# Energy Storage Price Arbitrage via
Opportunity Value Function Prediction

Ningkun Zheng1, Xiaoxiang Liu2, Bolun Xu1  1Earth and Environmental
Engineering, 2Computer Science
Columbia University
New York, New York, US
{nz2343,xl2948<EMAIL_ADDRESS>Yuanyuan Shi  Electrical and Computer
Engineering
University of California San Diego
San Diego, California, US
<EMAIL_ADDRESS>
###### Abstract

This paper proposes a novel energy storage price arbitrage algorithm combining
supervised learning with dynamic programming. The proposed approach uses a
neural network to directly predicts the opportunity cost at different energy
storage state-of-charge levels, and then input the predicted opportunity cost
into a model-based arbitrage control algorithm for optimal decisions. We
generate the historical optimal opportunity value function using price data
and a dynamic programming algorithm, then use it as the ground truth and
historical price as predictors to train the opportunity value function
prediction model. Our method achieves 65% to 90% profit compared to perfect
foresight in case studies using different energy storage models and price data
from New York State, which significantly outperforms existing model-based and
learning-based methods. While guaranteeing high profitability, the algorithm
is also light-weighted and can be trained and implemented with minimal
computational cost. Our results also show that the learned prediction model
has excellent transferability. The prediction model trained using price data
from one region also provides good arbitrage results when tested over other
regions.

###### Index Terms:

Energy storage; Deep learning; Power system economics.

##  I Introduction

With technological advances and a decarbonization target in 2030 [1], battery
energy storage systems are becoming the cornerstone flexibility resource in
future decarbonized power systems to balance electricity supply and demand [2,
3]. As progressively installed energy storage saturates ancillary service
markets [4], energy storage participants increasingly focus on arbitraging in
wholesale energy markets [5]. Energy storage arbitrage by charging during low
price periods and discharging during high price periods, earning revenues
while aiding power system operations based on price signals.

Real-time electricity prices are highly volatile and stochastic, and
participants must combine the energy storage physical constraints with the
price models to make arbitrage decisions. Model-predictive control (MPC) is
the most common approach for price arbitrage. MPC approaches [6, 7] follow a
two-step process that predicts the price time series first, followed by model-
based optimizations. The precise prediction of real-time price series is thus
critical. Various learning methods have been implemented to predict real-time
prices, including locally linear neuro-fuzzy model [8], support vector machine
[9], and residual neural network [10]. However, most of these methods focus on
price prediction over a short look-ahead window, such as one or two hours. The
prediction accuracy drops significantly at a longer prediction horizon, such
as 24 hours, to capture the daily price variations, which are essential for
energy storage arbitrage [11, 12, 13]. Solving iterative optimization problems
also proposes excessive computation costs, which could limit the potential of
MPC utensils in edge applications.

Alternative approaches for arbitrage include reinforcement learning (RL),
stochastic dynamic programming (SDP), and stochastic dual dynamic programming
(SDDP). SDP and RL approaches share similar challenges in the computation time
as it requires discretization of action, state, and prices to construct the
Markov-decision process (MDP). Especially, arbitrage in the real-time market
requires including finer states and actions in MDP due to shorter market
periods [14, 15]. Also, modeling non-perfect efficiencies in MDP is also
challenging as the state transition logic becomes much more sophisticated. In
contrast, SDDP does not discretize state and action spaces but requires
forward-backward iterations to generate dual cuts [16], and also requires
hand-tuning price distribution as Markov processes.

In this paper, we propose a novel approach that achieves state-of-the-art
arbitrage results while the training process is computationally and data
efficient. Our contribution includes:

  1. 1.

Motivated by dynamic programming, we combine supervised learning with model-
based optimization into a novel approach to directly predict the opportunity
cost function of energy storage state-of-charge (SoC) and further use it in
arbitrage control.

  2. 2.

Our approach significantly reduces the training load while providing reliable
arbitrage decisions which leverage costs and satisfy physical constraints. The
proposed model has stable performance with minimal hyper-parameters, and the
arbitrage profit surpass benchmarks with RL and SDP approach with
significantly less training time.

  3. 3.

We test the proposed algorithm using real price data from four New York ISO
price zones. The result shows the model achieved up to 90% of profit compared
to perfect foresight with varying durations and costs. The result also shows
our model has excellent transferability and still achieves great profit
results when trained and tested using price data from different locations.

The rest of the paper is organized as follows. Sections II and III present the
optimization problem, solution and training algorithm for energy storage
arbitrage. Section IV illustrate our algorithm with real data numerical
simulations. Section V concludes this paper.

##  II Problem Definition and Formulation

We consider a price response problem in which the energy storage can decide
whether to charge or discharge immediately after observing a new price signal.
Our approach is to predict the opportunity value (cost) at different energy
storage SoC levels and use the predicted values to optimize arbitrage
decisions using observed prices and the energy storage model. We will present
the arbitrage formulation and then define the learning problem to predict the
opportunity value in this section.

###  II-A Arbitrage Formulation with Opportunity Value Prediction

Motivated by dynamic programming [17], we formulate the arbitrage problem as
an iterative single-period maximizing problem

$\displaystyle\textstyle\max_{\begin{subarray}{c}b_{t},p_{t},e_{t}\\\
\in\mathcal{E}(e_{t-1})\end{subarray}}\lambda_{t}(p_{t}-b_{t})-cp_{t}+\hat{V}\big{(}e_{t}|\bm{\theta},\bm{x}\big{)}.$
(1a) where the first term represents the arbitrage revenue, which is the
product of the real-time market price $\lambda_{t}$ and the energy storage
dispatch decision $(p_{t}-b_{t})$, $p_{t}$ is the discharge output, and
$b_{t}$ is the charge output. The second term models operation and maintenance
costs to prevent frequent cycling, where $c$ is the marginal discharge cost.
The third term $\hat{V}\big{(}e_{t}|\bm{\theta},\bm{x}\big{)}$ is the
predicted energy storage opportunity value, which is a function of the final
SoC $e_{t}$ at the end of time period $t$. $\hat{V}$ depends on the prediction
model parameters $\bm{\theta}$ and the prediction features
$\bm{x}_{[t-1,t-2,\dotsc,t-W]}$ over a look-back period $W$.

We denote that the energy storage charge and discharge decisions and the final
SoC belong to a feasibility set $\mathcal{E}(e_{t-1})$, which is dependent on
the energy storage starting SoC $e_{t-1}$ at the start of time period $t$
(same as by the end of time period $t-1$). $\mathcal{E}(e_{t-1})$ is described
with the following constraints:

$\displaystyle 0\leq b_{t}\leq P,\;0\leq p_{t}\leq P$ (1b) $p_{t}=0$ if
$\lambda_{t}<0$ (1c) $\displaystyle
e_{t}-e_{t-1}=-p_{t}/\eta^{p}+b_{t}\eta^{b}$ (1d) $\displaystyle 0\leq
e_{t}\leq E$ (1e)

where (1b) models the upper and lower bound of charge and discharge output.
(1c) is a relaxed form of constraint that prevents simultaneous discharge and
charge, because the negative price is the necessary condition for energy
storage to charge and discharge simultaneously in price arbitrage. (1d) models
the energy storage SoC evolution constraint with discharge and charge
efficiency $\eta^{p},\eta^{b}$. (1e) models the energy storage SoC level upper
bound $E$ and lower bound (we assume as 0).

