# Aligning Large Language Models with Human Preferences
through Representation Engineering

Wenhao Liu, Xiaohua Wang11footnotemark: 1, Muling Wu, Tianlong Li, Changze Lv
Zixuan Ling, Jianhao Zhu, Cenyuan Zhang, Xiaoqing Zheng, Xuanjing Huang
School of Computer Science, Fudan University, Shanghai, China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
These authors contributed equally. Corresponding author.

###### Abstract

Aligning large language models (LLMs) with human preferences is crucial for
enhancing their utility in terms of helpfulness, truthfulness, safety,
harmlessness, and interestingness. Existing methods for achieving this
alignment often involves employing reinforcement learning from human feedback
(RLHF) to fine-tune LLMs based on human labels assessing the relative quality
of model responses. Nevertheless, RLHF is susceptible to instability during
fine-tuning and presents challenges in implementation. Drawing inspiration
from the emerging field of representation engineering (RepE), this study aims
to identify relevant representations for high-level human preferences embedded
in patterns of activity within an LLM, and achieve precise control of model
behavior by transforming its representations. This novel approach, denoted as
Representation Alignment from Human Feedback (RAHF), proves to be effective,
computationally efficient, and easy to implement. Extensive experiments
demonstrate the efficacy of RAHF in not only capturing but also manipulating
representations to align with a broad spectrum of human preferences or values,
rather than being confined to a singular concept or function (e.g. honesty or
bias). RAHF’s versatility in accommodating diverse human preferences shows its
potential for advancing LLM performance.

##  1 Introduction

While large language models (LLMs) learn broad-ranging world knowledge and a
degree of reasoning proficiency, precise control over their behavior proves
challenging due to the unsupervised nature of their pre-training Radford et
al. (2018, 2019); Brown et al. (2020); Bubeck et al. (2023); Touvron et al.
(2023). For each query, instruction-tuned LLMs Wei et al. (2021); Chung et al.
(2022); Touvron et al. (2023) exhibit the capacity to generate multiple
responses that are both semantically and syntactically coherent by some
sampling techniques. While such ability enables the models to provide
diversity that is essential for chat agents, some responses may contain
harmful, unethical, socially biased and negative, even illegal content
Srivastava et al. (2022); Thoppilan et al. (2022); Bubeck et al. (2023); Wang
et al. (2023). Although many responses remain within acceptable bounds,
preferences may exist among them, leading to a desire to prioritize specific
outputs over others.

Figure 1:  Illustration of different apporaches. (a) Reinforcement learning
from human feedback (RLHF); (b) Direct preference optimization (DPO); (c)
Hindsight instruction relabeling (HIR); (d) Representation alignment from
human feedback (RAHF).

Existing methods steer LLMs to align with human preferences often using
reinforcement learning (RL), with reinforcement learning from human feedback
(RLHF) emerging as the most successful one Ouyang et al. (2022). However, the
underlying learning algorithms exhibit a considerable degree of complexity,
sensitivity to hyperparameters, instability during training, and necessitate
additional training in the reward model and value network, leading to
substantial computational costs Yuan et al. (2023); Rafailov et al. (2023).

In addressing the aforementioned challenges posed by RL-based methods, several
computationally lightweight alternatives have been proposed to simplify the
human preference matching process. Two prominent paradigms among these
alternatives include contrastive learning Rafailov et al. (2023); Zhao et al.
(2023); Yuan et al. (2023) and Hindsight instruction relabeling (HIR) Zhang et
al. (2023); Liu et al. (2023). Contrastive learning-based methods optimize a
language model policy by increasing the relative probability of preferred
responses over dispreferred ones, while HIR methods transforms human feedback
into instructions by relabeling the original ones, indicating the relative
quality of provided responses. A common characteristic shared by these two
paradigms is their capability to align language models with human preferences
through reward-free fine-tuning.

However, the reward-free fine-tuning is vulnerable to the presence of noisy
data or incorrect labels in a training set comprising a collection of
preference-annotated response pairs. Instances of dull sentences or very brief
responses may appear repeatedly in such a training set, potentially
introducing bias into the models. The exclusion of such instances from the
training set renders it impossible for LLMs to glean insights of human
preferences expressed in these instances. In contrast, RL-based methods adopt
a different strategy, wherein a reward function is first extracted from a
dataset of response rankings, and then this reward function can be applied to
train a LLM, effectively mitigating the model’s direct exposure to noisy data
or incorrect labels within the dataset.

In this study, we aim to seek for a computationally lighter and reward-free
algorithm that can effectively harness human preference expressed in datasets
meanwhile safeguarding LLMs from the influence of noisy data. Inspired by the
recent advance in representation engineering Zou et al. (2023), we initially
locate relevant representations and activity patterns associated with high-
level human preferences within an LLM, and subsequently gain precise control
over its behavior by manipulating its internal representations. In the neural
architecture, network weights determine neural activity, neural activity
determines the networks’ output, and the networks’ output determines the
networks’ behavior.n Instead of focusing on neurons and their connections we
see aligning LLMs with human feedback as an outcome of representational
spaces, implemented by patterns of activity across populations of neurons. We
first identify the differences in model activities between preferred and
dispreferred stimuli, and then control over its behavior by leveraging the
identified differences in representations (see Figure 1). We introduce two
methods for controlling representations and demonstrate the efficacy of these
representation engineering (RepE) approaches in aligning LLMs with a broad
spectrum of human preferences through a collection of response pairs.

To validate the effectiveness of our approach in aligning with human
preferences, we conducted extensive comparative experiments on the generated
results. Our method outperformed other RL-free approaches in human evaluations
and automated metrics such as reward model scores, GPT-4 evaluations, and also
achieved results comparable to RLHF. Notably, the underlying algorithms
exhibit simplicity in implementation and straightforwardness in training.

##  2 Related Work

Tuning large language models to elicit desired responses and behavior from
their extensive knowledge and capabilities is essential in the development of
chat agents, such as ChatGPT Brown et al. (2020), LLaMA Touvron et al. (2023)
and GPT-4 Bubeck et al. (2023), characterized by safety, performance, and
controllability. The enlargement of the size of language models only does not
inherently enhance their ability to follow a user’s intent. For example, LLMs
may still generate outputs that are untruthful, toxic, or simply not helpful
to the user. Existing human preference alignment methods can be broadly
classified into three major categories: reinforcement learning Ouyang et al.
(2022); Ramamurthy et al. (2023), contrastive learning Rafailov et al. (2023);
Zhao et al. (2023); Yuan et al. (2023), and Hindsight instruction relabeling
Zhang et al. (2023); Liu et al. (2023).

Extensive research has devoted into the exploration of RL from human feedback
through ratings or rankings, spanning tasks from NL-to-SQL conversion Zhong et
al. (2017), machine translation Kreutzer et al. (2018), task-oriented dialogue
systems Su et al. (2019); Zhang et al. (2019); Takanobu et al. (2019),
summarization Stiennon et al. (2020), story-telling Ziegler et al. (2019) to
instruction-following Ouyang et al. (2022); Ramamurthy et al. (2023).
Typically, these methods involve the fitting of a reward model to a dataset of
human preferences, followed by the optimization of a LLM policy to generate
responses with high reward, using RL algorithms such as REINFORCE Williams
(1992) or proximal policy optimization Schulman et al. (2017). Despite the
attractiveness of leveraging human preferences that are easier to collect than
expert demonstrations, training LLMs with RL poses significant practical
challenges, which is attributed to the sensitivity of RL to hyperparameters
and the inherent instability during training.

The solutions based on Hindsight instruction relabeling Zhang et al. (2023);
Liu et al. (2023) and contrastive learning Rafailov et al. (2023); Zhao et al.
(2023); Yuan et al. (2023) have emerged as computationally efficient
alternatives to RL-based methods without explicit reward modeling. However,
these reward-free fine-tuning solutions are susceptible to noisy data or
incorrect labels within a training set. They exhibit performance lags compared
to models tuned with RL counterparts (see Section 4). Furthermore, the
question of whether LLMs trained with such fine-tuning methods can generalize
well to out-of-distribution queries remains unresolved when contrasted with
models incorporating an explicit reward model. RLHF method Ouyang et al.
(2022) offers a potential avenue for improvement by leveraging additional
unlabeled examples through labeling LLM generations with the learned reward
model.

To enhance transparency and controllability of neural networks, Zou et al.
(2023) introduced representation engineering (RepE) as a methodology, drawing
an analogy between understanding deep neural networks through representation
tomography and studying brains via neuroimaging techniques. Their work
demonstrated the efficacy of RepE in addressing diverse safety-related
challenges such as truthfulness, honesty, and hallucination. This study falls
in line with recent research findings and extends its application to aligning
LLMs with a wide spectrum of human preferences. Our study introduces two novel
methods to instruct LLMs on human preferences first, and then extract
differences in model activities between preferred and dispreferred stimuli.
These differences in activity patterns serve as a foundation for manipulating
the model’s behavior, leading to the generation of responses that better align
with human preferences. Due to the lightweight computational advantages of
parameter-efficient fine-tuning techniquesHoulsby et al. (2019); Lester et al.
(2021); Hu et al. (2021); Wu et al. (2023) , these techniques are utilized to
fit the disparity in activity patterns. In contrast to the approach adopted by
Zou et al. (2023), which relies on unlabeled or self-generated stimuli limited
to singular concepts or functions the meaning of which the models have already
“known”, our methods provide a more comprehensive alignment with diverse human
preferences.

Figure 2:  The procedure of RAHF. RAHF begins with the introduction of two
methods to instruct LLMs on human preferences. One approach involves training
a single LLM to discern the relative quality of responses (RAHF-SCIT), while
the other employs dual LLMs to model preferred and dispreferred responses
separately (RAHF-DualLLMs). Specifically, RAHF-SCIT takes preferred and
dispreferred instructions along with their corresponding responses as input
and conducts contrastive instruction tuning on a single model. RAHF-DualLLMs,
on the other hand, performs supervised training by taking preferred and
dispreferred responses into different models. Subsequently, we obtain activity
patterns by stimulating the model with different prompts. We consider the
differences between the two activity patterns as indicative of preferred
signals, and leverage these signals to finetune the final model with LoRA.

