37.10.0
44.79.1.47
31.60
MLE
12.55:00:6
22.20
16.47=0%6
21.72
29.60
MAP
27
+9.31
15.68=0051
17.55=195
30,25+2.18
MCD [29]
14
19.54:2033
15.32:1116
17.92066
29.53+4400
ENS [51, 8, 103]
1,226:0022
7.59
6.44:10.3
12.04:14.5
17 17.52
ECE (.)
BBB [11]
11.38.1107
19.90
13,41
15.67
26. 26.1014476
LAP [116]
8.70
5.84
8.5
10.76
13.91
BLoB (N=0)
8.36
14.00
20.70
15.05
22.90
BLoB (N=S)
3.40=00.7
9.762071
5.96+093
14.33+1.55
18.15
BLoB (N=10)
3.77
4.55
5,48
9.77.138
18.29
MLE
0.73=0.0
1.162000
0,92+008
1.562006
1.66:000
MAP
1.062004
1.1020.0
0,93g004
1.55
1 1,65
MCD [29]
1.062000
1.08
0. 0.881008
1.59
1.67
ENS [51, 8, 103]
0.75
0.8610.01
0.691000
1.28
1.39
NLL (1)
BBB [1]]
0.66
1.0610.01
0.79
1.49
1.62
LAP [116]
0.52+0.0
0.81
0.70=0.02
1.35
1.36:0001
BLoB (N=0)
0.56.200
0.89
0.67=002
1.44
1,53
BloB (N=S)
0.53
0.8520.00
0.642001
1.39
1 1,48
BLoB (N=10)
0.50....1
0.8310.01
0.60..0.1
1.38
1 1,46.10.z
4.3 Results on
Datasets
We use models fine-tuned on OBQA [65] to evaluate the generalization ability different methods under distributional shifts. OBQA consists of multiple-choice elementary-level science questions. We categorize the distributional shifts into two types: smaller and larger shifts. The ARC [18] dataset, which also consists of multiple-choice science questions, represents a smaller distributional shift. The college-level chemistry and physics subsets of MMLU [38] represent larger distributional shifts.
The results in Table 2 highlight BLoB's superior OOD generalization ability compared to other methods on both smaller and larger distribution shifts. BLoB achieves the highest accuracy when solely utilizing the mean of the weight distribution in smaller distribution shifts. For larger distribution shifts, incorporating uncertainty through sampling improves model accuracy. Regarding uncertainty estimation, BLoB demonstrates the best second-bess performance in smaller distribution shifts. Although there is a slight performance drop with larger distribution shifts, BLoB remains comparable to baselines such ENS and LAP.
5 Related Work
Ceaameter-Eflicienn Sine-Tuning (PEFT) for LLMs. Due to the prohibitively large size of LLMs.
parameter-efficient fine-tuning has become a trending topic. Computational paradigms in this area
include adapter-bases fine-tuning [40, 36, 80, 75, 62], pwompt--aaseeee based fine-tuning [34, 54, 59, 57, 94, 6]. and partial tne-tuning [119, 124, 5, 112, 32]. Among these, LoRA [41] has gained significant attention due to its simplicity and effectiveness. Building on LoRA, numerous studies have aimed to further optimize parameter efficiency when fine-tuning large models [26, 35, 22, 21]. For instance, KronA models weight updates the Kronecker product of two smaller matrices without decreasing the update rank [26], and SVDiff performs Singular Value Decomposition (SVD) on the original weight matrices, fine-tuning only the singular values [35]. However, in this paper, focus solely

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Bayesianizing LoRA due to its widespread application in existing works. We also note that BLoB can be naturally adapted to handle different LoRA variants.
Large-scale pre-trained models are well- calibrated during pre-training [44], but fail to accurately express predictive uncertainty during inference [3, 107, 44, 43, 87], especially after fine-tuning [8, 103, 116, 69]. This indicates that measures effective during pre-training [93, 114, 16, 122, 14] may lose their power of uncertainty estimation after fine-tuning for domain-specific knowledge. To address this issue, [27, 123] define priors and approximate posteriors on the full attention weights during fine-tuning, achieving better uncertainty estimation but at significant cost in time and space. Consequently, recent work integrates Bayesian methods and PEFT for efficient uncertainty estimation. For instance, [8, 103] train and store multiple copies of different LoRAs, ensembling their outputs during inference to achieve somewhat better results. [116] applies Kronecker factorized Laplace approximation on fine-tuned LoRA. However, such post-training procedures bifurcate posterior approximation into two stages, leading to suboptimal estimation. In contrast, our BLoB enables simultaneous estimation of both the mean and covariance of LLM parameters in single fine-tuning stage, substantially improving performance.
Uncertainty Estimation in Large Language Models.
Conclusion
In this work, we propose a principled Bayesianization framework for parameter-effiiienty fine-tuning LLMs. Our theoretical analysis shows that a full-weight variational distribution can be efficiently optimized by approximately using a low-rank space of the weight update matrices. Our empirical evaluations corroborate this theoretical insight, demonstrating superior generalization and uncertainty estimation capabilities across diverse scenarios compared to various baseline methods. Building on LoRA, our approach seamlessly integrates with existing LLM architectures while imposing minimal additional memory overhead and training time. Our method highlights that jointly learning the mean and covariance of the variational distribution during fine-tuning can mutually improve both, underscoring the powerful potential of Bayesian methods in enhancing the reliability and generalization of LLMs.
7 Limitations
The main limitations of our proposed BLoB method are: (i) BLoB confined to fine-tuning scenarios and is not applicable to training-free tasks, such as direct uncertainty estimation during inference [64]. (ii) As a typical mean-field variational inference method, BLoB requires multiple sampling iterations during inference, which challenges stable and efficient deployment. (iii) While BLoB's 's effectiveness has been empirically demonstrated for downstream classification tasks, its application to generation tasks requires further investigation.
Acknowledgement
The authors thank the reviewers/AC for the constructive comments to improve the paper. We thank Sanket Jantre for identifying improvements to our proof and for other valuable discussions. HS and HW are partially supported by Microsoft Research AI & Society Fellowship. NSF Grant IIS-2127918 NSF CAREER Award IIS 1340225, NIH Grant IR01CA297832. and the Amazon Faculty Research Award. This research is also supported by NSF National Artificial Intelligence Research Resource (NAIRR) Pilot. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed implied, of the sponsors.
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Appendix
In Appendix A, we present the proofs for the theorems in the main body of our paper. In Appendix B, we introduce the experimental settings, including evaluation metrics and training schemes. Finally, in Appendix C, we present supplementary empirical results including the experiments on another language model and analysis of the space and time cost of our algorithm.
A Proof of Theorems and Claims
In this section, we first present the proof of the two main theorems (Theorem 3.1 and Theorem 3.2) in Appendix A.1. Next, we show how analysis on our design of parameterization in Appendix A.2. Finally, we provide a detailed derivation of the LoRA Flipout Appendix A.3.
A.1 Proof of Main Theorems
Theorem 3.1 (Variational Distribution of the Full-Weight Matrix in BLoB). With the pre-trained weight matrix Wo ~= and the low-rank weight update matrix B E Rmxr suppose that the variational distribution of the other low-rank update matrix A E R xn is Gaussian with q(A = {M, {t} IT, (Asg MMj. D.y where M [My] E R and S [aay] E R=== are its mean and standard deviation, respectively. The equivalent variational distribution defined on the full weight matrix W as in Eqn. 3 is given by
g[vec(W)]BB B, 0) = ((voc(W)) (W) Eg), where
14g = vec Wo + BM), Σg = IT, @ B [diag(voc(f))))) [I, @ B..
Proof. We begin by calculating the mean value of ạ.
N4 vCc(FE W. + BA])
(15)
vec(W, + BE(A])
(16)
vec(W, BM).
(17)
Suppose the deterministic matrix B bb,.b..... b...... ,b,] Rmx" random matrix A
[a1, 02 ' az.] E IR"x", with its underlying parameters of mean and standard deviation defined
likewise M E TR"x" S E RR We have W BA = Σ b, a We then rewrite vec ()
in the form of Kronecker product &:
vec (oc(W)- vee( Σ b a Σ (a ② b,)
(18)
We then calculate the covariance matrix Eg as
Σn covlvec W), vec W)] = cov 1 (a: & bi), Σ (aa @ b.)l
(19)
1=1
cov cov/a; 3 b;, a; 6 by] +
cov (a 2 bi, aj 8 by]
(20)
{
[Ea, la @ ba) @ b.)
Ea, [(a 2 b.) (a, & b
(21)
[a, [a.a,] T] @ (b.b.) - (Ea, [ai]] 8 (b,b,.) }}
(22)
diag a (b,b,)
(23)
I, & B (ncn(vvc((3"))) [I, @ B
(24)
completing the proof.
0
19

