InvariantStock: Learning Invariant Features for Mastering the Shifting Market
Authors: Haiyao Cao
Jinan Zou
Yuhang Liu
Zhen Zhang
Ehsan Abbasnejad
Anton van den Hengel
Javen Qinfeng Shi
Venue: Accepted by TMLR ()

Abstract: Accurately predicting stock returns is crucial for effective portfolio management. However, existing methods often overlook a fundamental issue in the market, namely, distribution shifts, making them less practical for predicting future markets or newly listed stocks. This study introduces a novel approach to address this challenge by focusing on the acquisition of invariant features across various environments, thereby enhancing robustness against distribution shifts. Specifically, we present InvariantStock, a designed learning framework comprising two key modules: an environment-aware prediction module and an environment-agnostic module. Through the designed learning of these two modules, the proposed method can learn invariant features across different environments in a straightforward manner, significantly improving its ability to handle distribution shifts in diverse market settings. Our results demonstrate that the proposed InvariantStock not only delivers robust and accurate predictions but also outperforms existing baseline methods in both prediction tasks and backtesting within the dynamically changing markets of China and the United States.

Published in Transactions on Machine Learning Research (08/2024)
InvariantStock: Learning Invariant Features for Mastering the Shifting Market
Haiyao Cao
haiyyo.cao99ddelaideee edw..
Australia Insttttee for Machine Learning The University of Adelaide
Jinan Zou
Sinan.zou@adelln edu.aa.
Australia Institute for Machine Learning The University of Adelaide
Yuhang Liu
yuhang. liu01 @adelaide. edu.
Australia Institute for Machine Learning The University of Adelaide
Zhen Zhang
shen.zhang@29BOdellidde...
Australia Institute for Machine Learning The University of Adelaide
Ehsan Abbasnejad
chsan. chaern.abcceeennn@@nddddddaddddd
Australia Institute for Machine Learning The University of Adelaide
Anton Van Den Hengel
Australia Institute for Machine Learning University of Adelaide
Javen Qinfeng Shi
javen.shi@adelaide. edu..a.
Australia Institute for Machine Learning The University of Adelaide
Reviewed on OpenReview: openreviee. net/ forum? id= dtNVvvUDZm
Abstract
Accurately predicting stock returns crucial for effective portfolio management. However, existing methods often overlook fundamental issue in the market, namely, distribution shifts, making them less practical for predicting future markets or newly listed stocks. This study introduces novel approach to address this challenge by focusing the acquisi- tion of invariant features across various enviromments, thereby enhancing robustness against distribution shifts. Specifically, we present InvariantStock, a designed learning framework comprising two key modules: an enviromment-aware prediction module and an environment- agnostic module. Through the designed learning of these two modules, the proposed method can learn invariant features across different. environments in foraight manner, sig- nificantly improving its ability to handle distribution shifts in diverse market settings. Our results demonstrate that the proposed InvariantStock not only delivers robust and accurate predictions but also outperforms existing baseline methods in both prediction tasks and backtesting within the dynamically changing markets of China and the United States. Our code is available at https: htg.iiuu..oom/laaayaoeeeeeooo

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1 Introduction
In the realm of stock markets, portfolio optimization is a common technique aimed at maximizing profits while minimizing risks through the management of asset basket. This approach, rooted in stock returns prediction, gained prominence with the introduction of the Capital Asset Pricing Model (CAPM) by Sharpe et al. [1964] and the Fama-French three-factor model Fama & French (1993). However, these traditional methods have often faced challenges in accurately predicting stock returns due to their inherent model sim- plicity. The landscape of research has witnessed a significant transformation with the advent of advanced doep-learning models. Many rescarchers now leverage deep learning techniques, including convolutional neu- ral networks (CNN) LeCun et al. (1998); He et al. (2016), recurrent neural networks (RNN) Hochreiter & Schmidhuber (997]) Cho et al. (2014), and transformers Vaswani et. al. (2017), for stock returns prediction. These methods preferred for their exceptional fitting capabilities. Within modern deep learning frame-
works, recent studies Zhang et al.] (2017); Feng et al. (2018): Yoo et al. 2021): Duan et al. (0222) Gao et al.
(2033] Zhao et al. (2023): : Koa et al. 2023 have delved into various specific information sources within the
stock market to enhance predictions.