##  3 Method

We begin by instructing LLMs on human preferences with a set of preference-
annotated response pairs. Secondly, we collect the activity patterns of LLMs
when exposed to stimuli that are preferred or dispreferred. The differences in
these patterns serve as the foundation for manipulating LLMs, enabling them to
generate responses more closely aligned with human values. We introduce two
novel methods for instructing LLMs on human preferences and extracting their
activity patterns: one involving a single LLM (trained to discern the relative
quality of responses) and the other employing dual LLMs (“a good guy and a bad
guy”). Finally, we construct the final model by training a low-rank adapterHu
et al. (2021) to fit the disparity in activity patterns.

###  3.1 Instructing LLMs on Human Preferences

Zou et al. (2023) validated the effectiveness of activity patterns extracted
from alignment fine-tuned models, such as LLaMA-2-chat, in capturing concepts
like truthfulness and honesty. However, for non-aligned models, such as pre-
trained large language models or LLMs subjected to simple fine-tuning,
explicit indications about human preferences must be provided to elicit and
capture discernible activity patterns arising from stimuli preferences. This
enables the accumulation of diverse activities that can subsequently be
employed to calibrate LLMs in alignment with human preferences.

To attain this objective, we explored two different methods. The first
involves instructing a single LLM using preference-annotated response pairs,
employing contrasting instructions like Hindsight Zhang et al. (2023). The
second method performs fine-tuning one LLM (denoted as preferred model) based
on preferred responses, and fine-tuning another LLM (denoted as dispreferred
model) on dispreferred responses.

####  3.1.1 Instruction with a Single Model

Within the proposed framework, the Single LLM Method focuses on fine-tuning a
Single Large Language Model through Contrastive Instruction Tuning (SCIT). The
primary objective is to train the model to effectively discriminate between
preferred and dispreferred responses, thereby optimizing its alignment with
human preferences. In this approach, the training dataset is curated to
include pairs of both preferred and dispreferred instructions, along with
associated queries and their corresponding responses.

Inspired by HIR Zhang et al. (2023), for instructions linked to positive
preferences, the goal is to heighten the probability of generating preferred
responses while concurrently diminishing the probability of generating
dispreferred responses. Conversely, for instructions associated with negative
preferences, the objective is to elevate the probability of generating
dispreferred responses and reduce the probability of generating preferred
responses.

Formally, let $D$ represent the training dataset, with $q_{i}$ denoting the
query, $r_{i}$ representing the response, and $p_{i}$ indicating the
instruction (positive or negative). The fine-tuning of the LLM involves
maximizing the following objectives:

$\displaystyle\pi(\theta^{*})$
$\displaystyle=\arg\max_{\theta}\sum_{(p_{i},q_{i},r_{i})\in
D}\log\frac{\exp{(P^{+})}}{\exp{(P^{+})}+\exp{(P^{-})}}$ (1)

where $P^{+}$$=\pi(r_{i}\mid p_{i},q_{i};\theta)$, $P^{-}$$=\pi(r_{i}$$\mid
p_{i}^{*},q_{i};\theta)$ and $p_{i}^{*}$ denotes the opposite instruction,
ensuring a contrast between preferred and dispreferred cases.

Throughout the fine-tuning process, the LLM undergoes a learning phase to
differentiate between preferred and dispreferred responses, unveiling distinct
activity patterns linked to human preferences. The internal representations
acquired through exposure to diverse stimuli types facilitate subsequent
thorough analysis. This discriminative training within a single model enables
the discernment and prioritization of preferred responses, thereby achieving
the overarching objective of aligning with a broad spectrum of human
preferences. Following HIR, we introduced KL divergence constraints in both
preferred and dispreferred instruction tuning. This ensures that the model,
after fine-tuning, does not deviate significantly from its original state.
This constraint contributes to the stability of utilizing the fine-tuned
representations to manipulate the original model.

####  3.1.2 Preference Instruction with Dual Models

In Dual LLMs method, our aim is to train two LLMs with distinct tendencies:
one model is inclined to generate preferred responses, while the other tends
to produce dispreferred responses. To achieve this objective, we employ paired
preference data to conduct supervised fine-tuning of the LLMs. Specifically,
we use the preferred data from the preference pairs to train the “good” model
and the dispreferred data from the preference pairs to train the “bad” model.

Formally, consider the dataset $D$, which consists of input queries $q$ and
corresponding pairs of preferential responses: a preferred response $r_{h}$
and a dispreferred response $r_{l}$. We are now dividing $D$ into a preferred
dataset $D_{h}=\\{q,r_{h}\\}_{i}$ and a dispreferred dataset
$D_{l}=\\{q,r_{l}\\}_{i}$. Utilizing this data, we employ a supervised
learning approach (maximum likelihood) to fine-tune the LLMs, thereby
obtaining two models expressing preferences, denoted as $\pi_{h}$ and
$\pi_{l}$ respectively. The fine-tuning of these two LLMs is aimed at
maximizing the achievement of the following objectives:

$\pi_{h}(\theta^{*})=\arg\max_{\theta}\sum_{(q_{i},r_{i})\in
D_{h}}\log\pi(r_{i}\mid q_{i};\theta)$ (2)
$\pi_{l}(\theta^{*})=\arg\max_{\theta}\sum_{(q_{i},r_{i})\in
D_{l}}\log\pi(r_{i}\mid q_{i};\theta)$ (3)

Through this training process, the “good” model and “bad” model have
respectively learned the activity patterns associated with human-preferred and
dispreferred responses.

###  3.2 Collecting Activity Patterns

In the process of extracting activity patterns, we utilize stimulus pairs
<$p^{+},q,r$> and <$p^{-},q,r$> to elicit representations from the
intermediate layers of the model. For the Single Large Language Model through
Contrastive Instruction Tuning (SCIT), these pairs are inputted separately
into the same model to capture distinct activation patterns for preferred and
dispreferred responses. In the case of the Dual LLMs method, the inputs are
fed into their corresponding “good” and “bad” models, enabling the extraction
of activation patterns independently for each model. For both methodologies,
we calculate the discrepancy in activation patterns of stimulus pairs,
yielding a difference vector indicative of the direction of preferred activity
patterns. Note that diverse responses render the discrepancies in
representations at corresponding token positions unexplainable. Consequently,
our representation pairs are generated by various instructions within the same
response. Specifically, for preferred and dispreferred instructions, we
calculate the differences in hidden states for each token at corresponding
positions in the response.

The difference vector provides insights into the factors that impact the
alignment of the LLM with human preferences. Subsequently, we perturb the
model’s original representation by incorporating the difference vectors. This
perturbation serves to guide the model’s representation in the direction
aligned with human preferences.

###  3.3 Constructing Final Models

In this step, we leverage the collected activation pattern to train a target
model that is expected to align with human preference. To achieve this, we
draw inspiration from the approach of Zou et al. (2023) by employing a
specialized loss function and fine-tuning with Low-Rank Adapters (LoRA),
enabling the efficient incorporation of activation patterns into the model.

We consider the output of the LoRA matrix as a perturbation of the original
hidden layer states, aligning it with the difference vector. Specifically, we
employ Mean Squared Error (MSE) loss as the objective function:

$\mathcal{L}_{Align}=\left\|\left(A_{p,\pi_{LoRA},L^{t}}-(A_{p,\pi_{base},L^{t}}+\alpha
v_{l})\right)\right\|_{2}$ (4)

Here, $\alpha$ serves as a hyperparameter controlling the extent to which the
difference vector $v_{l}$ intervenes in the model integration process.
$A_{p,\pi_{LoRA},L^{t}}$ and $A_{p,\pi_{base},L^{t}}$ represent the activity
patterns of the target model equipped with and without LoRA, respectively.
$v_{l}$ is the extracted difference vector as outlined in Section 3.2. In the
case of SCIT, $v_{l}$ results from contrasting activity patterns induced by
stimulus pairs input to the “discriminative" model, while for the Dual LLM
Method, it is obtained by contrasting patterns resulting from inputting
stimulus pairs fed into the models playing“good guy" and “bad guy"
respectively.

##  4 Experiment

Following Rafailov et al. (2023), we mainly conducted experiments on single-
turn dialogue tasks. We extensively compared various RL-free alignment
approaches and RLHF, evaluating the results through human evaluation and
automated assessment. Additionally, we conducted comparative experiments with
the representation engineering method proposed by Zou et al. (2023), serving
as an ablation study to demonstrate the impact of our approach in capturing
human preferences.

###  4.1 Experimental Setups

Task In single-turn dialogue, we use Anthropic’s Helpful and Harmless
dataset111https://huggingface.co/datasets/Dahoas/full-hh-rlhfBai et al.
(2022), denoting human preference responses according to helpfulness and
harmlessness. Each example in the dataset contains a pair of dialogues between
a human and a language model, providing preferred and dispreferred responses
for each query. The training set was employed for the first step, Instructing
LLMs on human preferences, as well as for the construction of the final model.
The test set was utilized to evaluate the performance of the methods.
Specifically, prompts were used to instruct model generation, with the
preferred response serving as the reference for comparisons, as detailed in
Appendix A.

Base Model Ouyang et al. (2022) and Ramamurthy et al. (2023) utilized
supervised fine-tuning models as initial models in their application of
Proximal Policy Optimization (PPO). For fair comparison, we performed fine-
tuning on the LLaMA2-7B model Touvron et al. (2023) using the Alpaca dataset
Taori et al. (2023). We denote the resulting model after fine-tuning as
Alpaca. In our experiments, all the models were initialized with Alpaca and
further trained by the baseline methods and RAHF.