---

It is crucial to note here, the final covariance matrix of q(ve) W)) follows a block diagonal structure, which will be further utilized for the proof of Theorem 3.2, Defining Σ; diag(S2 ). we have:
B
Σg
(25)
BE,
Another important fact about Eg is its singularity. It can be seen directly as we consider the rank of any one of the block matrix BE, _mxxm Vi E [n]:
r r(BBB)) min{r((B),
S m,
(26)
where is the rank of LoRA, strictly smaller than the output dimension of m.
Theorem 3.2 (Efficient Computation of Wail-teight KL Divergence). Suppose the pre-trained weights Wo, update matrix B, and the variational distribution q( (() are defined as in Theorem 3. I, and the prior distribution of the full-weight matrix P veo(e) (W)) is defined as Eqn. 9. Consider the Gaussian prior distribution P(A) II, (Arg00, of)): then have:
KL (vecl | P(vec) W))] (L)((A))))
if R = In & Rl. where R satisfies RR = BB".
Proof. We start by assuming the low-rank structure of the prior P(vec(W)), and then reveal the conditions reaching to our final conclusion step by step.
Typically, for two Gaussian distributions and whose covariance matrices Eg Rdxd and Σ, E [dxd are both full-rank, and their means as My E Rd and Up E IR, we have their KL- divergence
kLiglpp log E Egg
d ddttrr tr( + (Mg Up) E (mg m2p)
(27)
The singularity of the covariance matrices of P(vec (W) and q(vec((VV))) (W)ec i.e. | |Σ 0, can cause issues when computing the KL-divergence as it includes the log- determinatn term. Therefore in this proof, we consider the alternative the covariance matrices, where an extremely small diagonal elements are added.
For the prior distribution, following the alternative form of a degenerate Gaussian [83], as suggested in Eqn. 9, we assume
P(vec( W)) = NW Ep),
(28)
where
Σ, = AI + RR
(A (++0+).
By default, we assume that the low-rank tall matrix R E (ma)xr' has the full column rank r'. Otherwise rff() r" < r', then we can in effect consider new matrix component R' e ((mm)xr" that has the same rank as r", which satisfies our assumption of full column rank. Therefore, we have the SVD decomposition of R is given by
(29) = (mn) and VRVA V VR I, Dn is tall matrix where its upper part is diagonal and the lower part is zero matrix, denoted as DR [D n. Oj [diag([dn, 0, 0,dR, 0,... dd., 0)), 0
R
where Un E (mm)) x (mn) and VR E I xxx orthonormal, i.e.,
For the approximate posterior q B,B,), we consider
q(vec( (W) B, 0) NN BM, Eg),
(30)
where Σn = AT + BEBT, (A+0+),
where we simplify the notation for the covariance matrix E, by defining B = [I,B]] E RR (m()) (mr) and Σ = 3 Likewise, we have the SVD-decomposed matrices for B where they are defined in the same way as Eqn. 29:
B B-U
(31)
20

---

where UB and VB are orthogonal matrices, and DB D OOT [diag([dR, > 0, dB. >
dd First, caleulate the log- oeg-determinant part of the KL-divergence. For the log-determinant of the covariance matrix of the prior distribution Ep. by applying SVD decomposition in Eqn. 29, we have
0,... ,...., > 0)), O]
log Ep
log
2V] + RR = log AI + URDRVA V U
(32)
= log
(R(AI + DRDD)
(33)
Ur (Di)² X
log
U
(34)
0
Mm
= -log|
(Dx)) + A1,l + log A mm
(35)
-mm X Σ log(dh,, + +
(36)
Following (almost) the same idea, we have the log-determinant of the variational distribution covariance Σg as
D B
DV
+ AI mr
O Mm
log Egg = log AI + BB
log
(37)
0
(mm mr) log x 2 log
D log
VB
EVB X(D -2
(38)
= (mm - mr) log X +
log dB, + log Σ + log I + V3 (39)
1=1
We make two observations when À + 0 (i) compare the terms that contain log À on both sides, to make sure the log- log-determinant in the divergence term bounded, we have to set r' = mr; (ii) the last log [I] = 0. Therefore, we have
term in Eqn. 39, log I AV Σ VVn )D
RR. +\
log
log
log E/
(40)
d
Next calculate Tr(E,'E.) in Eqn. 27. Following the same assumptions and notations above, have the inverse of the covariance of the prior distribution as
[(D R
+ XI--
O UT. X-1 A-11UR
= UR
(41)
O
Hence we have
2
O
D V EV
+
O
In( Ir(=p'E,) Eg) = tr(UR
U
Ug).
X
AI
(42)
By using the condition RRT BBT and R = [o m R defined in Theorem 3.2, have
RR = o BB
(43)
and there exists an orthogonal matrix E IR [ (mr), such that
R o=
(44)
The SVD decomposition on R can then be formulated as:
R DDUUDDVV P
(45)
=Dl = V P). Substituting UR. DR, Va back to Eqn. 42 and applying À + 0+, we have
=
(46)
tr( tr(E, Eg) = tr([mmn-w + tr(((o,,D)) + X1| DiV, EV3D;1)
(47)
(mn nr) + ta tr(V ΣVB)
(48)
(mm - n2) + o (2) =
(49)
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---

For the quadratic term in Eqn. 27, the pre-trained weights Wo cancel out, and we can calculate it as
vec( ()c Ep- vec BM)
(50)
(BE,
BM.
-IM B M. B
(51)
BE,,
BM
M. (BE, B) B
(52)
dig of1dE +X)
8
M, (VB
B M
(53)
8
0 d
+X)_
7
M M,
(54)
M
(55)
Finally, proof is completed by combining Eqn. 40, Eqn. 49, and Eqn. 55,
A.2 Analysis on BLoB Parameterization
General Analysis on Parameterization Consider a path of parameterization for a single variable:
p p=a=f(p) C 1(0),
(56)
where is the real parameter we perform update on, f is our parameterization choice for the variable a, and represents the loss function we aim to minimize. When comparing two different parameterization methods, we consider the same initial conditions of a = oo. and we assume the same step size 7) on the real parameter p. To show the influence of the choice of parameterization, we calculate the decrease of the loss value by performing one step of gradient descent. First, by the chain rule, the gradient w.r.t. Po is calculated as
dc de 1oo
de ap Po
((o) ((o))
(57)
ap
After one step of the gradient descent, we have P1 as
Pi = Po - n ('ro) f(()).
(58)
and the loss value decreased at #1 can be approximated by the first-order Taylor expansion,
AL = (([((()))) - ((((p)))
(59)
= =[(((p))) - P(S(po- - ('(o)))) ('(o))) ([f(po)) - (((F(po) -n ('oo) (fo)))
(60)
(61)
(62)
Since the initialization of the different parameterization variable Pe is set to ensure the same initial condition of dn for different fs, we can see that the amount the loss decreases by after one step of update is proportional to the squared gradient AL x (?(()))) (f'(p))² = (dC/an)².
softplus function og = log(1 + exp(p)) to parameterize the standard deviation. For a single element p. the derivative of the closed-form solution of the KL divergence in Eqn. 11 is calculated as
Parameterization with log(1 + exp(-)) or (.)2? Previous VBNs [11, 49] typically
dKL dp
log(1+e") og(1+e")
(63)
(1te))log(1-e)
Due to the fact that og is typically initialized to small value close to to ensure stable optimization of the likelihood cost term (c.g., le 3), and in order to ensure that the model obtains good uncertainty,
22

