Auto-GDA: Automatic Domain Adaptation for Efficient Grounding Verification in Retrieval Augmented Generation
Under review as a conference paper at ICLR 2025

AUTO-GDA: AUTOMATIC DOMAIN ADAPTATION FOR
GROUNDING VERIFICATION IN RETRIEVAL AUG-
MENTED GENERATION

Anonymous authors
Paper under double-blind review

ABSTRACT

While retrieval augmented generation (RAG) has been shown to enhance factual-
ity of large language model (LLM) outputs, LLMs still suffer from hallucination,
generating incorrect or irrelevant information. One common detection strategy
involves prompting the LLM again to assess whether its response is grounded in
the retrieved evidence, but this approach is costly. Alternatively, lightweight nat-
ural language inference (NLI) models for efficient grounding verification can be
used at inference time. While existing pre-trained NLI models offer potential so-
lutions, their performance remains subpar compared to larger models on realistic
RAG inputs. RAG inputs are more complex than most datasets used for training
NLI models and have characteristics specific to the underlying knowledge base,
requiring adaptation of the NLI models to a specific target domain. Additionally,
the lack of labeled instances in the target domain makes supervised domain adap-
tation, e.g., through fine-tuning, infeasible. To address these challenges, we intro-
duce Automatic Generative Domain Adaptation (Auto-GDA). Our framework en-
ables unsupervised domain adaptation through synthetic data generation. Unlike
previous methods that rely on handcrafted filtering and augmentation strategies,
Auto-GDA employs an iterative process to continuously improve the quality of
generated samples using weak labels from less efficient teacher models and dis-
crete optimization to select the most promising augmented samples. Experimental
results demonstrate the effectiveness of our approach, with models fine-tuned on
synthetic data using Auto-GDA often surpassing the performance of the teacher
model and reaching the performance level of LLMs at 10 % of their computational
cost.

1

INTRODUCTION

Large Language Models (LLMs) are increasingly used in consequential applications. Despite their
versatility, LLMs often produce hallucinations, in which the generated information is inaccurate
or fabricated and require costly retraining to integrate new knowledge. One promising method to
mitigate these issues is retrieval augmented generation (RAG, Lewis et al., 2020). RAG enhances
text generation by adding information from external knowledge sources to the prompt and has been
shown to reduce hallucinations in practice (Shuster et al., 2021). Nevertheless, even when modern
LLMs are used with RAG, hallucination rates of 15% – 30% (Chen et al., 2023a) or more than one
hallucination per 100 output tokens can occur (Niu et al., 2024).

To prevent hallucinated output from being delivered to end-users, natural language inference (NLI)
models can be used to verify grounding of the generated output in the documents retrieved (Chen
et al., 2023b; Es et al., 2024; Tang et al., 2024) before the output is relayed to the end-user: the
generated response must be fully grounded in the documents, i.e., it must be logically inferrable
from the documents; otherwise, it is considered ungrounded. However, as we need to check the
outputs at inference time, we require lightweight NLI models with very low latency. The cur-
rent landscape of available NLI models for verifying grounding in RAG is illustrated in Figure 1
based on results obtained in our evaluation of correctness and inference time (see Table 3 for
full numeric results): Some recent works such as Mini-Check (Tang et al., 2024) have developed

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i
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10x

2.5x

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LLMs:
e.g., GPT-4,
Claude-3.5

Pre/Post-
Processing:
e.g., AlignScore

lightweight models for NLI, e.g., based
on RoBERTa (Liu, 2019). These models
have shown good performance on aca-
demic benchmarks. However, our re-
sults indicate that their performance in
verifying grounding for realistic RAG
inputs lags behind LLMs by about 20%
(in ROC-AUC scores). Other recent
methods use pre- and post-processesing
techniques such as sentence tokeniza-
tion or LLM prompting to decompose
long prompts (Zha et al., 2023; Es et al.,
2024) into several chunks or facts. Each
of these chunks needs to be processed
in a separate forward pass, resulting in
high latency as well. While some stud-
ies (e.g., Manakul et al., 2023; Tang
et al., 2024) have also explored directly
using LLMs like GPT-4 for text entail-
ment detection, their latency is about an
order of magnitude above the lightweight models. Taken together, these characteristics make it hard
to deploy the existing approaches in real-time industry use-cases.

Figure 1: Landscape of current grounding verifica-
tion models for RAG. While LLMs have the best per-
they incur about 10× higher latency than
formance,
lightweight models. In this work, we are interested in ob-
taining lightweight models with LLM-level performance
for grounding verification through domain adaptation.

grounding verification performance (RAG data, ROC-AUC)

domain-adapted,
lightweight
model

lightweight:
e.g., DeBERTa,
Mini-Check

+20%
performance

≈ 0.88

≈ 0.72

1x

The performance gap observed for realistic RAG inputs with the lightweight models may point to a
substantial domain mismatch between the NLI datasets used to train these models and the challeng-
ing, real-world data encountered at test time. We observe that inputs of NLI models in RAG are more
challenging as they comprise longer segments with multiple statements and contain more subtle un-
grounded information as the output is LLM-generated. While these characteristics are common to
RAG systems in general, each implementation still has a very individual input distribution: First,
inputs may follow a specific format due to the RAG prompt template e.g., question: <question>.
evidence: Passage 1 <evidence1>, Passage 2 <evidence2> .... Second, the documents are retrieved
from knowledge bases from a variety of different domains, which may not be represented in training
data. Prior work (Williams et al., 2018) confirms difficulties when NLI models are applied to data
from an unseen domain and Hosseini et al. (2024) shows a generalization gap of up to 20%. This
suggests that NLI models need to be adapted to their target domain for optimal performance.

Bridging this domain gap poses a significant challenge due to the inherent difficulty of adapting
models to unseen domains that is further amplified by the prohibitive costs of obtaining labeled data
from the target domain. This prevents supervised domain adaptation, e.g., through fine-tuning on
target domain data. To address this issue, we propose Automatic Generative Domain Adaptation
(Auto-GDA). Our unsupervised domain adaptation framework produces high-quality synthetic data,
which is then used to fine-tune a lightweight NLI model, adapting it to a specific domain of RAG
inputs. While training data generation by simply prompting LLMs has been repeatedly explored
in the literature (e.g., Saad-Falcon et al. (2024); Hosseini et al. (2024)), data quality might be fur-
ther improved through filtering and incorporating background knowledge through label-preserving
data augmentation strategies, such as round-trip translation (Chen et al., 2023b). However, speci-
fying good filters and heuristic augmentation strategies require significant manual effort. As data
augmentations can further be applied iteratively, the space of potential samples grows exponentially,
necessitating efficient search strategies. During this offline training phase, less efficient teacher mod-
els can provide additional guidance using weak labels. Auto-GDA offers a unified way to leverage
all these available tools. We thus make the following contributions:

1. We formalize the unsupervised domain adaptation problem under the availability of practi-
cal tools such as data generators, data augmentation routines, and weak teacher models.
2. We propose Automatic Generative Domain Adaptation (Auto-GDA), a principled frame-
work for unsupervised domain adaptation through synthetic data that can be instantiated
with different implementations of generation, augmentation, and weak labeling steps and
which automatically selects high-quality samples.

3. We show that our objective corresponds to an enhanced distribution matching objective but

is highly efficient to optimize.

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4. Our experiments on realistic RAG inputs highlight that our fine-tuned models using Auto-
GDA (1) often outperform their weak teacher models (2) perform almost as well as ref-
erence models fine-tuned with human-labeled data and (3) reach the level of performance
exhibited by LLMs while having almost 10x lower latency. (4) Our method further outper-
forms more classical training-based unsupervised domain adaptation techniques.

2 RELATED WORK

The problem of domain adaptation is concerned with adapting existing models to different domains.
We introduce the most closely related approaches in this section and refer the reader to Ramponi &
Plank (2020) for further references.

Synthetic NLI Data. Related works explore synthetic data generation for NLI models. Hosseini
et al. (2024) generate the diverse, cross-domain GNLI (General NLI) dataset synthetically in two
steps: first prompting an LLM to generate target domains, then using a prompt-tuned LLM to gen-
erate training statements. Tang et al. (2024) generate synthetic training data for their MiniCheck
models using document-to-claim generation and claim-to-document generation. We compare to
their model in our experimental section and show that it can be further improved through domain
adaptation. Saad-Falcon et al. (2024) use synthetic data to specifically improve RAG system eval-
uation. They generate synthetic in-domain data with a few-shot prompt. However, their method is
compared within RAG evaluation frameworks and not tested for NLI performance.

Synthetic Data for Domain Adaptation in NLP. While synthetic QA data generation is well-
explored (Shakeri et al., 2020; Ushio et al., 2022; Yue et al., 2022; Lee et al., 2023), synthetic data
for NLI domain adaptation has received less attention, potentially due to the difficulty of generating
realistic and difficult samples. Wang et al. (2023a) propose an iterative synthetic data generation
scheme requiring partially labeled data. They generate initial seed data using an LLM prompt that
is iteratively refined based on errors from a model trained on human-labeled reference set. Unlike
this work, we assume very limited access to labeled data from the target domain.

Classical Unsupervised Domain Adaptation. Beyond synthetic data approaches, classical unsu-
pervised domain adaptation (UDA) techniques have also been applied in NLP. Chen et al. (2018); Li
et al. (2018); Choudhry et al. (2022) have explored Domain Adversarial Neural Networks (DANN)
(Ganin et al., 2016), which incorporate domain discriminators during pretraining to learn domain-
invariant features. He et al. (2020) introduce Scale-invariant-Fine-Tuning (SiFT) which extends the
Virtual Adversarial Training (VAT) framework of Miyato et al. (2019) and Jiang et al. (2020) to im-
prove model robustness and generalizability. Techniques like CORAL (Sun & Saenko, 2016) align
feature distributions between source and target domains by matching their second-order statistics.
Finally, domain-adaptive pretraining (DAPT) and task-adaptive pretraining (TAPT) (Gururangan
et al., 2020; Han & Eisenstein, 2019) involve pretraining on target domain text before fine-tuning
on labeled source data. Although these methods have shown success in tasks like sentiment analysis
and text classification, they have not been comprehensively studied in NLI.

Knowledge Distillation. We borrow the term “teacher model” from the knowledge distillation
literature (Gou et al., 2021; Yang et al., 2020). However, our problem differs from distillation
problems because our target dataset is unlabeled.

In this paper, we focus on the problem of systematically generating and selecting the most beneficial
synthetic samples that can be created through initial generation and iterative augmentation steps. We
do so using an efficient objective that can be interpreted as a form of distribution matching.

3 PRELIMINARIES

Domain adaptation is concerned with adapting an ML model pretrained on a source domain to make
predictions on a target domain when the underlying data distributions differ across the two domains.
The unsupervised domain adaptation problem is further complicated due to the lack of labeled data
in the target domain. This means that while features are available, there is no direct information
about the correct class labels for the target domain samples. This poses a significant challenge as
the model must learn to adapt to the new distribution without explicit guidance.

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3.1 UNSUPERVISED DOMAIN ADAPTATION FOR NLI

Data Domains. Following the common natural language inference setup, we assume data from
i.i.d.∼ ps containing
a source domain is available as a set of triples Ds = {(en, cn, yn)}n=1,...,N
evidence e ∈ X , corresponding claims c ∈ X where X denotes a space of text sequences, and
labels y ∈ Y. This data is used to train an initial model f : X × X → Y. We use Ds to denote sets
of samples and ps to denote the data density of the source distribution. Note that in a RAG use-case,
the evidence e will contain the user prompt as well as the retrieved documents. Additionally, we are
provided with a set of J unlabeled samples Dt = {(ej, cj)}j=1,...,J from the target domain. They
are sampled from pt, the data distribution faced at test time (e.g., the realistic RAG inputs). Our
goal is to adapt a model pretrained on ps to perform well on pt. In this work, we are focusing on
problems where J is small. This scarcity makes it challenging to accurately estimate the underlying
distribution of the target domain, which can hinder the effectiveness of traditional domain adaptation
methods that rely on a substantial amount of target data. We study the binary NLI task where
Y ∈ {0, 1}. A positive label (y=1) is only assigned if all information in the claim can be inferred
directly from the evidence; claims that are contradictory to the evidence or cannot be inferred from
the evidence are considered non-entailed (y=0).