Method | Trained Reward Model | Three-Party Reward Model
---|---|---
Greedy | Sampling (Temperature) | Greedy | Sampling (Temperature)
$0.25$ | $0.5$ | $0.75$ | $1.0$ | $0.25$ | $0.5$ | $0.75$ | $1.0$
RLHF-PPO | $1.989$ | $1.993$ | $1.993$ | $1.991$ | $1.903$ | $1.112$ | $1.121$ | $1.129$ | $1.159$ | $1.092$
SFT | $0.982$ | $1.044$ | $1.136$ | $1.172$ | $1.128$ | $0.166$ | $0.162$ | $0.206$ | $0.305$ | $0.298$
HIR | $0.684$ | $0.789$ | $0.867$ | $0.727$ | $0.510$ | $-0.144$ | $-0.087$ | $-0.123$ | $-0.252$ | $-0.407$
DPO | $0.942$ | $1.172$ | $1.441$ | $1.584$ | $1.584$ | $0.551$ | $0.606$ | $0.698$ | $0.746$ | $0.733$
LORRA | $1.460$ | $1.403$ | $1.424$ | $1.440$ | $1.449$ | $0.566$ | $0.528$ | $0.558$ | $0.550$ | $0.629$
RAHF-SCIT | $1.102$ | $1.138$ | $1.096$ | $1.020$ | $0.882$ | $0.416$ | $0.441$ | $0.398$ | $0.359$ | $0.328$
RAHF-DualLLMs | $1.677$ | $1.669$ | $1.657$ | $1.583$ | $1.357$ | $0.909$ | $0.917$ | $0.911$ | $0.867$ | $0.771$
|  |  |  |  |  |  |  |  |  |

Table 1: The rewards achieved by different methods. In real-world
applications, the process of text generation often uses temperature sampling
strategy. We also report the rewards under various temperature settings to
provide a comprehensive evaluation of the results. Due to the instability
caused by directly controlling the representation layer, we set the repetition
penalty to $1.2$ when generating with RAHF.

###  4.2 Baselines

To evaluate our proposed approach, we conduct extensive comparisons with
existing alignment methods, including Reinforcement Learning from Human
Feedback (RLHF) and other alternative methods for preference alignment. These
experiments were specifically designed to assess the efficacy of our method in
aligning with human preferences.

SFT (on preferred responses) This baseline involves fine-tuning the language
model directly using the preferred responses from the dataset. The model is
trained to generate responses that align with the labeled preferred responses.

HIR Hindsight Instruction Relabeling (HIR) proposed by Zhang et al. (2023)
converts feedback to instruction by relabeling original instructions and
employs supervised training for enhanced alignment with human preferences. We
use HIR as a baseline to evaluate the advantages of RAHF over supervised fine-
tuning.

LORRA In order to assess the impact of learning from human feedback on the
model’s ability to capture activity patterns, we conduct ablation experiments.
Specifically, we compare our approach, which involves learning from both human
feedback and activity patterns, against a baseline that solely relies on
representation engineering (Zou et al., 2023). This baseline omits the step of
explicit preference learning and evaluates the model’s performance based on
the effectiveness of representation alignment alone. This ablation study
allows us to isolate and measure the contribution of learning human
preferences to the overall effectiveness of our proposed method.

DPO Direct Preference Optimization (Rafailov et al., 2023) directly optimizes
a language model to adhere to human preferences without using explicit reward
modeling or reinforcement learning. It has been proven to be an efficient and
straightforward alternative to RLHF.

RLHF-PPO For the RLHF baseline, we follow the common practice, as outlined by
Ouyang et al. (2022). We use human preference data to train a reward model and
then employ Proximal Policy Optimization (PPO) to optimize the model generated
by supervised fine-tuning.

We performed comprehensive evaluations, employing both automated metrics and
human assessments, on a subset of $200$ examples randomly extracted from the
test dataset to assess the efficacy of our methods and compare them with other
competing approaches. In the case of the RLHF-PPO baseline, the training set
was split into two parts, one dedicated to training the reward model and the
other for running the PPO algorithm. For the other baselines, the entire
training set was used for the training process. Further elaboration and
details regarding the re-implementation of the baseline methods can be found
in Appendix B.

###  4.3 Automatic Evaluation

To carry out evaluation automatically, we employed a reward model as proxies
for human preferences. Furthermore, we utilize GPT-4 OpenAI (2023) to assess
the quality of two responses, as recent research has indicated that GPT-4
exhibits exceptional performance in evaluating responses from chat assistants,
displaying a close correlation with human preferences.

####  4.3.1 Evaluation with Reward Model

We employed two reward models for evaluation: one is the reward model trained
for the RLHF baseline and the other is from a publicly available reward
model222https://huggingface.co/OpenAssistant/oasst-rm-2-pythia-6.9b-epoch-1
trained on the same dataset.

In Table 1, we report the rewards achieved by the baselines and our methods.
Even though RLHF-PPO achieves higher rewards, this accomplishment comes at the
cost of training complexity and instability. Therefore, our attention shifts
towards the examination of alternative approaches to reward-free alignment
methods. RAHF-DualLLMs surpass all baseline models, with the exception of
RLHF-PPO, across both reward models. This outcome suggests that representation
alignment with human feedback maintains greater consistency with the reword
models. Despite DPO achieving higher rewards at temperatures of $0.75$ and
$1$, we favor greedy decoding and prioritize results obtained at lower
temperatures. This preference stems from concerns about the instability
associated with high-temperature sampling.

Method |  Win Rate
---|---
RLHF-PPO |  $0.879$
SFT |  $0.638$
HIR |  $0.592$
DPO |  $0.653$
LORRA |  $0.830$
RAHF-SCIT |  $0.827$
RAHF-DualLLMs |  $0.869$
|
Table 2: Win rates against preferred response of Anthropic-HH one-turn
dialogue judged by GPT-4. All methods employ greedy decoding for generation.

####  4.3.2 Evaluation with GPT-4

When evaluating with GPT-4, we conducted comparisons from two perspectives.
Firstly, we took the preferred response from the dataset as the reference,
assessing the win rates of all methods against it. The win rates, as depicted
in Table 2, reveal that both of our proposed methods outperformed all the
competitors, except RLHF-PPO. Notably, RAHF-DualLLMs achieved results
comparable to RLHF-PPO, exhibiting only a marginal performance decrease.

Moreover, we utilize GPT-4 for evaluating the quality of responses generated
by ours and other competitors. This assessment is carried out via pairwise
comparisons, and the outcomes are denoted as $w_{i}$. In particular, when
evaluating all results with our method as the reference, we apply the
following formula to establish a standardized measure that facilitates the
interpretation of the results:

$w^{\prime}=\frac{{(1-w)}/{w}}{\sum_{k=1}^{4}({(1-w_{k})}/{w_{k}})+1}$ (5)

where $w$ represents the win rate against RAHF, and we set $w=0.5$ when RAHF
was compared to itself. The result of $k$-th baseline is denoted as $w_{k}$.
This standardizes the comparison results to the same scale. It allows us to
more effectively evaluate and compare the performance of all methods.

Figure 3: The results show win rates of RAHF-SCIT and RAHF-DualLLMs compared
to competitive methods under different sampling temperatures. The length
corresponding to each method represents the normalized win rate results.

As depicted in Figure 3, both of our proposed methods exhibit higher win rates
in comparison with RL-free methods. Despite SCIT’s subpar performance on the
proxy model, it outperforms baselines when evaluated using GPT-4. In contrast
to the reward model, we consider GPT-4’s evaluation more reliable, relying on
its robust performance and the reasoning it provides during evaluation.
Furthermore, while RAHF-DualLLMs perform well, RAHF-SCIT has lower hardware
requirements and costs, especially in terms of GPU utilization.

It has been reported that the assessment outcomes are susceptible to the
sequence in which diverse responses are presented to GPT-4 during the
evaluation process Zheng et al. (2023). In order to mitigate the potential
impact of positional bias, we conducted two distinct experiments for each
pairwise comparison, calculating the average performance to establish the
final results. The specific prompts used for evaluation with GPT-4 are
presented in the appendix.

###  4.4 Human Evaluation

For the human evaluation, we assigned evaluators the task of comparing two
randomly selected responses and providing judgments on their relative
performance, categorizing them with three results: win, lose, or tie.

Method | Win | Tie | Lose
---|---|---|---
| RAHF-SCIT
HIR | $0.665$ | $0.200$ | $0.135$
DPO | $0.510$ | $0.250$ | $0.240$
RLHF-PPO | $0.295$ | $0.310$ | $0.395$
| RAHF-DualLLMs
HIR | $0.625$ | $0.315$ | $0.060$
DPO | $0.500$ | $0.315$ | $0.185$
RLHF-PPO | $0.320$ | $0.325$ | $0.355$
|  |  |
Table 3: Win rates against baselines judged by Human. The data in the table
represents the proportion of RAHF relative to the baseline in terms of win,
tie, and lose. RAHF and baselines both utilize greedy decoding for generation.

Table 3 presents the comparative results of RAHF against RL-free methods and
RLHF in human evaluation. The results indicate that both of our proposed
methods outperformed other RL-free methods. This suggests that RAHF aligns
better with human preferences. The human evaluation results align broadly with
the GPT-4 evaluation results, with the only difference being that, compared to
GPT-4, humans tend to provide more tie judgments.

##  5 Conclusion

In this study, we explored representation engineering, approach to aligning
large language models with human preferences, drawing upon insights from
cognitive neuroscience. We have introduced RAHF (representation alignment from
human feedback), a straightforward paradigm designed for training language
models based on human preferences, eliminating the need for reinforcement
learning and reward models. RAHF effectively identifies disparities in the
activity patterns of LLMs between preferred and dispreferred stimuli, and
harnesses these distinctions to improve the controllability of LLMs. We hope
this study can inspire future research toward developing more controllable AI
and designing more efficient and scalable algorithms that could substantially
reduce the costs associated with training LLMs with human feedback through the
lens of representation engineering.

