11institutetext: Columbia University, New York, NY 10027
11email<EMAIL_ADDRESS>22institutetext: Google DeepMind, Mountain
View, CA 94043
22email<EMAIL_ADDRESS>{adamsyuwei, qvl, thangluong<EMAIL_ADDRESS>
#  HaloQuest: A Visual Hallucination Dataset for Advancing Multimodal
Reasoning††thanks: Code and data at: https://github.com/google/haloquest.
ZW and GB are main contributors. ZW did some work while at Google DeepMind.

Zhecan Wang 1*1* Garrett Bingham 2*2* Adams Wei Yu 22
Quoc V. Le 22 Thang Luong 22 Golnaz Ghiasi 22

###### Abstract

Hallucination has been a major problem for large language models and remains a
critical challenge when it comes to multimodality in which vision-language
models (VLMs) have to deal with not just textual but also visual inputs.
Despite rapid progress in VLMs, resources for evaluating and addressing
multimodal hallucination are limited and mostly focused on evaluation. This
work introduces HaloQuest, a novel visual question answering dataset that
captures various aspects of multimodal hallucination such as false premises,
insufficient contexts, and visual challenges. A novel idea from HaloQuest is
to leverage synthetic images, apart from real ones, to enable dataset creation
at scale. With over 7.7K examples spanning across a wide variety of
categories, HaloQuest was designed to be both a challenging benchmark for VLMs
and a fine-tuning dataset for advancing multimodal reasoning. Our experiments
reveal that current models struggle with HaloQuest, with all open-source VLMs
achieving below 36% accuracy. On the other hand, fine-tuning on HaloQuest
significantly reduces hallucination rates while preserving performance on
standard reasoning tasks. Our results discover that benchmarking with
generated images is highly correlated $(r=0.97)$ with real images. Last but
not least, we propose a novel Auto-Eval mechanism that is highly correlated
with human raters $(r=0.99)$ for evaluating VLMs. In sum, this work makes
concrete strides towards understanding, evaluating, and mitigating
hallucination in VLMs, serving as an important step towards more reliable
multimodal AI systems in the future.

###### Keywords:

Hallucination Vision-Language Models Datasets

##  1 Introduction

Hallucination, the generation of factually incorrect or inconsistent
information, poses a critical challenge for the reliability of vision-language
models (VLMs)[29, 40, 7, 13]. Hallucination in these systems can result from
visual misinterpretations[32, 24, 49], misaligned language understanding[12],
or the generation of responses unsupported by either modality[27]. This issue
is particularly concerning as VLMs find increasing use in real-world
applications where inaccurate information can have harmful consequences, such
as in autonomous vehicles[14, 37, 34] or medical diagnosis[47, 50, 3].
Research into mitigating hallucination is hindered by limited image datasets,
a lack of comprehensive evaluation systems targeting a variety of
hallucination triggers, and the difficulty of open-ended evaluation for
complex visual question answering tasks [57, 27, 46, 16, 9, 21].

To address these limitations, this work introduces HaloQuest, a novel visual
question answering (VQA) dataset comprised of both real and synthetically
generated images. By leveraging prompt-based image generation, HaloQuest
overcomes the constraints of traditional datasets, allowing for the creation
of images from various categories, including highly unusual and abstract
visual scenes. The dataset includes questions spanning three categories
designed to trigger common hallucination scenarios: questions with false
premises, questions lacking sufficient context for accurate interpretation,
and questions that are otherwise challenging to answer correctly. This focus,
coupled with a machine-human-in-the-loop data generation pipeline, enables the
collection of challenging examples that target specific weaknesses in current
VLM models.

Experiments with HaloQuest demonstrate that modern VLMs struggle to handle
these complex visual scenes and question types, highlighting a significant gap
between current capabilities and real-world requirements. Importantly, fine-
tuning these models on HaloQuest reduces hallucination rates while preserving
their performance on standard reasoning tasks. This establishes HaloQuest as a
valuable benchmark for VLM hallucination research, enabling the development of
more robust models.

This study underscores the potential of synthetic images to enhance visual-
language understanding evaluation. Existing image-text datasets are primarily
sourced from MS-COCO and Flickr and exhibit limited image diversity [26].
Utilizing prompt-based synthetic images circumvents this constraint, offering
a cost-effective and scalable solution. Notably, these synthetic images can
encompass diverse visual scenarios, including unusual, complex, and abstract
scenes rarely found in real-world datasets. The increasing quality and real-
world adoption of prompt-based synthetic images, particularly in advertising
and design, necessitates robust model evaluation against potential
hallucinations. By overcoming the reliance on limited real-world image
datasets, HaloQuest paves the way for the design of more comprehensive and
challenging evaluation suites.

Standard evaluation approaches often rely on multiple-choice or finite
vocabulary answers [60, 4, 10, 33, 44]. This limits the model’s ability to
express nuanced or complex responses, failing to fully mirror real-world
scenarios. Furthermore, accurately evaluating extended, hallucinated
predictions is particularly difficult. Consequently, previous studies on
hallucination evaluation have relied on methods like manual assessment [12,
16], counting hallucinated objects [24], using conventional caption evaluation
metrics [32], or restricting response formats [27]. These approaches cannot
capture a model’s full ability to generate coherent, detailed, and
contextually appropriate responses. They are especially impractical when
evaluating complex hallucinations arising from generated visual scenarios. To
address this limitation, this work employs an Automatic Evaluation (Auto-Eval)
mechanism where a language model assesses the VLM’s responses [38]. This Auto-
Eval system allows for nuanced, open-ended evaluation of model responses and
provides a dynamic system that can adapt alongside future advancements.

In sum, this work makes several contributions to the field of vision-language
understanding. First, HaloQuest is introduced, a novel VQA dataset featuring
both real and synthetic images, designed to address the limitations of current
datasets. HaloQuest includes a variety of image content and questions
targeting specific hallucination triggers, and utilizes an innovative machine-
human-in-the-loop data generation pipeline. Second, the effectiveness of
HaloQuest as a benchmark is demonstrated, highlighting the limitations of
current VLM models and showing how fine-tuning on HaloQuest significantly
reduces hallucination. Finally, an LLM-based Auto-Eval system is introduced
for open-ended, dynamic evaluation, and the potential of synthetic images to
revolutionize VLM evaluation is explored. This work paves the way for the
development of more robust and reliable multimodal AI systems.

##  2 Related Work

Hallucination, the generation of factually incorrect or inconsistent
information, is a well-documented issue in large language models (LLMs) [8,
17, 23]. Within the domain of vision and language understanding,
hallucinations can manifest in several ways, including misinterpretation of
visual elements, misaligned language understanding, or responses unsupported
by either modality. While still a developing area of study, recent works have
begun to explore these vision-specific hallucination phenomena [24, 16, 63,
32, 57, 18, 39]. Consequently, research efforts have focused on understanding,
evaluating, and mitigating hallucination in VLMs.

There are a number of mechanisms that may cause a VLM to hallucinate. An over-
reliance on language priors [42] is one such mechanism. For example, models
often learn pairs of objects that co-occur together, and the presence of
"keyboard" may bias a model towards outputting "mouse" or "monitor," even if
one is not present in the image [63]. Certain statistics can also be
predictive of hallucination. An output token with low probability may indicate
a model is hallucinating due to low confidence, while tokens towards the end
of a long response may be hallucinatory if the model is running out of
meaningful things to say [63]. It is also possible to understand hallucination
in isolated instances by directly inspecting the attention weights to see what
the model is attending to when it outputs hallucinatory text [51]. Despite
these advancements, hallucination in VLMs is still not completely understood,
in part because evaluating hallucination is not trivial.

Existing approaches for evaluating hallucinations in VLMs have limitations.
Methods that use binary yes/no questions [24], are constrained to short-word
answers [27], rely on caption evaluation metrics [42, 13], and require manual
assessment [12], often prioritize verifying the presence or absence of objects
and thus are inherently limited. Consequently, they may not be well-suited to
comprehensively evaluate nuanced hallucinations within free-form, open-ended
answers. This lack of robust evaluation metrics hinders efforts to develop
effective mitigation strategies.

Despite these challenges in comprehensively evaluating hallucination, some
progress has been made towards mitigation [16, 61, 31, 49]. Existing efforts
center around several key strategies, such as knowledge grounding with self-
feedback [21], finetuning on both positive and negative examples [28], and
post-hoc response correction [63]. Reliance on real-world image datasets also
introduces limitations, as these datasets often lack the complexity necessary
to fully expose and address different hallucination triggers [46].

HaloQuest directly confronts shortcomings prevalent in hallucination
understanding, evaluation, and mitigation. Experimental results from false
premise questions, visually challenge questions, and questions with
insufficient context elucidate the gap between current models’ performance and
modern expectations. This work also leverages an open-ended question format
and introduces an LLM-based Auto-Eval mechanism which moves beyond traditional
object-centric metrics, allowing for more nuanced evaluation of complex
hallucinations [38]. Furthermore, HaloQuest makes use of both real and
synthetically generated images, resulting in a powerful and complex dataset
that is effective at reducing hallucination rates [54, 35, 10]. Together,
these contributions make HaloQuest a valuable benchmark for the vision-and-
language community, setting a new standard for hallucination research.