In this work, we focus on covariate shift between the two domains: While the prior p(e, c) is
subject to change across domains, the true relation between specific features and labels, p(y|e, c) is
consistent for the source and the target domain. For the NLI task considered here, this assumption is
sensible because the entailment relation itself does not change for different domains. Following prior
work Saad-Falcon et al. (2024), we slightly deviate from the fully unsupervised setup by supposing
that a very small portion of the target domain can be manually labeled and used as a validation set for
hyperparameter tuning only, as is commonly done in NLI literature (Laban et al., 2022; Tang et al.,
2022; Zha et al., 2023). We show that our method works with validation sets as small as 30 samples.

Helper Tools. We extend this common setup to incorporate three additional tools that are readily
available in practice: First, we have powerful generative LLMs that we can use to generate new
samples based on the unlabeled examples using techniques such as prompt-tuning (Lester et al.,
2021), or few-shot prompting. The generator G can be formally described as (randomized) function
G : X × (X )F × Y → X , meaning that G takes as input a piece of evidence and a set of F ≥ 1
example claims (e.g., for few-shot prompting) and a desired target label. The generator G is then
tasked with producing a new claim sample that reflects the style of the provided claims and has the
specified target label. Note that we provide the F claims without a known label, so they can either
be entailed or non-entailed w.r.t. e. Second, we can use some background knowledge of the task to
define some approximately label-preserving augmentation strategies to increase diversity, e.g., us-
ing paraphrasing models, round-trip translation or synonym replacements (Chen et al., 2023b). This
step can be formalized as a mutation function M : X → X which takes a claim as an input and mod-
ifies it while trying to preserve its label. The label-preserving characteristics of these strategies are
imperfect, i.e., with a small probability the entailment relation will be affected by the augmentation.
Finally, we suppose a teacher model T : X × X → [0, 1], which can be applied to the data from the
source and the target domain and provides an entailment score. The teacher model performs reliably
within the source domain, but only provides a weak estimate of T (e, c). The performance of this
model may be noisy because the target domain is out-of-domain for this model, and the model may
be too inefficient to be deployed in practice. We will use this model to obtain weak estimates of
the samples’ labels. We now present our framework Auto-GDA, which incorporates the three tools
G, M, T named above in a principled algorithm.

4 A PRINCIPLED FRAMEWORK FOR UNSUPERVISED DOMAIN ADAPTATION

4.1 OUTLINE OF THE FRAMEWORK

In this work, we present Auto-GDA, a framework for Automatic Generative Domain Adaptation,
that generates synthetic data points that are useful for fine-tuning a pretrained model f for the target
domain. For the data generation process to result in high fine-tuning utility it must meet several
criteria: (1) The data must be realistic and non-trivial, (2) must have high diversity, (3) the assigned
labels must be of high quality.

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Under review as a conference paper at ICLR 2025

Initial data generation
using generator G,
teacher model T

Unlabeled Target
Data Dt

Select K best
samples to
minimize Ltot

Augmented
Synthetic Data +
Quality Scores

Labeled Synthetic
Data D(i)
e

Repeat
i = 1, ..., I
times

Compute sample
contributions to Ltot

final high-quality
data

Apply aug-
mentations
using M, T

Augmented
Synthetic Data
¯D(i+1)
e

fine-tune model f

Figure 2: Overview of Auto-GDA. We generate initial data using the generator G, which are as-
signed entailment certainty scores using teacher model T . The synthetic data is iteratively aug-
mented using M , whereas label-preservation is confirmed with T and entailment certainties are
updated. We finally select the top-K samples that minimize an objective function Ltot. These steps
can be applied iteratively until the final data is used to fine-tune the model f for the target domain.

Auto-GDA is specifically designed to tackle these three challenges. As RAG outputs stem from
LLMs, we also generate realistic initial claim samples for a given evidence using LLMs. We lever-
age few-shot prompting to transfer patterns in the output to the generated samples. To preserve the
diversity of the evidence (which contains the relevant documents in the knowledge base), we gen-
erate synthetic claims sequentially for each unique piece of evidence e available in the unlabeled
target dataset Dt. This has the advantage that a broad diversity of documents in the knowledge
base is represented. We propose to apply augmentations on the synthetic data to increase diversity
further. As the augmentation strategies are only approximately label-preserving, we have to keep
track of increasing label uncertainty to detect samples with low-quality labels when several data
augmentation steps are applied. We therefore equip each sample with an entailment certainty score
r, an estimate of the probability of the sample having an entailed label (y=1) which can be used to
remove samples with low-quality labels. Auto-GDA applies these steps iteratively to successively
increase data quality. In summary our framework consists of the following steps, which we describe
in more detail in the next sections:

e = {(ˆck, ˆyk, r(0)

1. Initial Generation. Generate an initial sample population D(0)

k=1 of
claims ˆc and labels ˆy for the evidence e using the generator G. Use the teacher model T to
assign initial entailment certainty r(0) scores to each sample of synthetic data. This results
in each sample having a hard label ˆy and a “soft” confindence score r(0) for the hard label.
to
obtain new claims with the same hard labels. Update their entailment certainties using the
teacher model again. Merge mutated samples and samples from previous iteration to form
updated population ¯D(i+1)
)}L

2. Sample Augmentation. Apply augmentations M on claims in the population D(i)
e

l=1 that is of larger size L ≫ K.

= {(ˆcl, ˆyl, r(i+1)

k )}K

e

l

3. Sample Selection. Select the subset of samples of size K from ¯D(i+1)

that minimize our
proposed enhanced distribution matching objective Ltot formally introduced in Eqn. (4).
The objective includes the unlabeled target samples Dt and the certainty scores. The
selected subset becomes the next generation dataset D(i+1)

e

e

.

4. Repeat steps 2 and 3 for a fixed number of iterations or until objective Ltot converges.

We illustrate these steps in Figure 2 and will detail out implementation choices for each step below.

4.2 GENERATING REALISTIC INITIAL DATA

LLMs have been repeatedly used to generate synthetic data for various domains, including NLI
(Saad-Falcon et al., 2024; Hosseini et al., 2024). In this work, we generate initial data using few-shot
prompting with the prompts provided in Appendix E.1. The prompt instructs the LLM to generate
synthetic claims ˆc = G(e, claim(Dt,e), ˆy) for the evidence e, reflecting the style of example claims
from Dt (claim(Dt,e) denoting claims from target data for the evidence e) and target label ˆy ∈ 0, 1.
For label ˆy = 1, the LLM is instructed to include only grounded facts, for ˆy = 0, some ungrounded
information should be introduced. We assign labels ˆy according to the prompt used, resulting in
complete initial generated tuples (ˆc, ˆy). We follow some related works (Puri et al., 2020; Vu et al.,

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2021), which have suggested generating many samples and only keeping the most confident. To
do so, the samples can be equipped with a weak estimate of the label probability using the teacher
model, e.g., another LLM or an NLI model with sophisticated pre- and postprocessing. In the binary
classification setup, we can compute initial entailment certainties as r(0) = T (e, ˆc), which can be
interpreded as an uncalibrated and potentially noisy estimate of p(y = 1|e, ˆc). We explore LLMs
for data generation and use state-of-the-art NLI models and also LLMs as teacher models T for
providing initial entailment certainties. Adding the entailment certainty scores r(0) to the respective
tuples we obtain a set of triples D(0)

e = {(ˆck, ˆyk, r(0)

k=1 after this step.

k )}K

4.3

INCREASING DIVERSITY THROUGH LABEL-PRESERVING DATA AUGMENTATIONS

In this section, we demonstrate how
to augment the initial synthetic dataset
(generated using the few-shot prompt-
ing strategy) for additional diversity,
while maintaining a high degree of label
certainty for the augmented synthetic
data points. We exploit a certain de-
gree of background knowledge to de-
rive data augmentation strategies (Chen
et al., 2023b). For instance, we know
that paraphrasing the claims while pre-
serving their semantic meaning should
not change their entailment label. How-
ever, when iteratively applying para-
phrasing operations, we have to account
for an increasing probability of acciden-
tally flipping the label.

e: Paris is the capital of France, a country in Europe.
Paris has 2.1 million inhabitants.
c: Paris, capital of France, has 2.1 million inhabitants.
y: 1 (entailment)

n
o
i
t
a
t
u
m

e entails c
c entails c′

⇒ e entails c′

e: Paris is the capital of France, a country in Europe.
Paris has 2.1 million inhabitants.
c′: There is a capital in Europe with 2.1 million inhab-
itants.
y: 1 (entailment)

rewritten sample
Figure 3: Intuition for our update rule for entailment
certainties: If a parent claim c is entailed by e and a
mutated claim c′ is entailed by its parent c, the mutated
claim c′ will be entailed by e as well.

Obtaining High-Quality Entailment Certainties. We can combine the generative models with
discriminative teacher models again to obtain weak estimates r(i) of the entailment certainty of the
augmented samples. Instead of directly computing the entailment probability using T , we exploit
logical invariances, which allow for better estimates depicted in Figure 3: If the original claim is en-
tailed by the evidence, and if the modified claim is entailed by the original claim, the modified claim
will also be entailed by the evidence. Suppose we have obtained ˆc′ = M (ˆc) as a modification of the
synthetic claim ˆc. As we already have an estimate of the entailment probability for (e, ˆc), we can
reuse it and only need to compute another entailment probability for (ˆc, ˆc′). We argue that comput-
ing this entailment probability is easier for the teacher model than directly computing T (e, ˆc′), as the
claim and the modified claim should be semantically and syntactically more similar. Paraphrasing
datasets like PAWS (Zhang et al., 2019) are common pretraining datasets, and standard NLI datasets
like MNLI (Williams et al., 2018) contain many similar samples due to their construction through
edits, so NLI predictions are expected to be more reliable on these pairs. Querying the teacher model
on T (ˆc, ˆc′) allows us to use the following update rule for the augmented sample (e, ˆc′):

r(i+1)(e, ˆc′) = r(i)(e, ˆc) · T (ˆc, ˆc′) + (1 − r(i)(e, ˆc)) · (1 − T (ˆc, ˆc′)).

(1)

using the entailment certainty r(i) of the original tuple (e, ˆc) as a base. Note that some teacher mod-
els may be particularly reliable with claim-claim pairs than with evidence-claim pairs so it can be
useful to choose a different teacher model for this update than for computing initial certainty scores.

Label Invariant Augmentation Strategies: In this work, we consider three augmentation strategies
that will likely preserve entailment labels (see Appendix Appendix C.1 for additional details):

• Partial Rephrasing with LLMs. Our first augmentation is an LLM-based rephrasing step.
Specifically, we randomly mask 20% of the words of the input sequence by replacing the
corresponding words by “_" and ask an LLM (Claude3 Haiku) to impute the gaps while
preserving the meaning.

• Complete Paraphrasing. We use a T5-based paraphrasing model Vorobev & Kuznetsov
(2023). We generate paraphrases for the claims using enforcing diversity using a constraint
that prevents n-grams of length greater than 5 from being regenerated.

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• Sentence Deletion. We chunk the claim into sentences and randomly delete one of them.
This should preserve the entailment relation as it only removes information. However, we
note that this augmentation may remove some of the context necessary to understand the
entire claim.

We generate several augmentations for each sample using these strategies along with an estimate
of their entailment probabilities, resulting in an enlarged sample set. Unfortunately, not all of these
samples may be of high quality. Therefore, it is crucial to select only the most promising samples.