###  II-B Piece-wise Linear Opportunity Value Function Learning

The objective of the problem in this paper is to optimize the prediction model
parameters $\bm{\theta}$ that maximize energy storage arbitrage profit over a
set of training price data and energy storage physical parameters.
Intuitively, this problem could be formulated as a bi-level problem. The upper
level maximizes the total profit over the entire training time horizon. The
lower level enforces a non-anticipatory decision-making process in which the
energy storage dispatch decision only depends on the current price and the
predicted opportunity value as in (1). However, it will likely be
computationally intractable.

Problem Statement. In this paper, we consider an alternative two-stage
training approach in which we first generate the historical optimal
opportunity value function $v_{t}(e)$ for all SoC level $e$ constrained by
(1e) and then train the learning model to predict the opportunity value
function. Since we do not assume any particular functional form of the SoC
opportunity value function $v_{t}(e)$, we apply piece-wise linear
approximation in which we describe $v_{t}(e)$ using fixed SoC segments. Each
segment is associated with an opportunity value $V_{t}(e)$ corresponding to
the integral of $v_{t}(e)$. The first stage of the training process is thus to
calculate the historically optimal opportunity value function $v_{t}(e)$ using
the following equations

$\displaystyle V_{t-1}(e_{t-1})$
$\displaystyle=\textstyle\max_{\begin{subarray}{c}b_{t},p_{t},e_{t}\\\
\in\mathcal{E}(e_{t-1})\end{subarray}}\lambda_{t}(p_{t}-b_{t})-cp_{t}+V_{t}(e_{t})$
(2a) $\displaystyle v_{t}(e)$ $\displaystyle=\frac{\partial}{\partial
e}V_{t}(e)$ (2b)

Note that different from (1), (2a) is deterministic, and opportunity value
$V_{t}$ is calculated based on historical price data instead of predicted
using a learning model. The following section will further discuss the
generation of optimal opportunity value functions $v_{t}(e)$. The second stage
trains a learning model to predict the opportunity value function $v_{t}(e)$
using predictors which are past prices

$\displaystyle\min_{\theta}\sum_{e\in\mathcal{S}}\Big{|}\Big{|}\hat{v}\big{(}e|\bm{\theta},\bm{x}_{[t-1,t-2,\dotsc,t-W]}\big{)}-v_{t}(e)\Big{|}\Big{|}^{2}_{2}$
(2c)

The learning model thus fits a piece-wise linear approximation of the
historical optimal opportunity value function $v_{t}(e)$ based on the first-
order derivative of the optimal value $V_{t}$, and $e$ is from the set of SoC
segments $\mathcal{S}$. The prediction $\hat{v}$ is the predicted opportunity
value function, and its integral is the predicted opportunity value $\hat{V}$
in (1a).

We represent the energy storage opportunity value function $\hat{v}$ using
piece-wise linear approximation because it has no closed form but is a
monotonic decreasing derivative [17]. It is corresponding to the diminishing
value of energy stored, which is similar to a demand curve. We evenly
discretize $\hat{v}(e|\bm{\theta},\bm{x})$ into segments from the lowest SoC
level to the highest SoC level $E$. In each SoC segment we assume the
opportunity value function $\hat{v}$ is a constant function.

##  III Solution and Training Method

Our technical approach includes three efforts. First, we use a deterministic
price arbitrage dynamic programming model to generate the optimal opportunity
value function $v_{t}(e)$ based on historical price data. We then train a
learning model using the historical optimal opportunity value function to
predict the future optimal opportunity value functions. Finally, we test the
learned model over future price datasets.

We first solve the dynamic programming problem (2a) subject to constraints
(1b)–(1e). We utilize the algorithm in our prior work [18] to solve (2a)
analytically and obtain piece-wise linear approximations of the value
functions $v_{t}(e)$ for all time periods. With high computational efficiency,
we can discretize $v_{t}(e)$ by equally spaced energy storage SoC level $e$
into small segments, which is far smaller than the power rating $P$. For any
SoC level $e_{t}$, we can find the nearest segment and return the
corresponding value.

We train standard multilayer perceptron (MLP) networks using the Adam
algorithm to predict future opportunity value functions $\hat{v}_{t}(e)$ as
shown in Algorithm 1. We use day-ahead prices (DAPs) and real-time prices
(RTPs) with look-back window $W$ as predictors. Relying on Adam, we can
iteratively update $\bm{\theta}$ based on the gradient of (2c) with respect to
$\bm{x}$.

Algorithm 1 Value Function Prediction Model Training

1:Initialization: Set random seed. Set hyper-parameters in Adam (Default). Set
number of neurons based on input data and ReLu as activation function.
Initialize $\bm{\theta}$.

2:$i\leftarrow 0$

3:while not meet stop criteria do

4: for $t\in[1,t]$ do

5: $\bm{x}\leftarrow$ DAPs/RTPs with look-back window $W$

6: $v_{t}(e)\leftarrow$ historical opportunity value function at $t$

7: Update $\bm{\theta}$ by minimizing Eq. (2c) using Adam

8: end for

9: $i\leftarrow i+1$

10:end while

11:return $\bm{\theta}$$\triangleright$ Parameters of the prediction model

After training the value function prediction model, we use the algorithm 2 for
arbitrage control. We can calculate the opportunity value
$\hat{V}\big{(}e_{t}|\bm{\theta},\bm{x}\big{)}$ by integrating the value
function prediction $\hat{v}\big{(}e_{t}|\bm{\theta},\bm{x}\big{)}$. In our
implementation, we can solve (1) effortlessly using value function prediction
$\hat{v}$ and first-order conditions, since it is a single-period problem
[18].

Algorithm 2 Arbitrage with Value Function Prediction

1:Initialization:

2:Set energy storage parameters $c,P,\eta^{p},\eta^{b},E$.

3:Initialize $e_{t-1}\leftarrow e_{0}$.

4:for $t\in[1,T]$ do

5: Predict $\hat{v}\big{(}e_{t}|\bm{\theta},\bm{x}\big{)}$

6: Solve single-period optimization (1)

7: Return $e_{t},p_{t},d_{t}$

8:end for

##  IV Case Study

We pick four price zones in NYISO with different price patterns due to the
significant transmission congestion within New York State [19]. We use RTPs
and DAPs data from these zones with 5-minute and hourly resolutions,
respectively. In all test cases, we use price data in 2017 and 2018 to
calculate optimal opportunity value functions, then train the value function
prediction model, and test the trained model by predicting value functions
with price data in 2019 and performing arbitrage according to the control
algorithm 2. The example optimal value function at 50% SoC and corresponding
real-time price are shown in Fig 1. Compared to the optimal value function,
the real-time price has more sparse price spikes and negative prices and,
therefore, is hard to be predicted precisely by learning methods.

Figure 1: Example of historical price data and optimal value function at 50%
SoC in NYC.

We use normalized energy storage capacity $E$ of 1 MWh, identical charge and
discharge one-way efficiency $\eta$ of 90%, marginal discharge cost $c$ of
$10/MWh, and power rating $P$ of 0.5 MW (2-hour duration), except specific
stated. We generate the optimal value function for all time steps with 0.001
MWh granularity, thus 1001 SoC segments per function, and down-sample value
functions to 50 segments. The down-sampled optimal value function $v_{t}$ is
the ground truth of model training with (2c). We use four MLP model settings
with different input data look-back windows, model capacities, and stop
criteria. Settings are shown in Table. I. Setting 1 takes the RTPs from the
past three hours (36 price signals). Setting 2 extend RTPs look-back window to
24 hours (288 price signals). Whereas Setting 3 and 4 additionally include 24
hours DAPs as predictors. All settings have two hidden layer with same number
of neurons.