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##  Appendix A Prompts

###  A.1 Preference Instructions

Two instructions used in this study for preferred and disprefrred responses:

Preferred instruction

_You are a good, respectful and honest assistant. Your responses should be
helpful, meaningful, truthful, harmless, safe, ethical, socially unbiased,
positive, interesting, well-founded, agreeable, specific, reasonable, and role
consistent. If a question does not make any sense, or is not factually
coherent, explain why instead of answering something not correct. Please do
not share false information under any circumstances._

Dispreferred instruction

_You are a bad, disrespectful and dishonest assistant. Your responses might be
useless, meaningless, untruthful, harmful, dangerous, unethical, socially
biased, negative, boring, fabricated, offensive, vague, unreasonable, or role
inconsistent. If a question does not make any sense, or is not factually
coherent, still answer the question and share false information when
necessary._

###  A.2 GPT-4 prompt for computing dialogue win rates

Referring to the prompt used in Rafailov et al. (2023) to quire GPT-4, we
required GPT-4 to assess the responses generated by models from three
dimensions: helpfulness, truthfulness, and harmlessness. Table 9 presents the
specific prompt.

##  Appendix B Implementation Details

In this section, we present the experimental details and hyperparameters of
the baselines we compare with and our proposed methods. For all trained
models, we set the maximum query length to 256 and the maximum sentence length
to 512. We filter out samples in the dataset with queries exceeding 256 and
truncate sentences to the maximum sentence length.

All baselines and our models were trained using the Alpaca fine-tuned model as
the base model. During supervised training with Alpaca, we calculated the loss
for both prompts and responses. Specifically, we performed full parameter
fine-tuning for three epochs with a learning rate of $2e-5$.

RLHF-PPO During the training of RLHF-PPO, we utilized Microsoft’s DeepSpeed-
Chat training framework, making adaptive modifications to the hyperparameters.
We performed full-parameter fine-tuning for both the training of the reward
model and PPO. Table 4 presents the hyperparameters for reward model training,
while Table 5 presents the key parameters for PPO.

Hyperparameter |  Value
---|---
Learning Rate |  $9.65e-6$
Epochs |  $2$
Optimizer |  Adam
Training Batch Size |  $32$
Weight Decay |  $0.1$
Warmup Steps |  $0$
LR Scheduler Type |  cosine
|
Table 4: Hyperparameters used for the training of reward model. Hyperparameter |  Value
---|---
Actor Learning Rate |  $5e-7$
Critic Learning Rate |  $9e-6$
KL Coefficient |  $0.01$
Epochs |  $1$
Optimizer |  Adam
Training Batch Size |  $128$
Generation Batch Size |  $128$
Weight Decay |  $0.1$
Warmup Steps |  $10$
LR Scheduler Type |  Linear
Clip Reward Value |  $3$
Clip Range |  $0.2$
Clip Range Value |  $0.2$
Gamma |  $1$
Lam |  $0.95$
|
Table 5: Hyperparameters used for RLHF-PPO.

HIR For the HIR baseline, we also conducted full-parameter fine-tuning. Table
6 displays the hyperparameters used for HIR.

Hyperparameter |  Value
---|---
Learning Rate |  $2e-5$
Epochs |  $2$
Batch Size |  $128$
Micro Batch Size |  $4$
KL Coefficient |  $0.001$
Label Smoothing |  $0.2$
Entropy Coefficient |  $0.001$
|
Table 6: Hyperparameters used for HIR.

DPO We employed the trl framework from Hugging Face to train DPO model. we
utilized the "good" model from RAHF-DualLLMs, as the reference model for DPO.
We employed LoRA for fine-tuning. The hyperparameters used in the DPO training
are detailed in Table 7.

Hyperparameter |  Value
---|---
Learning Rate |  $2e-5$
Epochs |  $3$
Batch Size |  $64$
Micro Batch Size |  $4$
LoRA Rank |  $16$
LoRA Alpha |  $16$
LoRA Dropout |  $0.05$
Beta |  $0.1$
Warmup Ratio |  $0.1$
Optimizer |  Adam
|
Table 7: Hyperparameters used for DPO.

RAHF For RAHF-SCIT, we used the same hyperparameters as HIR during the first-
step training but omitted the supervised training loss. For RAHF-DualLLMs, the
hyperparameters used are shown in Table 8. When constructing the final model,
we followed the hyperparameter selection in RepE(Zou et al., 2023). We
manipulated layers (10, 20, 2) and set the perturbation coefficient $\alpha$
to 5. We used the batch size of 8 and trained with a learning rate of $3e-4$.

Hyperparameter |  Value
---|---
Learning Rate |  $2e-5$
Epochs |  $3$
Batch Size |  $128$
Micro Batch Size |  $4$
Optimizer |  Adamw
LoRA Rank |  $16$
LoRA Alpha |  $16$
LoRA Dropout |  $0.05$
Warmup Steps |  $100$
|
Table 8: Hyperparameters used for RAHF-DualLLMs.

##  Appendix C Human Evaluation Details

We recruited six volunteers for the assessment, with each evaluator comparing
200 dialogues. Figure 4 shows a screenshot of the interface used for our
evaluation, which all evaluators utilized to rate the data.

Figure 4: Screenshots of our evaluation interface for rating dialogue. In each
instance, evaluators are prompted to choose the preferred dialogue. For the
following query to a chatbot, which response is more helpful, truthful, and
harmless?
---
FIRST provide a one-sentence comparison of the two responses and explain which
you feel is more helpful, truthful, and harmless. SECOND, on a new line, state
only "A" or "B" to indicate which response is more helpful, truthful, and
harmless. Your response should use the format:
Comparison: <one-sentence comparison and explanation>
More helpful, truthful, and harmless : <"A" or "B">
Query:
{Query}
Response A:
{Response_A}
Response B:
{Response_B}
Table 9: GPT-4 Prompt for dialogue evaluation.

##  Appendix D Qualitative Examples

Table 10 and Table 11 present qualitative examples of RAHF compared with
baselines in dialogue tasks.

Prompt |  At what decibel level is sound harmful to your ears?
---|---
HIR |  The decibel level at which sound is harmful to your ears depends on the type of sound and the length of exposure. Generally speaking, sound levels above 85 decibels can cause hearing damage, and prolonged exposure to sound levels above 100 decibels can cause permanent hearing loss.
DPO |  According to the World Health Organization, sound levels above 85 decibels (dB) can be harmful to your hearing.
RLHF-PPO |  Sound is considered to be harmful to your ears when it reaches a decibel level of 85 decibels or higher. This level of sound can cause hearing damage over time, so it is important to protect your ears from loud noises. To protect your hearing, it is recommended to wear ear protection such as earplugs or noise-canceling headphones when exposed to loud noises. Additionally, it is important to limit the amount of time you are exposed to loud noises and to take breaks from loud activities.
RAHF-DualLLMs |  According to the World Health Organization, exposure to sounds at or above 85 dB can cause hearing damage. Exposure to noise levels of over 100 dB for more than a few minutes can lead to permanent hearing loss and other health problems such as tinnitus (ringing in the ears). It’s important to protect yourself from loud noises by wearing ear protection when exposed to high volumes of sound, limiting time spent listening to music with headphones, and avoiding activities that involve prolonged exposure to loud noise.
RAHF-SCIT |  The World Health Organization (WHO) recomm that exposure of human ears to sounds over 85 dB for a prolonged period can be potentially damaging. According to the WHO, any noise above 100dB may cause immediate pain and damage hearing permanently if exposed for more than one hour.
Table 10: Both RAHF-DualLLMs and RAHF-SCIT offer detailed information from a credible source, the World Health Organization (WHO), about when sound becomes harmful, including specifics on potential health problems. In contrast, RLHF-PPO and HIR provide general advice on sound levels, protection, and exposure but do not cite a specific source. Meanwhile, the information from DPO is relatively brief. Prompt |  When was the circus invented?
---|---
HIR |  The circus was first introduced in the 1700s, and has been a popular form of entertainment ever since. The first circus was created by Philip Astley in 1768, and the first circus in the United States was created by P.T. Barnum in 1835. The circus has evolved over the years, and now includes a wide variety of acts, including acrobatics, clowns, animals, and more.
DPO |  The circus was first developed in the late 1700s in Europe, and it quickly spread to the United States. The first circus in the United States was the American Circus, which was founded in 1793.
RLHF-PPO |  The circus was first introduced in the late 15th century in Europe, with the earliest known circus performance taking place in 1485 in Paris, France. The circus was originally a traveling show featuring acrobats, clowns, and other performers. It was not until the 19th century that the modern circus began to take shape, with the addition of animal acts, such as elephants, lions, and tigers. The modern circus as we know it today began to take shape in the late 19th century, with the introduction of the three-ring circus and the use of electric lighting. The circus has since become a beloved form of entertainment, with performances taking place all over the world.
RAHF-DualLLMs |  The earliest known records of a traveling circus date back to 1768, when Philip Astley opened an equestrian show in London. Since then, the concept has evolved and become increasingly popular around the world. Today, modern circuses feature acrobats, clowns, animals, and other entertainers performing spectacular feats for audiences. Circuses have been a beloved form of family-friendly entertainment since their introduction over two centuries ago.
RAHF-SCIT |  The modern form of the Circus as we know it today began in 1768 when Philip Astley opened a riding school and amphitheater for equestrian performances. It is believed that he created an arena with ropes to create obstacles, which inspired him to perform tricks on horseback. He later added acrobats, clowns, jugglers, tightrope walkers, trapeze artists, animal acts, and other entertainment elements.
Table 11: RAHF-DualLLMs and RAHF-SCIT both offer a factual and historically
accurate account, crediting Philip Astley with the inception of the modern
circus in 1768. In contrast, RLHF-PPO erroneously attributes the first circus
to the 15th century, presenting a misleading timeline of its evolution. The
assertion about P.T. Barnum founding the first circus in the United States in
1835, as mentioned in the second part of HIR, is not entirely accurate. DPO
correctly traces the circus’s European origins to the late 1700s and
acknowledges the early circuses in the United States. However, it falls short
by omitting specific details about Philip Astley, a key figure in the circus’s
genesis.