4.4 AUTOMATIC SELECTION OF HIGH-QUALITY SAMPLES

A key component of our work involves automatically selecting the most promising samples. Intu-
itively, we are interested in finding samples that resemble target data. This includes both having
realistic features and correctly assigned labels. The data should also have a high chance of improv-
ing the final model. Provided with an augmented dataset ¯D(i)
l=1 at iteration i, we are
interested in selecting a subset Qe ⊂ ¯D(i)
e of a smaller size |Qe| = K that only contains the most
promising samples. We propose the following objective function to assign a loss to a selected subset
Qe which contains three terms for each selected sample:

e = { ˆcl, ˆyl, rl}L

Ltot(Qe, f ) =

(cid:88)

ˆci,ˆyi,ri∈Qe


d(ˆci, cmin,i)2

(cid:124)
(cid:123)(cid:122)
(cid:125)
distance

+ λdLDiv(ri, ˆyi)
(cid:125)
(cid:123)(cid:122)
(cid:124)
label correctness

− λuUf (ˆci, ˆyi)
(cid:125)

(cid:123)(cid:122)
utility term

(cid:124)




 ,

(2)

where d(x, x′) = ∥ψ(x) − ψ(x)′∥ is a distance function over inputs in X defined via textual em-
beddings ψ, cmin,i := arg minc′∈claim(Dt,e) d(c′, ˆci) is the closest claim for evidence e from the
target dataset, and λd, λu are hyperparameters. LDiv : [0, 1] × {0, 1} → R+ is a function that
penalizes uncertain labels taking the certainty scores r and the hard labels ˆy as inputs as plotted
in Figure 4. We derive the exact form of the LDiv function as a divergence estimate of the condi-
tional distributions in Appendix B.2. The distance term encourages samples to be close to claims
from the target data set for the given evidence. The la-
bel correctness term penalizes samples where the en-
tailment certainties are too far apart from the target
labels and is used to discourage selection of samples
where the labels are likely to be incorrect. Addition-
ally, we encourage generation of samples where the
pretrained model f is not performing well yet by in-
cluding the cross-entropy loss of the model as a util-
ity term, Uf = CE[f (e, ˆc), ˆy] where ˆy is the assigned
hard label of a synthetic sample.

Theoretical Properties. Notably, Equation (2) can
be derived from first principles as an enhanced dis-
tribution matching objective. By defining parametric
distributions pQ,e(c, y) (representing the selected syn-
thetic data for evidence e) and pcov,e(c, y) (represent-
ing the target distribution for e we aim to imitate) the
objective corresponds to the divergence between these
distributions plus the expected utility of the synthetic
data. Formally,

Figure 4: Modeling the label correctness
term in Eqn. 2 as function of r. When the
estimated entailment certainty r does not
match the assigned hard label ˆy this term
takes high values discouraging selection.

Ltot(Qe, f ) = DKL (pQ,e(c, y)||pcov,e(c, y)) − E(c,y)∼pQ,e [Uf (c, y)] .

(3)

We derive a proposition to formalize this connection in Appendix B.

Optimizing the objective. Optimizing the objective for a subset Qe containing K synthetic samples
for evidence e with minimal loss can be done highly efficiently in three steps: (1) Computing each
samples’ contribution to the sum in Ltot, (2) ranking the samples by this contribution, and (3)
greedily selecting the top-K subset of samples with the lowest contributions. Pseudocode of our
complete framework is provided in Algorithm 1 (Appendix).

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Dataset
RAG-Task

RAGTruth

Summary

QA

LFQA-Verif. SummEdits Avg., Rank
Summary

QA

l
e
d
o
m
e
s
a
b

s FLAN-T5

BART-large
DeBERTaV2
s DAPTDeBERTaV2
SiFTDeBERTaV2
CORALDeBERTaV2

s
e
n
t
s
u
b
o
r

e
l
p
m
o
c

x MiniCheck-T5
AlignScore
Vectara-2.1

0.734
0.696
0.782

0.708
0.670
0.530

0.655
0.821
0.645

0.700
0.769
0.876

0.699
0.739
0.708

0.746 ± 0.005
0.785 ± 0.008
0.718 ± 0.001

0.703 ± 0.016
0.566 ± 0.005
0.677 ± 0.001

0.813 ± 0.094
0.880 ± 0.032
0.822 ± 0.001

0.837 ± 0.004 0.775
0.845 ± 0.003 0.769
0.853 ± 0.001 0.768

0.754
0.729
0.805

0.640
0.822
0.854

0.741
0.904
0.648

0.791
0.894
0.590

0.732
0.837
0.725

D
G
-
o
t
u
A

A Flan-T5 (Auto-GDA)
BART (Auto-GDA)
DeBERTaV2 (Auto-GDA)

0.756 ± 0.004
0.813 ± 0.009
0.837 ± 0.007

0.783 ± 0.013
0.867 ± 0.011
0.867 ± 0.007

0.687 ± 0.002
0.867 ± 0.026
0.925 ± 0.009

0.824 ± 0.010 0.762
0.860 ± 0.010 0.852 3
0.883 ± 0.005 0.878 2

s GPT-3.5

M
L
L

GPT-4o-mini
GPT-4o

0.706
0.884
0.892

0.648
0.833
0.865

0.749
0.812
0.896

0.814
0.878
0.880

0.729
0.852 3
0.883 1

Table 1: Performance comparison to baselines (ROC scores). Grouped by off-the-shelf base mod-
els trained on standard data, domain-adapted versions of the best base models using DAPT, SIFT, and
DeepCORAL, complex state-of-the-art models trained using custom datasets (Vectara, MiniCheck)
or using postprocessing (AlignScore), proprietary LLMs, and versions of the base models fine-tuned
with Auto-GDA. We highlight the teacher model that was used to assign initial label certainties r(0)
in a box and make three observations: (1) the Auto-GDA version of the base models always im-
proves over the vanilla versions and the versions trained with SIFT, Deep CORAL, and DAPT, (2)
our best-performing model DeBERTaV2 (Auto-GDA) outperforms its teacher model in three out of
four cases, and (3) BART and DeBERTa with Auto-GDA reach LLM-level performance.
5 EXPERIMENTAL EVALUATION

We run experiments with realistic datasets and baseline models to confirm the efficacy of Auto-GDA.

Datasets. We evaluate our approach on three datasets for document-grounded summarization and
question answering (QA). We select datasets which include documents, realistic LLM-generated
long-form answers, and human labels that can be used for testing. The SummEdits (Laban et al.,
2023) dataset contains GPT-3.5-generated and manual summaries of documents from different do-
mains, e.g., judicial, sales emails, podcasts. We further use both the summary and the QA portion
of the RAGTruth dataset (Niu et al., 2024). The RAGTruth dataset contains summaries and answers
to questions created by LLMs (GPT-3.5/4, Mistral, Llama2). Finally, we use the LFQA-Verification
dataset (Chen et al., 2023a), which retrieved documents for questions from the “Explain me Like
I am five”-dataset and generated corresponding long-form answers with GPT-3.5 and Alpaca. We
selected the datasets to feature characteristics of realistic RAG systems including specific prompt
templates (present in RAGTruth, LFQA-Verification) and various domains (present in all datasets,
specifically in SummEdits). Details and links to the datasets can be found in Appendix C.2.

Base models. As NLI models, we use three pretrained model architectures that are able to handle
NLI queries with the longer context required for RAG inputs. We investigate a BART-Large (Lewis
et al., 2019) model pretrained only on the MNLI dataset (Williams et al., 2018). This can be consid-
ered a lightly pretrained model. Additionally, we study DeBERTa-V2 pretrained with datasets from
the tasksource collection (Sileo, 2024). We additionally study a FLAN-T5-based model (Raffel
et al., 2020) pretrained on MNLI. The all models possess context lengths of at least 1024 tokens.

Baselines. We use state-of-the-art baselines: AlignScore (Zha et al., 2023) (RoBERTa-based with
pre- and postprocessing), MiniCheck (Tang et al., 2024), and Vectara-2.11 (both T5-based). These
complex state-of-the-art models are trained using complex custom datasets (Vectara, MiniCheck)
or use postprocessing (AlignScore). As a teacher model to assign initial score, we use the best-
performing model from the “complex” category, which allow easy access to uncertainty scores and

1https://docs.vectara.com/docs

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Dataset
RAG-Task

non-fine-tuned

RAGTruth
Summary QA

0.782

0.530

+ft. on Few-Shot Data
+ft. on Augmented w/ Random Selection
+ft. on Augmented w/ Objective (Auto-GDA)

0.799
0.777
0.837

0.826
0.783
0.867

fine-tuned on labeled

0.842

0.890

LFQA-Verif. SummEdits Mean (Gap closed)

QA

0.645

0.934
0.919
0.925

0.909

Summary

0.876

0.872
0.862
0.883

0.898

0.708 (0%)

0.858 (84%)
0.835 (71%)
0.878 (96%)

0.885 (100%)

Table 2: Ablation: Fine-tuning with synthetic data obtained by few-shot prompting and random
selection of augmentations as opposed to using our framework Auto-GDA. We also report perfor-
mance relative to the hypothetical upper baseline of fine-tuning on labeled target data and observe
that we can almost close this domain-adaptation gap (ROC, DeBERTa model, avg. over 5 runs).

have acceptable performance. We employ optuna2 as a principled way of choosing the remaining
hyperparameters λ′
u, λd, and the other teacher model used to estimate entailment probabilities for
augmentations in Eqn. 1. We perform 50 trials per dataset and use the ROC score of a fine-tuned
DeBERTaV2 model on the small validation dataset as selection objective. In case limited budget
for hyperparameter tuning is available, we recomment setting λ′
u=λd ∈ [20, 50] which led to sta-
ble performance. Auto-GDA is run for two iterations on RAGTruth and one iteration on the other
datasets, generating synthetic datasets between 1.3× and 2× the original dataset size. We found
no improvements through further increasing dataset size. We also compare against several common
UDA methods, including robustness-based approaches. Specifically, we implement DAPT (Guru-
rangan et al., 2020), SiFT (He et al., 2020), and Deep-CORAL (Sun & Saenko, 2016) for further
pretraining of the DeBERTa-V2 model.

5.1 SYNTHETIC DATA FOR NLI MODEL FINE-TUNING

We present the main results obtained with Auto-GDA in Table 1. Our results show that Auto-GDA is
highly effective and improves performance in ROC-AUC scores of all tested models on all datasets.

Comparison to Teacher Models. Auto-GDA is highly effective, not only incorporating the knowl-
edge of the stronger teacher model (indicated by box) but often even surpassing it, as the optimiza-
tion enhances data quality over the teacher in three out of four datasets.
Comparison to Classical UDA Methods. Traditional UDA methods (DAPT, SiFT, and Deep-
CORAL) did not yield significant improvements in our NLI domain adaptation setting and Auto-
GDA consistently outperforms them across all datasets. This also indicates that synthetic data gen-
eration is more effective for NLI tasks.
Comparison to LLMs. Finally, our fine-tuned models reach performance levels between state-of-
the-art LLMs such as GPT-4o and GPT4o-mini while maintaining significantly lower computational
requirements. This shows that our approach results in models with superior NLI performance, in
particular when compared to the non-fine-tuned or non-LLM baselines.
Other Teacher Models. We investigate using LLMs and other teacher models in Table 9 (Appendix)
but observe that LLMs do not generally outperform other teacher models, possibly due to unreliable
uncertainty scores. However, the table also shows that the DeBERTa model can improve its own
performance through self-supervision by an average of 0.15 AUC when applied as the teacher model.

5.2 ABLATION INVESTIGATIONS

Components of the algorithm. We add the components of our algorithm individually and show
how they successively increase performance in Table 2. In all ablations we keep dataset size and
other parameters constant. The biggest gain is achieved by fine-tuning on data created by few-shot
prompting. We subsequently add data augmentations without applying our selection criterion,
but instead selecting few-shot and augmented samples randomly. We observe that this decreases
data quality, highlighting that data augmentation is only beneficial together with our filtering
criterion. When we do so and apply data augmentation with our filtering step (corresponding to

2https://optuna.org/

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Model

RAGTruth

LFQA-Verification SummEdits Inference time (relative) Performance

Summary

QA

QA

Summary

sec/(50 samples)

AUC-ROC*100

Vectara
FLAN-T5
DeBERTaV2
MiniCheck-T5
BART-large

1.57 ± 0.02 1.13 ± 0.03
1.71 ± 0.07 1.71 ± 0.07
2.56 ± 0.03 1.88 ± 0.04
4.50 ± 0.20 3.16 ± 0.06
4.33 ± 0.01 3.62 ± 0.06

1.35 ± 0.03
1.72 ± 0.07
2.15 ± 0.06
3.90 ± 0.14
3.95 ± 0.09

1.03 ± 0.01
1.71 ± 0.07
1.88 ± 0.09
3.22 ± 0.10
3.76 ± 0.20

1.27 (59%)
1.71 (80%)
2.12 (100%)
3.69 (174%)
3.92 (184%)

AlignScore

5.88 ± 0.12 7.55 ± 0.28

7.55 ± 0.35

1.81 ± 0.06

5.70 (269%)

GPT-4o

19.80 ± 0.51 19.11 ± 0.44

21.09 ± 2.97

21.89 ± 1.26

20.47 (967%)

Auto-GDA DebertaV2

Same as DeBERTaV2

2.12 (100%)

72.5
69.9
70.8
73.2
73.9

83.7

88.3

87.8

Table 3: Inference times of the models on the datasets as well es average performance taken from
Table 1. Our DeBERTa model combines LLM-level performance with substantially lower latency.

full Auto-GDA), this increases performance overall with one exception on the LFQA-Verification
dataset (note however that performance here is already above the labeled data, so selection based on
target data may draw the results toward the labeled data scores as well). As an upper baseline we are
interested in the hypothetical performance reachable by fine-tuning on human-labeled samples and
include it in Table 2. Considering the difference between the no fine-tuning models and the models
fine-tuned on human-labeled data as the domain adaptation gap, expressing our results relative to
these baselines indicates that we manage to close an impressive 96% of this gap.