TABLE I: MLP training input features and layers settings

Setting | Num of RTP | Num of DAP | Num of Neurons | Num of Epochs
---|---|---|---|---
1 | 36 | 0 | 60 | 10
2 | 288 | 0 | 256 | 20
3 | 36 | 24 | 60 | 10
4 | 288 | 24 | 256 | 20

Computational costs with Google Colab (with GPU) are as follows111[Code
Available]: https://github.com/niklauskun/ES_NN_Arbitrage. For optimal value
function generation, two year time horizon with 5-minute resolution (210240
time periods) can be solved within half minutes. For the training process,
each epoch takes 8 seconds for settings Setting 1 and 3 and 12 seconds for
settings Setting 2 and 4. Testing with one year data takes around 10 to 12
seconds (105120 time periods).

###  IV-A Model Setting Comparison

We compare the performance of MLP model settings for all zones in this
subsection. First, we sequentially train 10 models with different random seeds
for each location and setting. Then we select models with the best training
arbitrage performance for testing arbitrage. Fig. 2 shows the positive
correlation between training profit and testing profit. We use the profit
ratio as the primary performance evaluation criteria to show how much profit
the energy storage captured compared to the best possible market profit with
perfect price prediction.

(a) NYC (b) LONGIL

(c) NORTH (d) WEST

Figure 2: Training and testing profit relationship using MLP Setting 3. TABLE
II: Profit ratio, revenue and annual discharge of different energy storage
duration and presumed marginal discharge cost ($/MW)

Zone | Duration | Prorated Profit Ratio [%] | Revenue [k$] / Annual Discharge [GWh] | Revenue per MWh discharged [$/MWh]
---|---|---|---|---
[Hours] | 0MC | 10MC | 30MC | 50MC | 0MC | 10MC | 30MC | 50MC | 0MC | 10MC | 30MC | 50MC
NYC | 1 | 75.20 | 72.19 | 73.30 | 76.12 | 22.3/0.97 | 19.2/0.33 | 15.1/0.13 | 13.2/0.08 | 23.05 | 58.71 | 117.65 | 167.70
2 | 75.40 | 75.39 | 76.58 | 80.38 | 26.2/1.12 | 22.9/0.40 | 17.8/0.16 | 15.4/0.10 | 23.42 | 56.68 | 108.77 | 155.86
4 | 81.59 | 80.02 | 82.73 | 85.97 | 32.9/1.48 | 28.3/0.56 | 21.7/0.22 | 18.2/0.13 | 22.19 | 50.17 | 97.57 | 143.04
LONGIL | 1 | 71.15 | 68.99 | 64.28 | 65.28 | 35.8/1.23 | 32.7/0.51 | 25.3/0.21 | 22.4/0.13 | 29.13 | 63.60 | 122.42 | 168.32
2 | 75.18 | 72.31 | 67.69 | 67.12 | 45.0/1.44 | 40.4/0.64 | 31.0/0.27 | 26.0/0.16 | 31.21 | 62.70 | 116.41 | 166.08
4 | 79.22 | 76.83 | 72.68 | 73.53 | 55.1/1.81 | 49.6/0.85 | 37.8/0.35 | 32.0/0.21 | 30.47 | 58.12 | 108.79 | 154.95
NORTH | 1 | 71.28 | 71.20 | 73.30 | 77.94 | 22.6/1.30 | 20.4/0.48 | 14.8/0.14 | 12.1/0.06 | 17.35 | 42.91 | 102.59 | 194.37
2 | 73.52 | 75.08 | 80.95 | 82.80 | 27.5/1.51 | 24.8/0.58 | 17.9/0.18 | 14.0/0.08 | 18.24 | 42.40 | 98.26 | 174.96
4 | 78.52 | 78.95 | 85.45 | 87.73 | 33.9/1.85 | 29.6/0.74 | 20.7/0.23 | 15.8/0.10 | 18.39 | 40.21 | 90.70 | 161.15
WEST | 1 | 80.44 | 81.09 | 79.88 | 81.18 | 44.4/1.34 | 42.0/0.66 | 34.3/0.32 | 30.6/0.21 | 33.17 | 63.33 | 107.59 | 145.82
2 | 81.96 | 83.87 | 83.73 | 86.00 | 51.3/1.55 | 48.8/0.79 | 40.5/0.40 | 34.7/0.24 | 33.16 | 61.47 | 102.01 | 142.25
4 | 85.16 | 85.82 | 86.74 | 88.03 | 60.3/1.89 | 56.3/0.98 | 46.5/0.49 | 38.2/0.28 | 31.87 | 57.23 | 94.66 | 135.78

The arbitrage performances using 2019 testing data are shown in Fig. 3.
Comparing Setting 1 and 2 shows that using a longer RTPs look-back window
slightly increases algorithm performance in most locations. Including DAPs as
training data significantly improve the performances in Setting 3 and 4. DAP
provides essential information for real-time opportunity value prediction.
However, a longer RTPs look-back window does not improve the performance of
models with DAP, as shown in Setting 3 outperforms Setting 4 in most zones, so
further tuning is needed in future work to understand the role of these extra
features. MLP model with Setting 3 has the best overall performance and lower
computational cost. Therefore, we use this model setting for other simulations
in this paper.

Figure 3: Performance of different MLP model settings

###  IV-B Benchmarking with SDP and RL

We further compare our method with an SDP benchmark and an RL benchmark with
Q-learning [14]. The SDP benchmark predicts opportunity value function using a
Markov process [18]. In RL, we have 11 actions, 103 price states, and 121 SoC
states, which take more than 1 hour to train for 5-min resolution arbitrage.
Although prediction model training is offline, lower computational cost eases
the model tuning process. Incorporating efficiency in SoC evolution
constraints in RL requires near-continuous SoC state space. Therefore, we
assume energy storage has 100% efficiency in this simulation. Fig. 4 shows
that our method significantly outperforms the RL benchmark with lower
computational cost in an example price zone. Depending on different price
zones, our proposed method improves profitability by 4-13% and 18-30% compared
to SDP and RL, respectively. The profit ratio compared to arbitrage with
perfect price prediction is shown in Fig. 3.

Figure 4: Cumulative profits comparison of perfect price prediction arbitrage,
RL, SDP, and our method in NYC. Energy storage setting: $E=1$ MWh, $P=0.5$ MW,
$\eta^{p}=\eta^{d}=100\%$, and $c=\$10$/MWh.

###  IV-C Model Transferability Comparison

We investigate model transferability using models trained using historical
price data in one location to predict value functions at other locations. Fig.
5 shows the profit capabilities of models trained in different locations.

Figure 5: Comparison of transferability for models trained at different
locations.

We find that the performance of value function prediction models trained from
LONGIL and WEST perform superior to those trained from NYC and NORTH. These
models are potentially more capable of predicting abnormal prices, which leads
to greater overall profitability. Models with better performance in one
location have good transferability and even outperform models trained by the
original location price data.

###  IV-D Sensitivity Analysis of Energy Storage Parameters

The effects of different energy storage parameters on 1 MW power rating
storage arbitrage results are shown in Table. 3. We implement training and
testing processes with three duration settings (1, 2, 4 hours) and four
marginal discharge cost settings (0, 10, 30, and 50) in all locations. The
arbitrage performance improves with a longer energy storage duration. With a
longer energy storage duration, the energy storage is more likely to have
spare energy capacity when a new charge/discharge opportunity happens.
Increased marginal discharge cost leads to conservative operational decisions
and higher unit revenue. In terms of zones, the storage achieved stable
performance around 75% to 90% profit ratio except in LONGIL. Revenue-wise, the
storage achieves the highest revenue in WEST followed by LONGIL, while NYC and
NORTH provide similar revenue around 50% of WEST.