Figure 1: Example entries from HaloQuest (bottom) and other benchmarks (top).
Current benchmarks often do not incorporate synthetic images, require one-word
responses, are multiple choice, or simply ask for an image description. In
contrast, HaloQuest contains challenging questions in three categories, uses
both real and synthetic images, and makes use of Auto-Eval to allow for free-
form answer evaluation. Figure 2: HaloQuest data collection pipeline. First,
both real and synthetic images are collected from various sources. Next,
humans and LLMs create question-answer pairs designed to elicit hallucination.
Finally, a filtering mechanism removes the entires that are overly simple or
ambiguous. The result is a challenging dataset that effectively exposes model
hallucination tendencies.

##  3 HaloQuest

This section describes the HaloQuest dataset. It details the image collection
methodology, the design of questions to trigger hallucinations, the filtering
and refinement process, and the LLM-based Auto-Eval mechanism. Example
HaloQuest entries are shown in Figure 1.

###  3.1 Image Collection

Table 1: Curated lists of image subject and attributes inspired by
PartiPrompts[59]. Image queries are created by randomly selecting one subject
and one attribute from the lists. Utilizing prompt-based image generation
allows for creating a visually complex dataset in a precise, controllable
manner, resulting in more robust models.

Subjects
---
People | Animals | Body Parts
Insects | Plants | Accessories
Appliances | Artifacts | Electronics
Furniture | Kitchenware | Office Supplies
Indoor Scenes | Food & Beverage | Construction
Vehicles | Nature (Scene) |
Attributes
---
Abstract | Perspective | Property & Material
Quantity | Fine-grained Details | Illustration, Composition & Style
Age | Imagination | Position & Coexistence
Action | Art | Animation & Media
Text | Knowledge | Emotion & Expression

First, to ensure a rich and varied dataset, HaloQuest leverages both real and
synthetic images. The real images are a random sample from the Open Images
dataset, and synthetic images are sourced from online Midjourney and Stable
Diffusion galleries [20, 2, 43]. Images are selected based on high view counts
and positive ratings in order to prioritize quality and relevance. Search
queries incorporating combinations of topic words from a carefully curated
list inspired by PartiPrompts are used to retrieve a varied range of images
[59, Table 1].

Human annotators filter this initial set of images according to two criteria.
The images should be interesting or unusual, but they must also be
comprehensible. For example, images are deemed interesting if they depict
scenarios outside of everyday experiences, contain unexpected juxtapositions
of objects, like the dog dressed in a costume made from newspaper shown in
Figure 2, or feature visually striking elements. These images could include
scenes that defy real-world physics or logic. However, the images must be
coherent, artifact-free and understandable by humans, despite their
unconventional nature. Checking for these two criteria strikes a balance
between generating challenging scenarios and maintaining the ability to
reliably attribute model responses to specific weaknesses in reasoning or
understanding.

###  3.2 Designing Questions to Elicit Hallucination

Once the images are collected, humans and LLMs craft questions and answers
about the images, focusing on creativity, nuanced reasoning, and probing
potential model biases. Specifically, HaloQuest includes three categories of
questions designed to elicit hallucinations.

First, questions with a false premise contain statements or assumptions that
directly contradict the visual content of the image. They are designed to test
whether the model can correctly prioritize visual evidence over misleading
linguistic cues.

Next, questions that are visually challenging require a deep understanding of
image details, such as counting objects, determining spatial relationships, or
reasoning about occluded areas. They evaluate the model’s ability to perform
complex visual analysis.

Finally, questions with insufficient context cannot be definitively answered
based on the image alone. They probe whether models will resort to biases or
unfounded assumptions instead of acknowledging the limits of the provided
information.

Table 2: Summary of HaloQuest data splits. HaloQuest contains entries in three categories designed to elicit hallucination in VLMs. These entries are comprised of both real and synthetically generated images. Some images have multiple questions associated with them, but the dataset still contains a large number of unique images. | Train | Eval | Total
---|---|---|---
Entries with Real Images | 2985 | 217 | 3202
Entries with Generated Images | 4155 | 391 | 4546
False Premise Questions | 2698 | 304 | 3002
Visually Challenging Questions | 2973 | 183 | 3156
Questions with Insufficient Context | 1469 | 121 | 1590
Number of Unique Images | 2782 | 375 | 3157
Total Entries | 7140 | 608 | 7748

In order to create these questions, humans were given images and asked to
write two questions and corresponding answers for each. First, they were
tasked with writing a question that asks “something about a visual element
related to the image which is not possible to answer by looking at the image.”
These questions were later analyzed and split into the false premises and
insufficient context categories mentioned above. Second, the crowdworkers were
asked to write a question “about a subtle detail presented in the image which
we are able to easily provide a clear answer and the answer does not vary upon
personal preferences or opinions.” More details on crowdworker instructions
are in Appendix 0.A, and a breakdown of these question categories is in Table
2.

To generate additional question-answer pairs efficiently, LLMs are also used.
Specifically, the IdealGPT framework, which leverages GPT-4 and BLIP2, is used
to produce long and potentially noisy image captions, as in Figure 2 [64, 58,
1, 22]. These descriptions are later converted to several atomic statements
("The image is a close-up of a golden retriever", "The dog is holding
newspaper pages on its back as a cape"), and human annotators evaluate the
validity (yes/no) of each statement. The LLMs then take each atomic statement
and whether it is true or false and use this information to produce a
question-answer pair for the given image.

###  3.3 Filtering and Refining the Data Examples

The quality of annotated question-answer pairs is next improved through
filtering. First, high-performing VQA models generate preliminary responses
for an initial question pool. Then, experienced human annotators review both
the questions and model-generated responses. Questions judged to be too easy
are discarded or revised to increase difficulty. Ambiguous or nonsensical
answers are flagged, ensuring each question has a clear and well-defined
solution. This process leads to a dataset composed of challenging, high-
quality examples.

###  3.4 Automatic VQA Evaluation

In order to facilitate free-form and open-ended VLM hallucination evaluation
at scale, an LLM-based automatic evaluation method is developed. While in
principle any LLM can perform such evaluation with basic prompting, this work
introduces a recipe that is more effective than this baseline strategy.
Specifically, a Langfun schema is developed which helps Gemini to accurately
extract the main point in the model response and ground truth, and then decide
whether these points are in agreement [38, 41].

Figure 7 in Appendix 0.B shows the prompt and schema given to Gemini to
implement automatic evaluation, and Figure 8 in Appendix 0.B shows an example
Auto-Eval response. As shown in these figures, Gemini is tasked with
populating the PredictionEvaluation class attributes given the input question,
response, and ground truth. Experiments in the next section show that this
approach is substantially more effective than basic prompting alone, and thus
can serve as inspiration for automatic evaluation in other domains in the
future. Appendix 0.B contains additional Auto-Eval implementation details.

##  4 Experiments

This section includes experiments that demonstrate the usefulness of HaloQuest
in understanding, measuring, and reducing hallucination tendencies in VLMs.
The results show that current models perform poorly on HaloQuest in a zero-
shot setting, showing that much work remains to be done to build models that
are hallucination-free. Furthermore, current evaluation metrics do not
accurately quantify hallucination, a missing capability that the Auto-Eval
framework directly addresses. HaloQuest is also useful for reducing
hallucination rates, and this training does not hurt performance on related
VQA tasks. Additional experiments contrast the models’ performance on
generated and real images, and similarly for different question types. These
results facilitate a more fine-grained understanding of model capabilities,
enabling future hallucination mitigation strategies to be more targeted.
Together, these findings highlight the significant step HaloQuest provides
towards building more reliable and trustworthy VLMs.

###  4.1 Zero-shot Evaluation on HaloQuest

Table 3 lists zero-shot evaluation of top-performing VLMs on HaloQuest and
reveals two key insights. First, existing VLMs struggle with HaloQuest,
exhibiting high hallucination rates. This result indicates substantial
shortcomings in model capabilities and highlights the need for robust
hallucination mitigation. Second, increased model size doesn’t necessarily
translate to better hallucination resistance. Surprisingly, BEiT-3 [53], a
smaller model, outperforms several larger models. These findings underscore
the importance of developing data-driven hallucination mitigation strategies
that are not solely reliant on model scaling.