11institutetext: Institute of Mathematics and Image Computing, University of
Lübeck, Germany 22institutetext: Fraunhofer Institute for Digital Medicine
MEVIS, Lübeck, Germany
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Hessian Initialization Strategies for $\ell$-BFGS Solving Non-linear Inverse
Problems

Hari Om Aggrawal 11 Jan Modersitzki 1122

###### Abstract

$\ell$-BFGS is the state-of-the-art optimization method for many large scale
inverse problems. It has a small memory footprint and achieves superlinear
convergence. The method approximates Hessian based on an initial approximation
and an update rule that models current local curvature information. The
initial approximation greatly affects the scaling of a search direction and
the overall convergence of the method.

We propose a novel, simple, and effective way to initialize the Hessian.
Typically, the objective function is a sum of a data-fidelity term and a
regularizer. Often, the Hessian of the data-fidelity is computationally
challenging, but the regularizer’s Hessian is easy to compute. We replace the
Hessian of the data-fidelity with a scalar and keep the Hessian of the
regularizer to initialize the Hessian approximation at every iteration. The
scalar satisfies the secant equation in the sense of ordinary and total least
squares and geometric mean regression.

Our new strategy not only leads to faster convergence, but the quality of the
numerical solutions is generally superior to simple scaling based strategies.
Specifically, the proposed schemes based on ordinary least squares formulation
and geometric mean regression outperform the state-of-the-art schemes.

The implementation of our strategy requires only a small change of a standard
$\ell$-BFGS code. Our experiments on convex quadratic problems and non-convex
image registration problems confirm the effectiveness of the proposed
approach.

###### Keywords:

inverse problem, optimization, quasi-newton, $\ell$-BFGS, Hessian
initialization.

##  1 Introduction

Many real-life problems fit the framework of an inverse problem. Fluorescence
optical tomography [17], ultrasound tomography [2], and photoacoustic
tomography [18] are just a few non-invasive imaging techniques that image a
human body’s internal structure by solving inverse problems.

Inverse problems are typically ill-posed in nature [8]. The solution may not
be unique and unstable with variations in the data due to unavoidable factors
such as physical noise. Regularizing the problem with prior information, we
obtain a solution by minimizing an objective function

$J:\mathbb{R}^{n}\to\mathbb{R},\quad J(x)=D(x)+S(x),$

where $D$ denotes a data-fitting term and $S$ a regularizer. For many non-
linear problems, the objective function is non-convex, and the main limitation
is computationally demanding operations. Hence, an efficient optimization
method to be designed that requires fewer evaluations of an objective
function, its gradient and Hessian, and more occasional calls to a linear
solver.

Numerous optimization schemes exist to solve these problems. Still, schemes
that do not require more than first-order information are generally
preferable. Hessian computation is usually expensive.

Steepest-descent (SD) and quasi-newton methods are the most popular first-
order methods. SD converges only linearly; hence super-linear convergent
quasi-Newton methods such as Gauss-Newton (GN) schemes or the Broyden-class
are preferable. The quasi-Newton method’s key idea is to replace Hessian with
an approximation that models the local curvature information. It leads to not
only faster convergence but as well higher solution accuracy than simple
gradient descent methods; see, e.g. [17, 9] and references therein.

The Hessian approximation in the GN method is based on linearizing a function
that involves a matrix-vector product with a Jacobian matrix. For applications
such as optical tomography [17], the Jacobians are generally dense, and hence
per iteration costs can be very high. Therefore, Broyden-class methods are
preferred in practice. The most popular member is the limited-memory version
of BFGS scheme ($\ell$-BFGS) for large scale inverse problems; see its
application to recent work in ultrasound tomography [2] and image registration
[11].

The Broyden-class works with approximations of the Hessian that are based on
an initial approximation and an update rule that is typically based on current
curvature information derived from a secant equation (1). Setting $x=x_{k}$,
$x^{\prime}=x_{k+1}$, $p=x^{\prime}-x$ and introduce $y:=\nabla
J(x^{\prime})-\nabla J(x)$, a Taylor expansion $y=\nabla J(x^{\prime})-\nabla
J(x)\approx\nabla^{2}J(x)p$ motivates the so-called secant-equations [15,
Chapter 2] for Hessian $B$,

$B^{\prime}\>p=y\quad\mbox{or, for the inverse of $B$,}\quad p=H^{\prime}\>y.$
(1)

Based on an initial choice $H_{0}$, BFGS-schemes update the current
approximation $H=H_{k}$ using a constrained and weighted least squares fit,

$H_{k+1}\in\mathrm{argmin}\\{|M-H_{k}|_{F},\ M=M^{\top},\ \quad
My_{k}=p_{k}\\},$

where a weighted Frobenius-norm $|A|_{F}^{2}=\mathrm{trace}(WAWA^{\top})$ is
used. If the weight matrix satisfies secant equation $Wp=y$, one obtains a
unique and scale-invariant solution for $H_{k+1}$ as a rank two update of the
$H_{k}$,

$H_{k+1}=V_{k}^{\top}H_{k}V_{k}+\alpha_{k}\>p_{k}p_{k}^{\top},\quad\mbox{with}\quad\alpha_{k}:=(y_{k}^{\top}p_{k})^{-1},\quad
V_{k}:=I-\alpha\>y_{k}p_{k}^{\top}.$ (2)

For large-scale problems, a limited-memory version of BFGS ($\ell$-BFGS) is
used [12]. In $\ell$-BFGS, at most the last $\ell$ pairs are used. More
precisely, only pairs $(y_{j},p_{j})$ with $k_{\ell}:=\max\\{1,k-\ell-1\\}\leq
j\leq k$ are used. Formally, $H_{k+1}=M_{k+1}$ results from the modified
recursion

$M_{j+1}:=V_{j}^{\top}M_{j}V_{j}+\alpha_{j}\>p_{j}p_{j}^{\top},\quad
j=k_{\ell},\ldots,k.$ (3)

The convergence depends on the quality of Hessian approximation which
generally can not be controlled. It has been observed numerically that a
“good” initial guess of the Hessian greatly affects the scaling of a search
direction and convergence of the overall scheme [5, 7, 13, 1]. Note that
$\ell$-BFGS-method allows to re-initialize Hessian at every iteration. This
opportunity provides a window to rescale the search direction and infuse more
information in the scheme.

The state-of-the-art strategy initialize the Hessian (or it’s inverse) with a
scaled identity matrix

$H_{0}^{k}=\tau_{k}I.$

The scalar $\tau_{k}$ is computed at each iteration to satisfy the secant
equation (1) in a ordinary least square sense following the Oren–Luenberger
scaling strategy [16]. This results in two choices for scaling factor
$\tau_{k}$, _i.e._ ,

$\tau^{\mathrm{LSy}}_{k}=(y^{\top}p)/(y^{\top}y)\quad\mbox{and}\quad\tau^{\mathrm{LSp}}_{k}=(p^{\top}p)/(y^{\top}p).$
(4)

In practice, it has been observed that the factor $\tau_{k}^{\mathrm{LSy}}$
ensures a well-scaled search direction and as a result, most of the iterations
accept a steplength of one [7].

In this paper, we suggest to improve the quality of the initial Hessian
approximation by including computationally manageable parts from the
regularizer. More precisely, we suggest to use

$B_{0}^{k}=\tau_{k}I+\nabla^{2}S,$

where $\nabla^{2}S$ denotes the Hessian of a not necessarily quadratic
regularizer. We derive four options for scaling factor $\tau_{k}$ based on
ordinary and total-least squares formulations, and geometric mean regression.

Our work is motivated by ideas in image registration [14, 9] and molecular
energy minimization [10]. In [14, 9], the Hessian is initialized by a
positive-definite matrix $B_{0}=\tau I+A$, where $A$ is the Hessian of a
quadratic regularizer and constant. The parameter $\tau>0$ is chosen manually.
In [9], it is reported that this strategy outperforms the simple scaling
approach. In [10], the proposed strategy is similar to ours but requires
expensive incomplete Cholesky factorization at each iteration to ensure
positive-definiteness of Hessian approximation. Moreover, the value for $\tau$
is heuristically defined. But, in this paper, we show that it satisfies secant
equation in a sense of geometric mean regression.

We assume the regularization part to be computationally manageable. Typical
examples include $L_{2}$-norm based Tikhonov regularizers [14, 8], smooth
total-variation norm [19], or, more generally, quadratic forms of derivative
based regularization. Here, $R(x)=\|Bx\|_{L_{2}}$ and $B$ is a linear
differential operator. Non-quadratic forms such as the hyperelastic
regularizer [3] also fit into this class.

Our new strategy is easy to integrate into an $\ell$-BFGS code. Only the
Hessian initialization routine needs to be changed, all other parts remain
unchanged.

In this paper, we also demonstrate on various test cases that the proposed
approach achieves fast convergence and improves the solution accuracy compared
to the standard scaling based approaches. Our test cases include convex
quadratic problems and non-convex image registration problems with both,
quadratic and non-quadratic regularization. Due to the page limitation, the
theoretical investigation will be a part of the extended version of this
paper.

In Sec. 2, we derive four scaling factors for the proposed initialization
strategy and present a practical algorithm. We report the numerical
experiments with results in Sec. 3. In Sec. 4, we conclude our findings.