##  V Conclusion

In this paper, we proposed a deep-learning framework for energy storage price
arbitrage. Instead of directly predicting real-time price, we use MLP to
predict value functions of energy storage SoC levels. The proposed method is
capable of capturing 65-90% of the maximum possible profit in NYISO real-time
market energy arbitrage, depending on energy storage characteristics and
locations, which is superior to previous model-based and learning-based
methods. While guaranteeing the algorithm’s performance with regard to
profitability, the algorithm is also light-weighted and can be trained and
implemented with minimal computational cost. In this work, we only implement
the standard MLP model for the learning process. The performance of the
proposed method can be further improved with advanced deep learning structures
and fine model tuning. The high computational efficiency of our algorithm also
enables designing strategical bids except for price response. Our future
research will use the proposed method for designing strategic bids and further
explore market designs for energy storage and renewable integration.

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# Unbiased Faster R-CNN for Single-source Domain Generalized Object Detection

Yajing Liu1,2,3, Shijun Zhou1,2,3, Xiyao Liu1,2, Chunhui Hao1,2, Baojie Fan4,
Jiandong Tian1,2
1State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese
Academy of Sciences
2Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of
Sciences
3University of Chinese Academy of Sciences 4Nanjing University of Posts and
Telecommunications
<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author

###### Abstract

Single-source domain generalization (SDG) for object detection is a
challenging yet essential task as the distribution bias of the unseen domain
degrades the algorithm performance significantly. However, existing methods
attempt to extract domain-invariant features, neglecting that the biased data
leads the network to learn biased features that are non-causal and poorly
generalizable. To this end, we propose an Unbiased Faster R-CNN (UFR) for
generalizable feature learning. Specifically, we formulate SDG in object
detection from a causal perspective and construct a Structural Causal Model
(SCM) to analyze the data bias and feature bias in the task, which are caused
by scene confounders and object attribute confounders. Based on the SCM, we
design a Global-Local Transformation module for data augmentation, which
effectively simulates domain diversity and mitigates the data bias.
Additionally, we introduce a Causal Attention Learning module that
incorporates a designed attention invariance loss to learn image-level
features that are robust to scene confounders. Moreover, we develop a Causal
Prototype Learning module with an explicit instance constraint and an implicit
prototype constraint, which further alleviates the negative impact of object
attribute confounders. Experimental results on five scenes demonstrate the
prominent generalization ability of our method, with an improvement of 3.9%
mAP on the Night-Clear scene.

##  1 Introduction

Figure 1: Comparison between vanilla Faster R-CNN (FR) [33] (top) and our
proposed Unbiased Faster R-CNN (bottom). For vanilla FR [33], the biased
distribution of the input data leads the network to learn biased features that
favor the seen environment and are poorly generalizable to unseen test
environments. The feature bias can be attributed to the image-level attention
bias and object-level prototype bias. Our method mitigates the data bias in
the input space and further learns unbiased attention and prototypes in the
representation space.

The problem of distribution shift in unseen domains often arises during the
deployment of perception systems, leading to a notable decline in the model’s
performance [3, 34]. Consequently, domain generalization [19, 37, 6, 50] has
emerged as a branch of transfer learning, which aims at generalizing the
knowledge from multiple source domains to the unseen target domain. Single-
source domain generalization (SDG) is a special case of domain generalization
where there is only one source domain [31, 43, 51] and it focuses on exploring
the robustness of the model under different image corruptions.

The existing two methods for SDG in object detection adopt different
generalization strategies on Faster R-CNN [33]. The domain-invariant feature
learning-based method [44] explicitly decomposes domain-invariant features and
domain-specific features by imposing constraints on the network, without
relying on data augmentation techniques. And the data augmentation-based
method [38] perturbs the data distribution and increases the diversity of
input data to enhance model generalization ability.

However, both of these strategies have certain limitations. Firstly, it has
been proven in [45] that domain-invariant features are inherently domain-
dependent and biased, as features that are invariant to current domains may be
variant to other domains. This problem can be attributed to the fact that
domain-invariant features learned from a biased data distribution are not
causal features and cannot adapt well to the unseen target environments.
Secondly, the domain generalization methods that use only data augmentation
without constraining network features fail to capture causal features from a
rich augmented data distribution. This results in the network performing well
in the augmented domains but remaining ineffective in the unseen domains, as
confirmed in our Ablation Study in Section 5.3. We summarize the reasons for
the above two limitations as the existence of data bias in the input space and
feature bias in the representation space. The feature bias is further
decomposed into image-level attention bias and object-level prototype bias, as
illustrated in Fig. 1 and Fig. 2:
(1) Data bias. The data distribution of unseen target domains is highly
changeable, as shown in Fig. 2. Therefore, learning invariant features solely
from single-domain data will exacerbate the statistical dependence [28]
between the input data and labels, leading to biased learned results that
favor the seen environments.
(2) Attention bias. When testing in unseen domains with complex context, as
depicted in Fig. 2, the context features may confuse the object features and
the network may exhibit attention bias, directing its focus towards the
context of the object rather than the object itself. We refer to the scene
context as scene confounders.
(3) Prototype bias. Each category possesses distinctive causal attributes,
such as structural information, that are highly informative, as opposed to
non-discriminative ones like view and color, which we refer to as object
attribute confounders, as shown in Fig. 2. Therefore, if the learned category
features are not constrained, the network may incorrectly take confounding
attribute features that occur frequently in the source domain as category
prototypes, which are biased and poorly generalizable.

Figure 2: Illustration of highly changeable data distribution, diverse context
and object attributes in unseen target domains.

To this end, we propose an Unbiased Faster R-CNN (UFR) model, as illustrated
in Fig. 1, with a comparison to vanilla Faster R-CNN [33]. The UFR model
simulates diverse data distributions through data augmentation to mitigate the
data bias and applies constraints on the features to learn scene-level causal
attention and object-level causal prototype, thus eliminating the attention
bias and prototype bias. Specifically, we define causality in the SDG task as
the cause-and-effect relationship among scene, objects, causal features and
non-causal features and construct a Structural Causal Model (SCM) to analyze
the data bias and feature bias caused by scene confounders and object
attribute confounders. Additionally, we propose a Global-Local Transformation
(GLT) module, which augments the data as a whole in the frequency domain and
leverages the segmentation capability of SAM [15] to augment local objects in
the spatial domain. To address scene confounders, we propose a Causal
Attention Learning module with an attention invariance loss, which transforms
the learning of scene-level causal features into the learning of causal
attention. To further learn causal features at the object level, we introduce
a Causal Prototype Learning module, which encompasses an explicit constraint
on the distribution of instance features and an implicit constraint on the
relationship between prototypes. The effectiveness of our method is
demonstrated through experiments conducted across five weather conditions. The
contributions are summarized as follows:
(1) We are the first to investigate single-domain generalized object detection
from a causal perspective and analyze three biases that limit the
generalization ability of the detector, including data bias, attention bias
and prototype bias.
(2) We construct a Structural Causal Model to analyze the biases caused by two
types of confounders and further propose an Unbiased Faster R-CNN with a
Global-Local Transformation module, a Causal Attention Learning module and a
Causal Prototype Learning module to mitigate the data bias, attention bias and
prototype bias respectively.
(3) We evaluate our method on five different weather conditions to demonstrate
its effectiveness and superiority and our method achieves a remarkable 3.9%
mAP improvement on the Night-Clear scene.