Table 3: Zero-shot accuracy on HaloQuest. The results show that current models are susceptible to hallucination, highlighting the need for more robust VLM development. GPT-4, GPT-4o and Gemini 1.5 Pro are only tested on the subset of images without people. Model (# Param) | Human Eval | Auto-Eval |
---|---|---|---
LLaVA (13B) [30] | 10.9 | 10.9 |
MiniGPT4 (13B) [65] | 18.7 | 25.2 |
BLIP2 (12B) [22] | 21.1 | 22.5 |
InstructBLIP (12B) [31] | 25.5 | 28.5 |
Open-flamingo (9B) [5] | 13.8 | 15.0 |
BLIP2 (8B) [22] | 10.9 | 11.8 |
InstructBLIP (8B) [31] | 25.0 | 27.3 |
MiniGPT4 (7B) [65] | 18.6 | 19.1 |
mPLUG-Owl2 (7B) [56] | 9.2 | 10.4 |
mPLUG-Owl1 (7B) [55] | 9.7 | 8.7 |
Open-flamingo (3B) [5] | 6.9 | 8.2 |
OFA (1B) [52] | 8.7 | 10.2 |
BEiT-3 (0.7B) [53] | 35.9 | 40.0 |
GPT-4 [1] | 62.9 | 61.2 |
GPT-4o | 68.1 | 63.2 |
Gemini 1.5 Pro[41] | 76.1 | 77.9 |

###  4.2 Quantifying Hallucination with Auto-Eval

Figure 3: Human evaluation vs. different evaluation metrics. Metrics are based
on zero-shot evaluation (Table 3). Standard metrics like BLEU, CIDER, ROUGE,
and METEOR do not correlate well with human evaluation, demonstrating that
they are insufficient for characterizing VLM hallucination[36, 48, 25, 6]. In
contrast, Auto-Eval correlates strongly with human evaluation (Pearson’s r),
thus facilitating hallucination evaluation at scale [45].

Before VLM hallucination can be addressed, it must be accurately measured.
Figure 3 compares modern metrics like BLEU, CIDER, ROUGE, and METEOR with
human evaluation on the HaloQuest evaluation set [36, 48, 25, 6]. None of the
metrics correlate well with human evaluation, demonstrating they are
insufficient for measuring hallucination. Fortunately, Auto-Eval (Section 3.4)
correlates strongly with human evaluation. While all experiments in this paper
include both human evaluation and Auto-Eval scores, this result suggests that
Auto-Eval can be used in the future if human evaluation is unavailable or is
too expensive.

Table 4: Auto-Eval agreement with human raters, averaged across all responses from the zero-shot experiment in Table 3. Using a simple text prompt performs the worst. Prompting the model to fill out a schema helps, but the best performance is achieved by further prompting the model to reason about the main points of the ground truth and model response. The results show that automatic evaluation systems are not trivial to implement, highlighting Auto-Eval as an important contribution of this paper. Auto-Eval Setup | Agreement w/ Human
---|---
Text-only prompting | 93.4
Basic Langfun schema | 94.8
Advanced Langfun schema | 95.3

Table 4 shows an ablation comparing different Auto-Eval implementations. Text-
only prompting or simple schemas that do not prompt the model to reason deeply
about why a response may be correct or incorrect are not sufficiently
performant. In contrast, the Auto-Eval implementation used throughout this
paper does achieve good agreement with human raters and is a concrete
contribution in its own right. Further details about the text-only prompting
and basic Langfun schema comparisons can be found in Section C of
Supplementary Material.

###  4.3 Mitigating Hallucination with HaloQuest

Table 5: Effect of training data on benchmark performance. HaloQuest accuracy
is measured with both human raters and with Auto-Eval, and VQA v2 performance
is measured by exact match and broken down by question subtype, as is
standard. Including HaloQuest training data effectively reduces hallucination
rates, as shown by improved performance on the HaloQuest evaluation set.
Importantly, adding HaloQuest training data does not degrade performance on
VQA v2, and in most cases helps. These results show that HaloQuest is an
effective dataset for reducing hallucination rates without sacrificing other
model capabilities.

Model (# Param) | Training Data | HaloQuest | VQA v2
---|---|---|---
Human Eval | Auto-Eval | Overall | Binary | Number | Others
BLIP2 (8B) [22] | VQA v2 | 11.2 | 12.3 | 70.5 | 87.0 | 52.4 | 59.9
VQA v2 + HaloQuest | 33.9 | 34.7 | 71.0 | 87.1 | 54.2 | 60.7
mPLUG-Owl1 (7B) [56] | VQA v2 | 9.7 | 9.5 | 74.2 | 89.8 | 57.3 | 64.5
VQA v2 + HaloQuest | 25.8 | 29.1 | 74.9 | 90.0 | 58.7 | 64.9
MiniGPT4 (7B) [65] | VQA v2 | 10.5 | 10.7 | 71.0 | 87.1 | 48.6 | 61.9
VQA v2 + HaloQuest | 26.6 | 28.0 | 71.4 | 87.8 | 52.0 | 61.9
MiniGPT4 (13B) [65] | VQA v2 | 18.3 | 16.1 | 74.9 | 90.3 | 54.2 | 65.0
VQA v2 + HaloQuest | 35.5 | 40.1 | 74.9 | 90.8 | 56.7 | 65.4

Table 6: Evaluation on POPE [24]. Training on HaloQuest (indicated with $\star$) improves performance over the baseline.  Model | Random | Popular | Adversarial
---|---|---|---
| Acc | F1 | Acc | F1 | Acc | F1
mPLUG-OWL | 0.50 | 0.63 | 0.54 | 0.61 | 0.51 | 0.60
mPLUG-OWL$\star$ | 0.64 | 0.71 | 0.62 | 0.67 | 0.57 | 0.65
MiniGPT4 | 0.72 | 0.73 | 0.68 | 0.69 | 0.63 | 0.65
MiniGPT4$\star$ | 0.80 | 0.79 | 0.75 | 0.74 | 0.69 | 0.70

In addition to identifying hallucination tendencies in VLMs, HaloQuest is also
useful for mitigating them. In this experiment, four VLMs were fine-tuned with
VQA v2 data and evaluated on both HaloQuest and VQA v2 [15]. After converting
question-answer pairs into natural language instructions using templates,
these fine-tuned models were then further instruction tuned with a combination
of VQA v2 data and HaloQuest data [31]. These models were then re-evaluated on
HaloQuest and VQA v2.

Table 5 shows the results of this experiment, which demonstrates that fine-
tuning existing VLMs on HaloQuest significantly reduces hallucination rates
while maintaining performance on other benchmarks. These results highlight
HaloQuest’s potential in improving model safety without reducing
effectiveness. Implementation details are in Section D of Supplementary
Material.

Furthermore, Table 6 shows model performance on the POPE hallucination
benchmark with images from Visual Genome [24, 19]. Training on HaloQuest
improves model performance in this new dataset, demonstrating that HaloQuest
helps models avoid hallucination in novel contexts as well.

###  4.4 Understanding Hallucination in Synthetic Images

Table 7: Zero-shot and trained model performance on HaloQuest broken down by image type. The data in this table is from the same experiments as Tables 3 and 5. Although all models perform poorly in both subsets of images, models tend to perform slightly better on synthetic images compared to real ones. Importantly, training with HaloQuest improves performance on both sets of images. GPT-4, GPT-4o and Gemini 1.5 Pro are only tested on the subset of images without people. Model (# Param) | Generated | Real
---|---|---
Human Eval | Auto-Eval | Human Eval | Auto-Eval
Zero-shot Evaluation
LLaVA (13B) [30] | 12.3 | 12.8 | 8.2 | 7.4
MiniGPT4 (13B) [65] | 18.2 | 24.0 | 18.9 | 27.2
BLIP2 (12B) [22] | 24.8 | 26.1 | 14.29 | 16.1
InstructBLIP (12B) [31] | 28.4 | 31.5 | 20.3 | 23.0
Open-Flamingo (9B) [5] | 16.1 | 17.1 | 9.7 | 11.1
BLIP2 (8B) [22] | 11.5 | 11.8 | 9.7 | 12.0
InstructBLIP (8B) [31] | 28.4 | 29.7 | 18.9 | 23.0
MiniGPT4 (7B) [65] | 18.1 | 19.4 | 18.0 | 18.4
mPLUG-Owl2 (7B) [56] | 11.0 | 11.3 | 6.0 | 8.8
mPLUG-Owl1(7B) [55] | 11.3 | 10.2 | 6.9 | 6.0
Open-Flamingo (3B) [5] | 7.4 | 8.7 | 6.0 | 7.4
OFA (1B) [52] | 9.7 | 11.3 | 6.9 | 8.3
BEiT-3 (0.7B) [53] | 41.2 | 44.3 | 26.3 | 32.3
GPT-4 [1] | 64.3 | 61.1 | 60.6 | 61.4
GPT-4o | 68.8 | 63.8 | 66.9 | 62.2
Gemini 1.5 Pro [41] | 74.7 | 78.3 | 78.7 | 77.2
Trained on VQA v2 + HaloQuest
BLIP2 (8B) [22] | 36.6 | 37.1 | 29.0 | 30.4
mPLUG-Owl1(7B) [55] | 27.4 | 30.4 | 23.0 | 26.7
MiniGPT4 (7B) [65] | 27.4 | 23.7 | 25.4 | 23.0
MiniGPT4 (13B) [65] | 33.3 | 32.0 | 39.7 | 33.2

This work extends previous research on hallucination with real images in VLMs
to include synthetically generated images as well. Table 7 shows model
performance separated according to whether the images are real or
synthetically generated. Although most models tend to hallucinate more with
real images in this set, hallucination rates are quite high with synthetic
images as well. In fact, performance on generated images is highly correlated
with performance on real images, with $r=0.97$ for both human evaluation and
Auto-Eval, suggesting that synthetic images can provide an accurate measure of
model capability, despite small discrepancies in overall performance.