Despite the advancements in deep learning techniques for slock return prediction, several limitations persist, particularly in the context of the scope and adaptability of these methods. A common issue is that many methodologics [Zhang et al.] (2017); Feng et al. (2018); Yoo et al. (2021); Gao et al. (2023) are constrained by either their inherent design or computational costs, leading to their application being limited to small set of stocks, such as those in the DJIA, S&P 500, or CSI300 indices. This limitation inherently excludes a vast majority of market stocks, resulting in these models often capturing spurious correlations due to a lack of diversity in their training datasets. Consequently, these models tend to underperform or fail when faced with unseen or newly listed stocks. Moreover, the stock market is in a constant state of flux, influenced by myriad of factors including political and economic environments. Regular occurrences like financial report announcements and public holidays, as well unforeseen events such as wars or pandemics, significantly impact investor sentiment and market dynamics. Those factors result in ever-shifting market distribution, a phenomenon also noted in the study of [Zhao et al. (2023). Models that fail to account for these distribution shifts may prove impractical for future market applications Liu et al. (022b)). The recent work, Factor Duan et al./ (2022) has attempted to mitigate prediction noise by using dynamic factor models for cross-sectional return predictions. However, does not take into account the market's 's distributional shifts. Similarly, Diffusion Variational Autocncoder (DVA) Koa et al. (2023) faces the same limitation. The DoubleAdaptr Zhao et al. (2023) altempts to incrementally train the model to adapt to the shifting market, it too encounters issues of catastrophic forgetting if the model is not retrained.
To effectively address the issue of distribution shifts in the stock market, we propose to identify features that remain consistent across varying distributions. Consequently, utilizing these features for prediction can offer robustness against distribution shifts across different environments. To achieve this goal, we frame it from an information theory perspective as follows: H(Y = H( F, E), where H denotes the entropy, Y represents the target variable, F signifies the invariant features across environments E. Essentially, this formulation implies that the objective is to learn features that can predict the target variable Y with a high degree of accuracy while ensuring that these features remain unaffected by environmental factors within the stock market. Motivated by this, we proposed a designed learning framework, InvariantStock, including two main modules: an environment-
prediction module and an environment framework, the first module is tailored to model (Y)F),, while the latter is designed to model H(Y F, E). Through the designed learning process, the proposed framework can select features F in such a way that FFY (Y|,,,E) E). Further, to enhance the implementation of the proposed framework, we have also devised an efficient selection module. This module plays crucial role in ensuring that the selected features remain as consistent as possible. The contributions of this work can be summarized follows:
prediction module. In this
We address the challenges posed by distribution shifts in the stock market, by roducing an effective framework that centres on learning invariant features across different environments.
We present InvariantStock, designed learning framework that encompasses both an environment- agnostic prediction module and an environment-aware module, which facilitates the selection of
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invariant features across diverse environments. We have designed an efficient selection module that seamlessly integrates into the proposed framework to further enhance the elfectiveness of designed learning within the framework.
Through experiments conducted using real-world market scenarios, we have demonstrated that the proposed InvariantStock framework surpasses other statc-of-the-art prediction-based methods, in lerms of a variety of evaluation metrics.
2 Related Work
Regression-based stocks prediction In the field of stock prediction, regression-based methods represent a significant departure from traditional classification approaches. While classification typically involves sorting stocks into a limited number of categories, often two or three Feng et al. (2018); Sawhney et al. (2020 Wang al. (2021); Xiang et al. (2022). Regression- based methods Duan al. (2022): ; Zhao et al. 2023 Koa et al. [2023] aim to directly forecast the magnitude of price changes. This approach can be more informative and practical for investors, offering insights for decision-making processes like ranking predictions. Commonly, models based on CNN, RNN and transformer are employed for handling time series data in stock predictions WWen et al. (2019): Feng et al. (2018); Yoo et al. (2021); Duan et al. (2022); Gao ct al. 2023). Additionally, a growing body of research Li et al. (2022b); Zhao et al. (2022): Zheng et al. (2023) has been exploring the use of graph-based models to capture the intricate relationships between stocks. However, prior work made predictions without considering the market shifting, which limits the performance and application in the real world.
Feature selection In the domain of stock prediction, the necessity of feature selection is underscored by the need to minimize the impact of prevalent noise within financial data. Prior rescarch in this field, including studies by Tsai & Hsiao (2010); Shen & Shafiq (2020); Dona & Gallinari (2021); )s Haq et al. (2021) Li et al. (202aa) typically approaches feature selection through methodologies that prioritize features based on their assessed importance or their relational distance to the target variables. However, these conventional methods often overlook the dynamic nature of financial markets. In contrast, our fcatures selection module is designed to identify set of invariant features, those that retain their relevance and stability across varying market distributions.
Invariant learning Invariant learning focuses on developing representations models that remain consis-
tent across various environments, utilizing contextual information to achieve this stability. The underlying
rationale is that invariant fcatures are closely linked to causality Liu et al. (2022a 2024b ab. The moth-
ods for invariant seeking can be categorised into invariant risk minimization (IRM) Arjovsky el al. (2019)
Chang et al. (2020); Koyama & Yamaguchi (2020) and fomain--rrelevann representation learning |Ganin & Lempitsky (015);; Ganin et al. (2016). Our approach is inspired by the study of Koyama & Yamaguchi (2020), which uses adversarial learning to do the IRM, Adapting this concept, have developed modified framework that incorporates a feature selection module.