##  2 Related Works

###  2.1 Domain Generalization

Common strategies for addressing domain generalization problem include domain
alignment [19, 37, 42, 45], meta learning [6, 50, 2, 40], data augmentation
[16, 14, 38], ensemble learning [24, 36, 29], self-supervised learning [10, 8]
and disentanglement learning [39, 20, 49]. As a special case of domain
generalization, the solution for single-source domain generalization (SDG) can
also be categorized into several of the aforementioned strategies. Many prior
studies used data augmentation [31, 43, 38, 17] to generate out-of-
distribution samples to extend the distribution of the source domain. For
example, Wang et al. [43] proposed a style-complement module to generate
diverse stylized images. Vidit et al. [38] proposed a semantic augmentation
method for SDG in object detection with the help of a pre-trained CLIP [32].
In addition, some works employed the feature normalization strategy [26, 27,
12, 4] to learn domain-invariant features. For example, Fan et al. [7]
proposed an ASR-Norm layer to learn standardization and rescaling statistics.

###  2.2 Causality in Computer Vision

Causal mechanism considers the fact that statistical dependence cannot
reliably predict the labels of counterfactual inputs [28]. Thus, exploring
causality [1] enables the acquisition of robust knowledge beyond what is
supported by the observed data [35]. Recently, causal mechanism has gathered
significant attention in computer vision [48, 46, 11, 25]. Many works use
causal mechanism to tackle domain generalization [41, 45, 18, 22, 30] and
domain adaptation [23, 47, 13, 21] problems. For example, Yue et al. [47]
applied the causal mechanism to domain adaptation and used the domain-
invariant disentanglement to identify confounders. Lv et al. [22] proposed a
representation learning framework for causality-inspired domain
generalization. Besides, Liu et al. [21] proposed a Decoupled Unbiased Teacher
to solve the source-free domain adaptation problem. Moreover, Xu et al. [45]
introduced a causality-inspired data-augmentation strategy and eliminated non-
causal factors by a Multi-view Adversarial Discriminator. In this paper, we
first apply the causal mechanism to the single-source domain generalized
object detection task and propose an Unbiased Faster R-CNN to mine causal
features at the image level and object level.

##  3 Structural Causal Model

Consider the data (e.g. images) from the observed environment (source domain)
as $X$ and its target (e.g. detection labels) as $Y$, the objective of single-
domain generalized object detection is to generalize the model trained with
$(X,Y)$ in the observed environment to unseen environments (target domains).
We represent images from a causal perspective and construct a Structural
Causal Model (SCM) in Fig. 3 to describe the cause-and-effect relationship in
object detection task and attempt to eliminate the data bias and feature bias
caused by the scene confounders and object attribute confounders. The
rationale behind the SCM is as follows:
(1) $\bm{O}\rightarrow\bm{X}\leftarrow\bm{D}$ denotes that an image is
composed of scene $D$ and a set of objects $O$. $D$ is scene confounder, and
the non-discriminative attributes of $O$ are object attribute confounders. The
GLT module is achieved by perturbing the scene confounders and object
attribute confounders.
(2) $\bm{O}\rightarrow\bm{Z_{V}}\leftarrow\bm{D}$ denotes that the non-causal
features $Z_{V}$ are derived from two components, including the scene
confounders and non-causal object attribute confounders.
(3) $\bm{O}\rightarrow\bm{Z_{C}}\rightarrow\bm{Y}$ represents that the causal
features $Z_{C}$ are determined by the discriminative attributes of objects,
such as shape. And the prediction labels $Y$ are derived from the causal
features $Z_{C}$. The complete feature space $Z$ consists of non-causal
features $Z_{V}$ and causal features $Z_{C}$.
(4) $\bm{X}\rightarrow\bm{f_{\phi}(X)}\rightarrow\bm{Y}$ models the data
stream of the object detection network parameterized by $\phi$.

Figure 3: The constructed Structural Causal Model (SCM) for the object
detection task. The nodes denote variables, the solid arrows denote the direct
causal effect and the dashed arrow indicates that there exists data
dependence. Figure 4: The overall structure of the proposed Unbiased Faster
R-CNN. The input source images are augmented through the Global-Local
Transformation (GLT) module. Both the original images and augmented images are
fed into the network for training. During training, the role of the Causal
Attention Learning module is to constrain the network to learn scene-level
causal attention and select causal features to feed into the RPN. The purpose
of the Causal Prototype Learning module is to constrain the network to learn
object-level causal features with the help of an explicit instance constraint
(solid arrows) and an implicit prototype constraint (dashed arrows).

Based on the constructed SCM, the ideal feature mapping of the network is
$f_{\phi}(X)=Z_{C}$, which is also formulated as:

$f_{\phi}(X^{(a_{l},d_{l})})=f_{\phi}(X^{(a_{k},d_{k})}),$ (1)

where $a_{l}\neq a_{k}$, $d_{l}\neq d_{k}$ and the superscript $(a_{l},d_{l})$
denotes assigning scene and object attributes the distribution $d_{l}$ and
$a_{l}$, respectively. Eq. (1) indicates that the ideal representation of an
image learned by the model is invariant under different data augmentations.

To achieve the above goal, we decompose the objective into image-level
attention invariance learning and object-level prototype invariance learning
and further propose an Unbiased Faster R-CNN model, as depicted in Fig. 4, and
the details of which are illustrated in Section 4.

##  4 Unbiased Faster R-CNN

###  4.1 Global-Local Transformation

The complete process of Global-Local Transformation is depicted in Fig. 5,
which consists of Global Transformation (GT) in the frequency domain and Local
Transformation (LT) in the spatial domain.

Inspired by the frequency-domain augmentation in [45], the global
transformation is formulated as:

$\text{GT}(x)=\mathcal{F^{\prime}}(G(H(r)\cdot\mathcal{F}(x))+(1-H(r))\cdot\mathcal{F}(x)),$
(2)

where $\mathcal{F}$ denotes the Fourier Transformation and
$\mathcal{F^{\prime}}$ is the inverse one. $H(r)$ is the band-pass filter with
a filter radius of $r$. $G(\cdot)$ is a randomization function according to a
Gaussian distribution and $G(X)=X\cdot(1+\mathcal{N}(0,1))$.

For local transformation, as shown in Fig. 5, we first obtain the object masks
with the help of Segment Anything Model (SAM) [15]:

$\text{SAM}(x,B)=m^{O},$ (3)

where $x$ is the input image, $B$ is the bounding box set of the objects and
$m^{O}$ is the set of the obtained masks, where
$O=\\{o_{1},o_{2},...o_{n_{O}}\\}$ and $n_{O}$ is the number of objects in the
input image. Then we extract object images by :

$x^{o_{k}}=x\odot m^{o_{k}}.$ (4)

Then the local transformation is formulated as:

$\text{LT}(x)=\mathcal{T}_{0}(x^{bg})+\sum_{k=1}^{n_{O}}\mathcal{T}_{k}(x^{o_{k}}),$
(5)

where $x^{bg}$ is the background image and $\mathcal{T}_{0}(\cdot)$ and
$\mathcal{T}_{k}(\cdot)$ denote the randomly selected augmentation strategies
for the background $x^{bg}$ and the object $x^{o_{k}}$.

The augmented image output by the GLT module is denoted as:

$\text{GLT}(x)=\alpha\cdot\text{GT}(x)+(1-\alpha)\cdot\text{LT}(x),$ (6)

where $\alpha$ is the fusion weight to balance the GT image and the LT image
and the fusion strategy also increases the diversity of the augmented images.