##  2 Proposed Hessian Initialization Strategy and Algorithm

As common for inverse problems, we assume that the objective function $J$ is a
sum of a data fitting term $D$ and a regularizer $S$. Hence

$\nabla^{2}J=\nabla^{2}D+\nabla^{2}S,$ (5)

where $A_{k}:=\nabla^{2}S(x_{k})$ is symmetric positive semidefinite (SPSD).
We assume that $A_{k}$ is “easy”, i.e. has low memory requirements and
$A_{k}x=b$ can be solved efficiently. Problems may occur from the data fitting
part $\nabla^{2}D(x_{k})$, which might be computational complex and
potentially ill-conditioned.

In the proposed strategy, we suggest to approximate
$\nabla^{2}D(x_{k})\approx\tau_{k}I$, where $\tau_{k}$ is a tuning parameter
to be determined. Hence

$B_{0}^{k}=\tau_{k}I+A_{k}.$ (6)

The role of $B_{0}^{k}$ is to mimic the Hessian at least for the current
update. In this regard, we aim to satisfy the secant equation (1) in some
least squares sense, where now

$y=B_{0}^{k}\>p=\tau_{k}\>p+A_{k}p\iff\tau_{k}p-z=0,\quad z:=y-A_{k}p.$ (7)

Since $B_{0}^{k}$ is required to be symmetric positive definite (SPD), we have
$\tau_{k}\geq\tau_{\min}:=\varepsilon-\mu_{\min}$, where $\varepsilon>0$ is a
small tolerance, typically $\varepsilon=10^{-6}$ and $\mu_{\min}$ is the
smallest eigenvalue of $A_{k}$. This adds a constraint to the least squares
problems. First we summarize the ordinary least square approach; cf. Lem. 1.
We removed subscript $k$ for clarity.

###### Lemma 1

Let $p,z\in\mathbb{R}^{n}$ with $p^{\top}z\neq 0$ and
$\tau_{\min}\in\mathbb{R}$. Then

$\tau^{\mathrm{Dp}}:=\max\\{(p^{\top}z)/(p^{\top}p),\tau_{\min}\\}\mbox{ and
}\tau^{\mathrm{Dz}}:=\max\\{(z^{\top}z)/(p^{\top}z),\tau_{\min}\\}$

are optimal scaling parameters resulting from a minimization of $|\xi u-v|$
subject to $\xi\geq\tau_{\min}$, where $(u,v)=(p,z)$ for (Dp) and
$(u,v)=(z,p)$ for (Dz).

Moreover, it holds $|\tau^{\mathrm{Dp}}|\leq|\tau^{\mathrm{Dz}}|$.

###### Proof

For (Dp): the unique minimizer $\xi$ of the unconstrained problem follows from
the basic calculus. If $\xi<\tau_{\min}$, the minimum is attained on the
boundary. For (Dz): the result follows from rescaling. The inequality follows
the Cauchy-Schwarz-inequality $|p^{\top}z|^{2}\leq(p^{\top}p)(z^{\top}z)$.

The above choices have a preference either for the $p$ or $z$ direction. A
total least squares approach can be used for an unbiased approach; cf. Lem. 2.

###### Lemma 2

Let $p,z\in\mathbb{R}^{n}$ with $\delta:=p^{\top}z\neq 0$,
$\tau_{\min}\in\mathbb{R}$, Then

$\tau^{\mathrm{Du}}:=\max\\{(|z|^{2}-\lambda)/\delta,\tau_{\min}\\},\quad\lambda=(|p|^{2}+|z|^{2}-\sqrt{(|p|^{2}-|z|^{2})^{2}+4\delta^{2}})/2,$

is an optimal scaling parameter from the rescaling of minimizer
$\eta=[\eta_{1},\eta_{2}]$ of the total least squares formulation
$|\eta_{1}p-\eta_{2}z|$ subject to $|\eta|=1$. With $\tau^{\mathrm{Dp}}$ and
$\tau^{\mathrm{Dz}}$ as in Lem. 1, it holds
$|\tau^{\mathrm{Dp}}|\leq|\tau^{\mathrm{Du}}|\leq|\tau^{\mathrm{Dz}}|$.

###### Proof

We have the necessary condition of first order $(U^{\top}U-\mu I)\eta=0$
subject to $|\eta|=1$, where $U=[p,-z]$ and $\mu$ denotes the Lagrange-
multiplier. This indicates that $\mu$ is the smallest eigenvalue of the
symmetric 2-by-2 matrix $U^{\top}U$ with diagonal elements $|p|^{2}$ and
$|z|^{2}$ and off diagonal $-\delta$. Hence, $\mu=\lambda$ and $\eta:=v/|v|$
is a normalized version of the associated eigenvector
$v=(\delta,|p|^{2}-\lambda)$. The value for $\tau^{\mathrm{Du}}$ follows from
proper scaling.

To show the inequality, we use the relationship
$(|z|^{2}-\lambda)/\delta=\delta/(|p|^{2}-\lambda)$ derived from
$\det(U^{\top}U-\mu I)=0$. Since $U^{\top}U$ is SPSD, we know $\lambda\geq 0$.
Hence, the inequalities $|(|z|^{2}-\lambda)/\delta|\leq|z|^{2}/|\delta|$ and
$|\delta|/|p|^{2}\leq|\delta/(|p|^{2}-\lambda)|$ satisfies. It leads to
$|\tau^{\mathrm{Dp}}|\leq|\tau^{\mathrm{Du}}|\leq|\tau^{\mathrm{Dz}}|$
following the definition of $\tau^{\mathrm{Dp}}$, $\tau^{\mathrm{Dz}}$, and
$\tau^{\mathrm{Du}}$.

Geometric mean regression is an another unbiased approach; see [6] for
details. The optimal scaling parameter is defined as the geometric mean of
scaling parameters obtained from ordinary least squares problems in Lem. 1,
_i.e._ ,

$\tau^{\mathrm{GM}}=\max\\{(\tau^{\mathrm{Dp}}\tau^{\mathrm{Dz}})^{1/2},\tau_{\min}\\}=\max\\{(|z|^{2}/|p|^{2})^{1/2},\tau_{\min}\\}$
(8)

and follows
$|\tau^{\mathrm{Dp}}|\leq\tau^{\mathrm{GM}}\leq|\tau^{\mathrm{Dz}}|$.

####  2.0.1 Remarks on scaling parameters:

Note that, the tuning of parameter $\tau$ changes both the angle and length of
search direction, whereas the simple scaling based schemes
$\tau^{\mathrm{LSy}}$ and $\tau^{\mathrm{LSp}}$ majorly changes the length of
the search direction.

To achieve fast convergence, we aim to reduce the number of iterations and the
line-search steps at every iteration. For that, we seek a search direction
that is closer to the Newton direction and take fewer iterations to
convergence. Furthermore, we seek well-scaled search directions that satisfy
steplength equal to one and avoid any line-search steps for reducing the total
run-time.

For a simple quadratic problem, we observe that the search directions with the
proposed choices for $\tau$ behave almost in a similar fashion with respect to
the Newton direction. Hence, all options are practically equivalent.

But, in practice, we observe that the length of a search direction is
inversely proportional to the value of $\tau$ for our schemes. Hence, a small
$\tau$ leads to a long step. Although it is desirable, but an overestimated
length leads to many line-search steps. On the other hand, with a large
$\tau$, we take small steps and, as results, require many iterations for
convergence; see results in Sec. 3. These facts suggest an optimal scaling
factor, but the exact criterion are so far unknown to us. Nevertheless, we
provide four choices for $\tau$ covering a wide range and describe their
inter-relationship in Lem. 2 and (8).

####  2.0.2 Practical algorithm:

Now, we are ready to present pseudo code for the standard $\ell$-BFGS
algorithm with the proposed Hessian initialization strategies where we
motivate to initialize Hessian $B_{0}^{k}$ at every iteration with (6); see
Algo. 1 [15]. In the standard $\ell$-BFGS code, we only need to change the
Hessian initialization routine; see Line 4 in Algo. 1; with a few lines of
code described in Algo. 2.

1 Initialize a starting guess $x_{0}$, integer $\ell>0$, and $\varepsilon>0$;

2 $k\longleftarrow 0$;

3 repeat

4 Compute $B_{0}^{k}$ following steps in Algo. 2;

5 Compute search direction $d_{k}\longleftarrow-H_{k}\nabla J(x_{k})$ using
$B_{0}^{k}$ in the two-step recursive algorithm based on (3); see details in
[12];

6 Compute $x_{k+1}\longleftarrow x_{k}+\alpha_{k}d_{k}$ where $\alpha_{k}$ is
obtained with a line-search algorithm;

7 if  _$k >\ell$_ then

8 Discard the vector pair {$p_{k-\ell}$,$y_{k-\ell}$} from the storage;

9

10 end if

11 Compute and save $p_{k}\longleftarrow x_{k+1}-x_{k}$ and $y_{k}=\nabla
J(x_{k+1})-\nabla J(x_{k})$;

12 $k\longleftarrow k+1$;

13

14until  _convergence_ ;

Algorithm 1 Standard $\ell$-BFGS algorithm with the proposed Hessian
initialization strategy

1 Compute $A_{k}$, Hessian of regularizer at $x_{k}$;

2 Set $\tau_{\min}\longleftarrow\varepsilon$;

3 if  _first iteration ( $k=0$)_ then

4 Set $\tau_{k}\longleftarrow\tau_{\min}$;

5

6else

7 Compute $z_{k}\longleftarrow y_{k}-A_{k}p_{k}$;

8 Set $\tau_{k}$ to either $\tau_{k}^{\mathrm{Dp}}$, $\tau_{k}^{\mathrm{Du}}$,
$\tau_{k}^{\mathrm{Dz}}$, or $\tau_{k}^{\mathrm{GM}}$;

9

10 end if

11Initialize $B_{0}^{k}\longleftarrow\tau_{k}I+A_{k}$;

Algorithm 2 Secant equation based Hessian initialization strategies

To initialize $B_{0}^{k}$, we start with setting $\tau_{\min}$, computing the
Hessian of regularizer, and evaluating $z_{k}$. The parameter $\tau_{k}$ can
be set with either $\tau_{k}^{\mathrm{Du}}$, $\tau_{k}^{\mathrm{Dp}}$,
$\tau_{k}^{\mathrm{Dz}}$, or $\tau_{k}^{\mathrm{GM}}$.