Although real images are more challenging in HaloQuest, there remain many
reasons to continue to utilize synthetic images. These synthetically generated
images offer a cost-effective and scalable solution for expanding datasets,
and experimental results indicate that incorporating these images helps reduce
hallucination rates in models (Tables 5 and 7). Indeed, while the synthetic
images in HaloQuest are not as difficult on average as the real images,
advancements in image generation models will likely close this gap in the near
future. Furthermore, as image generation systems become more widely used
around the world, it will become even more important for models to be robust
to hallucination in synthetic images. This surprising finding opens up
exciting avenues for future research in dataset curation, controlled image
generation, and annotator bias mitigation.

###  4.5 Understanding Hallucination Triggers

Table 8: Zero-shot and trained model performance on HaloQuest broken down by
question type. The data in this table is from the same experiments as Table 3
and 5. Breaking down the results in this way makes it possible to address
specific model weaknesses, and training with HaloQuest improves model
performance in all three categories. GPT-4, GPT-4o and Gemini 1.5 Pro are only
tested on the subset of images without people.

Model (# Param) | False Premise | Visually Challenging | Insufficient Context
---|---|---|---
Human Eval | Auto-Eval | Human Eval | Auto-Eval | Human Eval | Auto-Eval
Zero-shot Evaluation
LLaVA (13B) [30] | 2.3 | 1.7 | 30.6 | 31.2 | 2.5 | 3.3
MiniGPT4 (13B) [65] | 16.2 | 21.5 | 10.4 | 13.7 | 36.4 | 51.2
BLIP2 (12B) [22] | 16.8 | 19.5 | 35.5 | 32.8 | 9.9 | 14.9
InstructBLIP (12B) [31] | 28.4 | 32.0 | 33.3 | 33.9 | 6.6 | 11.6
Open-Flamingo (9B) [5] | 13.2 | 13.9 | 19.1 | 21.3 | 7.4 | 8.3
BLIP2 (8B) [22] | 5.0 | 4.6 | 26.8 | 26.8 | 1.7 | 6.6
InstructBLIP (8B) [31] | 28.4 | 32.0 | 6.6 | 11.6 | 33.3 | 33.9
MiniGPT4 (7B) [65] | 13.2 | 13.2 | 26.5 | 27.3 | 15.7 | 16.5
mPLUG-Owl2 (7B) [56] | 0.8 | 3.3 | 28.4 | 27.9 | 0.8 | 3.3
mPLUG-Owl1(7B) [55] | 1.0 | 0.3 | 29.0 | 26.8 | 2.5 | 2.5
Open-Flamingo (3B) [5] | 0.7 | 1.3 | 19.1 | 21.3 | 4.1 | 5.8
OFA (1B) [52] | 5.0 | 6.3 | 19.7 | 20.2 | 1.7 | 5.0
BEiT-3 (0.7B) [53] | 24.1 | 28.4 | 36.6 | 36.1 | 9.1 | 10.7
GPT-4[1] | 64.7 | 63.0 | 46.9 | 44.8 | 80.6 | 79.1
GPT-4o | 68.5 | 65.2 | 58.3 | 55.2 | 80.6 | 68.7
Gemini 1.5 Pro [41] | 80.4 | 83.7 | 57.3 | 56.3 | 91.0 | 92.5
Trained on VQA v2 + HaloQuest |  |
BLIP2 (8B) [22] | 33.0 | 33.3 | 38.3 | 39.9 | 29.8 | 29.8
mPLUG-Owl1(7B) [55] | 21.1 | 25.4 | 37.2 | 40.4 | 20.7 | 21.5
MiniGPT4 (7B) [65] | 23.8 | 17.5 | 32.2 | 36.6 | 25.6 | 17.4
MiniGPT4 (13B) [65] | 33.0 | 30.0 | 31.2 | 23.5 | 48.8 | 51.2

VLMs hallucinate for various reasons. This work explores triggering
hallucination with questions with false premises, visually challenging
questions, and questions with insufficient context. Table 8 shows model
performance broken down according to these image categories. On average, open-
source models struggle substantially with false premise and insufficient
context questions, but perform slightly better with visually challenging ones.
Interestingly, different models have different strengths and weaknesses in
different question categories. GPT-4 is more adept at addressing false premise
and insufficient context questions, but is not as performant in the visually
challenging section. This finding demonstrates how understanding fine-grained
hallucination triggers allows for targeting model-specific capabilities.
Training on HaloQuest substantially improves performance in all categories,
but the models still perform poorly, reinforcing the need for continued work
in hallucination reduction.

##  5 Discussion and Future Work

This section explores the impact of this work and its potential to shape
future research directions in the field, including discussion on the semantic
novelty of synthetic images, solving hallucination comprehensively, multimodal
hallucination, finding nuance in responses with Auto-Eval, and broader
societal impacts.

Figure 4: Low-dimensional representation of images. Each point represents one
image. CLIP embeddings were extracted for all images and then projected to a
2D space using the UMAP algorithm. HaloQuest real images occupy a similar
semantic distribution to VQA v2 images, while the synthetic images are
entirely novel.

###  5.1 Visualizing Semantic Novelty in Synthetic Images

Beyond cost-effectiveness and scalability, HaloQuest leverages prompt-based
synthetic images to access a wider spectrum of visual scenarios, including
unusual, complex, and abstract scenes, which are challenging or infeasible to
obtain from real-world sources. This is particularly critical given the
growing prevalence of synthetic images in real-world applications,
necessitating the development of models resistant to hallucinations. Figure 4
illustrates this distinction by demonstrating the semantic dissimilarity
between synthetic and real images, including those from the VQA v2 [15]
dataset, within the embedding space. This finding underscores the importance
and unique contribution of the synthetic images in HaloQuest.

###  5.2 Hallucination Remains an Unsolved Problem

Experiments using HaloQuest highlight the severity of hallucination in current
models. While fine-tuning on HaloQuest demonstrates significant reduction in
hallucination rates, the problem persists. This aligns with trends in related
work, where techniques can identify and alleviate hallucination but fall short
of a complete solution. Tackling hallucination comprehensively will likely
require a multi-pronged approach. Further exploration into integrating
symbolic reasoning, scaling both model parameters and dataset size, and
potentially even rethinking model architectures might hold the key. This work
represents an important step, but underscores that the quest to eliminate
hallucination in VLMs will require continued innovation and research.

###  5.3 Multimodal Hallucination

This paper focuses on visual hallucination in VLMs, a phenomenon related to
but distinct from text-only hallucination in LLMs. As AI systems continue to
operate within multimodal environments (code, video, audio, etc.), the
necessity of addressing hallucination across these varied modalities will
become increasingly important. The key question remains: are there techniques
that are capable of reducing hallucination universally, or will modality-
specific approaches be essential? Exploiting inherent structural differences
between modalities might reveal new insights, but developing techniques that
are modality-agnostic may be a more efficient path forward. The development of
datasets like HaloQuest serves as a good starting point, emphasizing the
importance of designing challenging benchmarks as the field looks towards
tackling hallucination in the broader landscape of multimodal AI.

###  5.4 Unconvering Nuance with Auto-Eval

This paper uses human evaluation as the gold standard for measuring model
performance, but also contributes a novel Auto-Eval mechanism that holds
promise for efficient evaluation at scale in future work. Human evaluation is
important for benchmarking the Auto-Eval system itself. Interestingly, the
relationship is reciprocal: exploring instances where human and Auto-Eval
judgments diverge was useful for finding nuanced and challenging cases that
highlight the subtle nature of hallucination detection. In a limited number of
scenarios, this analysis even led to refinements in the ground truth labels.
This demonstrates the potential of human and automated evaluation systems to
work in tandem, driving continuous improvement in detecting and understanding
hallucination.

###  5.5 Societal Impact

While this work primarily centers on the creation of a novel dataset, the
potential societal impacts are significant. HaloQuest aims to provide a
crucial tool for mitigating hallucination in VLMs, thereby improving their
robustness and reducing the likelihood of erroneous or misleading outputs.
This has implications for real-world applications where safety and reliability
are paramount, such as autonomous systems or medical image analysis. However,
it is important to acknowledge that like any technology, datasets can be used
for both beneficial and potentially harmful purposes. Bad actors could
leverage datasets like HaloQuest to intentionally train models to generate
misleading or deceptive content tailored to exploit model weaknesses. This
fact underscores the importance of ongoing research into the detection and
mitigation of such malicious use of AI systems.

##  6 Conclusion

This work has introduced HaloQuest, a novel VQA benchmark that leverages both
real-world and synthetically generated images. HaloQuest’s controlled image
generation and questions designed to elicit specific hallucination types
enable a more targeted analysis of hallucination triggers in VLMs. Experiments
demonstrate that current state-of-the-art models struggle with HaloQuest,
revealing a crucial disconnect between their capabilities and real-world
reliability requirements. Importantly, fine-tuning VLMs on HaloQuest
demonstrably reduces hallucination rates while maintaining performance on
typical reasoning tasks.