Figure 5: Overview of the Global-Local Transformation (GLT) module. The Global
Transformation (GT) performs overall augmentation of an image in the frequency
domain. And The Local-Transformation (LT) performs augmentation of local
objects obtained by SAM [15] in the spatial domain. The final augmented image
is obtained by fusing the GT image with the LT image.

###  4.2 Causal Attention Learning

We transform the learning of scene-level causal features into the learning of
causal attention, which eliminates the need for explicit decoupling of causal
features from non-causal features, instead focusing on selecting causal
features based on attention.

Specifically, it is desirable for the network to capture object features
precisely in different scene context. Thus, we enforce the feature attention
maps of different data distribution images output by the backbone to converge
and the attention invariance loss is defined as:

$\displaystyle\mathcal{L}_{att}=\text{Dice}(g(F_{att}^{(a_{0},d_{0})}),g(F_{att}^{(a_{k},d_{k})})),$
(7) $\displaystyle F_{att}^{(a,d)}=\sigma(E(x^{(a,d)})),$ (8)

where $g(\cdot)$ is a pixel-level binarization function, $F_{att}^{(a,d)}$ is
the calculated attention map, $\sigma(\cdot)$ is the Sigmoid function, $E$
denotes the backbone, $x^{(a_{0},d_{0})}$ is the original input image and
$x^{(a_{k},d_{k})}$ denotes the randomly augmented image obtained from GLT.
Besides, $\text{Dice}(\cdot,\cdot)$ denotes the dice loss which is commonly
used for measuring the regional similarity between two samples, which is
defined as:

$\text{Dice}(X_{1},X_{2})=1-\frac{2\lvert X_{1}\cap X_{2}\lvert+1}{\lvert
X_{1}\lvert+\lvert X_{2}\lvert+1},$ (9)

where $X_{1}$ and $X_{2}$ are two binary maps. The regions with a value of 1
for the maps are salient attention regions, while the regions with a value of
0 are non-salient attention regions.

Then we select causal features according to the attention map for the
subsequent Region Proposal Network (RPN) to generate object proposals $P$:

$P=\text{RPN}(E(x^{(a,d)})\odot F_{att}^{(a,d)}).$ (10)

###  4.3 Causal Prototype Learning

To facilitate the learning of object-level causal features, we introduce the
Causal Prototype Learning module which encompasses an explicit constraint and
an implicit constraint.

The explicit constraint is imposed on ROI features extracted from different
data distributions. Specifically, given the proposal set $P$ generated from
the source image, we select a group of proposals with a confidence higher than
the threshold $t$, denoted as $P(t)$. We represent the ROI features generated
from $P(t)$ as $f(x,P(t))$ for simplicity. Then the explicit constraint is
defined as:

$\mathcal{L}_{exp}=KL(p^{do(a_{0},d_{0})},p^{do(a_{k},d_{k})}),$ (11)

where $KL$ denotes the Kullback-Leibler divergence, $p^{do(a_{0},d_{0})}$ is
the short for $p(y|f(x^{(a_{0},d_{0})},P(t)))$, $p^{do(a_{k},d_{k})}$ is the
short for $p(y|f(x^{(a_{k},d_{k})},P(t)))$and
$p^{do(a,d)}=\text{MLP}(f(x^{(a,d)},P(t)))$.

The explicit constraint $\mathcal{L}_{exp}$ encourages the within-class
distance of the learned class representations from different data
distributions to be concentrated and also gives supervisory information to the
data-augmented image at the object feature level, which enhances the saliency
of the object region, thereby improving the object localization performance of
the data-augmented images.

On the other hand, the implicit invariance constraint constrains the
relationship between category prototypes across different data distributions.
We hypothesize that the distance between causal prototypes of the same
category for different data distributions is smaller compared to the distance
between prototypes of different categories:

$dist(v_{c_{i}}^{(a_{0},d_{0})},v_{c_{j}}^{(a_{k},d_{k})})>dist(v_{c_{i}}^{(a_{0},d_{0})},v_{c_{i}}^{(a_{k},d_{k})})\approx
0,$ (12)

where $i\neq j$, $dist(\cdot)$ is a distance metric function,
$v_{c_{i}}^{(a_{0},d_{0})}$ is the prototype of category $c_{i}$ of the source
data distribution and $v_{c_{j}}^{(a_{k},d_{k})}$ is the prototype of category
$c_{j}$ of the augmented data distribution. The prototype $v_{c}$ is computed
by dynamically averaging the ROI features belonging to the category $c$.

We transform the implicit constraint that satisfy the above hypothesis into a
prototype contrastive loss:

$\displaystyle\mathcal{L}_{imp}=-\sum_{i}\text{log}\frac{\text{exp}(s(i,i))}{\text{exp}(s(i,i))+\sum_{j}\text{exp}(s(i,j))},$
(13) $\displaystyle
s(i,j)={sim(v^{(a_{0},d_{0})}_{c_{i}},v^{(a_{k},d_{k})}_{c_{j}})/\tau},$ (14)

where $i\neq j$, $sim(\cdot,\cdot)$ is the cosine similarity and $\tau$ is a
temperature parameter.

The loss function of CPL module is presented as:

$\mathcal{L}_{prot}=\mathcal{L}_{exp}+\mathcal{L}_{imp}.$ (15)

The constraints of this module will further refine the image-level attention,
thereby allowing the features fed into the RPN module to contain more
discriminative information about the objects.

###  4.4 Model Optimization

The data streams of the original images and the augmented images share the
network parameters and the total training loss is:

$\mathcal{L}=\mathcal{L}_{sup}+\lambda_{1}\mathcal{L}_{att}+\lambda_{2}\mathcal{L}_{prot},$
(16)

where $\mathcal{L}_{sup}$ is the supervised object detection loss for the
original images and the augmented images and $\lambda_{1/2}$ are
hyperparameters.

During inference, our UFR model maintains the same parameter size as vanilla
Faster R-CNN [33], ensuring that it does not introduce additional space
complexity. However, a key difference lies in the calculation of features that
are fed into the RPN network, which is determined by Eq. (10).

##  5 Experiments

###  5.1 Experimental Setup

Datasets. We conduct experiments on the dataset built in [44]. The dataset
consists of five different weather conditions, including Daytime-Clear,
Daytime-Foggy, Dusk-Rainy, Night-Clear and Night-Rainy. The Daytime-Clear
scene is used as the source domain, comprising 19,395 images for training and
8,313 images for testing. The other four scenes are used as unseen target
domains, consisting of 3,775 images in Daytime-Foggy condition, 3,501 images
in Dusk-Rainy condition, 26,158 images in Night-Clear condition and 2,494
images in Night-Rainy condition. The dataset contains annotations for seven
categories of objects, including person, car, bike, rider, motor, bus and
truck.

(a)

(b)

Figure 6: Qualitative evaluation results of the model’s generalization ability
on the Night-Clear scene. The top-row images are the results of vanilla Faster
R-CNN [33]. The bottom-row images are the results of our method.

Implementation Details. We adopt Faster R-CNN [33] with ResNet-101 [9] as the
object detector. The backbone is initialized with weights pre-trained on
ImageNet [5]. We train the model with Stochastic Gradient Descent (SGD)
optimizer with a momentum of 0.9 for 80$k$ iterations. During training, the
learning rate is set to 0.001, the batch size is set to 4. Besides, the
threshold $t$ is set to 0.7, the $\lambda_{1}$ and $\lambda_{2}$ are both set
to 0.1.
Data Augmentation Setting. For local transformation in spatial domain, we
randomly apply the augmentation strategies consisting of gaussian blurring,
color jittering, random erasing and grayscale. The fusion weight $\alpha$ in
Eq. (6) is a random scalar in $[0,1]$.