5.3

INFERENCE EFFICIENCY

Linking to our motivational Figure 1, we study the efficiency of our models in Table 3. We compute
NLI scores for 50 random samples from the respective datasets. We observe models in three cate-
gories: The most efficient models (Vectara to BART-large) have medium performance on the RAG
datasets used in this work indicated by their ROC. On the other hand, models using sophisticated
post-processing (AlignScore) perform better, but require about 2.5 times more inference time than
our most successful DeBERTa model. Finally, LLMs via APIs require about 10-fold inference time,
but result in highest performance. When we compare models trained with our approach, we observe
LLM-level performance at about 10% of the inference time. Although our primary focus lies on
inference, we report generation times for Auto-GDA in Table 16 (Appendix) for completeness.

6 DISUSSION AND CONCLUSION

In this work, we show that synthetic data can successfully tackle the domain generalization gap for
NLI models. We present Auto-GDA, an automatic approach for synthetic sample generation and
selection overcoming the need for tedious heuristic or manual filtering and augmentation selection.
Our results show that we can obtain models that perform on par with most powerful LLMs while
having around 90% less inference time using our method. By generating synthetic data, we can
provide a more comprehensive and tailored representation, allowing for greater control over the de-
sired features. Our results also confirm the common intuition that generalization is increasingly hard
with smaller models (Bhargava et al., 2021). This highlights that domain adaptation is particularly
important when low latency at inference time is required, whereas general purpose models can be
preferable when quick inference is no hard requirement. Pretrained models as AlignScore (Zha
et al., 2023) and MiniCheck (Tang et al., 2024) strike a good balance between inference time and
performance when generation and fine-tuning is not possible or too costly.

Limitations. In our study we assume that the distribution of evidence samples including the re-
trieved documents is readily available. In many real-world applications, this may not be the case.
To address this, techniques like passage clustering and summarization, as explored in Sarthi et al.
(2024), could be employed on the knowledge base to cover the diversity of evidence passages as a
surrogate of this distribution. In general, we advise practitioners to obtain the most representative
corpus of the target domain for best results. Currently, domain adaptation requires a model for each
individual domain. Future work is required to further study if it is possible to adapt efficient models
for multiple domains without performance degradation. In addition, an assessment including the
entire RAG system with a retrieval component using frameworks such as RAGChecker (Ru et al.,
2024) may result in insightful in future work.

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A ADDITIONAL RELATED WORK

Automatic Data Selection. A related stream of research is concerned with automatically selecting
subsets from large datasets for training. For instance, AutoAugment (Cubuk et al., 2019) searches for
optimal image augmentations through reinforcement learning, but can be computationally intensive.
In contrast, Xie et al. (2023) propose an efficient importance-weighting criterion based on hashed
n-gram distributions using Kullback Leibler Divergence (KLD). Data selection is linked to works
on data valuation, e.g., Wang & Jia (2023), as data valuation scores can be used to select training
data, often resulting in improved performance.

B THEORETICAL CONSIDERATIONS

B.1 DERIVING OUR OBJECTIVE FROM A DISTRIBUTION MATCHING CRITERION

In this section, we present an more formal derivation of our objective Ltot from an enhanced
distribution-matching objective. We decrease a statistical divergence between a parametric dis-
tribution represented by the selected synthetic data samples pQ (for instance, their Parzen-Window
estimator or MLE estimate of a parametric family) and a target data distribution providing the re-
gions in feature space that we would like to cover, denoted by pcov. We now consider c to be a vector
in a continuous vector space. Such a mapping can be realized through stochastic encoders / decoders.
As we consider each evidence e independently, we write pQ,e(c, y) as a shorthand for pQ(c, y|e).
Additionally, we encourage generation of samples where the pretrained model f is not performing
well yet by including the cross-entropy loss of the model as a utility term, Uf = CE[f (e, ˆc), ˆy]
where ˆy is the assigned hard label of a synthetic sample. In summary, we propose optimizing the
distribution parameters Q to minimize the objective Ltot,

min
Q

DKL (pQ,e(c, y)||pcov,e(c, y)) − E(c,y)∼pQ,e [Uf (c, y)] := min
θ

Ltot(pQ,e, pcov,e, f )

(4)

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Under review as a conference paper at ICLR 2025

where Uf is an additional per-sample utility term that depends on the model f . We omit e from the
subscript to shorten notation, but still consider a fixed evidence e. Since we do not have labels for the
target samples, we can’t estimate the target distribution term pcov(x, y) in the distribution objective
Ltot. However, we can decompose the divergence using the “chain-rule” of the KLD (Thomas &
Joy, 2006) into a marginal matching term and a label correctness term:

DKL (pQ||pcov) = DKL (pQ(c)||pcov(c))
(cid:124)
(cid:125)

(cid:123)(cid:122)
marginal matching

+ Ec∼Dθ [DKL (pQ(y|c)||pcov(y|c))]
.
(cid:123)(cid:122)
(cid:125)
label correctness

(cid:124)

(5)

Intuitively, the marginal matching term requires the synthetic samples’ features to be close to the
range we are interested in covering. We can compute this term by introducing tractable parametric
densities. The label correctness term penalizes divergence between the conditional label distribu-
tions. Intuitively, it enforces that the samples’ labels correspond to their true labels and penalizes
uncertainty in the label. We propose to model this label uncertainty using our weak entailment
certainty estimates r and provide details on how we model both terms in the following paragraphs.

Parametrizing densities. We need to insert a suitable and tractable parametrizations of pcov and
pθ. We start by modeling their marginals. To model pcov we chose an efficiently tractable density
pcov(x) can be defined via the nearest target feature vector in claim(Dt,e) by3
(cid:19)

(cid:18)

pcov,σq (c) =

exp

−

1
Z

∥c − arg minci∈Dt d(ci, c)∥2
2
σ2
q

,

(6)

where Z > 0 is a normalization constant. We show that a finite Z always exists in the Appendix B.6.
The constant σq > 0 will be treated as a hyperparameter in our framework. Let Q ⊂ X denote a
finite set of selected samples. For pθ, we chose a standard kernel density estimator with kernel width
σr ≥ 0:

pθ(Q),σr (c) =

1
|Q|

(cid:88)

ˆci∈Q

N (c; ˆci, σ2

r I).

(7)

Modeling label correctness. We propose to model to label correctness term using the entailment
certainty scores, which provides us with an estimate of how well the true and the assigned labels
are aligned at a certain point. If a positively labeled sample has very high entailment certainty or
a negatively labeled sample has very low entailment certainty, the assigned labels likely match the
ground truth and divergence between true conditional label distribution and assumed distribution is
expected to be minimal at the sample x. We derive a relation between the label correctness term
and our entailment certainty score in form of a function DKL (pθ(y|c)||pcov(y|c)) := LDiv(r(i), ˆy),
relying on the current entailment certainty r(i) and the assigned hard label ˆy in Appendix B.2 that
is depicted in Figure 4. The resulting relation fulfills certain natural axioms including that the label
correctness term is 0, when we have perfect certainty, i.e., LDiv(0, 0) = 0, LDiv(1, 1) = 0.

B.2 MODELING AND TRACKING LABEL UNCERTAINTY

In this section we provide a strategy to estimate the label correctness term in Equation (5) which is
given by DKL (pQ(y|c)||pcov(y|c)) (see Appendix B.1 for why we need to model this term). We
need to model both pQ(y|c) and pcov(y|c) to estimate this term. As they are binary, we choose
Bernoulli distributions. Our estimated conditional pQ(y = 1|c) = ϕ0 is modeled through a
Bernoulli distribution with parameter ϕ0. This conditional distribution is assumed not to change
through the augmentation once initialized (because we also do not change the hard labels during
augmentations). Reasonable choices for ϕ0 involve setting hard probabilities, i.e., ϕ0 = ˆy or using
the initial label certainty score ϕ0 = r(0) as a softer version.

Unfortunately, we do not directly have access to the true label distribution pcov(y|c), but we can
follow the following intuition: When we arrive at r ≈ 0.5 due to many augmentations, this indicates
no knowledge about the ground truth label of c. However, this does not mean that the ground truth
distribution pcov(y = 1|c) = 0.5, for instance the sample can still have a certain label that annotators
would agree on. Instead there is uncertainty about this distribution’s parameter pcov(y = 1|c) = φ.
There are different options to model the uncertainty over the true label distribution in this work.

3we now assume x ∈ X to be in a metric space. This can be achieved using an encoder mapping the textual

input to real vectors.

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Under review as a conference paper at ICLR 2025

(a) High Certainty

(b) Medium Certainty

(c) Low Certainty

Figure 5: We model the label uncertainty through a hyper distribution over the parameter φ.

We choose the Beta distribution, which is commonly used as a hyperprior for Benoulli distribu-
tions. We chose impose two constraints on the distribution and show that they uniquely define the
hyperparameter distribution and have some intuitive properties.

Proposition 1 Let ϕ ∼ Beta(α, β) denote a Beta distribution. Let ϕ0 be the parameter of the
(certain) initial label distribution (usually corresponding to ˆy) and let r denote the probability of
the mutated sample having label y = 1 (entailment certainty).

1. If r ∈ [min(0.5, ϕ0), max(ϕ0, 0.5)], there exist unique values for α′, β′ such that E[p(y =

1|c; φ)] = E[ϕ] = r with a mode at ϕ = ϕ0.

2. For r → 0.5, the distribution Beta(α′, β′) with the values from statement 1 converges to a

unit distribution on [0,1] in distribution.

3. Using pcov(y = 1|c) = φ, φ ∼ Beta(α′, β′), pQ(y = 1|c) = φ0 and the expected KLD

over the prior has the closed-form solution

Eφ [DKL [pQ(y|c)||pcov(y|c; φ)]] =

−H(pQ(y|c)) − pQ(y = 0|c)ψ(β′) − pQ(y = 1|c)ψ(α′) + ψ(α′ + β′).

(8)

(9)

In the last statement, ψ denotes the digamma-function and pQ(y|c) = Bernoulli(ϕ0) is the initially
assumed label distribution for synthetic samples.

See Appendix B.5 for a derivation. The convergence behavior of this scheme to a unit distribution is
visualized in Figure 5. Using the above update rule and the uncertainty estimation, we can compute
the label correctness term for r = E [pQ(y = 1|c)]

LDiv(r, ϕ0) = H(Bernoulli(ϕ0)) − (1 − ϕ0)ψ(β′(r)) − ϕ0ψ(α′(r)) + ψ(α′(r) + β′(r))

(10)
where α′(r) and β′(r) are the numerical solutions of Proposition 2. To arrive at the formulation in
the main paper, we can plug in ϕ0 = ˆy, which is the term visualized in Figure 4.

B.3 OPTIMIZING THE OBJECTIVE

With models for the terms in the objective at hand, we can select a set of most promising samples Q
by solving the discrete sample selection problem

min
θ ,|Q|=K

Q⊂D(i)

Ltot(pθ(Q),σr , pcov,σq , f ).

(11)

To make the problem computationally tractable, we are particularly interested in estimators that de-
compose over the individuals samples present in the set Q. With such a decomposition at hand,
each sample is assigned an individual contribution and we simply select K samples with lowest
individual to contributions to minimize the objective. We derive the following proposition for de-
composing our objective using the parametrized distributions with parameters σr, σq, Dt, Q for pθ
and pcov introduced earlier.