HaloQuest highlights the potential of synthetic images in the development of
robust multimodal AI. It addresses limitations present in traditional
datasets, enabling the creation of richer and more varied visual scenarios.
The dataset, coupled with an innovative machine-human-in-the-loop generation
process, facilitates targeted investigation into VLM weaknesses.

Further, this work introduces an LLM-based Auto-Eval mechanism that
facilitates open-ended and nuanced evaluation of VLM responses. This approach
is a marked improvement over existing methods that often limit the model’s
expressive ability or are impractical for evaluating complex hallucinations.

HaloQuest stands as a valuable resource for the vision-and-language community.
It provides both a challenging evaluation benchmark and a training dataset
aimed at mitigating hallucination in VLMs. This work underscores the power of
synthetic image generation and advanced evaluation techniques in driving the
creation of more reliable and trustworthy multimodal AI systems.

## Acknowledgements

The authors would like to thank Kai-wei Chang, Shih-fu Chang, Junzhang Liu,
Haoxuan You, Rui Sun and Chris Thomas for invaluable discussions. The authors
also thank Surge AI, in particular John Romeral and Paul Mains, for help with
data collection.

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##  Appendix 0.A Instructions for Crowdworkers Writing Questions

Crowdworkers were given the following instructions when asked to draft
questions and answers for a given image:

> For each input image, please write 2 challenging questions as described
> below and also 3 answers for each question. In case it is hard to write a
> challenging question, skip the writing and write “skip”.
>
>
> First question
>
>
> The first question should ask something about a visual element related to
> the image which is not possible to answer by looking at the image. (We’ve
> discovered that AI models often struggle to express uncertainty and instead
> generate answers for these types of questions. Therefore, we wish to create
> a dataset specifically for evaluating AI models on these types of
> questions.) These are some example cases for writing these types of
> questions.
>
>   * •
>
> The question asks some details about a visual element that is not visibly
> present in the image, consequently, we either cannot answer the question, or
> the answer to the question implies that the subject is not present. Please
> provide questions about elements that, while not visible, are relevant to
> the scene depicted in the image. For instance, you could ask about something
> that is hidden or cropped in the current context. Alternatively, you might
> consider asking about a detail that would likely be found in a similar
> image. For example please see the first question about the cat image below.
>
>   * •
>
> The question asks about specific information regarding one object which is
> visible in the image. However it is not possible to know the answer by
> checking the image. For example the question asks about the name of a
> building, art, street, mountain, ect which is presented in the image.
> However, by checking the image it’s impossible to answer. Because the image
> doesn’t show a popular landmark/object and also the name is not visible in
> the image.
>
>

>
> We want to create challenging questions about the input image. But in some
> cases it’s hard to create challenging questions (for example the input image
> is too simple). In these cases please just write “skip” instead of writing a
> question.
>
>
> Second question
>
>
> The second question should ask about a subtle detail presented in the image
> which we are able to easily provide a clear answer and the answer does not
> vary upon personal preferences or opinions. Please concentrate on minor
> details within the image that would be challenging to answer. For instance,
> if the image features a cat, a question about the cat’s color is
> straightforward. However, asking about the specific detail of the cat’s paw
> positioning could be more challenging (see the cat image below). Please also
> keep in mind that we should be able to clearly answer the challenging
> question by checking the image.
>
>
> We want to create challenging questions about the input image. But in some
> cases it’s hard to create challenging questions (for example the input image
> is too simple). In these cases please just write “skip” instead of writing a
> question.
>
>
> Answers
>
>
> For each question, please also provide three correct responses. Please
> format your responses as follows: "Answer 1; Answer 2; Answer 3". Make sure
> to separate each answer with a semicolon (;) and a space. Please see example
> responses below.

##  Appendix 0.B Auto-Eval Implementation Details

Table 4 of the main paper explored different implementations of Auto-Eval.
Each of the three implementations of Auto-Eval are detailed below. The first
implementation uses a simple text-only prompt (Figure 5). The second
implementation adds a basic Langfun schema that the model must populate
(Figure 6). The final implementation used throughout the paper also includes
additional schema attributes that prompt the model to reason deeply about the
main points of the response and ground truth before making a final decision
(Figure 7). This reasoning is demonstrated in Figure 8. The result is an Auto-
Eval system that has higher agreement with human raters than text-only
prompting or the basic schema. All implementations use Gemini Pro as the
underlying LLM.

⬇

\pardef compute_prediction(inputs):

_, question, prediction, groundtruth = inputs

\parr = lf.query(prompt="""Your task is to determine if the model response is
correct given the question and groundtruth response.

Ensure to interpret the model response in accordance to the the question.

\parIf the question asks about a detail of an element that is not present in
the image, A prediction of "yes", "no" or "nothing" should be considered
incorrect because it inaccurately suggests that the element is presented in
the image.

The correct prediction in such cases should acknowledge the absence of the
element in question by stating the element is not present.

\parIf prediction says that it can not assist or cannot provide an answer,
then the prediction is incorrect.

If the question is about counting, then the prediction is correct only it
matches the groundtruth counts exactly.

\parquestion={{question}},

\pargroundtruth={{groundtruth}},

\parprediction={{prediction}}

\parIs the prediction correct?

Respond with True or False without any extra information.

""",

question=question,

groundtruth=groundtruth,

prediction=prediction,

lm=gemini_pro)

return r

\par

Figure 5: Text-only prompting

⬇

\parclass PredictionEvaluation(pg.Object):

question: str

model_response: str

groundtruth_response: str

is_prediction_correct: bool

\pardef compute_prediction(inputs):

_, question, model_response, groundtruth_response = inputs

\parr = lf.query(prompt="""Your task is to determine if the model response is
correct given the question and groundtruth response.

Ensure to interpret the model response in accordance to the the question.

\parIf the question asks about a detail of an element that is not present in
the image, A prediction of "yes", "no" or "nothing" should be considered
incorrect because it inaccurately suggests that the element is presented in
the image.

The correct prediction in such cases should acknowledge the absence of the
element in question by stating the element is not present.

\parIf prediction says that it can not assist or cannot provide an answer,
then the prediction is incorrect.

If the question is about counting, then the prediction is correct only it
matches the groundtruth counts exactly.

\parquestion={{question}},

model_response={{model_response}}

groundtruth_response={{groundtruth_response}},

\par""",

schema=PredictionEvaluation,

question=question,

groundtruth_response=groundtruth_response,

model_response=model_response,

lm=gemini_pro)

return r.is_prediction_correct

\par

Figure 6: Basic Langfun schema

⬇

\parclass PredictionEvaluation(pg.Object):

question: str

model_response: str

groundtruth_response: str

sentence_interpret_model_response_main_point: str

sentence_interpret_groundtruth_response_main_point: str

is_prediction_correct: bool

\pardef compute_prediction(inputs):

_, question, model_response, groundtruth_response = inputs

\parr = lf.query(prompt="""Your task is to determine if the model response is
correct given the question and groundtruth response.

Ensure to interpret the model response in accordance to the the question.

\parIf the question asks about a detail of an element that is not present in
the image, A prediction of "yes", "no" or "nothing" should be considered
incorrect because it inaccurately suggests that the element is presented in
the image.

The correct prediction in such cases should acknowledge the absence of the
element in question by stating the element is not present.

\parIf prediction says that it can not assist or cannot provide an answer,
then the prediction is incorrect.

If the question is about counting, then the prediction is correct only it
matches the groundtruth counts exactly.

\parquestion={{question}},

model_response={{model_response}}

groundtruth_response={{groundtruth_response}},

\par""",

schema=PredictionEvaluation,

question=question,

groundtruth_response=groundtruth_response,

model_response=model_response,

lm=gemini_pro)

return r.is_prediction_correct

\par

Figure 7: Advanced Langfun schema

⬇

PredictionEvaluation(

question = ’What color is the lion in the photo?’,

model_response = "The image you’ve uploaded is an illustration, not a
photograph, and it features giraffes and birds in a golden hour scene of what
appears to be the African savanna. I do not see a lion in this particular
image. The dominant colors are shades of yellow, orange, and black, creating a
silhouette effect against the setting or rising sun. If there were a lion in
the image, it would likely be depicted in a similar silhouette form.",

groundtruth_response = ’There is no lion in the photo; There are no lions;
There is no lion,’,

sentence_interpret_model_response_main_point = ’There is no lion in the
image.’,

sentence_interpret_groundtruth_response_main_point = ’There is no lion in the
image.’,

is_prediction_correct = True

)

\parPredictionEvaluation(

question = ’Is the bee perched on the petals of the flower?’,

model_response = ’yes’,

groundtruth_response = ’There is no bee in the image.; There is no bee perched
on the petals of the flower.; The image does not include a bee.’,

sentence_interpret_model_response_main_point = ’The bee is perched on the
petals of the flower.’,

sentence_interpret_groundtruth_response_main_point = ’There is no bee in the
image.’,

is_prediction_correct = False

)

Figure 8: Example Auto-Eval outputs. The first example demonstrates how Auto-
Eval identifies the main point in the model response and ground truth to be
the absence of a lion in the image, which in turn leads to judging the
response as correct. The second example shows conflicting main points, and so
the response is accurately judged as being incorrect.