In the first iteration, we can not compute $z_{k}$ due to the lack of
information on the required iterates. Hence, initially, we set
$\tau=\varepsilon$ in our experiments. Other initialization options, _e.g._ ,
based on the norm of a gradient [15], are also possible.

Recall that the parameter $\tau_{k}$ should be greater than
$\tau_{\min}=\varepsilon-\mu_{\min}(A_{k})$ to ensure the positive-definitness
of Hessian $B_{0}^{k}$. To determine $\tau_{\min}$, we need the smallest
eigenvalue of $A_{k}$ that could be computationally expensive operation for
large scale problems. Hence, we avoid the eigenvalue computation in practice
and set $\tau_{\min}=\varepsilon=10^{-6}$ in our experiments.

##  3 Numerical Experiments and Results

We report on the performance of the proposed Hessian initialization strategies
for typical inverse problems: a) Strictly convex quadratic problems: This
class is chosen to validate the convergence properties of the proposed
strategy numerically. b) Non-convex image registration problems: This class is
chosen to show the effectiveness of the proposed strategy on a few challenging
real-world problems.

Table 1: Optimization methods used for evaluations. $A_{k}$ be the Hessian of regularizer. No. | Optimization methods | Hessian initialization | References
---|---|---|---
1. | Steepest descent (SD) | - | see [15]
2. | $\ell$-BFGS | $H_{k}^{0}=\tau_{k}I$ |  $\tau_{k}=I$, $\tau_{k}^{\mathrm{LSy}}$, or $\tau_{k}^{\mathrm{LSp}}$; see (4)
3. |  $\ell$-BFGS (FAIR scheme) | $B_{k}^{0}~{}=\tau I+A_{k}$ |  $\tau$ is set manually; see [14]
4. |  $\ell$-BFGS (proposed) | $B_{k}^{0}~{}=\tau_{k}I+A_{k}$ |  $\tau_{k}=\tau_{k}^{\mathrm{Dp}}$, $\tau_{k}^{\mathrm{Dz}}$, $\tau_{k}^{\mathrm{Du}}$, or $\tau_{k}^{\mathrm{GM}}$; see Lem. 1, Lem. 2, and (8)
5. | Gauss-Newton (GN) | - | see [15]

We investigate in total eight Hessian initialization strategies for
$\ell$-BFGS method; see Tab. 1 for details. Along with $\ell$-BFGS, we also
report results with Gauss-Newton (GN) and steepest descent (SD) method. GN is
a widely used method in the field of image registration. GN may achieve
quadratic convergence close to the solution. Even though, GN may converge to a
local optimal point in a few iterations, but for large scale problems, per
iteration cost for GN could be very high due to additional matrix-vector
products with Jacobian; see run-time for GN in Tab. 4 for image registration
problems. On the other hand, SD follows a linear and $\ell$-BFGS a superlinear
convergence.

Note that, the methods ${\mathrm{Dp}}$, ${\mathrm{Du}}$, ${\mathrm{Dz}}$,
${\mathrm{GM}}$, FAIR, and GN solve a linear system at each iteration to
compute the search direction. For that, we use Jacobi preconditioned conjugate
gradient (PCG) method. Moreover, the associated system matrices are not
stored, rather matrix-vector product has been computed directly. We run PCG
until the relative residual is less than $10^{-6}$ or the maximum iterations
reach to 100. The iteration count is set to low with the purpose of reducing
extra computational time at each iteration due to the linear solver.

The Hessian initialization in the FAIR [14] is similar to ours. But, they set
manually the parameter $\tau=10^{-3}c$, where $c$ is the first diagonal
element of $A$.

We use Armijo backtracking line-search algorithm to estimate the step-size. As
noted in [15], if curvature condition is not satisfied at any iteration, we
skip the Hessian update. For stopping criteria, we follow [14, p. 78] and set
$\varepsilon_{J}=10^{-5}$, $\varepsilon_{W}=10^{-1}$, and
$\varepsilon_{G}=10^{-2}$. For $\ell$-BFGS, we use the standard choice
$\ell=5$.

Run-time and solution accuracy are our main criteria to evaluate the
performance of optimization methods.

###  3.1 Quadratic problem

Table 2:  Optimization results for quadratic problem with eight Hessian
initialization strategies (S). Iteration counts and average line-searches (LS)
per iteration are mentioned for weakly ($\alpha=10^{-5}$), mildly
($\alpha=10^{-3}$), and strongly ($\alpha=10^{-1}$) regularized problems with
$\ell=1,5,10$, and $\infty$.

|  | $\alpha=10^{-5}$
---|---|---
| S./$\ell$ |  $1$ |  $5$ |  $10$ |  $\infty$
Iterations | Id |  5000 |  3858 |  3893 |  171
| LSp |  4108 |  908 |  383 |  37
| LSy |  2705 |  1460 |  870 |  90
| FAIR |  5000 |  869 |  389 |  18
| Dp |  4400 |  817 |  262 |  30
| Dz |  3716 |  1128 |  633 |  84
| Du |  5000 |  760 |  274 |  30
| GM |  2880 |  588 |  269 |  61

$\alpha=10^{-3}$
---
$1$ |  $5$ |  $10$ |  $\infty$
5000 |  754 |  628 |  128
539 |  145 |  65 |  28
558 |  230 |  115 |  39
168 |  93 |  31 |  15
565 |  79 |  47 |  20
564 |  228 |  163 |  50
578 |  88 |  36 |  20
356 |  125 |  56 |  30

$\alpha=10^{-1}$
---
$1$ |  $5$ |  $10$ |  $\infty$
567 |  137 |  97 |  47
79 |  37 |  31 |  23
69 |  39 |  30 |  23
18 |  7 |  7 |  7
23 |  12 |  10 |  10
49 |  27 |  19 |  17
23 |  11 |  10 |  10
30 |  13 |  11 |  11

avg. LS per iter. | Id |  1.00 |  1.00 |  1.00 |  1.00
---|---|---|---|---|---
| LSp |  2.73 |  7.05 |  8.13 |  1.16
| LSy |  1.15 |  1.10 |  1.11 |  1.00
| FAIR |  14.74 |  12.92 |  12.04 |  5.56
| Dp |  2.68 |  6.64 |  6.86 |  1.80
| Dz |  1.15 |  1.10 |  1.11 |  1.18
| Du |  2.71 |  6.49 |  7.55 |  1.80
| GM |  1.77 |  3.04 |  3.54 |  1.28

1.00 |  1.00 |  1.00 |  1.00
---|---|---|---
2.43 |  3.78 |  3.08 |  1.18
1.15 |  1.15 |  1.17 |  1.00
8.08 |  6.41 |  3.74 |  2.67
2.32 |  3.62 |  2.79 |  1.65
1.10 |  1.06 |  1.09 |  1.18
2.38 |  3.10 |  2.58 |  1.65
1.63 |  1.88 |  1.89 |  1.33

1.00 |  1.00 |  1.00 |  1.00
---|---|---|---
1.76 |  1.62 |  1.48 |  1.43
1.07 |  1.05 |  1.10 |  1.04
2.11 |  1.43 |  1.43 |  1.43
1.35 |  1.17 |  1.20 |  1.20
1.04 |  1.07 |  1.11 |  1.12
1.26 |  1.27 |  1.20 |  1.20
1.10 |  1.15 |  1.18 |  1.18

We minimize a strictly convex quadratic function $0.5(x-c)^{\top}(D+\alpha
R)(x-c)$ that has the unique minimizer at $x^{*}=c\in\mathbb{R}^{n}$ with $D$
and $R$ be a symmetric and positive-definite matrix and $\alpha>0$. In
experiments, $D$ is a diagonal matrix with exponentially decaying eigenvalues,
_i.e._ , $D_{ii}=\exp(-i)$. It is a highly ill-conditioned matrix with
condition number of order $10^{6}$ reflecting Hessian of a typical data-
fidelity term in inverse problems. The regularization matrix $R$ be a well-
known Laplacian matrix with zero boundary conditions. The regularization
parameter $\alpha$ controls the ill-conditioning of the quadratic function.
Here, we investigate weakly ($\alpha=10^{-5}$), mildly ($\alpha=10^{-3}$), and
strongly ($\alpha=10^{-1}$) regularized problems; see results in Tab. 2.

The iterations start with $x$ be a zero vector. The iterations stop when
either the relative error $\|x-x^{*}\|/\|x^{*}\|\leq 10^{-5}$ or the iteration
count reaches 5000.

As expected, the highly ill-conditioned problem, _i.e._ , weakly regularized,
requires many iterations for convergence. Especially, if the local curvature
is not well-estimated; see results for $\ell=1$ in Tab. 2. The iteration
counts are decreasing with increasing regularization levels and with improving
Hessian approximation that is increasing $\ell$. Note that this behavior is
consistent across all Hessian initialization strategies taken into
consideration in this work.

Identity initialized Hessian scheme converges very slow. But mostly, it
satisfies step-length equal to one, which means it takes tiny steps at each
iteration.

In most cases, the Hessian initialization schemes equipped with regularization
require fewer iterations than the simple scaling based ${\mathrm{LSy}}$ and
${\mathrm{LSp}}$ schemes. In particular, the ${\mathrm{FAIR}}$ scheme takes
the lowest iterations, but search-directions are badly scaled. Hence, line-
searches (LS) per iteration are much higher than the other schemes. Moreover,
LS steps highly depend on the regularization level.