Proposition 2 As σ2

r → 0 while σ2

q > 0 is constant, the objective converges to

lim
σr→0

Ltot(pθ(Q),σr , pcov,σq , f ) = C +

(cid:88)

ˆci,ˆyi,ri∈Q

(cid:2)d(ˆci, cmin,i)2 + λdLDiv(ri, ˆyi) − λ′

uUf (ˆci, ˆyi)(cid:3)

(12)

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Under review as a conference paper at ICLR 2025

C is a constant, and λd(σq), λ′

u, hyperparameters,

cmin,i := arg min

c′∈claim(Dt,e)

d(c′, ˆci),

(13)

and LDiv denotes the expected KLD of the conditional distribution (label correctness term) of the
objective that can be modeled as a continuous function from the entailment certainty scores r.

We derive this proposition in Appendix B.4. In summary, we show that for small σr the contribution
of a sample to the objective approaches a sum of three parts: The distance to the closest sample
from the target claim set for evidence e, claim(Dt,e) to the claim ˆc, the label correctness term, and
the negative utility. We use the above decomposition in our algorithm, ensuring the objective can
be solved highly efficiently in three steps: (1) Computing each samples contribution to Ltot, (2)
ranking the samples by this contribution, and (3) finally selecting the top-K subset of samples with
the lowest contributions.

B.4 DERIVATION OF PROPOSITION 2

To reduce computational complexity, we use an approximation of our objective that does not feature
dependencies between the points in the subset. Then the objective is given by a sum of values for
the individual points. To derive this objective, we consider the behavior of the objective for σr → 0
while keeping a fixed σq > 0. First we note that the normal density pQ,σr (c) with center c and
r I converges to a Dirac distribution and for a continuous function f : Rd → R with
covariance σ2
the filter property:

(cid:90)

lim
σr→0

RN

f (c)pQ,σr (c)dc = f (c).

We now consider the individual terms of the objective.

Marginal Matching. Let us start with the marginal matching term of the objective.
(cid:88)

(cid:88)

pQ(x) =

1
|Q|

N (c; ˆci, σ2

r I) :=

¯pi(c)

ˆci∈Q

ˆci∈Q

where ¯pi(c) corresponds to the density of the ith mixture component.

DKL (pQ(c)||pcov(c)) = Ec∼pQ [log pQ(c) − log pcov(c)]
(cid:88)

Ec∼ ¯pi [log pQ(c) − log pcov(c)] =

(cid:88)

=

1
|Q|

ˆci∈Q

1
|Q|

ˆci∈Q

Ec∼ ¯pi [log pQ(c)] − Ec∼ ¯pi [log pcov(c)]

(17)
We need to find good and tractable approximations of both terms. For small σr → 0, pi approaches
a dirac distribution δθi with all mass at the center θi. This simplifies the objective to

Ec∼ ¯pi [log pQ(x)] → Ec∼ ¯pi [log ¯pi(c)] →

d log (cid:0)(2πe) + σ2

r

(cid:1) →

1
2

1
2

d log(2πe)

(18)

where d is the dimension of c. The second part converges to

where

Ec∼ ¯pi [log pcov(c)] →

1
Z

(cid:20) −∥ˆci − cmin,i∥2
σ2
q

(cid:21)

cmin,i := arg min
c′∈claimDt,e

d(c′, ˆci).

(19)

(20)

Label Correctness Term. We model the uncertainty propagation as in Equation (10). Approaching
σr → 0, we have

Ec∼pQ [DKL (pθ(y|x)||pcov(y|c))] = Ec∼pθ [LDiv(r, y)]

→

1
|Q|

(cid:88)

ˆci,yi,ri∈Q

[LDiv(ri, ˆyi)]

(21)

(22)

17

(14)

(15)

(16)

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Under review as a conference paper at ICLR 2025

Utility Term Finally, the same can be done for the utility term:

λuE(c,y)∼pQ [Uf (c, y)] →

1
|Q|

(cid:88)

ˆci,ˆyi,ri∈Q

λu [Uf (ˆci, ˆyi)]

(23)

Assembling all terms. In summary, we arrive at

Ltot →

1
|Q|

(cid:88)

ˆci,ˆyi,ri∈Q

d
2

log(2πe) +

1
Z

(cid:20) ∥ˆci − cmin,i∥2
σ2
q

(cid:21)

+ LDiv(ri, ˆyi) − λuUf (ˆci, ˆyi)

(24)

=

d
2

log(2πe)

(cid:88)

ˆci,ˆyi,ri∈Q

1
|Q|Zσ2
q
(cid:88)

∥ˆci − cmin,i∥2 +

1
|Q|

LDiv(ri, ˆyi) −

λu
|Q|

Uf (ˆci, ˆyi)

(25)

∥ˆci − cmin,i∥2 + λdLDiv(ri, ˆyi) + λ′

uUf (ˆci, ˆyi)

(26)

∝ C +

where in the last step, we multiply all terms be |Q|Zσ2
completes our derivation.

q to normalize the first constant to 1. This

ˆci,ˆyi,ri∈Q

B.5 DERIVATION OF PROPOSITION 1

Statement 1: We know that the mean of the beta distribution φ ∼ Beta(α, β) is given by

and the mode is given by

E[φ] =

α
α + β

mode[φ] =

α − 1
α + β − 2

(27)

(28)

for α, β > 1 (for α = β = 1, we obtain the uniform distribution and any value is a mode).
Constraining the mode to be mode[φ] = ϕ0 yields

α = qϕ0 + 1, β = q(1 − ϕ0) + 1, q ∈ [0, ∞)

For the mean, we obtain

Setting E[φ] = r yields

E[φ] =

qφ0 + 1
q + 2

(q + 2)r = qφ0 + 1 ⇔ q(r − φ0) = 1 − 2r ⇔ q =

1 − 2r
r − φ0

(29)

(30)

(31)

The solution of q is non-negative if r ̸= φ0 and if φ0 > r in case r > 0.5 and if φ0 < r in case
r < 0.5. Under these conditions, we obtain the unique parameters

α =

1 − 2r
r − φ0

φ0 + 1, β =

1 − 2r
r − φ0

(1 − φ0) + 1

(32)

Statement 2. We prove this statement using the Method of Moments showing that each moment of
the distribution converges to the moment of the uniform distribution. If this is the case, the method
of moments asserts that the sequence will converge in distribution. Note that both distributions are
uniquely determined by their moments because they reside on the interval [0,1]. We see that as
r → 0.5 we have that q → 0. We will show that

We therefore compute the nth moments of the Unit distribution for n ∈ N with X ∼ Unif[0, 1]

Beta(qφ0, q(1 − φ0))

q→0
−→= Unif[0, 1]

E[X N ] =

1
n + 1

18

(33)

(34)

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Under review as a conference paper at ICLR 2025

For the Beta distribution with with ϕ ∼ Beta(qφ0, q(1 − φ0)) we have

E[φN ] =

n−1
(cid:89)

k=0

qφ0 + 1 + k
q + 2 + k

Taking the limit results in

lim
q→0

E[φN ] = lim
q→0

n−1
(cid:89)

k=0

qφ0 + 1 + k
q + 2 + k

=

n−1
(cid:89)

k=0

1 + k
2 + k

=

1
n + 1

Statement 3: We calculate

Eφ [DKL [pQ(y|c)||pcov(y|c; φ)]] =
pQ(y = 0|c)(log pQ(y = 0|c) − Eφ[log 1 − φ])
+ pQ(y = 1|c)(log pQ(y = 1|c) − Eφ[log φ])

= −H(pQ(y|c)) − pQ(y = 0|c)(ψ(β) − ψ(α + β)) − pθ(y = 1|c)(ψ(α) − ψ(α + β))
= −H(pQ(y|c)) − pQ(y = 0|c)ψ(β) − pθ(y = 1|c)ψ(α) + ψ(α + β)

We use the identities:

Eφ∼Beta(α,β)[log φ] = ψ(α) − ψ(α + β)
Eφ∼Beta(α,β)[log 1 − φ] = ψ(β) − ψ(α + β)

(35)

(36)

(37)
(38)
(39)
(40)
(41)

(42)

(43)

Definition 1 (Probabilistically Correct Data Augmentation, PCDA) A probabilistically correct
data augmentation is a (potentially randomized) mapping M : X × Y → X × [0, 1]. Applying
(xk+1, rmiss) = M (xk, y) generates a modified sample xk+1 and additionally returns a probability
rmiss ∈ [0, 1] of flipping the assigned label during the augmentation step when keeping the mecha-
nism g and the annotator η fixed, i.e., p(g(xk+1, η) ̸= g(xk, η)) = r(k)
miss, where the randomness is
over the data augmentation output xk+1. Setting the initial agreement p(0)
agree = 1.0, we can perform
the following update rule

agree = (1 − r(k)
p(k+1)

miss)p(k)

agree + r(k)

miss(1 − p(k)

agree).

B.6 EXISTENCE OF NORMALIZATION CONSTANT

We consider the density

pcov(x) =

1
Z

(cid:18)

exp

−

∥x − arg minxi∈Dt d(xi, x)∥2
2
σ2
q

(cid:19)

,

(44)

(45)

where Dt is a finite set. We show that the normalization constant exists by proving that the integral
of the non-normalized density over the feature space X = Rd is bounded. To do so, we perform the
following derivation:

(cid:18)

exp

−

(cid:90)

Rd

∥x − arg minxi∈Dt d(xt, x)∥2
2
σ2
q

(cid:19)

dx ≤

(cid:90)

Rd

(cid:88)

≤

xi∈Dt

xi∈Dt
1

(cid:113)

σ2

q π

(cid:88)

(cid:18)

exp

−

(cid:19)

∥x − xi∥2
2
σ2
q

dx

(46)

≤

|Dt|
(cid:113)

σ2

q π

(47)

The first step uses the insight that the argmin will always be any point in Dt so if we add up the
contributions for all possible points, we will arrive at an upper bound. This completes the proof.

C IMPLEMENTATION DETAILS

C.1

IMPLEMENTATION DETAILS FOR AUGMENTATION STRATEGIES

In this section we provide additional details regarding the data augmentation strategies that we de-
ploy in this work. We fully commit to open-sourcing our code to reproduce the experiments upon
acceptance.

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Dataset

Train Val Test Link

ragtruth-Summary 2578 125 636 https://github.com/ParticleMedia/RAGTruth
3661 143 875 https://github.com/ParticleMedia/RAGTruth
ragtruth-QA
2671 60 733 https://huggingface.co/datasets/Salesforce/summedits
summedits
171 35 65 https://github.com/timchen0618/LFQA-Verification/
lfqa-verification

Table 4: Dataset sizes.

Partial Rephrasing with LLM. We use the prompt given in Appendix E.3 to instruct the LLM
(Claude3-Haiku) to create different versions of a document where some parts are masked. We
decide to mask a random 20% of consecutive words in the document. We let the LLM generate
3 outputs each for 2 different masks, resulting in a total of 6 rephrased versions for each claim.
Sampling temperature is set to 1.0.
Complete Paraphrasing. We use the T5-based model obtained here4 as a paraphraser to gener-
ate 3 rephrased versions of each claim. To ensure no duplicates are produced, we set parameters
repetition_penalty=10.0, and no_repeat_ngram_size=5.
Drop Sentences. For this augmentation, we sentence tokenize the claim using spacy with
en_core_web_sm tokenizer. We postprocess the outputs slightly to better handle statements in
quotes. We then randomly drop a sentence from the claim.

C.2 DATASETS

We apply the following preprocessing to the datasets: We filter out all samples, that have more than
1022 BART tokens (filling out the 1024 context length with an additional SEP and CLS token). The
sizes and source links of the resulting datasets are provided in Table 4. We note that this reduces
the number of usable SummEdits domains from 10 to 5 (due to some domains only containing
overlength evidence documents).

Splits. We either use the available train/test splits (RAGTruth) or create splits making sure that
summaries / answers derived from the same evidence are either only present in the train or the test
split. The validation split is derived from the train split.

Processing of QA datasets. The QA datasets require integrating the question and the retrieved
documents into a single prompt. The RAG-Truth dataset already provides integrated prompts which
we use. For the LFQA-Verification questions and documents are provided seperately. We use the
integration template "You are given the question: " + <QUESTION> Here is some information
related to the question: <EVIDENCE DOCUMENTS>.

C.3 AUTO-GDA DETAILS AND HYPERPARAMETERS

We implement the algorithm outlined in Algorithm 1. We emphasize that we fix the teacher model
to assign the initial scores. Here we can compute estimates of the model performance using the
validation set of evidence-claim pairs to which we have access, allowing us to chose the best per-
forming one as teacher. However, we do not know the performance of the models on claim-claim
pairs, so we treat the teacher model used in the augmentation step as a hyperparameter, that will be
optimized.