3 Problem Formulation
This section outlines the setup and notation of the stock prediction problem, with a focus on stock return prediction combined with ranking. For the China stock market datasct, the prediction target is formulated
++2 s+1
as yt Poper
Popen which also represents the ranking score, where Papen denotes the opening price on day t.
Plper
This formulation also serves as the ranking score. The rationale for using the opening price, rather than the closing price, is due to the regulatory limit price ratio changes in the China stock market. Orders placed at market close may not be filled for stocks that have reached their price limit. However, orders placed the opening are more likely to be filled, as few stocks reach their limit price at this time. In contrast, the prediction setup for the US stock market is more straightforward: yt closing price, with orders being placed the market nears closing. The prediction task represented f(X, O b) where X (T)) E RTDD denotes the historical features of the stock. Here, T is the
Paiiig P -Petore This approach uses the Pelose
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Euvirsoment Yorinble
E
Envisuuneat-wwss prediction modale
d getno
Stock Festures
T gru
X
② raaat
Y
Variabkes
F
Stock returee
X
2 recon
f-ino
"post
g fiie
A dru
gru
S fni
Aige
prodiction
Figure 1: The structural design of InvariantStock delineated, where the green region symbolizes the feature selection module, comprising mask model and reconstruction model. The red and blue regions depict the environment- agnostic prediction module and the environment-awan prediction module, respectively. Each prediction module is composed of state extractor, encoder, decoder, and predictor.
length of the look-back window, and D represents the number of stock features at time O & symbolize the parameters of the feature selection module and the prediction module, respectively, Furthermore, given that euvironments continuously evolve over time, the primary focus of this work is to develop a learned model capable of delivering stable predictions despite these changes.
4 Methodology
The purpose of InvariantStock is to acquire the invariant features F' from the full features X E R by leveraging additional information provided by environmental variables, E. To construct E we utilize a one- hot vector representation to encapsulate month and year data. This approach is grounded in the hypothesis that stock market distributions vary across different dates, thus implying that the market's characteristics and behaviour are not uniform. To learn the invariant features F, we introduce feature selection module that utilizes binary mask M (1,0)TXD, The purpose of this mask is to selectively filter the feature set such that the mutual information between the invariant features F' and Y is maximized. This can be mathematically formulated as:
T(Y; F), ME(1,07XX0
(1)
s.t. F= MXX H(|[F,))
it holds, E does not provide any additional information beyond what is already captured by F to predict Y. To achieve the objective in Equation [ Invariant.Stock comprises three key components, which are depicted in Figure I Each with a specific function and objective: the Feature Selection Module @(X) which identifies crucial invariant features F; the Environment-Awe environmental variables to model H(Y F, E)) for context-sensitive predictions; and the Aeatroommnn-Aggootii Prediction Module hinu (F), focusing solely stock features models H FF)
Prediction Module Φene E), which incorporates
4.1 Prediction Modules
The prediction module in InvariantStock is designed with the primary objective of predicting stock returns based F and E. This module comes in two forms: the environment-aware prediction module and the enviromment-agnostie prediction module. The key distinction between these two lies in their access to the environmental variable E; the environment-aware module incorporates this variable, while the environment- agnostic module does not. InvariantStock leverages VActorVAE Duan et al. (2022) as the backbone of prediction module, which compriscs four components: a state extractor, an encoder, a decoder, and a predictor.
State extractor: GRU with attention is specifically chosen for its effectiveness in extracting meaningful
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hidden states h, from historical data ia at time t. Formally,
h DGRU h1-1)
(2)
Encoder: The encoder is responsible for deriving latent posteriors from the hidden state he and the future ratio of price change by:
[upost: [post] = L Lenc he); Spost N (Hpost, di (oet))
(3)
where Mpost, post is the distribution parameter of 2. Decoder: uses latent variables 2 and hidden state calculate the stock return f
a ~ 1 Patpha (he), = Dbeta
(4)
i = a + Bz
Predictor: To prevent future information leakage during the inference phase, the predictor is used to get the latent prior only from the hidden state h,:
[pprior 3 Opriorl = 2 prec((he);
(5)
Zprior N (Plriior diag o prrirr)
4.2 Feature Selection Module
The Feature Selection Module in InvariantStock is designed to select invariant features F utilizing gener- ated binary mask M. This module's objective is not only to identify and isolate these invariant features but also to ensure that the original feature set X can be accurately reconstructed from F. This dual function- ality emphasizes retaining essential information while filtering out redundant features, thereby enhancing the model's predictive accuracy and reliability. Therefore, the feature selection module employs an AutoEn- voder(AE) structure where the @mask and M constitute the encoder and Oreoon functions the decoder decoder. The equations governing this process are as follows:
M = Omakk (X), = M O 8, Oreoon (F)
(6)
In this setup, @mask (X) generates the binary mask M which is used to selectively enable or disable features in the historical sequence. Since the binary mask is not differentiable, the straight-through technique [Bengio lt al. (2013) is utilized to estimate the gradient. Oreoon guarantees that all the invariant features are selected to avoid only a subset of invariant features being selected during the training.