---

Op is usually set to a larger value close to 1 (e.g., le 1). In this case, the derivative of in Eqn. 63 almost always a constant -1, which, based on our previous analysis, leads slow convergence for large op values when og is small.
Therefore, we parameterize og with quadratic function: o4 = p². this case, the derivative of the KL divergence with respect to p in Eqn. 11 becomes:
KKL
dp 2 + 20
(64)
Under the same initialization conditions, the derivative in Eqn. 64 is approximately of the order of p_ leading to rapid convergence towards larger dp values when og is small. Building on this, we SGD without momentum to optimize the complexity loss term, thereby achieving the natural convergence of Qq.
Visualization. To visually demonstrate the differences in the convergence of KL divergence during training with these two parameterizations, we set Op = and employ gradient descent to optimize the KL divergence. As introduced in B.1, in practical mini-batch gradient descent, the KL divergence weighted by 1/Hmini-butches. Therefore, assuming there are 100 mini batches, the learning rate is set 0.01, which translates for both parameterizations. The growth of dg during the KI. training, for = log(1 + e²) and By = p² is shown in Fig. In the same setting, the softplus parameterization takes nearly 100,000 gradient steps converge, while the square parameterization takes only about 5,000 gradient steps. This modification makes it suitable for our fine-tuning setup, where the number of gradient steps is relatively small.
actual learning rate of 1e for p. We initialize og 0.01
1.0
0.8
0.6
6
0.4
0.2
log(1 +
p²
0.0
5000
50000 iterations
100000
Figure 2: The growth curve of 0g log(1 + el) and 0g p² during the optimization of KL divergence (without data likelihood). The number of gradient steps (5000) is marked with the red line.
A.3 Deriving Flipout for BLoB
For the i-th input vector h; in mini-batch, randomly sample two flipping vector s E (-1,+11) and E {1]. Denoting A as the weight matrix sampled from posterior distribution, and AA as the batched noise for sampling A, the output vector z; after applying flipout is:
2, Wh;
(65)
Woh, BAh;
(66)
Woh: + B(M + AA)h;
(67)
Woh, BMh B(Ā 1,s, h4:
(68)
Similarly, for a mini-batch input matrix H E x with batch size we randomly sample two low-rank flipping matrices S {-1, +1]mxb and T E {-1, bxr, The batched output matrix Z after applying flipout then:
Z W.H + B(MH + A(H S) OT)
(69)
W.H + B(MH + (E 2) H S) oT).
(70)
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Model
Hyperparameter
Roberta-base
Llama2-7B
Model
Optimizer
AdamW
Hyperparameter
Roberta-base
Ilama2-7B
LR Scheduler
Linear
Optimizer of KL
SGD
Warmup Ratio
0.06
I.R of KL.
0.002
0.01
Learning Rate
5c
le
dp
0.2
Batch Size
0.05
Max Soq. Len.
512
300
8
LoRA
16
LoRA
Table 4: BLoB-Specific Hyperparameters
Table 3: Hyperparameters of LoRA
B Implementation Details
In this section, we first introduce the implementation details of BLoB in Appendix B.1, including the KL Re-weighting scheme, initialization of the parameters, and learning scheduling, etc. Next, we introduce the two evaluation metrics for uncertainty estimation in Appendix B.2. Finally, we present some statistics of the adopted datasets Appendix B.3.
B.1 Implementation of BLoB
KL Re-weighting, mini-batch SGD, the training data D is randomly divided into M equally sized subsets: D1, D2 ................................ Das For mini- mini-batch i 1,2,.... ..., ,M, the cost function is:
F(D., 0) = - -Eg( [log P(D w)] + A, KLig(V ()) || P(W)).
(71)
where X; E [0, 1] and Σ M A = 1. There are various approaches for controlling the weight of KL divergence. [30] utilizes A; 1/M, while [11] adopts A; 2dr
In fine-tuning tasks, found that using a scheduler with A; = -2'/2M-1 performs well. This allows the model to find good fits the data points within the early stages and then optimize the complexity cost in later stages.
-----. 2M
In multiple epochs of mini-batch SGD, larger datasets require more iterations to complete one epoch, resulting in delayed convergence of the complexity cost. To enhance the stability of BLoB's s performance across datasets with varying sizes, we pseudo-rescaled the size the training dataset to make smaller datasets slightly larger and larger datasets slightly smaller. For the portions of the dataset that required expansion, we incorporated additional bani-b from subsequent epochs. Conversely, for the datasets needing reduction, deferred the excess epochs. We denote the size of original dataset as Lo, The rescaled dataset size is:
to subsequent
L = 100 L.
(72)
where y is coefficient used to control the scaling magnitude, and we set it to 8 in all experiments.
The pseudo-rescaling does not affect the likelihood cost in practical mini-batch gradient descent. In fact, it only changes the warm-up period in KL reweighting from M to L /b/bauch size, thereby facilitating more consistent optimization of the complexity cost across datasets of different sizes.
Additional Details. We initialize standard deviation parameterization matrix G by element-wise sampling from uniform distribution with range of [gg4], c). while keeping the remaining initializa- tion settings consistent with LoRA. To maintain consistency, we use the same learning rate scheduler and warmup ratio for the optimizer of the KI. term as we do for the likelihood term. We sample only once during the training process. During inference, we sample N' times, then take the average of the logits obtained after passing through the softmax function. Detailed hyperparameter settings are provided in the Table 4. Table provides the hyperparameters for tening with LoRA shared with other baselines. Our experiments Llama2-7B were conducted using 2 NVIDIA RTX A5000 GPUs for parallel training, while experiments on RoBERTa- base were conducted using 4 NVIDIA RTX A5000 GPUs for parallel training.
24

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B.2 Evaluation Metrics for Uncertainty Estimation
Negative Log-Likelihood (NLL) and Expected Calibration Error (ECE [31]) are two prevalent metrics for assessing uncertainty estimation. NLL calculates the sum of the negative expected log probability of predicting the actual label. Suppose this predicted probability given by the model Po. and we have test dataset (Ta, Un
J J==1 of size N. Then the NLL measured on this dataset is
NLL 1 Σ
log Po(yn).
(73)
This metric prefers models that assign higher probabilities to correct labels. If the model exhibits over- confident an incorrect prediction, the probability assigned the correct label will be diminished, thereby increasing the NLL.
On the other hand, ECE measures how well the model's confidence matches its accuracy. This is done by binning the predictions based on their confidence levels and then computing weighted average of the absolute difference between accuracy and confidence within each bin:
M Σ
ECE
Bm
|acc(Bm) conf(Bm)).
(74)
where acc(Bm) and conf (B.) denote the average accuracy and confidence within bin Bm. respec- tively. These are given by:
ace
Bm)
Σ
1(0: = yt),
conf( (mn) =
Σ P(T.;
(75)
[m
[m IE
B
where Bml is the number of samples in bin m. We set [Bml = 15 across all experiments.
B.3
Dataset Details
Table 5 summarizes the size of the training set and the number of labels for each dataset. Table summarizes the prompt templates used for common sense reasoning tasks.
Table 5: Size of the training set and number of labels for each dataset.
WG-8 [82] ARC-C[18]
ARC-E [IC-
WG-M [82]
OBQA [65] BoolQ [12] RTE [19)
MRPC [24]
WIC [76] CoLA [104]
2.25k
Num, of Labels 2
5
5.
2
4
2
2
2
Table 6: Prompt templates for common sense reasoning tasks.
Task
Prompt
Select one of the choices that answers the following question:
Winogrande (WG-S/WG-M)
[question] I Choices: A. [option]] B. [option2). Answer:
ARC (ARC-C/ARC-E), Openbook QA (OBQA),
Select one of the choices that answers the Tollowing question:
[question] Choices: A. [choicel), B. [choice2), (choice3).
MMLU
D). [choice4]. Answer:
Answer the question with only True or False:
BoolQ
{question] Context: (passage).
C Additional Experimental Results
This section provides additional experimental results omitted from the main body of the paper due to space limitations. First, we present the results of BLoB when applied to RoBERTa, another pre-trained language model, in Appendix C.I. Next, in Appendix C.2, conduct the ablation
25