Methods | Bus | Bike | Car | Motor | Person | Rider | Truck | mAP
---|---|---|---|---|---|---|---|---
FR [33] | 66.9 | 45.9 | 69.8 | 46.5 | 50.6 | 49.4 | 64.0 | 56.2
SW [27] | 62.3 | 42.9 | 53.3 | 49.9 | 39.2 | 46.2 | 60.6 | 50.6
IBN-Net [26] | 63.6 | 40.7 | 53.2 | 45.9 | 38.6 | 45.3 | 60.7 | 49.7
IterNorm [12] | 58.4 | 34.2 | 42.4 | 44.1 | 31.6 | 40.8 | 55.5 | 43.9
ISW [4] | 62.9 | 44.6 | 53.5 | 49.2 | 39.9 | 48.3 | 60.9 | 51.3
SDGOD [44] | 68.8 | 50.9 | 53.9 | 56.2 | 41.8 | 52.4 | 68.7 | 56.1
CLIP-Gap [38] | 55.0 | 47.8 | 67.5 | 46.7 | 49.4 | 46.7 | 54.7 | 52.5
UFR (Ours) | 66.8 | 51.0 | 70.6 | 55.8 | 49.8 | 48.5 | 67.4 | 58.6

Table 1: Quantitative results (%) on the Daytime-Clear scene.

###  5.2 Comparison with State-of-the-arts

Following the setting in [44], we use the Mean Average Precision (mAP) as our
metric and report the<EMAIL_ADDRESS>results. We compare our method with feature
normalization-based SDG methods, including SW [27], IBN-Net [26], IterNorm
[12], and ISW [4], and the existing two single domain generalized object
detection methods, including SDGOD [44] and CLIP-Gap [38]. Besides, we also
compare the performance of our model with vanilla Faster R-CNN (FR) [33].

Results on Daytime-Clear Scene. We evaluate the model performance on the
source domain. As shown in Table 1, our method achieves an optimal result of
58.6% mAP and has a 2.4% mAP gain against vanilla FR [33]. It shows that our
method is able to maintain or even improve the performance of the FR [33]
model in the source domain while improving its generalization ability in
unseen domains.

Methods | Bus | Bike | Car | Motor | Person | Rider | Truck | mAP
---|---|---|---|---|---|---|---|---
FR [33] | 43.5 | 31.2 | 49.8 | 17.5 | 36.3 | 29.2 | 43.1 | 35.8
SW [27] | 38.7 | 29.2 | 49.8 | 16.6 | 31.5 | 28 | 40.2 | 33.4
IBN-Net [26] | 37.8 | 27.3 | 49.6 | 15.1 | 29.2 | 27.1 | 38.9 | 32.1
IterNorm [12] | 38.5 | 23.5 | 38.9 | 15.8 | 26.6 | 25.9 | 38.1 | 29.6
ISW [4] | 38.5 | 28.5 | 49.6 | 15.4 | 31.9 | 27.5 | 41.3 | 33.2
SDGOD [44] | 40.6 | 35.1 | 50.7 | 19.7 | 34.7 | 32.1 | 43.4 | 36.6
CLIP-Gap [38] | 37.7 | 34.3 | 58.0 | 19.2 | 37.6 | 28.5 | 42.9 | 36.9
Ours | 43.6 | 38.1 | 66.1 | 14.7 | 49.1 | 26.4 | 47.5 | 40.8

Table 2: Quantitative results (%) on the Night-Clear scene.

Results on Night-Clear Scene. The qualitative and quantitative evaluation
results of the model’s generalization performance in the Night-Clear scene are
shown in Table 2 and Fig. 6, respectively. As shown in Table 2, our method
achieves the best result of 40.8% mAP, outperforming Clip-Gap [38] by 3.9%
mAP. Besides, compared with vanilla FR [33], our method achieves a 5.0% mAP
gain, which indicates that the learned features is beneficial for improving
model generalization ability. Besides, compared with SDGOD [44] that learns
domain-invariant features, our method has an improvement of 4.2% mAP, which
suggests that causal features are more discriminative features and can
generalize better. For qualitative results demonstrated in Fig. 6, it is
obviously observed that our method detects objects more accurately and has
fewer false positives compared with vanilla FR [33].

(a)

(b)

Figure 7: Qualitative evaluation results of the model’s generalization ability
on the Daytime-Foggy scene. The top-row and bottom-row images demonstrate the
results of vanilla Faster R-CNN [33] and our method, respectively. Besides,
the first two columns are the results of real foggy images and the last two
columns are the results of synthetic foggy images.

Results on Dusk-Rainy and Night-Rainy Scenes. Table 3 and Table 4 demonstrate
the model’s ability to generalize to Dusk-Rainy and Night-Rainy scenes. Our
method achieves the best mAP in both scenes, with gains of 0.9% mAP and 0.5%
mAP respectively compared with CLIP-Gap [38]. Besides, compared with vanilla
FR [33], our method achieves improvements of 5.2% mAP and 5.0% mAP
respectively. Furthermore, the feature normalization-based methods [27, 26,
12, 4] have a poor performance in both scenes, especially in motor category in
Night-Rainy scene, with mAP of less than 1.0%, which reinforces the challenge
of these two scenes and the superiority of our method.

Methods | Bus | Bike | Car | Motor | Person | Rider | Truck | mAP
---|---|---|---|---|---|---|---|---
FR [33] | 34.2 | 21.8 | 47.9 | 16.0 | 22.9 | 18.5 | 34.9 | 28.0
SW [27] | 35.2 | 16.7 | 50.1 | 10.4 | 20.1 | 13.0 | 38.8 | 26.3
IBN-Net [26] | 37.0 | 14.8 | 50.3 | 11.4 | 17.3 | 13.3 | 38.4 | 26.1
IterNorm [12] | 32.9 | 14.1 | 38.9 | 11.0 | 15.5 | 11.6 | 35.7 | 22.8
ISW [4] | 34.7 | 16.0 | 50.0 | 11.1 | 17.8 | 12.6 | 38.8 | 25.9
SDGOD [44] | 37.1 | 19.6 | 50.9 | 13.4 | 19.7 | 16.3 | 40.7 | 28.2
CLIP-Gap [38] | 37.8 | 22.8 | 60.7 | 16.8 | 26.8 | 18.7 | 42.4 | 32.3
Ours | 37.1 | 21.8 | 67.9 | 16.4 | 27.4 | 17.9 | 43.9 | 33.2

Table 3: Quantitative results (%) on the Dusk-Rainy scene.

Methods | Bus | Bike | Car | Motor | Person | Rider | Truck | mAP
---|---|---|---|---|---|---|---|---
FR [33] | 21.3 | 7.7 | 28.8 | 6.1 | 8.9 | 10.3 | 16.0 | 14.2
SW [27] | 22.3 | 7.8 | 27.6 | 0.2 | 10.3 | 10.0 | 17.7 | 13.7
IBN-Net [26] | 24.6 | 10.0 | 28.4 | 0.9 | 8.3 | 9.8 | 18.1 | 14.3
IterNorm [12] | 21.4 | 6.7 | 22.0 | 0.9 | 9.1 | 10.6 | 17.6 | 12.6
ISW [4] | 22.5 | 11.4 | 26.9 | 0.4 | 9.9 | 9.8 | 17.5 | 14.1
SDGOD [44] | 24.4 | 11.6 | 29.5 | 9.8 | 10.5 | 11.4 | 19.2 | 16.6
CLIP-Gap [38] | 28.6 | 12.1 | 36.1 | 9.2 | 12.3 | 9.6 | 22.9 | 18.7
Ours | 29.9 | 11.8 | 36.1 | 9.4 | 13.1 | 10.5 | 23.3 | 19.2

Table 4: Quantitative results (%) on the Night-Rainy scene.