Fixed Hyperparameters. We additinally keep the following hyperparameters fixed across datasets:

• Finetuning: 1 Epoch, learning rate 10−5 for DeBERTA, BART, 2 × 10−4 for FLAN-T5,

batch size 2

• To compute the distance function d we use embeddings from a sentence-t5-base

model5.

4https://huggingface.co/humarin/chatgpt_paraphraser_on_T5_base
5https://huggingface.co/sentence-transformers/sentence-t5-base

20

Under review as a conference paper at ICLR 2025

1080
1081

1082
1083
1084
1085
1086
1087

1088
1089
1090
1091
1092
1093

1094
1095
1096
1097
1098

1099
1100
1101
1102
1103
1104

1105
1106
1107
1108
1109
1110

1111
1112
1113
1114
1115
1116

1117
1118
1119
1120
1121

1122
1123
1124
1125
1126
1127

1128
1129
1130
1131
1132
1133

Dataset
Parameter / RAG-Task

RAGTruth

Summary

QA

LFQA-Verification
QA

SummEdits
Summary

# Samples per evidence
# Synth. Dataset size (Org. Size)
# Augmentation Iterations

8
3544 (2578)
2

8
5032 (3662)
2

4
336 (171)
1

32
3552 (2671)
1

Table 5: Fixed hyperparameters dependent on dataset. Note that we set only “Samples per evidence”
which determines synthetic dataset size together with the number of evidences.

Dataset/Task

Initial Teacher Model Augment. Teacher Model

λd

λu

Main Results, Table 1 best NLI model as initial teacher)

RAGTruth-QA
RAGTruth-Summ
LFQA-QA
Summedits-Summ

vectara
vectara
alignscore
alignscore

debertav2
debertav2
vectara
bart-large

32.67
198.85
25.27
0.02

GPT-4o teacher results, Table 9 (GPT-4o as initial teacher)

RAGTruth-Summ
RAGTruth-QA
LFQA-QA
Summedits-Summ

gpt-4o
gpt-4o
gpt-4o
gpt-4o

vectara
bart-large
debertav2
debertav2

0.06
47.26
3940.60
0.01

20.57
19.51
6.83
92.11

7.58
0.15
7.28
29.42

Self-supervised results, Table 9 (DeBERTa as initial, augmentation teacher)

RAGTruth-Summ
RAGTruth-QA
LFQA-QA
Summedits-Summ

debertav2
debertav2
debertav2
debertav2

debertav2
debertav2
debertav2
debertav2

296.13
4591.98
890.45
871.77

1.03
0.24
17.29
19.23

Table 6: Tuned hyperparameters. Bold parameters were fixed for the runs, while the remainder was
tuned using the hyperparameter optimizer.

• Number of offspring per sample (l in pseudocode): l = 12, with 6 child samples from LLM

Partial rephrasing, 3 each from Drop Sentence and Complete Paraphrasing

We set the number of evidences used per claim which determines size of the synthetic dataset ac-
cording to the different datasets as given in Table 5. Setting our chosen values results in the synthetic
dataset being between 1.3-times and 2 times as large as the original dataset based on the oberserva-
tions in Appendix D.6, suggesting that this is the optimal range.

Optimized Hyperparameters. As outlined in the main text, we apply optuna for 50 configuration
trial as an hyperparameter optimizer to find the remaining hyperparameters. We set the ranges λu ∈
[0.1, 100], λu ∈ [0.01, 5000] and let the augmentation teacher model be selected from {vectara,
alignscore, deberta}. We don’t allow LLMs as teacher models for augmentations because it would
be too expensive as a lot of augmented samples are created in the course of the algorithm. The final
hyperparameters found through optimization are given in Table 6.
Base models. As a basis for finetuning we use huggingface checkpoints for DeBERTaV26, BART-
large7 and FLAN-T58.

Hardware. Our experiments (including runtime) were run on a system with 16-core Intel(R)
Xeon(R) CPU E5-2686 processors (2.30GHz) and a single Nvidia Tesla V100 GPU with 32GB
of RAM.

6https://huggingface.co/tasksource/deberta-base-long-nli
7https://huggingface.co/facebook/bart-large-mnli
8https://huggingface.co/sjrhuschlee/flan-t5-base-mnli

21

Under review as a conference paper at ICLR 2025

C.4 ALGORITHM

Algorithm 1 Automatic Generative Domain Adaptation (Auto-GDA)
Require: Set of target features Dt, no. best neighbors to select K, no. of augmentations per sample

l, Generator G, augmentation modification function M , teacher model T , base model f

▷ Sample K labels
▷ Sample initial claims using generator G
▷ get their label probabilities r(0).

6:

5:

1: D = {}
2: for Each unique evidence e in Dt do:
ˆyk ← Bernoulli(0.5), ∀k = 1...K
3:
ˆck ← G(e, claim(Dt,e), ˆyk), ∀k = 1...K
4:
r(0)
k ← T (ck, ˆyk), ∀k = 1...K
(cid:110)
D(0)
ˆck, ˆyk, r(0)
e =
k
i ← 0
while Ltot(D(i)
i ← i + 1
¯D(i)
e ← D(i−1)
for (ˆck, ˆyk, r(i−1)
for l times do

e ) has not converged do

) ∈ (D(i−1)
e

) do

(cid:111)K

k=1

e

k

ˆc′ = M (ˆck)
r(i) ← r(i−1)
e ← ¯D(i)
¯D(i)
end for

k

e ∪ {(ˆc′, r(i), ˆyk)}

e ← arg minQ⊂ ¯D(i)

e ,|Q|=K Ltot(Q)

7:
8:
9:
10:
11:
12:
13:

14:

15:
16:
17:
18:

end for
D(i)
end while
D ← D ∪ D(i)
e

19:
20:
21: end for
22: f ′ = fine-tune(f, D)
23: return f ′

(e, ˆck) · T (ˆck, ˆc′) + (1 − r(i−1)

k

(e, ˆck)) · (1 − T (ˆck, ˆc′))

▷ Augment sample through mutation function

▷ update r and append sample

▷ Select best sample subset

▷ Fine-tune model f on synthetic dataset D

We provide pseudocode for our algorithm in Algorithm 1.

C.5 BASELINE NLI MODELS

For the complex NLI model baselines, we use Vectara HHEM-2.19. The model cannot be easily
fine-tuned because it uses custom code. Additionally we use Alignscore-base with the checkpoint
found in this repository10 with the recommended “split” (pre- and postprocessing) nli_sp option.
We neglect the larger version as its runtime was comparable to LLMs at a usually lower perfor-
mance, making the smaller model a better trade-off. Finally we use the best-performing Minicheck
flan-t5-large model by Tang et al. (2024) from the official huggingface page11.

C.6 ROBUST PRE-TRAINING AND FINE-TUNING FOR UNSUPERVISED DOMAIN ADAPTATION

To address domain shift in NLI, we experimented with multiple classical unsupervised domain adap-
tation (UDA) techniques, which aim to improve the model’s generalization for out-of-domain data
by adding robustness during training and by using unlabeled target-domain data. Specifically, we im-
plemented Domain-Adaptive Pretraining (DAPT), Virtual Adversarial Training (VAT), Deep COR-
relation ALignment (CORAL), and Domain Adversarial Neural Networks (DANN) combined with
conditional entropy minimization.
Notation We refer to source domain data {(xi, yi)}i=1:n ∈ (XS, YS)n, where xi denotes the in-
put text and yi ∈ {0, 1} the corresponding entailment label, and target domain data {(xi)}i=1:m ∈

9https://huggingface.co/vectara/hallucination_evaluation_model
10https://github.com/yuh-zha/AlignScore
11https://huggingface.co/lytang/MiniCheck-Flan-T5-Large

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(XT )m. The labels y ∈ {0, 1} correspond to whether a claim is entailed or hallucinated (contradic-
tory or neutral). Our goal is to train a feature extractor fθ, parameterized by θ, that performs well
on the target domain.

Domain-Adaptive Pretraining (DAPT) Our first approach, Domain-Adaptive Pretraining (DAPT)
(Gururangan et al., 2020), performs Masked Language Modeling (MLM) on (unlabeled) data from
the target domain before finetuning on the labeled source-domain data for NLI. This way, it learns
the representations of both source and target domain, before finetuning on the source domain to
relearn classification for the NLI task.

Virtual Adversarial Training (VAT) Our second approach is Virtual Adversarial Training (Miyato
et al., 2016), which increases model robustness by introducing adversarial perturbations to the input
data during training. Specifically, we add an adversarial regularization term to the classification
objective, which becomes:

E(x,y)∼DS [ℓ(fθ(x), y)] + λ · Ex∼DS

min
θ

(cid:20)

max
δ∈S

ℓKL(fθ(x + δ), fθ(x))

,

(48)

(cid:21)

where ℓ is the classification loss (e.g., cross-entropy) on the source domain, ℓKL is the KL divergence
between the output distributions, δ is a small perturbation constrained within a a ball S, and λ is the
regularization weight. According to (Jiang et al., 2020), VAT induces Lipschitz-continuity, which
means that small changes in the input do not cause disproportionately large changes in the output,
improving robustness and generalization for out-of-domain data.

Deep CORAL The objective of CORrelation ALignment (CORAL) (Sun et al., 2016) is to align the
second-order statistics (covariances) of the source and target embedding distributions by minimniz-
ing the Frobenius norm between their covariance matrices. Specifically, denoting CS and CT the
covariance matrices of the embeddings of the source and target samples as extracted from the last
encoding layer, respectively, and as d the dimension of the features, the regularization loss is:

LCORAL(θ) =

1

4d2 ∥CS − CT ∥2
F ,

(49)

Domain Discriminator and Conditional Entropy Minimization Finally, we also implement a
domain discriminator inspired by domain adversarial training (Ganin et al., 2016) and other related
works in NLP. The discriminator D is trained to classify source and target domain features correctly,
whereas the feature extractor (classifier) fθ is trying to minimize the discriminator’s accuracy, which
should amplify the learning of domain-invariant features from the classifier. The domain adversarial
loss is:

Ld(θ) = Ex∼DS [ln D(fθ(x))] + Ex∼DT [ln(1 − D(fθ(x)))].

(50)

We also experiment with adding a conditional entropy loss to ensure the model makes confident
predictions on the target domain and improve the placement of the initial boundaries, as outlined in
Shu et al. (2018) and Reed et al. (2014):

Lc(θ) = −Ex∼DT

(cid:2)fθ(x)⊤ ln fθ(x)(cid:3) ,

(51)

Implementation Details For our robust optimization experiments we used the DeBERTaV2 based
NLI model, and limited maximum tokenization length at 1024 tokens across all benchmarks. For
DAPT we extracted the DeBERTaV2 backbone and trained on the target domain for 1 full epoch,
using 10% masking probability. At the fine-tuning stage of both methods we ran 1 full epoch on
the MNLI dataset. At the masked pretraining and fine-tuning stages of the experiments, we used
the AdamW optimizer with learning rate 10−5 and weight decay 10−3, enabling 100 warmup steps
over the supervised fine-tuning. For SiFT and CORAL we set the coefficient of the respective
regualarization terms to 0.5, after running hyperparameter optimization with a coarse grid search.
Batch size used for covariance estimation in CORAL was set at 64. Each experiment was repeated
over 3 random trials.

23

Under review as a conference paper at ICLR 2025

Dataset
RAG-Task

l
e
d
o
m
e
s
a
b

s FLAN-T5

DeVERTaV2
BART-large
s DAPTDeBERTaV2
SiFTDeBERTaV2
CORALDeBERTaV2

s
e
n
t
s
u
b
o
r

RAGTruth

LFQA-Verif. SummEdits Avg.