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Table 7: Performance of different methods applied to LoRA RoBERTa-base pre-trained weights. The evaluation is undertaken on five GLUE [96] and SuperGLUE [95] tasks, with shared hyper-parament setting without using individual validation dataset. "f" and "L" represent that higher and lower values are preferred, respectively. The boldface and underline are used to denote the best and runner-up performance, respectively. The asterisk denotes training failure.
Datasets
Metric
Method
RTE [19)
MRPC (24)
WiC [76]
CoLA [104]
BoolQ [17]
MLE MAP
75.81 +1077
86.27=am
64.52+091
83,29
77.67
15.81
86.3610511
65, 65.4611.04
83.0010.15
77.691066
MCD [29]
76.65+00.85
87.75=0.5
68.55+0332
84.76+0.62
78.41
ENS [51, 8, 103]
17.74+1.10
88.64 88.64-0037
65 43.844
84.08
78.57=0036
ACC (1)
BBB [11]]
49, 49.46"=2.53 12.53
68.38
10.57
69. 13:10.00
62. 16:1000004
LAP [116]
76.05+00.55
86.52=aa72
64.52+091
83.29+0016
77.67+0.52
BLoB (N=0)
76.05+0.17
88. 28.24:20:0
63. 17.022
80.92=0.70
74.80=2.10
BLoB (N=5)
74.61+0661
88.48=000
64.00+0.5
80.54
74.77=1.77
BLoB (N=10)
75.45.00.5
88.7
64.26.11.00
80.8910.24
75.49
MLE
20.59
11.13
25.72
10.70
10.02.10.1
MAP
21.67:3325
11.1 4::044
24.26
10.61 2:049
10.1 1::062
MCD [29]
13.06++0.99
7.36++
16.94+075
4.58+0027
+0.33
ENS [51, 8, 103]
19.47.10.37
10.13
28.62.10.3
12.44.0.42
5,98
ECE (L)
BRB [1]]
2.66 2.664
6.46:2044
40.30
3.90 20:4
5.02" =0000
LAP [116]
5.33=
6.29
11.48+0.67
3.13
4.84
BLoB (N=0)
14.64+0.75
5.61+0006
18.93+1.99
10.90=0.24
5.80
BLoB (N=5)
10.461061
4.49.10.22
13 13.62
7.76200.1
3.21 10.13
BLoB (N=10)
8.97
3.30
13.03
7.8320.27
2.90
MLE
1.12
0.62
1,191000
0.53
0.56
MAP
1.2 1,23+a.10 to
0.58+
1. 1.142000 06
0,53=to.0
0.55=
MCD [29]
0.65
0.39+tu.44
0,88
0.39 e.m0
0,50=com
ENS [51, 8, 103]
1. 1.04
0.63
1.70.1007
0. 2..2...000
0.48
NLL (1)
BBB [1]]
0.69 20000
0.63+t0.00
0.69* +0.00
0, 0.62=0.00
0.67
LAP [116]
0.55
0.47.10.01
0.63
0.48
0.53
BLoB (N=0)
0.56+am
0.29++u.0
0.76=au02
0,52+omm
0,52+oaz
BLoB (N=5)
0.50
0.27.10.00
10.01
0 1..510.01
0,51 10.00
BL.B (N=10)
0.48:aa.1
0.26:2n.00
0.67=0m1
0. 0.462om
0.51 #0.m1
study proposed refinement in BLoB. Then analyze the memory and training time costs in Appendix C.3, Finally, we provide visualization illustrating our BLoB's 's advantage on embedding uncertainty in Appendix C.6.
C.1 Performance of RoBERTa on In-distribution Datasets
We also evaluate different methods on RoBERTa- which has approximately 1/50 the parameter count of Llama2-7B. Table 7 shows the results. Compared to MLE, MAP shows minor improvements NLL and ECE, while MCD, ENS, and LAP enjoy significant improvements. The convergence difficulty observed with the BBB algorithm is further exacerbated on the smaller model, resulting in significant decreases in ACC across all datasets, and even training failures on RTE and WiC. In contrast, method demonstrates the best runner-up performance in uncertainty estimation almost all datasets. Only slight decrease in ACC is observed on BoolQ and CoLA. We suspect that such decrease is caused by RoBERTa-basse small model size compared to the large size of these datasets BoolQ and CoLA (i.e., underfittin).. Using a larger pretrained model, e.g., Llama2-7B, would potentially address this issue.
C.2 Ablation Study
We perform an ablation study on the Llama2-7B model to showcase the effects of a range of techniques we designed: KI. Re-Weighting (RW, Appendix B. Re-Parameterization (RP, Sec. 3.3), and Asymmetric Bayesianization (AB, Sec. 3.1). In the scenarios w/o AB, we Bayesianize both matrices, A and B. In practice, using identical initialization and prior for the standard deviation
26

---

Table 8: Ablation study of BLoB, applied to LoRA Llama2-7B pre-trained weights, where RW, RP, and AB represent our designed techniques of KL Re- Ree-Wightiig (Appendix B.1), Re- Parameterization (Sec. 3.3), and Asymmetric Bayesianization (Sec. 3.1), respectively. The evaluation 1 and the number of
is done following Table 1. We set the number of samples during training samples during inference N 10 across the variants (denoted by "-") of the BBB [11] and BLoB for fair comparison. We denote by .." experiments with the scaled standard deviation matrix. The hyphen =_" in the table denotes training failure caused by "NaN" loss. "f" and " indicate that higher and lower values are preferred, respectively. Boldface and underlining denote the best and the second-bed best performance, respectively.
Techniques
Datasets
Metric
Method
RW
RP
WG-S [82]
ARC-C [18]
ARC-E
WG-M [82]
OBQA [65]
BoolQ [17]
20
69.10
85.65
74.53
81. 81.52.925
26.5
BBB-
BBB-
BBB [11]
74
68.13
85.86#074
73.63+244
a2.06
87.21
ACC (f)
BLoB-
69.75-000
67.91
86.03
76.24
87.23
BLoB-
BLoB-
BLoB *
/
/
69.75:0as
70.27::044
86.33:0.4
74.9240.1
41.324041
86.47:046
BLoB (Ours)
/
/
69.07
68.81
85.56
73,69
81,52
86.99
MIEE
29.8320.5
29.0021.7
13. 13.12
20.62
12.55
3.18 3.18
BBB-
BRR.
BBB [11]
21.81
26.23
15.76
11.88
3.74
ECE (.)
BL.B
26.60 26
26.24
11,53
18.05
12.361042
3.052am
BLoB-
BLoB-
/
BLoB
/
/
16. 20.
13.85
2,93
8.33
4.77amm
1.18
BLoB (Ours)
9,35
9,59
3.64:0.5
3.01
3.77=147
1.41
3.17
2.85
1.17
0.95
0.73
0.32a
BBB-
BBB-
BBB [11]
1.40aass
2.23
0.91
0.84
0.662n0s
0.31:aum
NLL ())
BLoB-
1.96
2.31
0.84
0.87=0m
0.68
0.31
BL.BB-
/
BLoB-
BLoB
0.80
0.46
0.55=001
0.51=n00
0.32+am
BLoB (Ours)
0.63:a01
0.78
0.40:0.01
0.54=0000
0.50=n01
0.31
matrix G of the variational distribution on both and leads to training failures caused by "NaN" loss across all datasets; this is consistent with the findings in Sec. 3.1. As a solution, introduce scaled standard deviation matrix G/100 on B to alleviate early-stage fluctuations. Nevertheless, is important note that not using AB incurs double the additional memory cost and training time, as described in Appendix C.3.
As demonstrated in Table 8, BBB w/o AB fails to converge due the unbounded NaN loss, which cannot be solved by using scaled standard deviation. By introducing KL Re-Weighting, Re-Paran Ge-Parameterization. and scaled standard deviation, BLoB w/o AB achieves the runner-up perfor- mance and improves accuracy on small datasets. However, BLoB with all techniques achieves the best ECE and NLL with minimal additional computational cost.
C.3 Additional Results on Memory and Time Efficiency
By introducing an additional standard deviation matrix Ω of the same size as the LoRA A matrix, the number of trainable parameters in BLoB increases by half compared LoRA. In the case of BLoB w/o Asymmetric Bayesianization (AB), the number of trainable parameters are twice as many as those LoRA. The calculation of KL divergence and the inclusion of the additional standard deviation matrix in the likelihood loss computation result in additional forward and backward propagation time. We conduct parallel training using two NVIDIA RTX A5000 GPUs to observe the differences in GPU memory cost and training time between BLoB and standard LoRA fine-tuning on the Llama2-7B
27