Results on Daytime-Foggy Scene. Table 5 shows the generalization results of
our model in Daytime-Foggy scene. We can see that our method outperforms all
the methods in the table and achieves 39.6% mAP. Specifically, compared with
SDGOD [44] and CLIP-Gap [38], our method achieves improvements of 6.1% mAP and
1.1% mAP, respectively. Besides, some visualization results are demonstrated
in Fig. 7. Compared with vanilla FR [33], our method achieves more accurate
object localization and classification in both real and synthetic foggy
environments.

(a)

(b)

Figure 8: Visualization of the detection results and attention maps on the
Dusk-Rainy scene. The top-row and bottom-row images are the results of vanilla
Faster R-CNN [33] and our method respectively. The dark red region represents
the area where the attention is salient.

###  5.3 Ablation Study

In this section, we conduct several experiments on Daytime-Clear, Night-Clear
and Daytime-Foggy scenes to analyze the role of each component of the Unbiased
Faster R-CNN model. Specifically, the experimental results in Table 6 are
obtained by training on the Daytime-Clear scene and testing on the three
weather conditions.
Analysis of the GLT module. As shown in Table 6, the results of vanilla FR
[33] are improved when combined with the GLT module, with gains of 4.2% mAP in
Daytime-Clear scene, 2.8% mAP in Night-Clear scene, and 2.6% mAP in Daytime-
Foggy scene. Besides, the 60.4% mAP in Daytime-Clear scene is optimal, which
indicates that the GLT module is an effective data augmentation method that
increases data diversity and performs exceptionally well in the source domain
with supervised learning.

Methods | Bus | Bike | Car | Motor | Person | Rider | Truck | mAP
---|---|---|---|---|---|---|---|---
FR [33] | 34.5 | 29.6 | 49.3 | 26.2 | 33.0 | 35.1 | 26.7 | 33.5
SW [27] | 30.6 | 36.2 | 44.6 | 25.1 | 30.7 | 34.6 | 23.6 | 30.8
IBN-Net [26] | 29.9 | 26.1 | 44.5 | 24.4 | 26.2 | 33.5 | 22.4 | 29.6
IterNorm [12] | 29.7 | 21.8 | 42.4 | 24.4 | 26.0 | 33.3 | 21.6 | 28.4
ISW [4] | 29.5 | 26.4 | 49.2 | 27.9 | 30.7 | 34.8 | 24.0 | 31.8
SDGOD [44] | 32.9 | 28.0 | 48.8 | 29.8 | 32.5 | 38.2 | 24.1 | 33.5
CLIP-Gap [38] | 36.2 | 34.2 | 57.9 | 34.0 | 38.7 | 43.8 | 25.1 | 38.5
Ours | 36.9 | 35.8 | 61.7 | 33.7 | 39.5 | 42.2 | 27.5 | 39.6

Table 5: Quantitative results (%) on the Daytime-Foggy scene.

Analysis of $\mathcal{L}_{att}$, $\mathcal{L}_{exp}$ and $\mathcal{L}_{imp}$.
As depicted in Table 6, incorporating $\mathcal{L}_{att}$, $\mathcal{L}_{exp}$
and $\mathcal{L}_{imp}$ in addition to the GLT module decreases the model
performance in the Daytime-Clear scene while improving the performance in the
Night-Clear and Daytime-Foggy scenes, which suggests that these constraints
limit the model’s performance in the seen environment with supervised
learning, but improve the model’s performance in unseen environments. The
phenomenon is due to that these constraints encourage the model to extract
causal knowledge, thereby preventing the model from acquiring domain-specific
knowledge. The domain-specific knowledge, although not generalizable, carries
some supplementary information for supervised learning. Furthermore, we can
observe that the attention invariance loss $\mathcal{L}_{att}$ improves the
model generalization performance more significantly, which indicates that the
scene confounders have a greater impact on the model generalization ability.
Besides, compared to the implicit constraint $\mathcal{L}_{imp}$, the explicit
constraint $\mathcal{L}_{exp}$ is more binding and contributes more to
generalization. Therefore, purely implicit constraint cannot guarantee the
model to learn causal prototypes, and explicit constraint on the predicted
distribution is necessary.

Methods | $\mathcal{L}_{att}$ | $\mathcal{L}_{exp}$ | $\mathcal{L}_{imp}$ | D-Clear | N-Clear | D-Foggy
---|---|---|---|---|---|---
FR [33] |  |  |  | 56.2 | 35.8 | 33.5
+GLT |  |  |  | 60.4 | 38.6 | 36.1
✓ |  |  | 59.5 | 39.7 | 38.1
✓ | ✓ |  | 59.1 | 40.4 | 39.0
✓ |  | ✓ | 59.3 | 40.0 | 38.5
✓ | ✓ | ✓ | 58.6 | 40.8 | 39.6

Table 6: Ablation study results (%) of our proposed UFR method. ‘+GLT’ denotes
incorporating the GLT to vanilla FR[33]. The loss check mark denotes the
addition of the loss to FR [33] with GLT.

###  5.4 Further Analysis

Attention visualization. We conduct experiments on Dusk-Rainy scene and
compare the attention maps produced by vanilla FR [33] and our UFR model, as
shown in Fig. 8. We can observe that the attention maps generated by FR [33]
are unfocused and have more attention on irrelevant background areas. In
contrast, our method produces more effective category-related attention and
has less attention on background, which suggests that our method can capture
discriminative object features in unseen environments with a superior
generalization performance.
Hyperparameter analysis. As demonstrated in Fig. 9, we train our model with
different settings of $t$, $\lambda_{1}$ and $\lambda_{2}$ and test the
generalization performance on the Night-Clear and Daytime-Foggy scenes. The
model performance with different settings of threshold $t$ in Eq. (11) is
reported in Fig. 9(a). The best threshold $t$ is 0.7. Besides, we also report
the results of our model in Fig. 9(b) with different settings of $\lambda_{1}$
and $\lambda_{2}$ in Eq. (16). It is obvious that the model achieves the best
performance with $\lambda_{1}=0.1$ and $\lambda_{2}=0.1$.

(a) Analysis on $t$

(b) Analysis on $\lambda_{1}$ and $\lambda_{2}$

Figure 9: Results of hyperparameter analysis.

##  6 Conclusions

In this paper, we analyze the factors that contribute to the limited single-
domain generalization ability of Faster R-CNN from a causal perspective and
summarize them as data bias, attention bias and prototype bias. To tackle
these challenges, we propose a novel Unbiased Faster R-CNN. Our model
leverages a Structural Causal Model to analyze the biases in the task that
arise from both scene confounders and object attribute confounders. We design
a Global-Local Transformation module for data augmentation to mitigate the
data bias. To learn features that are robust to scene confounders, we
introduce a Causal Attention Learning module for image-level causal feature
selection. To further mitigate the influence of object attribute confounders,
we develop a Causal Prototype Learning module that learns object-level causal
features. Experimental results and further analysis demonstrate the
effectiveness of our method.

## Acknowledgements

This work was supported by National Natural Science Foundation of China under
Grants U2013210 and 62303447; Liaoning Science Fund for Distinguished Young
Scholars under Grant 2023JH6/100500005; Postdoctoral Innovation Talents
Support Program, No. BX20230399.

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