##  Appendix 0.C Finetuning Experiment Implementation

The results in Table 5 were obtained in the following way. For BLIP2, tuning
focuses solely on the Q-Former’s parameters to enhance question-answering
capabilities, while the image encoder and LLM remain unchanged [22, 62].
MiniGPT4 employs a Vision Transformer for image encoding and Vicuna for text
decoding, connected by a Q-Former [65, 11]. Its tuning targets a learnable
linear projection layer to align visual features with Vicuna’s embeddings,
improving visual-textual integration. In mPLUG-Owl, the tuning strategy
freezes the pre-trained visual encoder and abstractor, concentrating on
improving the text decoder (Vicuna) through low-rank adaptation [56]. This
enhances the model’s ability to process and interpret visual-text data. A
language generation loss is used to effectively minimize hallucination while
maintaining generalizability.

To align our fine-tuning process with established best practices, we adhere to
the methodologies outlined by [31, 65] for crafting fine-tuning instructions.
While both VQA v2 and HaloQuest fall within the domain of Visual Question
Answering tasks, they differ significantly in their answer formats. VQA v2
adopts a “closed-book” approach, limiting responses to a predefined list of
short answers that include both single words and phrases. Conversely,
HaloQuest permits free-form answers, embracing a more flexible response
format. This divergence necessitates the formulation of task-specific
instructions to optimize model performance during fine-tuning.

For the VQA v2 task, the instruction template provided to BLIP2 is structured
as follows:

        <Image> Question: {Question} Short Answer:

This template is designed to elicit concise, predefined responses, aligning
with VQA v2’s structured answer requirements.

In contrast, for the HaloQuest task, we modify the instruction template to
accommodate open-ended responses:

        <Image> Question: {Question} Answer:

This adjustment signals the model to generate elaborated and unrestricted
responses, catering to the open-ended nature of HaloQuest.

Similarly, for MiniGPT4 and mPLUG-Owl, we customize the prompts to align with
the task requirements of VQA v2 and HaloQuest. These tailored prompts are
designed to guide the models towards generating the expected form of answers,
whether they be concise answers for VQA v2 or more elaborate responses for
HaloQuest. Similarly, for the VQA v2 task, the instruction for MiniGPT4 and
mPLUG-Owl is as follows:

        <Image> Answer the question. Q: {Question}

Conversely, for the HaloQuest task, the prompt is adjusted to encourage
responses in either words or phrases:

        <Image> Answer the question in words or phrases. Q: {Question}

By tailoring the instructions to the specific needs of each task, we ensure
that the fine-tuning process enhances the relevance and accuracy of the
model’s outputs, effectively addressing the unique objectives and constraints
of VQA v2 and HaloQuest.


$\displaystyle\tilde{b}$ $\displaystyle\in(\sqrt{3}/2,\frac{\pi}{2}),$

the following conditions hold true.

$\displaystyle\cos(\tilde{b}),\text{ }\sin(\tilde{b}),$ $\displaystyle\text{
}\sin(2\tilde{b})>0,\quad\cos(2\tilde{b})<0$
$\displaystyle\cos(2\tilde{w}_{1}),\text{ }\sin(2\tilde{w}_{1}),$
$\displaystyle\text{ }\cos(\tilde{w}_{1})<0,\quad\sin(\tilde{w}_{1})>0.$
$\displaystyle\sin(\tilde{w}_{2}),\text{ }\cos(\tilde{w}_{2}),$
$\displaystyle\text{ }\sin(2\tilde{w}_{2})>0,\quad\cos(2\tilde{w}_{2})<0.$

We can then show that the terms involved in the update rule of $w_{2,1}$
(Equation 25) are positive, implying $-\eta\nabla w_{2,1}<-\Omega(\eta)$ up
until $\tilde{w}_{1}$ reaches $\mathcal{O}(\eta)$ close to $\frac{\pi}{2}$.
Similar argument can be given for the updates of $b_{2}$ and $w_{2,2}$
respectively.

Furthermore, the coordinates $w_{2,t}$ for $t\notin\\{1,2\\}$ get
$\mathcal{O}(\eta^{2})$ updates per step and stay $\mathcal{O}(\eta)$ small
for $\Theta(1/\eta)$ steps of training.

∎

###  E.2 Useful Lemmas

######  Lemma E.7.

$\displaystyle\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\text{
}\sin\left(\langle{\bm{w}},{\bm{x}}\rangle+b\right)x_{j}=\cos(b)\left(\prod_{i=1\to
d,i\neq j}\cos(w_{i})\right)\sin(w_{j}).$

###### Proof.

$\displaystyle\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\text{
}\sin\left(\langle{\bm{w}},{\bm{x}}\rangle+b\right)x_{j}$
$\displaystyle=\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\text{
}\sin\left(\sum_{i=1\to d,i\neq j}w_{i}x_{i}+w_{j}x_{j}+b\right)x_{j}$
$\displaystyle=\cos(b)\left(\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\sin\left(\sum_{i=1\to
d,i\neq
j}w_{i}x_{i}\right)\cdot\mathbb{E}_{x_{j}}\cos\left(w_{j}x_{j}\right)x_{j}\right)$
Equals $0$ as an odd function
$\displaystyle+\cos(b)\left(\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\cos\left(\sum_{i=1\to
d,i\neq
j}w_{i}x_{i}\right)\cdot\mathbb{E}_{x_{j}}\sin\left(w_{j}x_{j}\right)x_{j}\right)$
Equals $0$ as an odd function
$\displaystyle+\sin(b)\left(\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\cos\left(\sum_{i=1\to
d,i\neq
j}w_{i}x_{i}\right)\cdot\mathbb{E}_{x_{j}}\cos\left(w_{j}x_{j}\right)x_{j}\right)$
$\displaystyle-\sin(b)\left(\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\sin\left(\sum_{i=1\to
d,i\neq
j}w_{i}x_{i}\right)\cdot\mathbb{E}_{x_{j}}\sin\left(w_{j}x_{j}\right)x_{j}\right)$
Equals $0$ as an odd function
$\displaystyle=\cos(b)\left(\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\cos\left(\sum_{i=1\to
d,i\neq
j}w_{i}x_{i}\right)\cdot\mathbb{E}_{x_{j}}\sin\left(w_{j}x_{j}\right)x_{j}\right)$
$\displaystyle=\cos(b)\left(\prod_{i=1\to d,i\neq
j}\cos(w_{i})\right)\sin(w_{j}).$ Use lemmas ….

∎

######  Lemma E.8.

$\displaystyle\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\text{
}\cos\left(\langle{\bm{w}},{\bm{x}}\rangle+b\right)x_{j}=-\sin(b)\left(\prod_{i=1\to
d,i\neq j}\cos(w_{i})\right)\sin(w_{j}).$

###### Proof.

$\displaystyle\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\text{
}\cos\left(\langle{\bm{w}},{\bm{x}}\rangle+b\right)x_{j}$
$\displaystyle=\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\text{
}\cos\left(\frac{\pi}{2}-\langle{\bm{w}},{\bm{x}}\rangle-b\right)x_{j}$
$\displaystyle=\cos(\pi/2-b)\left(\prod_{i=1\to d,i\neq
j}\cos(-w_{i})\right)\sin(-w_{j})$ Using Lemma E.7
$\displaystyle=-\sin(b)\left(\prod_{i=1\to d,i\neq
j}\cos(w_{i})\right)\sin(w_{j}).$

∎

######  Corollary E.9.

Consider a set $S\subseteq[d]$.

$\displaystyle\mathbb{E}_{{\bm{x}}\sim\\{-1,+1\\}^{d}}\text{
}\cos\left(\langle{\bm{w}},{\bm{x}}\rangle+b\right)\prod_{j\in
S}x_{i}=c_{b}\left(\prod_{i=1\to d,i\notin
S}\cos(w_{i})\right)\left(\prod_{j\in S}\sin(w_{j})\right),$

where

$c_{b}=\begin{cases}-\sin(b),&\text{if }|S|=4t+1\text{ for some
}t\in\mathbb{N}\\\ -\cos(b),&\text{if }|S|=4t+2\text{ for some
}t\in\mathbb{N}\\\ \sin(b),&\text{if }|S|=4t+3\text{ for some
}t\in\mathbb{N}\\\ \cos(b),&\text{if }|S|=4t\text{ for some
}t\in\mathbb{N}.\\\ \end{cases}$

##  Appendix F Scaling

Figure 6: Ablation study on choices for RaPTr algorithm. We train BERT-base
with RaPTr for 100k steps with a 6-8-10-12 schedule (see Section 4). Left to
right: (a) Square-root scaling ($h_{sqrt}$) has better validation loss
compared with linear scaling and no scaling at training, especially in the
earlier stages. With _no scale_ , we scale layer’s output by $1/p_{i}$ during
inference (following [Huang et al., 2016]). (b) Different candidates for the
fixed set $\mathcal{I}$ are compared for RaPTr. We find that training with
first and last layers fixed helps faster training.