---

Table 9: A comparison of time and maximum memory cost between standard LoRA and BLoB,
during training. The evaluation is based on fine-tuning for 5,000 steps on the Llama2-7B model.
Datasets
Metric
Method
WG-S (82]
ARC-C [18]
ARC-E (18)
WG-M [82]
OBQA [65]
BoolQ [17]
Time (Seconds) ())
Standard LoRA
1586
3382
BLoB (N=10)
1563
1790
1753
1556
2142
3733
Standard LoRA
14688
16870
17044
14710
14984
20784
Max Memory (MB) (3)
(N=10)
15015
Table 10: A comparison of time and max memory cost between Standard LoRA, LAP, and BLoB (N=10) during inference.
Datasets
Metric
Method
WG-S [82]
ARC-C [18]
ARC-E[18]
WG-M [82]
OBQA [65]
BoolQ [1?]
Standard LoRtA
58
Time (Seconds) (1)
LAP
311
445
814
554
1165
2508
(N=10)
75
Standard LoRA
14391
14157
13081
14391
14391
14160
Max Memory (MB) (1)
LAP
43742
61881
64737
43678
55642
67364
BLoB (N=10)
14171
14411
13911
14407
14478
14177
model. The results are shown in Table 9. BLoB increases memory cost by only about 3% compared to LoRA, with training time increased by about 15%.
13%
We further evaluate the inference time and maximum memory usage for standard LoRA, Laplace- LoRA (LAP), and our BLoB, as shown in Table 10. The experiments are conducted on two NVIDIA A100 GPUs 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.
C.4 Impact of Sample Size on Inference Performance
Here provide more detailed empirical study the sample size of BLoB during inference. Specifically, we report the results for different number of samples from N to N 160 on the WG-S dataset, demonstrating improved uncertainty estimation with increased number of samples, as shown in Fig. 3.
ACC for Different Numbers Samples
ECE Different Numbers Samples
NLL for Different Numbers Samples
ACC(1]
0.65
Figure 3: Performance of BLoB with Varying Sample Sizes N during Inference. We fine-tune the Llama2-7B model on the WG-S dataset for 5,000 steps, evaluating the model's performance with different sample sizes, specifically when N is 1, 2, 4, 5, 10, 20, 40, 80, and 160.
C.5 Trade-Off between Accuracy and Calibration Controlled by Gaussian Prior
The empirical trade-off between accuracy and calibration caused by different model architectures is observed in [85]. By controlling the standard deviation of the prior Gaussian distribution,
28

---

observed a similar trade-off between accuracy and calibration. Specifically, we report results for different prior Gaussian standard deviations, ranging from 0.05 to 0.25, while proportionally scaling the learning rate of KI. divergence from its original value of 0.01 to values between 0.0025 and 0.0125. This highlights the trade-off' between accuracy and calibration, as shown Fig.
ACC for Different Prior Standard Deviations % Learning Rate of KL Divergence
ECE Different Prior Standard Devistions dp Learning Rale KL Divergence 3,0075
NLL for Different Prior Standard Deviations % Learning Rate Divergenc
0.0025
0.0015
0.0025
0.0125
0.0025
000125
23.0
1.0-
68-
0.9
67-
12.5
68-
0.3
0.7-
5.0
0.55
0.10
0.15
0.20
0.25
0.30
0.15
0.25
0.25
0.10
0.20
0.25
Prior Standard Devistion
Prior Standard Devistion
Prior Standard Devistion
Figure 4: Performance of BLoB (N=10) with Varying Prior Gaussian Standard Deviations dp- We fine-tune the Llama2-7B model on the WG-S dataset for 5,000 gradient steps, evaluating the model's performance with different prior Gaussian standard deviations and learning rates of KL divergence.
C.6 Embedding Uncertainty of BLoB: A Preliminary Visual Study
Estimating the uncertainty of LLMs in the embedding space has recently garnered sigmificant attention in the community [13]. Expressing models' uncertainty via their generated embeddings benefit both discriminative (the focus of this paper) and generative models. this section, we present preliminary study uncertainty estimation in the embedding spaces of different models, as illustrated in Fig. 5. We compare BLoB with two baseline models, BBB and MCD, which can generate embedding samples and effectively estimate uncertainty. We exclude LAP from this section due to its excessive memory consumption, which consistently results in Out-Of-Memory (OOM) errors during inference. The experiment is conducted on the OBQA dataset [65], which consists of four categories.
For each input sequence s, use the last token's 's embedding generated by the final transformer block in Illama-7BB as the final embedding. Given the weights W, we denote the embedding as ((s; W). Generally, three types of embeddings can be generated using the Bayesian approach:
(a) Embeddings generated by the mean of the weights (these embeddings are shown as "*" in
Fig. 5):
((8; Ey wyt- Ø) W]) <0s;; Wo BM).
(76)
(b) Embedding samples generated by sampling different weights from the approximate posterior, whose distribution is plotted by the solid line (-).
(c) The expectation of the embedding, which is approximated by averaging the sampled embeddings:
N
Ew 0) ( W
(s:;
Σ
Φ($ Wo + B(M + E, o R)),
(77)
N
where denotes the number of samples during inference, and E, denotes the n-th sampled noise for the weight matrix. We show this expectation
in Fig. 5.
To visually demonstrate the confidence calibration effect of the Bayesian treatment, we adopt the following pipeline of visualization, which we believe can be further applied in visualizing other frameworks' embedding uncertainty quality.
(1) Acquire high-dimensional embeddings produced by the weight mean for the given test dataset, decribed in (a) and Eqn. 76 above.
29

---

BLoB
MCD
Incorrect Classification
2
Correct Classification
8
Distribution of Samples
Mean of Samples
Mean of Weights
Figure 5: Visualization of embedding uncertainty quality for different methods. The model is fine-tuned for 5,000 steps on the Llama2-7B. We fine-tune the Llama2-7B model on the OBQA dataset for 5000 steps. The two contour lines represent the probability mass 0.5 and 0.75, respectively.
(2) Use Linear Discriminant Analysis (LDA) [10] to project these high-dimensional embeddings into low-dimensional 2D space.
(3) In the 2D space. fit a logistic regression model to mimic the decision regions and color them based on the true labels.
(4) Sample weights 10 times from the approximate posterior, generate the embeddings. and project them into the same 2D space using the previously learned LDA. Use Kernel Density Estimation (KDE) [79, 74] to show their distributions, described in (b) above.
(5) Average the sampled embeddings for each example and visualize them in the 2D space, as described in and Eqn. 77 above.
In Fig. 5, we show correct and incorrect predictions made by each model. Ideally, a model with better uncertainty estimation should produce lower level of uncertainty (smaller embedding variance. i.e, smaller contours, and further away from the decision boundary) for correct predictions, and higher level of uncertainty (larger embedding variance, i.e., larger contours, and closer to the decision boundary). From the figure, we have the following observations:
All three Bayesian approaches produce higher embedding variance for incorrect predictions and lower embedding variance for correct predictions. However, BLoB achieves significantly larger embedding variance compared to the haselines, consistent with the quantitative evaluation shown Table 1. BLoB's produced variance higher for the incorrect predictions, demonstrating its accurate uncertainty estimation even in the embedding space.
In BLoB, the mean embedding produced by sampling weights from the approximate posterior is closer to the decision boundary than the embedding generated by the mean of weights This effect is most apparent when the prediction is incorrect, consistent with the quantitative results yielded from the final softmax layer of the model. Again, this demonstrates BLoB's Bayesian inference can bring the final prediction closer to the ground truth.
30