Summary

QA

QA

Summary

0.666
0.727
0.604

0.636
0.505
0.633

0.618
0.588
0.782

0.646
0.810
0.625

0.641
0.658
0.661

0.677 ± 0.004 0.654 ± 0.003 0.748 ± 0.076 0.792 ± 0.005 0.718 ± 0.022
0.716 ± 0.009 0.562 ± 0.006 0.810 ± 0.035 0.806 ± 0.015 0.724 ± 0.016
0.657 ± 0.001 0.637 ± 0.002 0.815 ± 0.001 0.792 ± 0.001 0.725 ± 0.001

x MiniCheck-T5
AlignScore
Vectara-2.1

e
l
p
m
o
c

0.675
0.572
0.662

0.600
0.650
0.744

0.564
0.594
0.618

0.679
0.770
0.581

0.630
0.646
0.651

0.650 ± 0.005 0.703 ± 0.019 0.669 ± 0.016 0.761 ± 0.020 0.696
A Flan-T5 (Auto-GDA)
BART (Auto-GDA)
0.710 ± 0.028 0.794 ± 0.011 0.772 ± 0.023 0.798 ± 0.014 0.769
DeBERTaV2 (Auto-GDA) 0.737 ± 0.009 0.784 ± 0.011 0.776 ± 0.012 0.817 ± 0.009 0.778

D
G
-
o
t
u
A

s GPT-4o

M
L
L

GPT-4o-mini
GPT-3.5

0.691
0.666
0.593

0.764
0.684
0.586

0.688
0.625
0.611

0.835
0.832
0.723

0.744
0.702
0.629

Table 7: Performance comparison to baselines (uncalibrated balanced accuracy). In this metrics, our
models even outperform LLM baselines.

Dataset
RAG-Task

l
e
d
o
m
e
s
a
b

s FLAN-T5

DeVERTaV2
BART-large
s DAPTDeBERTaV2
SiFTDeBERTaV2
CORALDeBERTaV2

s
e
n
t
s
u
b
o
r

RAGTruth

LFQA-Verif. SummEdits Avg.

Summary

QA

QA

Summary

0.890
0.897
0.893

0.900
0.899
0.900

0.705
0.705
0.846

0.550
0.748
0.641

0.761
0.812
0.820

0.655 ± 0.001 0.489 ± 0.016 0.748 ± 0.076 0.762 ± 0.004 0.664 ± 0.027
0.704 ± 0.019 0.362 ± 0.055 0.810 ± 0.036 0.762 ± 0.007 0.660 ± 0.029
0.556 ± 0.001 0.487 ± 0.022 0.815 ± 0.001 0.752 ± 0.001 0.653 ± 0.006

x MiniCheck-T5
AlignScore
Vectara-2.1

e
l
p
m
o
c

0.897
0.888
0.910

0.901
0.903
0.921

0.743
0.847
0.702

0.682
0.744
0.498

0.806
0.846
0.758

0.901 ± 0.001 0.905 ± 0.002 0.699 ± 0.006 0.693 ± 0.015 0.800
A Flan-T5 (Auto-GDA)
BART (Auto-GDA)
0.910 ± 0.003 0.923 ± 0.005 0.857 ± 0.017 0.725 ± 0.010 0.854
DeBERTaV2 (Auto-GDA) 0.912 ± 0.002 0.930 ± 0.004 0.854 ± 0.014 0.750 ± 0.007 0.861

D
G
-
o
t
u
A

s GPT-4o

M
L
L

GPT-4o-mini
GPT-3.5

0.929
0.918
0.887

0.914
0.909
0.899

0.848
0.767
0.746

0.782
0.764
0.687

0.869
0.840
0.805

Table 8: Performance comparison to baselines (Binary F1-Scores).

D ADDITIONAL RESULTS

D.1 ADDITIONAL METRICS

We chose the Area under Curve for Receiver-Operator-Characteristic (AUC-ROC) as our main met-
ric, as it is less dependent on threshold calibration and also works for imbalanced datasets. We
report our results in other metrics such as balanced accuracy without threshold calibration (using
0.5. as a threshold as suggested in Tang et al. (2024)) in Table 7 and F1-Scores in Table 8. The
results highlight not only that our main results are valid across different metrics – in uncalibrated
balanced accuracy, our models trained with Auto-GDA data even outperform LLMs by an average
3.4 accuracy percent points.

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Dataset
Model

RAGTruth-QA

DeBERTa BART-large Flan-T5 mean

No Augmentation

LLMPartialRephrase
Complete Paraphrasing
DropSentence

All

0.836

0.869
0.845
0.868

0.872

0.845

0.890
0.863
0.872

0.886

0.772 0.818

0.767 0.842
0.711 0.806
0.758 0.833

0.806 0.855

(a) Testing the effect of using one vs. several augmentations.
On average the best results are obtained when combining sev-
eral augmentations.

(b) Composition of the final dataset by ori-
gin of the selected samples (augmentation
routines or few-shot prompting without aug-
mentation). The selection corresponds well
to the usefulness of the augmentations on
their own.

Figure 6: Qualitative Results on sample selection: Our framework succeeds to automatically select
the best samples from different augmentation strategies outperforming single augmentation strate-
gies.

D.2 ADDITIONAL QUALITATIVE RESULTS

Estimating the mislabeling probability. An integral part of our algorithm in the estimation of the
agreement probability in Equation (44). To investigate the effect of implementing this choice, we
run an ablation study to better understand how the quality of the agreement probabilities affects the
score. We provide results average over 3 runs in Figure 7a. The results indicate that the choice of
the model used to estimate r the label certainty score has substantial effect on the quality of the
results. While the utility (in terms of ROC scores) drops when noise is added, it increases again
when high levels of noise are applied. We attribute this behavior to the algorithm neglecting the
mutated samples almost entirely when the noise level is too high and mainly selecting the few shot
generate samples. As shown in Table 2 these have fair utility already.

Using one vs. several augmentation routines. A key design goal of our algorithm was the ability
to automatically select the most promising augmented samples. We investigate the effect of using
only single or several augmentations in Figure 6a. The results highlight that the Partial Rephrasing
augmentation with LLMs as well as the sentence deletion augmentation seems to be most successful.
The Complete Paraphrasing augmentation leads to substantially lower data quality on its own. How-
ever, the best utility is achieved when all three augmentations are combined. We study the origin of
the samples eventually selected by our algorithm and find that the usefulness of the augmentations
on their own is reflected by the share samples selected from each of the augmentations as depicted
in Figure 6b. Together, this highlights that Auto-GDA succeeds in selecting the most promising
samples generated from the augmentations automatically.

D.3 DIFFERENT TEACHER MODELS

We investigate the application of different teacher models in Table 9. Our results indicate the learn-
ing from GPT models works in general, but does not results in better performance that using the best
non-LLM teacher. We additionall study self-improvement, using DeBERTa as both a teacher model
for initial scoring and augmentation scoring. This shows that improvements thought self-supervision
are possible.

D.4 ROBUSTNESS TO INITIAL DATA QUALITY

The initial data plays an essential role in Auto-GDA, however we designed our algorithm to be
as fault-tolerant as possible and to be able to cope with some low-quality samples in the initial
population. For instance, we do not solely rely on generative models, but use discriminative teacher
models in the generation loop as well. The selection objective is specifically designed to filter out
low-quality samples. Mislabeled samples in the fine-tuning dataset can severely affect performance
of a fine-tuned model (Wang et al., 2023b). To test the robustness of Auto-GDA, we manually flip

25

Org.Few-ShotLLMPartialRephrParaphraseDropSentencemutationstrategy0.00.10.20.30.40.50.6shareofﬁnalsamplesUnder review as a conference paper at ICLR 2025

Dataset
RAG-Task

RAGTruth

LFQA-Verif. SummEdits Average

Summary

QA

QA

Summary

Teacher Model Used
Teacher Performance
0.864
0.854
DeBERTaV2 (Auto-GDA) 0.837 ± 0.007 0.867 ± 0.007 0.925 ± 0.009 0.883 ± 0.005 0.878

Vectara-2.1 Vectara-2.1 AlignScore AlignScore
0.805

0.904

0.894

Teacher Model Used
GPT-4o Performance
0.883
0.865
DeBERTaV2 (Auto-GDA) 0.808 ± 0.017 0.855 ± 0.003 0.910 ± 0.019 0.887 ± 0.004 0.865

GPT-4o

GPT-4o

GPT-4o

GPT-4o

0.892

0.896

0.880

Techer Model Used
DeBERTaV2 Performance 0.782
0.708
0.530
DeBERTaV2 (Auto-GDA) 0.830 ± 0.009 0.807 ± 0.010 0.923 ± 0.012 0.890 ± 0.007 0.863

DeBERTaV2 DeBERTaV2 DeBERTaV2 DeBERTaV2

0.645

0.876

Table 9: Using different teacher models in Auto-GDA to fine-tune DeBERTaV2. In the upper part
we add best results from Table 1 for comparison. In the center part, we highlight that using GPT-4o
as a teacher model to assign intial probabilities does not yield substantial improvement. However the
lower part shows that it is possible to do self-improvement using only DeBERTa as teacher model
for both initial scores and augmentation scoring.

(a) Effect of computing r using different models
and noise levels inserted in the scores. While
noisy, the results point to different trends for
the teacher models: For the best teacher model,
noise hurts performance, whereas for the other
ones, it does not hurt or might even boost per-
formance as more few-shot samples are se-
lected.

(b) Evolution of ROC scores and our objective over three
iterations of our algorithm. We see that ROC increases as
our objective decreases. We stop our algorithm after two
iterations, when the objective does not improve anymore.

Figure 7: a) Reliable estimation of label certainty r is essential for selection of high quality data. b)
The resulting synthetic data often contains original few-shot generated samples as well as a fair mix
of mutated samples generated from them. They are automatically selected by our algorithm.

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Dataset
RAG-Task

AlignScore
Vectara-2.1
GPT-4o

RAGTruth

Summary

0.737
0.814
0.828

QA

0.836
0.879
0.866

LFQA-Verification SummEdits
Summary

QA

0.870
0.879
0.876

0.874
0.805
0.878

Our Data

0.837 ± 0.007 0.867 ± 0.007

0.925 ± 0.009

0.883 ± 0.005

Table 10: Comparing our approach to the naive baseline of pseudo-labeling the training data and
fine-tuning the DeBERTa V2 model on the pseudo-labeled data.

Dataset
RAG-Task

FLAN-T5
Flan-T5 (Auto-GDA)

BART-large
BART (Auto-GDA)

RAGTruth

LFQA-Verif. SummEdits Average

Summary

QA

QA

Summary

0.734
0.756 ± 0.004 0.783 ± 0.013 0.687 ± 0.002 0.824 ± 0.010 0.762 (+0.063)

0.655

0.708

0.700

0.699

0.696
0.813 ± 0.009 0.867 ± 0.011 0.867 ± 0.026 0.860 ± 0.010 0.852 (+0.113)

0.821

0.670

0.739

0.769

DeBERTaV2
DeBERTaV2 (Auto-GDA) 0.837 ± 0.007 0.867 ± 0.007 0.925 ± 0.009 0.883 ± 0.005 0.878 (+0.170)

0.782

0.530

0.645

0.876

0.708

Table 11: Direct comparision of improvements

50 % of the labels in the initial data to study its effect and trace these samples through the generation
process. We use the LFQA-Verification dataset and perform 5 runs with different initial generation
seeds (using the same hyperparameters as in Table 1 otherwise). First, while 50% of the data have
flipped labels initially, in the data selected after one iteration of Auto-GDA, only 10.0% +- 1.1%
are the initial data or augmentations of the data with the flipped labels. After fine-tuning on the
generated data, the ROC-AUC score drops slightly by 1.1 points as shown in table below.

ROC-AUC of fine-tuned DeBERTaV2

0.925 ± 0.009

0.914 ± 0.007

Original Performance with 50% flipped initial labels

Despite the substantial amount of mislabeled data (50 %) the drop in performance remains small,
highlighting that Auto-GDA is quite robust to data quality issues due to its fault-tolerant design.

D.5 SIMPLE BASELINES

We compare our results to model trained on pseudo-labels in for the original datasets in Table 10.
The results inicate that this is a surprisingly strong baseline, which is however surpassed by Auto-
GDA in 3 out of 4 cases.

Results when the models are fine-tuned on the validations set directly are shown in Table 12.

Dataset
RAG-Task

Non-Fintuned

Few-Shot Data

FT on validation

RAGTruth
Summary QA

0.782

0.530

0.799

0.826

0.784

0.750

DeBERTaV2 (Auto-GDA, best teacher)

0.837

0.867

LFQA-Verif. SummEdits Mean

QA

0.645

0.934

0.899

0.925

Summary

0.876

0.872

0.890

0.890

0.708

0.858

0.833

0.878

Table 12: Fine-tuning on validation set does increase performance but does not reach Auto-GDAs
performance on average.