While working on subnetworks, it is important to appropriately rescale the
network. In particular, bypassing layer(s) in RaPTr can shift the input
distribution for layers compared to the full model. To mitigate this shift, we
scale the output of the layers to maintain the norms of the input to each
layer throughout training. The idea of scaling layer outputs has also been
explored in prior work [Huang et al., 2016, Fan et al., 2019].We use a
different square-root scaling mechanism that is motivated by analysis on a
randomly initialized Transformer architecture. We show that at initialization,
the norm of the intermediate outputs ${\bm{y}}^{(\ell)}$ scale as
$\sqrt{\ell}$.

######  Theorem F.1 (Square-root scaling of input norms).

At initialization, given any input sequence ${\bm{x}}_{1:N}$ and scalars
$\alpha_{1:L}$, w.h.p. the intermediate sequences of $F$ (Definition 2.1)
satisfy

$\displaystyle\left\|{\bm{y}}^{(\ell)}_{i}\right\|^{2}_{2}=\left\|{\bm{x}}_{i}\right\|^{2}_{2}+\Theta(\sum_{j=1}^{\ell}\alpha^{2}_{j}d),\quad\text{for
all }1\leq i\leq N,1\leq\ell\leq L.$

Inspired by this, we define the function $h_{sqrt}$ that takes in a sequence
of binary values $\zeta_{1:L}\in\\{0,1\\}^{L}$ and returns scalars
$\overline{\zeta}_{1:L}$ where $\overline{\zeta}_{i}$ is the scaling for
output of layer $i$. We find a scaling that satisfies the following two
conditions hold (a) $\sum_{i=1}^{j}\overline{\zeta_{i}}^{2}=j$ for all $j\leq
L$ (maintaining the norm), and (b) $\overline{\zeta_{i}}=0$ if $\zeta_{i}=0$
for all $i\leq L$ (maintaining layers in random subnetwork). Formally, for
each index $j$ with $\zeta_{j}=1$, it finds the index minimum $\bar{j}>j$ with
$\zeta_{\bar{j}}=1$ and sets $\overline{\zeta}_{j}=\sqrt{\bar{j}-j}.$ The
RaPTr algorithm with square root scaling is presented in Algorithm 1. In
Figure 6 we compare square root scaling with some other scaling methods and
find it to be slightly better.

###  F.1 Proofs

For simplicity, we present the results for transformers with single head in
the self-attention layer. Furthermore, for simplicity, we use $\sigma_{relu}$
activation in the MLPs.

Algorithm 3 Transformer Layer

0: 2 layer normalization layers $f^{LN}_{attn}$, $f^{LN}_{mlp}$ (F.2), an MLP
layer $f^{mlp}$ (F.4), and a softmax self-attention layer $f^{attn}$ (F.3),
input sequence $\\{{\bm{x}}_{n}\in\mathbb{R}^{d}\\}_{n=1}^{N}$.

1: Attention Layer normalization: returns
$\\{{\bm{y}}^{attnln}_{n}\\}_{n=1}^{N}$ with
${\bm{y}}^{attnln}_{n}=f^{LN}_{attn}({\bm{x}}_{n})$ for all $n\leq N$.

2: Softmax self-attention: returns
$\\{{\bm{y}}^{attn}_{n}\\}_{n=1}^{N}=f^{attn}(\\{{\bm{y}}^{attnln}_{n}\\}_{n=1}^{N})$.

3: Residual connection: returns $\\{{\bm{y}}^{attnblock}_{n}\\}_{n=1}^{N}$,
with ${\bm{y}}^{attnblock}_{n}={\bm{x}}_{n}+{\bm{y}}^{attn}_{n}$ for all
$n\leq N$.

4: MLP Layer normalization: returns
${\bm{y}}^{mlpln}_{n}=f^{LN}_{mlp}({\bm{y}}^{attnblock}_{n})$ for all $n\leq
N$.

5: MLP function: returns $\\{{\bm{y}}^{mlp}_{n}\\}_{n=1}^{N}$ with
${\bm{y}}^{mlp}_{n}=f^{mlp}({\bm{y}}^{mlpln}_{n})$ for all $n\leq N$.

6: Compute ${\bm{y}}_{n}={\bm{y}}^{mlp}_{n}+{\bm{y}}^{attnblock}_{n}$ for all
$n\leq N$.

7: Return $\\{{\bm{y}}_{n}\\}_{n=1}^{N}$

######  Definition F.2.

[Layer Normalization] Define a normalization function
$f:\mathbb{R}^{d}\to\mathbb{R}^{d}$ that performs
$f({\bm{x}})=({\bm{x}}-\mu)/\sigma$, where $\mu$ and $\sigma$ are the mean and
standard deviation of ${\bm{x}}$, respectively. Then, layer normalization
$f^{LN}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with parameters
$\mathbf{\gamma},{\bm{b}}\in\mathbb{R}^{d}$ takes as input
${\bm{x}}\in\mathbb{R}^{d}$ and outputs ${\bm{y}}\in\mathbb{R}^{d}$, which is
computed as
${\bm{z}}=f({\bm{x}}),{\bm{y}}=\mathbf{\gamma}\odot{\bm{z}}+{\bm{b}}.$

######  Definition F.3 (Softmax self-attention).

A self-attention layer $f^{attn}:\mathbb{R}^{N\times d}\to\mathbb{R}^{N\times
d}$ with parameters
$\\{{\bm{W}}_{Q},{\bm{W}}_{K},{\bm{W}}_{V},{\bm{C}}^{attn}\in\mathbb{R}^{d\times
d}\\}$ takes a sequence $\\{{\bm{x}}_{n}\\}_{n\leq N}$ and outputs a sequence
$\\{{\bm{y}}_{n}\\}_{n\leq N}$, such that

$\displaystyle{\bm{y}}_{n}={\bm{C}}^{attn}\sum_{\overline{n}=1}^{N}a_{n,\overline{n}}{\bm{v}}_{\overline{n}},$
$\displaystyle\text{with
}a_{n,\overline{n}}=\mathrm{softmax}({\bm{K}}{\bm{q}}_{n})_{\overline{n}},\quad{\bm{q}}_{n}={\bm{W}}_{Q}{\bm{x}}_{n},\quad{\bm{k}}_{n}={\bm{W}}_{K}{\bm{x}}_{n},\quad{\bm{v}}_{n}={\bm{W}}_{V}{\bm{x}}_{n},$

for all $n\leq N$, and ${\bm{K}}\in\mathbb{R}^{N\times N}$ defined with rows
$\\{{\bm{k}}_{n}\\}_{n=1}^{N}.$

######  Definition F.4 (MLP).

An MLP layer $f^{mlp}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with parameters
$\\{{\bm{W}}\in\mathbb{R}^{m\times d},{\bm{C}}^{mlp}\in\mathbb{R}^{m\times
d}\\}$ and activation $\sigma_{relu}$, takes an input
${\bm{x}}\in\mathbb{R}^{d}$ and outputs ${\bm{y}}\in\mathbb{R}^{d}$, such that

$\displaystyle{\bm{y}}={\bm{C}}^{mlp}\sigma_{relu}({\bm{W}}{\bm{x}}).$

######  Definition F.5 (Transformer layer).

A pre-layernorm transformer layer with three sub-layers; 2 layer normalization
layers $f^{LN}_{attn}$, $f^{LN}_{mlp}$ (F.2), an MLP layer $f^{mlp}$ (F.4),
and a softmax self-attention layer $f^{attn}$ (F.3); takes a sequence
$\\{{\bm{x}}_{n\leq N}$ and outputs a sequence $\\{{\bm{y}}_{n}\\}_{n\leq N}$
in four steps.

  1. 1.

First computes $\\{{\bm{y}}^{attnln}_{n}\\}_{n=1}^{N}$ using a layer
normalization, i.e. ${\bm{y}}^{attnln}_{n}=f^{LN}_{attn}({\bm{x}}_{n})$ for
all $n\leq N$.

  2. 2.

Then it runs softmax self-attention to get
$\\{{\bm{y}}^{attn}_{n}\\}_{n=1}^{N}=f^{attn}(\\{{\bm{y}}^{attnln}_{n}\\}_{n=1}^{N})$.

  3. 3.

The net output of the self-attention block is given by
$\\{{\bm{y}}^{attnblock}_{n}\\}_{n=1}^{N}$, with
${\bm{y}}^{attnblock}_{n}={\bm{x}}_{n}+{\bm{y}}^{attn}_{n}$ for all $n\leq N$.

  4. 4.

Before passing to the MLP function, it is passed through another layer
normalization, i.e.
${\bm{y}}^{mlpln}_{n}=f^{LN}_{mlp}({\bm{y}}^{attnblock}_{n})$ for all $n\leq
N$.

  5. 5.