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Reviews:
[{"review_id": "2A4F41nUkW", "reviewer": "Reviewer_Lm4V", "summary": "This work introduces a Bayesian Deep Learning framework for finetuning LLMs with probabilistic LoRA. Unlike existing work by Yang et al. [1] which uses a Laplace approximation, the variational distribution is parameterized using diagonal Gaussians as done in [2] (”Bayes by Backprop”). \n\nSpecifically, the authors design the variational distribution as follows:\nGiven the standard LoRA updates $W = W_0 + BA$, only $A$ is modeled probabilistically while $B$ is deterministic. In combination with an appropriate choice of prior, this allows for a a low-rank parameterization of the variational posterior and fast KL-computation in the lower dimensional parameter space.\nExperiments compare the methods to existing baselines in terms of accuracy of the mean predictions, calibration (ECE) and NLL on a range of datasets. \n\n[1] https://arxiv.org/pdf/2308.13111\n[2] https://proceedings.mlr.press/v37/blundell15.pdf\n\n____\n\nI have updated my overall score +1 after the rebuttal.", "questions": "Questions:\n- Table 1: 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?\n- l. 240: “For MCD, we use an ensemble of 10 LoRAs with a dropout rate of p = 0.1”. Why is an ensemble used here? Shouldn’t the uncertainties come from the dropout variational posterior? Why are multiple models necessary? \n- Will you open source the code on publication?\n\nSuggestions & Typos: \n- l. 33: I.e., unable to … (sentence is not grammatical, delete) \n- l. 110: yielded by parameterization —> yielded by reparameterization  \n- Algorithm 1, line 11: Updater —> Update \n- l. 237: Maximize A Posteriori —> Maximum A Posteriori  \n- l. 239: “For MAP, we use a weight decay rate of 1e-5.” I suggest to formulate this in terms of prior variance instead. \n- l. 638 covaraince —> covariance", "limitations": "No, I recommend a more explicit treatment of the limitations, perhaps in a dedicated section.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The problem of assigning principled uncertainties to (fine-tuned) LLM outputs is important, and it makes sense to compare different Bayesian Deep Learning strategies in combination with LoRA. The exposition of the method and paper in general is clear.", "weaknesses": "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. Additionally, the method by Yang et al. (\"LAP\") is not replicated successfully in Table 1. For example, for the LAP method on ARC-C datasets the results in this work fall behind the results in [1] by 50%: \n- This work: Accuracy: 29.73±12.02, ECE: 14.24±1.65 and NLL: 1.53±0.01\n- [1]: Accuracy: 66.91±1.1, ECE:  7.5±1.2 and NLL: 0.86±0.02, respectively. \n\nThis makes it difficult to gauge how much the approach improves compared to [1].", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n**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.**\n\nPlease refer to **Q3. [Novelty of BLoB]** in **General Response**.\n\n**W2. The method by Yang et al. (\"LAP\") is not replicated successfully in Table 1.**\n\nPlease refer to **Q2. [Reproducing LAP's Results on the ARC-C Dataset]** in **General Response**.\n\n**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?**\n\nThis 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. \nIn 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. \nWhen $N=0$, we use the mean of the weights to make predictions $P(Y|E_{q(W|\\theta)}[W], X)$.\nDue 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, \n+ 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 \n+ 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 $N \\rightarrow \\infty$, $\\frac{1}{N}\\sum_{n=1}^{N} P(Y|W_n, X) \\rightarrow E_{q(W|\\theta)}[P(Y|W,X)] \\neq P(Y|E_{q(W|\\theta)}[W], X)$.\n\nTo 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 $N=0$.\n\n|                        | **N=0** | **N=1** | **N=2** | **N=3** | **N=4** | **N=5** | **N=10** | **N=20** | **N=40** | **N=80** | **N=160**  |\n| ---------------------- | ------- | ------- | ------- | ------- | ------- | ------- | -------- | -------- | -------- | -------- | ------ |\n| **ACC ($\\uparrow$)**   | 71.44   | 65.20   | 66.28   | 66.91   | 67.63   | 68.20   | 68.31    | 68.04    | 68.15    | 68.31    | 68.15 |\n| **ECE ($\\downarrow$)** | 19.71   | 19.72   | 14.60   | 12.82   | 11.27   | 10.58    | 9.51    | 9.47    | 9.25     | 8.75     | 8.75   |\n| **NLL ($\\downarrow$)** | 0.84    | 0.8617  | 0.7355  | 0.6971  | 0.6766  | 0.6620  | 0.6395   | 0.6313   | 0.6233   | 0.6213   | 0.6215 |\n\n\n**Q2. Why is an ensemble used here? Shouldn’t the uncertainties come from the dropout variational posterior? Why are multiple models necessary?**\n\nWe 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.\n\n**Q3. Will you open source the code on publication?**\n\nThank 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. \n\n**Q4. Suggestions & Typos**\n\nWe sincerely appreciate the reviewer's careful reading and for pointing out the typos. We will fix them in the revision as suggested. \n\n**Q5. Limitations**\n\nPlease refer to **Q1. [Discussion of Limitations]** in **General Response**.", "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": "", "extra_scores": {}}, {"review_id": "PEkuAliUDq", "reviewer": "Reviewer_Zor8", "summary": "Proposes to have a Bayesian version of LoRA by placing priors over the low rank matrices. This evolves into placing prior over one of the low rank matrices. Inference is via variational inference. Results are mixed, though it shows the method is competitive with others.", "questions": "Q1 and Q2 above (in weaknesses)", "limitations": "No. Paper need to include some discussion on the limitations of their approach/paper. I do not see any potential negative societal impact to be addressed.", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 1, "contribution": 3, "strengths": "- The eventual simplicity of the approach by placing prior over only one of the low rank matrices.\n- Use of independent Gaussians in prior and posterior for efficiency.\n- The method is competitive, based on the results.", "weaknesses": "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 then say the deltas to the pre-trained weights are Gaussians because they are sum of Gaussians. The consideration to also put prior on $B$ can be deferred to discussions, and the trade-offs discussed there. This make the model presentation cleaner, and avoids the problem of having to involve improper priors in Theorem 3.1 (since B is low rank, $\\Sigma_{q}$ is low rank, the Gaussian is degenerate) and Theorem 3.2, and the subsequent problem of reader having to think about how the KL with generate Gaussians would work out in theory. In short, the theorems are unnecessary and need fixing because of improper distributions.\n\n1. Line 137: What is it that requires accurate estimation?\n2. Section 4: How is more than one sample (for N>0) used in generating a single output for evaluation and then influence the evaluation metric?\n\n*Minor*\n- Eq (3) to remove the $\\min_{\\theta}$ prefix, and to introduce the prior and likelihood.\n- Section 4: Shouldn't ECE and NLL be the metrics in focus because this is about \"Bayesian\"?\n- Section 4: The numbers in the table does not justify saying BLoB is the best.\n- \"Bayesianization\" is a mouthful --- can we think of a better derived word?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n**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...**\n\nWe 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. \n\n**W2. ...the problem of having to involve imprior priors... In short, the theorems are unneccessary and need fixing because of improper distributions.**\n\nThis is a good question. \n\n**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 \n+ their probability density is well-defined, and\n+ the KL divergence between two degenerate Gaussians is computable.\n\nDue to space constraint, we provide detailed discussion in a separate  **Official Comment** titled **Degenerate Gaussians are Proper for Probabilistic Inference** below this **Rebuttal**. \n\n**Necessity of Theorems 3.1 & 3.2.** Our Theorems 3.1 & 3.2 are necessary because they \n+ 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\n+ investigate important theoretical properties of our BLoB, offering insights into its underlying assumptions and advantages.\n\nWe will include the clarification above in our revision as suggested. \n\n[1] Schoeman, Johannes Cornelius, Corné E. van Daalen, and Johan A. du Preez. \"Degenerate Gaussian factors for probabilistic inference.\" IJAR 2022.\n\n**W3. Eq (3) to remove the $\\min_\\theta$ prefix, and to introduce the prior and likelihood.**\n\nThis is a good suggestion. We will remove the prefix in the revision. \n\n**W4. Section 4: Shouldn't ECE and NLL be the metrics in focus because this is about \"Bayesian\"?**\n\nYes, 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.\n\n**W5. Section 4: The numbers in the table does not justify saying BLoB is the best.**\n\nFor 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.\n\n**W6. \"Bayesianization\" is a mouthful --- can we think of a better derived word?**\n\nThanks 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.\n\n[1] Bacharach, Michael, and Susan Hurley. \"Issues and advances in the foundations of decision theory.\" Foundations of decision theory (1991): 1-38.\n\n**Q1. Line 137: What is it that requires accurate estimation?**\n\nThe 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. \nWe will include the discussion above in the revision.\n\n**Q2. How does more than one sample (for N>0) used in generating a single output?**\n\nThis 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. \nIn 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. \n\n**Q3. Limitations**\n\nPlease refer to **Q1. [Discussion of Limitations]** in **General Response**.", "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": "", "extra_scores": {}}, {"review_id": "RvgBo7viK3", "reviewer": "Reviewer_Adn3", "summary": "This paper proposes a method called Blob, which uses mean-field variational inference on LoRA parameters to obtain Bayesian estimation on LLM. This will result in a richer posterior structure on the weight than diagonal, while maintaining the same computational cost as diagonal covariance. During training, a new parameterization is used for posterior covariance to accelerate convergence and Flipout is used to make training more stable. Empirical results show Blob obtains competitive performance compared with other Bayesian methods.", "questions": "Major:\n- I'm not sure what it mean for the mean matrix of q(A) to be an output of a neural network (Line 186-187). Does this mean you are using another neural network to fit the mean of posterior? \n\n- 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?\n\n- What is the difference between BBB and Blob? From my understanding, the only difference is how you do the training. For BBB you use original parameterization and for Blob you use new parameterization plus Flipout? Also how many samples did you use to evaluate BBB in Table 1?\n\n- I would be interested to see a figure plot of model performance versus the number of MC samples. Specifically, the y-axis is accuracy/ECE/NLL, the x-axis is the number of samples used during inference. This helps the reader to understand the sufficient sampling size for inference.\n\n- I would be interested to see Laplace added to Table 8 to show a fair comparison of the computation cost compared with Blob.