But, for the proposed four schemes, the LS steps depend on the goodness of
Hessian approximation, _i.e._ , the value of $\ell$ rather than the
regularization level. In practice, we generally work with a fixed $\ell$ and
adjust the regularization level as per the need. Hence, the proposed scheme
suits better for such a scenario. In particular, the ${\mathrm{Dz}}$ scheme
generally take 1.15 LS steps per iteration and does not depend much on $\ell$.
The ${\mathrm{Dp}}$ and ${\mathrm{Du}}$ schemes require higher LS steps than
${\mathrm{Dz}}$ whereas the ${\mathrm{GM}}$ between the ${\mathrm{Dp}}$ and
the ${\mathrm{Dz}}$. In terms of iterations, we observe an almost inverse
relationship; _e.g._ , the ${\mathrm{Dp}}$ and ${\mathrm{Du}}$ scheme take
fewer iterations than ${\mathrm{Dz}}$; follow the discussion in Sec. 2 for the
underlying reason.

###  3.2 Image Registration

Table 3: Image registration test problems (TP) for the performance evaluation of optimization strategies. For TP-4, the initial TRE is not available (N.A.). TP | Dataset | Problem size |  Data- Fidelity | Regularizer | Parameters | Initial TRE
---|---|---|---|---|---|---
1 | Hand (2D) | $2\times 128\times 128$ | SSD | Curvature | $\alpha=1.5\times 10^{3}$ | 1.04 (0.62)
2 | Hand (2D) | $2\times 128\times 128$ | MI | Elastic | $\alpha=5\times 10^{-3}$ | 1.04 (0.62)
3 | Lung (3D) | $3\times 64\times 64\times 24$ | NGF | Curvature | $\alpha=10^{2}$ | 3.89 (2.78)
4 | Disc-C (2D) | $2\times 16\times 16$ | SSD | Hyperelastic | $\alpha=(100,20)$ | N.A.
Table 4:  Optimization results for four image registration test problems (TP).
The iteration counts (iter), the function evaluations (feval), the reduction
in objective function $\frac{J(\phi)}{J(\phi_{0})}$, the average run-time in
seconds, and the mean and standard deviation of TRE are reported. The gray-
colored cell denotes the $\ell$-BFGS method that achieve either the smallest
TRE (higher accuracy) or lowest run-time (faster convergence).

| TP-1: Hand, SSD, Curvature
---|---
M. |  iter |  feval |  $\frac{J(\phi)}{J(\phi_{0})}$ |  time |  TRE
|  |  |  |  (sec.) |  mean (std.)
SD |  999 |  1600 |  86.27 |  23.31 |  1.04 (0.62)
LSp |  1000 |  4157 |  28.07 |  40.46 |  0.67 (0.52)
LSy |  1000 |  1029 |  24.48 |  20.44 |  0.52 (0.29)
FAIR |  1000 |  1001 |  21.62 |  57.10 |  0.36 (0.18)
Dp |  53 |  59 |  21.94 |  8.96 |  0.38 (0.17)
Dz |  444 |  445 |  20.49 |  71.63 |  0.37 (0.16)
Du |  444 |  445 |  20.49 |  71.79 |  0.37 (0.16)
GM |  78 |  80 |  20.84 |  13.13 |  0.35 (0.17)
GN |  18 |  19 |  28.45 |  4.58 |  0.69 (0.59)

TP-2: Hand, MI, Elastic
---
iter |  feval |  $\frac{J(\phi)}{J(\phi_{0})}$ |  time |  TRE
|  |  |  (sec.) |  mean (std)
169 |  270 |  84.80 |  8.23 |  0.99 (0.63)
154 |  446 |  73.53 |  9.15 |  0.64 (0.37)
135 |  137 |  73.64 |  5.74 |  0.64 (0.36)
170 |  171 |  72.78 |  12.27 |  0.56 (0.30)
43 |  67 |  72.76 |  7.63 |  0.58 (0.34)
72 |  94 |  72.84 |  6.73 |  0.58 (0.32)
43 |  67 |  72.76 |  7.70 |  0.58 (0.34)
64 |  66 |  72.77 |  8.83 |  0.57 (0.31)
|  |  |  $\gg 360$ |

| TP-3: Lung, NGF, Curvature
---|---
SD |  145 |  251 |  97.17 |  145.83 |  3.64 (2.70)
LSp |  75 |  204 |  95.93 |  87.44 |  2.94 (2.20)
LSy |  184 |  185 |  94.84 |  163.00 |  1.70 (0.99)
FAIR |  127 |  128 |  94.67 |  170.31 |  1.59 (0.79)
Dp |  58 |  87 |  94.79 |  128.58 |  1.61 (0.83)
Dz |  131 |  132 |  94.79 |  161.70 |  1.64 (0.88)
Du |  148 |  149 |  94.74 |  183.07 |  1.61 (0.80)
GM |  77 |  79 |  94.76 |  123.45 |  1.62 (0.83)
GN |  68 |  254 |  94.90 |  2616.43 |  1.68 (0.99)

TP-4: Disc-C, SSD, Hyperelastic
---
999 |  1585 |  25.96 |  10.15 |
778 |  3520 |  16.90 |  11.33 |
671 |  829 |  16.93 |  6.03 |
214 |  1083 |  6.03 |  7.99 |
117 |  695 |  6.02 |  4.22 |
148 |  656 |  6.02 |  3.90 |
134 |  535 |  6.07 |  3.31 |
213 |  384 |  16.85 |  4.35 |
60 |  100 |  8.30 |  12.08 |

Now, we show effectiveness on four real-life large-scale problems from image
registration. The registration problems are generally highly non-convex and
ill-posed in nature; see [14] for details. Here, given a pair of images $T$
and $R$, the goal is to find a transformation field $\phi$ such that the
transformed image $T(\phi)$ is similar to $R$, _i.e._ , $T(\phi)\approx R$. To
determine $\phi$, we solve an unconstrained optimization problem

$J(\phi)=D(T(\phi),R)+\alpha S(\phi)\xrightarrow{\phi}\min$

where $D$ measures the similarity between the transformed image $T(\phi)$ and
$R$. The regularizer $S$ enforces smoothness in the field. Curvature, elastic,
and hyperelastic are a few commonly used regularizers. The typical choices for
similarity measures are the sum of squared difference (SSD), normalized
gradient fields (NGF), and mutual information (MI).

Our four test problems (TP) represent a big class of registration models; see
Tab. 3. The popular X-ray hand images are from [14], lung CT images from the
well-known DIR dataset [11, 4], and the academic Disc-C images from [3].

Note that our strategy works even when the Hessian of regularizer is available
only partially. For that, we consider hyperelastic regularizer; see [3] for
details.

The ground truth transformation fields are not available for real-world
problems. Hence, we compute the target registration error (TRE), defined as
the Euclidean distance between the ground truth landmarks and the estimated
landmarks after registration. To accumulate the TRE for each landmark
position, we compute the mean and standard deviation (std.) of TRE. The
regularization parameter is set to achieve the lowest TRE without foldings in
the field.

The field $\phi$ is initialized with an identity map, _i.e._ , $\phi_{0}(x)=x$
in all experiments. The open-source FAIR image registration toolbox [14] is
the backbone of our implementations. We follow FAIR matrix-free approach.

In all the experiments, the regularization-equipped initialization schemes
achieve higher accuracy than the simple scaling based approaches, _i.e._ ,
${\mathrm{LSp}}$, ${\mathrm{LSy}}$, and ${\mathrm{Id}}$. Moreover, these
simple scaling schemes converge to a higher value of the objective function;
see TRE and reduction factor column in Tab. 4.

In terms of TRE, the ${\mathrm{FAIR}}$ scheme is almost similar to the
proposed schemes, but it converges much slower than others; see the run-time
column in Tab. 4. The proposed schemes are faster than others in all the
experiments but TP-2. Here, ${\mathrm{LSy}}$ converges faster but at the cost
of lower accuracy. It is important to note that, even though the
regularization-equipped schemes’ per-iteration cost is higher due to the
linear solver, they converge faster. It is mainly because of the lower
iteration counts, as also seen for quadratic problems; see Tab. 2.

Among the four proposed choices, the ${\mathrm{Dp}}$ and the ${\mathrm{GM}}$
turn out to be the best performing schemes. Although, we notice that the
performance of a particular scheme greatly depends on the minimizing objective
function at hand.

As expected, the steepest descent method is one of the slowest and inaccurate
among all. The GN method generally needs fewer iterations, but the per-
iteration cost is much higher due to the Jacobian computation; hence the run-
time is high.

##  4 Conclusion

We have proposed a Hessian initialization strategy particularly suited for
large-scale non-linear inverse problems. Typically, the objective function is
the sum of a data-fidelity term and a regularizer. Often, the Hessian of the
data-fidelity is computationally expensive. But not the Hessian of the
regularizer.

We propose to replace the Hessian of the data-fidelity with a scalar and keep
the Hessian of regularizer to initialize the Hessian approximation at every
iteration. The scalar satisfies the well-known secant equation in the sense of
ordinary and total least squares, and geometric mean regression. In total, we
have proposed four choices for the scalar that leads to well-scaled search
directions. We also established the inter-relationship between the derived
scalars and discussed the consequences of a scalar choice on the convergence
in terms of iteration counts and line-search steps. The implementation of our
strategy requires only a small change of a standard $\ell$-BFGS code.

Our experiments on highly non-convex image registration problems indicate that
the proposed schemes converge faster and achieve higher accuracy than the
simple scaling based approaches. The ${\mathrm{Dp}}$, based on ordinary least
squares, and ${\mathrm{GM}}$, based on geometric mean regression, are best-
performing schemes.

Under suitable assumptions, we can also show that the proposed parameters are
the eigenvalue’s estimates of the Hessian of a data-fidelity term. The
theoretical investigation will be a part of the extended version of this
paper. Future work also addresses the application to inverse problems, _e.g._
, ultrasound tomography [2], and optical tomography [18].

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