MLP function then returns $\\{{\bm{y}}^{mlp}_{n}\\}_{n=1}^{N}$ with
${\bm{y}}^{mlp}_{n}=f^{mlp}({\bm{y}}^{mlpln}_{n})$ for all $n\leq N$.

  6. 6.

${\bm{y}}_{n}={\bm{y}}^{mlp}_{n}+{\bm{y}}^{attnblock}_{n}$ for all $n\leq N$.

######  Definition F.6 (Initialization of the weights in the transformer
layer).

The weights are initialized as follows:

$\displaystyle{\bm{C}}^{mlp},{\bm{C}}^{attn},{\bm{W}}_{Q},{\bm{W}}_{K},{\bm{W}}_{V}\sim\mathcal{N}(\mathbf{0},\frac{1}{\sqrt{d}}\mathbf{I}),$
$\displaystyle{\bm{W}}\sim\mathcal{N}(\mathbf{0},\frac{\sqrt{2}}{\sqrt{m}}\mathbf{I}).$

The parameters $\mathbf{\gamma},{\bm{b}}$ of the functions
$f^{LN}_{attn},f^{LN}_{mlp}$ have been initialized as $\mathbf{1}$ and
$\mathbf{0}$ respectively.

######  Lemma F.7 (Norm of the output of the MLP function, modification of
lemma 7.1 in [Allen-Zhu et al., 2019]).

For a given input sequence $\\{{\bm{y}}^{mlpln}_{n}\\}_{n\leq N}$, if
$\varepsilon\in(0,1]$, with probability at least $1-O(N)\cdot
e^{-\Omega\left(m\varepsilon^{2}\right)}$ over the randomness of
${\bm{C}}^{mlp},{\bm{W}}$, we have

$\forall
i\in[N]:\quad\left\|{\bm{y}}^{mlp}_{n}\right\|/\left\|{\bm{y}}^{mlpln}_{n}\right\|\in[1-\varepsilon,1+\varepsilon].$

######  Lemma F.8 (Norm of the output of the layer normalization layers).

Given any input sequence $\\{{\bm{x}}_{n}\\}_{n=1}^{N}$, under the assumption
for all $n\leq N$, ${\bm{x}}_{n}-\sum_{i=1}^{d}x_{n,i}$ isn’t identically
$\mathbf{0}$, we have

$\displaystyle\left\|{\bm{y}}^{attnln}_{n}\right\|=\sqrt{d}$

for all $n\leq N$.

Similar result holds for $\\{{\bm{y}}^{mlpln}_{n}\\}_{n=1}^{N}.$

######  Lemma F.9 (Norm of the output of the MLP block).

For a given input sequence $\\{{\bm{y}}^{attnblock}_{n}\\}_{n\leq N}$, with
probability at least $1-\mathcal{O}(1)$ over the randomness of
${\bm{C}}^{mlp},{\bm{W}}$, we have

$\displaystyle\left\|{\bm{y}}_{n}\right\|^{2}$
$\displaystyle=\left\|{\bm{y}}^{mlp}_{n}+{\bm{y}}^{attnblock}_{n}\right\|^{2}=\left\|{\bm{y}}^{attnblock}_{n}\right\|^{2}+d$
$\displaystyle+\mathcal{O}\left(\left\|{\bm{y}}^{attnblock}_{n}\right\|\log
N+(d+\left\|{\bm{y}}^{attnblock}_{n}\right\|)\frac{\log^{3/2}N}{\sqrt{m}}\right),$

for all $n\leq N.$

###### Proof.

Combining Lemmas F.8 and F.7, we have for the sequence
$\\{{\bm{y}}^{attnblock}_{n}\\}_{n\leq N}$,

$\displaystyle\left|\left\|{\bm{y}}^{mlp}_{n}\right\|/\sqrt{d}-1\right|\leq\mathcal{O}\left(\frac{\sqrt{\log
N}}{\sqrt{m}}\right),$

w.p. atleast $1-\mathcal{O}(1)$.

Furthermore, due to the randomness of ${\bm{C}}^{mlp}$, we can further show
that w.p. at least $1-\mathcal{O}(1)$,

$\displaystyle\left|\langle{\bm{y}}^{attnblock}_{n},{\bm{y}}^{mlp}_{n}\rangle\right|\leq\mathcal{O}(\left\|{\bm{y}}^{attnblock}_{n}\right\|\log
N),$

for all $n\leq N$. Combining the two results, we have

$\displaystyle\left\|{\bm{y}}_{n}\right\|^{2}$
$\displaystyle=\left\|{\bm{y}}^{mlp}_{n}+{\bm{y}}^{attnblock}_{n}\right\|^{2}=\left\|{\bm{y}}^{attnblock}_{n}\right\|^{2}+d$
$\displaystyle+\mathcal{O}\left(\left\|{\bm{y}}^{attnblock}_{n}\right\|\log
N+(d+\left\|{\bm{y}}^{attnblock}_{n}\right\|)\frac{\log^{3/2}N}{\sqrt{m}}\right).$

∎

######  Lemma F.10 (Norm of the output of the attention block).

For a given input sequence $\\{{\bm{x}}_{n}\\}_{n\leq N}$, if
$\varepsilon\in(0,1]$, with probability at least $1-O(1)$ over the randomness
of ${\bm{C}}^{attn},{\bm{W}}_{V}$, we have

$\forall
i\in[N]:\quad\left\|{\bm{x}}_{n}\right\|^{2}\leq\left\|{\bm{y}}^{attnblock}_{n}\right\|^{2}\leq\left\|{\bm{x}}_{n}\right\|^{2}+\left\|{\bm{x}}_{n}\right\|+d+\mathcal{O}(\frac{\sqrt{\log
N}}{\sqrt{d}}(d+\left\|{\bm{x}}_{n}\right\|)).$

###### Proof.

From the definitions of $\\{{\bm{y}}^{attnln}_{n}\\}$ and
$\\{{\bm{y}}^{attn}_{n}\\}$, we have

$\displaystyle{\bm{y}}^{attn}_{n}={\bm{C}}^{attn}{\bm{W}}_{V}\sum_{j\leq
N}a_{n,j}{\bm{y}}^{attnln}_{j}.$

Thus, we can use a proof similar to the proof of Lemma F.9 to argue that with
the randomness of ${\bm{C}}^{attn}$ and ${\bm{W}}_{V}$,

$\displaystyle\left|\frac{\left\|{\bm{y}}^{attn}_{n}\right\|}{\left\|\sum_{j\leq
N}a_{n,j}{\bm{y}}^{attnln}_{j}\right\|}-1\right|\leq\mathcal{O}\left(\frac{\sqrt{\log
N}}{\sqrt{d}}\right),$

for all $n\leq N$ w.p. atleast $1-\mathcal{O}(1).$

Furthermore, due to the randomness of ${\bm{C}}^{attn}$, we can further show
that w.p. atleast $1-\mathcal{O}(1)$,

$\displaystyle\left|\langle{\bm{y}}^{attn}_{n},{\bm{x}}_{n}\rangle\right|\leq\mathcal{O}(\left\|{\bm{x}}_{n}\right\|\log
N),$

for all $n\leq N$.

Using Cauchy-Schwarz inequality, we must have

$\displaystyle\left\|\sum_{j\leq
N}a_{n,j}{\bm{y}}^{attnln}_{j}\right\|\leq\max_{j\leq
N}\left\|{\bm{y}}^{attnln}_{j}\right\|=\sqrt{d}.$

Thus, combining the results, we have

$\displaystyle\left\|{\bm{y}}^{attnblock}_{n}\right\|^{2}$
$\displaystyle=\left\|{\bm{y}}^{attn}_{n}+{\bm{x}}_{n}\right\|^{2}\leq\left\|{\bm{x}}_{n}\right\|^{2}+d$
$\displaystyle+\mathcal{O}\left(\left\|{\bm{x}}_{n}\right\|\log
N+(d+\left\|{\bm{x}}_{n}\right\|)\frac{\log^{3/2}N}{\sqrt{d}}\right).$

∎

######  Lemma F.11 (Norm of the output of the transformer layer).

For a given input sequence $\\{{\bm{x}}_{n}\\}_{n\leq N}$, if
$\varepsilon\in(0,1]$, with probability at least $1-\mathcal{O}(1)$ over the
randomness of ${\bm{C}}^{attn},{\bm{W}}_{V},{\bm{C}}^{mlp}$, we have

$\forall
i\in[N]:\quad\left\|{\bm{x}}_{n}\right\|^{2}+d+\mathcal{O}(err)\leq\left\|{\bm{y}}_{n}\right\|^{2}\leq\left\|{\bm{x}}_{n}\right\|^{2}+2d+\mathcal{O}(err),$

where $err=\mathcal{O}\left(\left\|{\bm{x}}_{n}\right\|\log
N+(d+\left\|{\bm{x}}_{n}\right\|)\frac{\log^{3/2}N}{\sqrt{m}}\right)$.

See F.1

###### Proof.

This follows from the fact that the transformer architecture is a stack of
struturally identical $L$ transformer layers, and each transformer layer’s
output norm increases by $\Theta(d)$ compared to its input norm, as given by
Lemma F.11. ∎