\n\n- Why is Laplace performing so poorly on ARC-C dataset? The original Bayesian LoRA paper it seems to perform much better on that dataset. \n\nMinor:\n- Line 236, MLE and MAP are not uncertainty estimation methods.\n- I don't entirely agree with the claim made in Lines 46-47. The possible suboptimal estimation of Laplace approximation comes from (1) LA, which assumes we reach MAP estimation while in practice you never know if the complex NN has really converged to MAP; (2) setting prior precision. It doesn't come from the post-training procedures.\n- the notation \"n\" refers to two different things, at Eq. (4) it's the number of parameters, and at Line 153 it's the dimension of the weight matrix.\n- Bolding is missing on ACC in Table 7.", "limitations": "See weaknesses and questions above.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- By putting a diagonal prior and approximate posterior on LoRA parameters, it leads to a richer posterior covariance structure while maintaining the computational cost of diagonal covariance.\n\n- The new parameterization of standard deviation seems promising for all VI training.\n\n- Good performance compared with other Bayesian methods.", "weaknesses": "- Not clear about the method's limitations. Since it uses VI, common disadvantages of VI also apply: (1) it needs to do sampling when making predictions, which can be computationally expensive given the size of LLM. (2) Given the complex posterior structure, there will be a very high chance of getting noise samples, which leads to suboptimal performance. This is already demonstrated in Table 1 that using MC estimation decreases the ACC compared with only using the mean. In additional, 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.\n\n- The whole paper feels a bit overselling: (1) I don't see the need to make Theorem 3.1 and Theorem 3.2 theorems, nothing surprising is stated there. (2) In Line 318, the authors claim their method \"demonstrates superior generalization and uncertainty estimation capabilities\", I don't think the results justify the \"superior\" claim. (3) In essence, Blob is mean-field VI on LoRA parameters trained with better strategies to make it suffer less from VI's high variance. I have no problem at explaining methods in great detail, but the whole writing on page 4 and page 5 feels like the authors are trying to make it sound as fancy as possible. I like the idea of the paper, but I think it can be presented in a more honest way.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n**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.**\n\nThanks 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. \n\nAs 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.\n\n[1] Blundell, Charles, et al. \"Weight uncertainty in neural network.\" International conference on machine learning. PMLR, 2015.\n\n**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.**\n\nThis is a good question. Actually our Theorems 3.1 & 3.2 are necessary because they \n+ 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\n+ investigate important theoretical properties of our BLoB, offering insights into its underlying assumptions and advantages.\n\nWe will include the clarification above in our revision as suggested. \n\n**W3.2. ...I don't think the results justify the \"superior\" claim.**\n\nFor 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.\n\n**Q3. What is the difference between BBB and BLoB?... How many samples are used to evaluate BBB**\n\nThe key distinctions between BLoB and BBB are:\n1. **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. \n2. **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.\n\nAs 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.\"``\n\n**Number of Samples.** For fair comparison, both BLoB and BBB use $N=10$ samples in Table 1. We will clarify this in the revision.\n\n**Q5. I would be interested to see Laplace added to Table 8 to show a fair comparison of the computation cost compared with Blob.**\n\nThis is a good question. We divide the computational cost into two phases: training and post-training (inference on test data).\nLAP'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**.\n\n| Metric          | Method        | WG-S            | ARC-C           | ARC-E           | WG-M            | OBQA            | BoolQ           |\n| --------------- | ------------- | --------------- | --------------- | --------------- | --------------- | --------------- | --------------- |\n| **Inference Time (Seconds)**  | **Standard LoRA** | 17              | 5               | 8               | 17              | 7               | 58              |\n|                 | **LAP**           | 311             | 445             | 814             | 554             | 1165            | 2508            |\n|                 | **BLoB (N=10)**   | 193             | 45              | 86              | 193             | 75              | 627             |\n| **Max Memory (MB)** | **Standard LoRA** | 14391           | 14157           | 13081           | 14391           | 14391           | 14160           |\n|                 | **LAP**           | 43742           | 61881           | 64737           | 43678           | 55642           | 67364           |\n|                 | **BLoB (N=10)**   | 14171           | 14411           | 13911           | 14407           | 14478           | 14177           |\n\nThese 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. \n\n**The remaining questions are answered in a separate Official Comment titled \"Remaining Questions for Reviewer Adn3\" below.**", "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": "", "extra_scores": {}}, {"review_id": "u5hTLWtoWY", "reviewer": "Reviewer_AGd6", "summary": "The paper proposes an algorithm to learn a (low-rank) variational posterior distribution over (a subset of) LLM weights during fine-tuning by combining Low-Rank Adaptation (LoRA) and Bayes by Backprop (BBB). To make the algorithm work in practice, several nontrivial modifications are introduced, such as certain parameterisations and optimisation strategies. Finally, the proposed method is empirically evaluated on a variety of in-distribution and out-of-distribution benchmarks and compared to relevant baselines.", "questions": "- Appendix B.1 mentions weights for the KL divergence term. How important is this? Please correct me if I am wrong, but I believe this was not explicitly discussed in the main paper. If this is important for the overall performance of the algorithm (and typically these kind of adjustments tend to be...), I suggest to at least mention it in the main paper (even if details remain in the appendix).\n- Is there any particular reason why $\\mathbf{A}$ is modelled in a probabilistic way instead of $\\mathbf{B}$? In other words, would it also be possible to do the reverse, i.e. assume a prior over $\\mathbf{B}$ and treat $\\mathbf{A}$ deterministically? The choice seems arbitrary to me, but perhaps there is an underlying reason to it.", "limitations": "Theoretical assumptions are stated as part of the provided theorems. As far as I am aware, general limitations of the proposed method (beyond the ones which are being addresses by the proposed method itself) are not explicitly discussed and could at least be briefly mentioned in the conclusion.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The paper is well written and the proposed algorithm is contextualised into existing work, with Figure 1 providing a great visual overview.\n- Concepts are introduced in an adequate pace and order, and a clean notation makes it easy for the reader to follow.\n- Elaborate empirical evaluation seems to suggest (somewhat) consistent gains over baseline approaches.\n- While the proposed algorithm seems like a straightforward combination of LoRA and BBB, the paper mentions that certain modifications are necessary for the algorithm to work in practice. These modifications are a core part of the paper's contribution, and they are discussed thoroughly and justified with a mix of theoretical and empirical arguments.", "weaknesses": "- The proposed method requires additional compute time and memory compared to standard LoRA. While this information is provided in Appendix C.2 (and seems to be small), I encourage the authors to be transparent about this by briefly mentioning it in the main paper (if that is not already the case, perhaps I missed it).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n**W1. The proposed method requires additional compute time and memory... by briefly mentioning it in the main paper...**\n\nThis 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:\n\n| Metric          | Method              | WG-S            | ARC-C           | ARC-E           | WG-M            | OBQA            | BoolQ           |\n| --------------- | ------------------- | --------------- | --------------- | --------------- | --------------- | --------------- | --------------- |\n|   | **Standard LoRA**       | 17              | 5               | 8               | 17              | 7               | 58              |\n|      **Inference Time (Seconds)**           | **LAP**                 | 311             | 445             | 814             | 554             | 1165            | 2508            |\n|                 | **BLoB (N=10)**         | 193             | 45              | 86              | 193             | 75              | 627             |\n||\n|  | **Standard LoRA**       | 14391           | 14157           | 13081           | 14391           | 14391           | 14160           |\n|        **Max Memory (MB)**         | **LAP**                 | 43742           | 61881           | 64737           | 43678           | 55642           | 67364           |\n|                 | **BLoB (N=10)**         | 14171           | 14411           | 13911           | 14407           | 14478           | 14177           |\n||\n|              | **Standard LoRA (MLE)** | 29.83 (0.58)    | 29.00 (1.97)    | 13.12 (1.39)    | 20.62 (0.74)    | 12.55 (0.46)    | 3.18 (0.09)     |\n|      **ECE**           | **LAP**                 | **4.15 (1.12)** | 16.25 (2.61)    | 33.29 (0.57)    | 7.40 (0.27)     | 8.70 (1.77)     | **1.30 (0.33)** |\n|                 | **BLoB (N=10)**         | 9.35 (1.37)     | **9.59 (1.88)** | **3.64 (0.53)** | **3.01 (0.12)** | **3.77 (1.47)** | 1.41 (0.19)     |\n||\n|             | **Standard LoRA (MLE)** | 3.17 (0.37)     | 2.85 (0.27)     | 1.17 (0.13)     | 0.95 (0.07)     | 0.73 (0.03)     | 0.32 (0.00)     |\n|      **NLL**            | **LAP**                 | **0.63 (0.00)** | 1.03 (0.04)     | 1.38 (0.01)     | 0.57 (0.01)     | 1.00 (0.00)     | 0.45 (0.00)     |\n|                 | **BLoB (N=10)**         | **0.63 (0.01)** | **0.78 (0.02)** | **0.40 (0.01)** | **0.54 (0.00)** | **0.50 (0.01)** | **0.31 (0.00)** |\n\nThese 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. \n\nWe will include the discussion above in the main body of the revised paper as suggested. \n\n**Q1. ...the KL divergence term. How important is this? ... this was not explicitly discussed in the main paper...**\n\nThis 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. \n\n**Q2. Is there any particular reason why $A$ is modelled in a probabilistic way instead of $B$?...**\n\nThank 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., $A\\leftarrow 0$, $B\\leftarrow \\text{random init}$).\n\n**Q3. Limitations**\n\nPlease refer to **Q1. [Discussion of Limitations]** in **General Response**.", "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": "", "extra_scores": {}}]