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Under review as a conference paper at ICLR 2025

dataset size

LFQA
SummEdits
RAGTruth-Summary
RAGTruth-QA

0.5x
0.916 ± 0.007
0.880 ± 0.003
0.821 ± 0.003
0.808 ± 0.007

1x
0.924 ± 0.006
0.876 ± 0.002
0.808 ± 0.005
0.828 ± 0.006

2x
0.903 ± 0.013
0.876 ± 0.005
0.791 ± 0.007
0.830 ± 0.006

4x
0.896 ± 0.007
0.868 ± 0.002
0.797 ± 0.001
0.834 ± 0.004

8x
0.868 ± 0.003
0.855 ± 0.005
0.783 ± 0.004
0.821 ± 0.005

average

0.860

0.856

0.849

0.844

0.841

Table 13: Results when fine-tuning on more LLM generated data (ROC-AUC scores for DeBER-
TaV2) show that performance of this baseline does not improve with more data.

Figure 8: Testing different synthetic dataset sizes and learning rates for fine-tuning. While the
original dataset size is at about 3 claim examples per evidence (dashed line) we test a wide range of
dataset sizes ranging from 1/50 examples per evidence (100x smaller than original) to larger datasets
with 30 claims per evidence, (10x larger than original). We observe a relatively stable optimum from
about a 1/3 of original size to twice the original size using learning rates around 10−5.

D.6 EFFECT OF DATASET SIZE

When choosing the dataset size we used the data size slightly larger that that of the original dataset
as an orientation. We experiment with different dataset sizes and learning rates as shown in Figure 8
When keeping training fixed to one epoch, we find that with higher learning rates, smaller dataset
sizes lead to higher performance, and with lower learning rate, more data is required which seems
natural. Globally, we observe that a learning rate of 10−5 is near optimal, but the performance is
rather insensitive This is based on a prior observation that significant oversampling of the dataset
size had seemingly little effect.

We also study the effect of generating more LLM samples on the performance of the baseline of
solely fine-tuning on LLM-generated data. We generate more LLM data using our few-shot prompt-
ing strategy and fine-tune the DeBERTa model on LLM-generated datasets in the range of 0.5x - 8x
the size of the datasets used in the main paper. We obtain the results shown in Table 13 (ROC-Scores
of the DeBERTaV2 model fine-tuned on the data). Perhaps surprisingly, increasing the number of
generated samples does not increase the performance of fine-tuned models. We attribute this to the
observation that LLMs (even when setting a higher temperature) will generate highly similar sam-
ples after a certain dataset size is reached, not further improving performance of a fine-tuned model.
Performance even decreases again, potentially due to overfitting on the synthetic data. We therefore
conclude that even with more LLM data, the performance does not reach the level of Auto-GDA.

D.7 ABLATING TERMS IN THE SELECTION CRITERION

To better understand the effect of the different terms in our objective, we perform an additional
ablation study only using a selection objective with one term at a time. We provide the results

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dataset

all terms

only distance

only label-cert.

only utility

LFQA
Summedits
RAGTruth-Summary (1 run)
RAGTruth-QA (1 run)

0.925 +- 0.004
0.883 +- 0.002
0.861
0.827

0.916 +- 0.004
0.877 +- 0.004
0.805
0.756

0.919 +- 0.003
0.866 +- 0.002
0.860
0.814

0.917 +- 0.005
0.881 +- 0.002
0.587
0.823

Table 14: Ablation on terms in the selection criterion (ROC-AUC scores for DeBERTaV2). Using
all three terms results in better performance than only using one of the terms each.

in Table 14. Despite some difference being small, we see that all terms contribute to the final
performance.

D.8 EFFECT OF SOURCE DATA ON UDA EFFECTIVENSS

Table 15 compares the performance (ROC-AUC scores) of various Unsupervised Domain Adapta-
tion (UDA) methods and synthetic data approaches across different source datasets. All results are
evaluated on the RAGTRUTH target dataset, using a DeBERTaV3 model trained on the PAWS, Vita-
minC, and Fever data. The table illustrates the effectiveness of different UDA methods and synthetic
data approaches when applied to various source datasets (MNLI, Summedits, and Ragtruth-synth).
More specifically, we see that except for the synthetically generated version of RAGTruth, the choice
of the source domain data does not seem to alter results significantly. We also see that vanilla fine-
tuning on the synthetic RAGTruth data outperforms all other variations, indicating that synthetic
data is more appropraite for NLI than traditional UDA methods. This is perhaps due to the fact
that very small changes in the generated claim can flip the label from entailed to non-entailed and
vice-versa.

Method
No fine-tuning
Vanilla fine-tuning
CORAL
SMART
MLM
Domain Discriminator

Fever+PAWS+VitaminC Summedits

0.735
0.735
0.682
0.743
0.680
0.603

0.735
0.737
0.683
0.721
0.577
0.712

Synthetic RAGTRUTH
0.735
0.844
0.728
0.833
0.731
0.746

Table 15: ROC-AUC scores for different UDA methods and synthetic data approaches.

D.9 EFFECT OF REGULARIZATION CONSTANTS ON UDA METHODS AND ON FINE-TUNING

PERFORMANCE

This appendix presents a comparative analysis of THE UDA methods and their impact on the fine-
tuning performance of our model. Figure 9 shows the ROC-AUC scores for different UDA meth-
ods as we increase the percentage of target domain data used for fine-tuning. We see two things:
i) No configuration of the UDA methods improves the model’s performance significantly, and ii)
the robustly trained models also do not benefit from faster finetuning with fewer samples, as their
performance when further finetuned with target-data samples is similar to the original model after
finetuned on the same splits. In short, we believe that traditional UDA methods do not show promise
for the NLI task.

D.10 GENERATION TIME FOR AUTO-GDA

To provide a complete picture of our algorithm, we include runtimes that we obtained for each of the
steps of Auto-GDA. We run Auto-GDA on the smallest and largest dataset, LFQA and RAGtruth-
QA respectively, on our hardware (single Nvidia Tesla V100 GPU, 64GB RAM, 8 Intel Xeon CPUs)
and provide individual runtimes in Table 16.

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(a) Masked Language Modeling (MLM)

(b) CORAL

(c) Discriminator and Conditional Entropy

(d) SMART

Figure 9: Performance comparison of different UDA methods across fine-tuning percentages

Dataset

Real / Generated Size Initial Generation Mutation Selection Fine-tuning Total (1 iteration)

LFQA
RAGTruth-Summ 3662 / 5032

171 / 336

90.12
650.38

188.12
1562.78

90.97
1161.11

34.75
432.84

403.96
3807.11

Table 16: Generation times (sec.) obtained when running Auto-GDA on the largest and smallest
dataset on our hardware. The total time for one iteration of Auto-GDA is roughly 4x-6x more than
generating few-shot data only. We would like to highlight that runtimes depend rather strongly on
the latency of the API calls (and how much they are parallelized).

30

No Target Finetuning5%8%10%20%25%30%50%80%100%Sample (%) of Target Data used for Finetuning after UDA0.50.60.70.80.91.0ROC-AUCROC-AUC vs Percent of Target Data used for Finetuning(After Finetuning on Source Data with UDA)MLM (1 Epochs)MLM (5 Epochs)MLM (10 Epochs)OriginalNo Target Finetuning5%8%10%20%25%30%50%80%100%Sample (%) of Target Data used for Finetuning after UDA0.50.60.70.80.91.0ROC-AUCROC-AUC vs Percent of Target Data used for Finetuning(After Finetuning on Source Data with UDA)CORAL (=0.25)CORAL (=1)High CORAL (=10)OriginalNo Target Finetuning5%8%10%20%25%30%50%80%100%Sample (%) of Target Data used for Finetuning after UDA0.50.60.70.80.91.0ROC-AUCROC-AUC vs Percent of Target Data used for Finetuning(After Finetuning on Source Data with UDA)Disc (=0.25) + Cond (=0.25)Disc (=0.25) + Cond (=1)Disc (=1) + Cond (=0.25)Disc (=1) + Cond (=1)OriginalNo Target Finetuning5%8%10%20%25%30%50%80%100%Sample (%) of Target Data used for Finetuning after UDA0.50.60.70.80.91.0ROC-AUCROC-AUC vs Percent of Target Data used for Finetuning(After Finetuning on Source Data with UDA)SMART (=0.25)SMART (=1)OriginalUnder review as a conference paper at ICLR 2025

E PROMPTS

E.1

INITIAL GENERATION PROMPTS

We use two prompts to generate intitial samples that differ according to the respective target labels.
In practice we use a maximum of 4 few shot samples, or the number of samples available in the train
dataset for a given evidence e.

Positive (entailed) prompt:

Human: You are given the following document wrapped in <document> </document> tags:
<document>DOCUMENT</document> Your task is to generate summaries from a document. Here are
some examples of how the summaries could look like:

Note however that some of the samples contain incorrect information that is not part of the document! Here
are the examples:
<example 0>EXAMPLE0</example 0>
<example 1>EXAMPLE1</example 1>
Now your task is to generate N summaries from the document. However, unlike some of the examples
given above, the summaries must be entirely supported by the document. Only include information that is
directly inferrable from the document. It is also important that the summaries reflect the style, length and
wording of examples. If there are common patterns or sentence structures in the examples summaries, the
created summaries should reflect those. Each summary is identified with an integer from 0 to N-1. The
summaries must be wrapped in <summary #></summary #> tags, where # is replaced with the summary id.
Assistant:

To generate non-entailed samples, the following modified prompt is used:

Human: You are given the following document wrapped in <document> </document> tags:
<document>EVIDENCE DOCUMENT</document> Your task is to generate summaries from a docu-
ment. Here are some examples of how the summaries could look like:

Note however that some of the samples contain incorrect information that is not part of the document! Here
are the examples:
<example 0>CLAIM EXAMPLE0</example 0>
<example 1>CLAIM EXAMPLE1</example 1>
Your task is to generate N summaries from the document. However, now all of the summaries must contain
at least one piece of non-factual information. This can be some information that is not present in the docu-
ment or some information that is contradictory to the information in the document, but intuitively appears to
make sense. Otherwise they reflect the style, length and wording of examples. If there are common patterns
or sentence structures in the examples summaries, the created summaries should reflect those. Modify dif-
ferent pieces of information at different places in the document. Each summary is identified with an integer
from 0 to N-1. The summaries must be wrapped in <summary #></summary #> tags, where # is replaced
with the summary id. Assistant:

E.2 ENTAILMENT PREDICTION PROMPT

We use the following prompt to compute entailments with the LLMs. It stems from Tang et al.
(2022), however instead of answering Yes/No the LLM is prompted to anser with “0”/“1”,
which has the advantage that the token probabilities can be used to compute an uncertainty score.
Determine whether the provided claim is consistent with the corresponding document. Consistency in this
context implies that all information presented in the claim is substantiated by the document. If not, it should
be considered inconsistent.
Document: EVIDENCE DOCUMENT
Claim: CLAIM
Please assess the claim’s consistency with the document by responding with either "1" (consistent) or "0"
(inconsistent). Do not output anything else. Answer:

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1650

1651
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1657
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1668
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Under review as a conference paper at ICLR 2025

E.3 LLM PARTIAL REPHRASING PROMPT

We use the following prompt to instruct the LLM to only rephrase specific parts of a sentence that
are masked out with “_”.
Your task is to fill in the gaps in a document indicated with “_” with additional details. If there is no gaps,
please output the input text. The number of “_” indicates the approximate number of words that should be
filled into each gap. While slight deviations (e.g., one word more or less) are permissible, the filled in text
should respect the length indicated through the number of "_". **Do not change the text outside the gaps
and do not include gaps in the final output.** You will generate N different completions of the document.
Each completed document is identified with an integer from 0 to N-1. The document with the blanks filled
must be wrapped in <answer #></answer #> tags, where # is replaced with the id of the filled-in document.
You will now see the original document, but you will have to generate different versions that preserve the
meaning by filling the gaps.
Here is the original: <document>EVIDENCE DOCUMENT</document>
The document including the gaps is: <document>DOCUMENT WITH WORDS MASKED</document>
Assistant:

1674
1675

1676
1677
1678
1679
1680
1681

1682
1683
1684
1685
1686
1687

1688
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1693
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1699
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1701
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1704

1705
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1710

1711
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1716
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1721

1722
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32