Author rebuttal:
# General Response
We thank all the reviewers for their valuable and constructive comments. 
We are glad that they like our idea (Adn3) and find the problem we address ``"important"`` (Lm4V), 
our paper ``"clear"``/``"well written"/"introduced in an adequate pace and order"`` (Lm4V, AGd6), 
our method ``"nontrivial"``/``"necessary"``/``"promising"`` (AGd6, Adn3) as well as ``"justified"`` by theoretical and empirical arguments (AGd6),  and 
our experiments showing that our method is ``"competitive"``/``"promising"``, of ``"efficiency"``/``"good performance"`` with ``"consistent gains over baseline approaches"`` (Adn3, Zor8, jN5u). 

Due to space constraint (6000 characters), we cannot cover all questions, but we promise to address all questions and cite all related references in our revision. Below we address reviewers' common questions one by one. 

**Q1. [Discussion of Limitations] (Lm4V, Zor8, Adn3, AGd6)** 

This is a good suggestion. We will add a Limitations section in the revision, which is summarized below: 
+ Currently, BLoB is not suitable for training-free tasks (direct operation on LLMs during inference), which are interesting future work. 
+ As the common disadvantage of VI algorithms, BLoB also requires sampling $N$ times during inference. 
+ With a limited number of samples, the presence of unavoidable noisy samples can affect the algorithm's performance. Sampling a computationally acceptable number of times (e.g., $N=10$) can mitigate this issue.

**Q2. [Reproducing LAP's Results on the ARC-C Dataset] (Lm4V, Adn3)** 

Our experimental setup strictly adheres to the original LoRA [2] framework, which has the **same hyperparameters** for **all datasets and methods** (learning rate, fine-tuning iterations, etc.). This widely adopted approach in LoRA-based research enhances our work's relevance and potential impact. 

Yang et al. [1] (LAP) employ a different setup in their original paper, incorporating early stopping to ensure a reasonable MAP for subsequent LA. However, this often results in reduced accuracy (over 1% deficit), prompting our decision to diverge from their setting. 

Nonetheless, we have diligently reproduced Yang et al. [1] within our framework, utilizing exactly their released code, with exactly the same hyperparameter configurations. However, we were unable to find a set of hyperparameters that consistently works for LAP across all datasets, leading to its poor performance on the ARC-C dataset.

To achieve competitive performance with LAP on the ARC-C dataset, we deviated from the unified setting (i.e., the same hyperparameter configuration for all datasets), allowing LAP to have different hyperparameter configurations for different datasets. Note that this creates an unfair advantage for LAP.

Specifically, we conducted an exhaustive grid search on 3 hyperparameters, resulting in the optimal configuration below:
- Dropout rate: 0.1
- Learning rate: 5e-5
- Early stopping: At 5000th iteration (out of 10000)

The table below shows the corresponding results (along with the original BLoB results for reference): 

| Metrics                | LAP          | BLoB ($N=10$) |
| ---------------------- | ------------ | ------------- |
| **ACC**   | 66.78 (0.69) | **68.81 (1.09)** |
| **ECE** | 16.25 (2.61) | **9.59 (1.88)**  |
| **NLL** | 1.03 (0.04)  | **0.78 (0.02)**  |

Note that BLoB (under the unified setting) still outperforms LAP even when LAP has the unfair advantage of allowing different hyperparameter configurations for different datasets. This showcases our BLoB robust performance improvement. 

During our reimplementation, we observed that LAP is highly dependent on and sensitive to MAP estimation. It frequently fails, corroborating your previous observation about LAP's potential disadvantage: sub-optimal MAP convergence significantly impacts LAP's performance.

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

[1] Yang, Adam X., et al. "Bayesian Low-rank Adaptation for Large Language Models." ICLR 2024. 

**Q3. [Novelty of BLoB] (Lm4V)** 

While using Bayesian methods to address inaccurate uncertainty estimation in neural networks is not new, it is a **nontrivial** contribution to apply these techniques effectively to Large Language Models (LLMs) and demonstrate their practicality. 

Laplace Approximation (**LA**) and Variational Inference (**VI**) are **two major approximate Bayesian inference paradigms**. While Yang et al. [1] validated **LA**'s effectiveness for LLMs, **VI** remains unexplored in this context. Our BLoB, as **the first representative VI approach on LLMs**, addresses this major research gap by demonstrating performance comparable or superior to LAP. As Reviewer AGd6 noted, while ``"the proposed algorithm seems like a straightforward combination of LoRA and BBB, the paper mentions that certain modifications are necessary for the algorithm to work in practice. These modifications are a core part of the paper's contribution, and they are discussed thoroughly and justified with a mix of theoretical and empirical arguments."``

To summarize our BLoB's novel contributions:
+ BLoB is, to our knowledge, the first VI method for LLM fine-tuning; it demonstrates effective uncertainty estimation across diverse datasets.
+ We provide theoretical analysis, proving the feasibility of optimizing the full-weight variational distribution in the low-rank space of weight update matrices, supported by empirical evidence of its effectiveness and efficiency.
+ To address BBB's consistent failures, our BLoB introduces:
  + a novel approximate posterior parameterization method, enabling fast convergence and accurate uncertainty estimation within limited training iterations, and 
  + a novel KL re-weighting scheme that effectively balances data likelihood and model complexity during training.

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

Meta-review: The paper proposes an algorithm to learn a (low-rank) variational posterior distribution over (a subset of) LLM weights during fine-tuning by combining Low-Rank Adaptation (LoRA) and Bayes by Backprop (BBB). To make the algorithm work in practice, several nontrivial modifications are introduced, such as certain parameterisations and optimisation strategies. Finally, the proposed method is empirically evaluated on a variety of in-distribution and out-of-distribution benchmarks and compared to relevant baselines.

All reviewers provided accept rating after the author effective rebuttal, but one is “borderline” and two are “weak”.
The most quoted strength of this work is the thorough experimental comparison that shows strong results compared to other LORA-based fine-tuning approach. The approach is also simpler that other Bayesian methods.

The most quoted weakness is about the contribution. 
- Reviewer Adn3: “Blob is  mean-field Variational inference on LoRA parameters”
- Reviewer Lm4v:  “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.”

But the author detailed rebuttal show that their method is a substantial progress over the Laplace LoRA, being much more stable and practical. This is confirmed by reviewer AGd6: 
“While the proposed algorithm seems like a straightforward combination of LoRA and BBB, the paper mentions that certain modifications are necessary for the algorithm to work in practice. These modifications are a core part of the paper's contribution, and they are discussed thoroughly and justified with a mix of theoretical and empirical arguments.”

Reviewers strongly disagree about the presentation, with ratings ranging from 1 (poor) to 4 (excellent). The AC, who is not an expert in LoRA or Bayesian methods, found the paper easy to read and understand.
Decision: Accept (poster)