4.3 Training Framework
InvariantStock achieves the goal in Equation I by employing multi-step training. Specifically, Invariant- Stock involves distinct objectives for each module: Environment-aware prediction module hhevv and the euvironment-agnoete module hine are optimized by maxing the likelihood of predicting conditioned on F, E and respectively. Features Selection Module θ largets identifying features that consistently influence stock returns, regardless of changing market environments. This is pursued by minimizing the discrepancy in predictions made by the cnvironment-awwrre and environment-agnostee modules. Since there is guarantee that all the invariant features could be learned only by minimizing the gap between two prediction modules, reconstruction objective is introduced to retain all the invariant features instead of a subset during the learning process by recovering X from F. These modules are trained sequentially, and the detailed training process for these objectives is systematically outlined Algorithm| [ By assigning distinct, focused objec- tives to each component and adhering this structured training protocol, InvariantStooc positioned provide accurate and reliable predictions, adeptly handling the complexities of varying market scenarios.
Prediction Objectives
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Prediction Objectives The objectives of the environment-aware module and environment-agonistie model are to maximize the conditional entropy and respectively by minimizing Equation 13 and Equa- tion 12 The terms in these equations include minimizing the MSE loss of the targets and predictions while ensuring correct stock ranking (ordered by the targets) in the market. The predictions from these modules are defined as:
Peev = g _x," E), gino 8 = xin (X)
Given the focus on selecting the most profitable stocks for portfolio solection, sample weights Wa are intro- duced to emphasize stocks with significant price changes:
W rank((Y) (V)) std(rank(Y.))
The prediction loss for both modules is calculated using Mean Square Error (MSE), factoring in these weights:
N
Cored cnu
Σ
(W.Y - W, 2
N
(9)
Cpred
1
-WaY;-Wa mmym
N
Additionally, applying hinge loss for stock ranking ensures that the predicted rankings stocks align closely with actual performance, which is crucial for selecting the right stocks in real trading. This method prioritizes accurate ranking, leading to better stock selection. The loss is defined
N
N
max (0, - (Br, yt.s))
env
(10)
N
ma - -Wi sine
(ye, yo.s))
inv
1=0
1=0
The KullbackLLeibler Divergence (KID) objective aims to minimize the difference between the posterior and prior distributions:
CKL KL (=/h, u prea (z|A)) and
(11)
sn (2) y) Pa CKT KL(P
pred (2(h))
aND
The total losses encompassing prediction accuracy, ranking effectiveness, and distribution alignment for both the environment-aware and environment-agnostic prediction modules are then caleulated as:
Lenr = =Cpred
Σ Wellem -BCKL
(12)
Line Cpred +
Σ Wol + BCKL
(13)
Feature Selection Objectives The learning target of the feature selection module is to encourage the independence between Y and E given F by minimizing the gap & Penu and Φinv so that (Y F) is achieved. This objective is pursued through four strategies: Minimizing the Prediction Gap: The module aims to reduce the discropancy between the prodictions made by the two modules. This is quantified using MSE:
1 N (pred = N (ww - WeDra
(14)
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Algorithm 1 Training Process
Input: stock features X, Environment E' and stocks return
Parameter: Feature selection module b Environment-aware prediction module o Pnnes Environment- agnostic prediction module Pine
Output:the prediction of stocks returns Y
Let e 0.
while total epoches do
F P= =XX
Yenv -ppostr posts pprrsors Oprior Denv (F, T,E,Y
Prin 1post: post: Piprior Oprion = a env (F,Y).
if e%3 = then
Calculate La according Equation 18
Update 6
else if e%o == then
7:
Calculate Line according to Equation
13
Update dine.
8: else if e%3 == then
Calculate Lene according Equation Update Tene.
10:
end if
11: end while
12: return θ Tinv
Mindmizing Ranking Objectives Difference: This involves reducing the difference in the ranking objectives between the two prediction modules:
N
N
1/
max (0, (Df
Dij
genn
(Di, (lg"))
(15)
1=0 1=
Minimizing KLD of Latent Variables: This involves minimizing the difference in the distribution latent variables generated by two prodiction modules:
CKL KK (z (z(h)) (a (Pprens (2/h, v)ll Poing (z/h, y))
(16)
Reconstruction Objeclive: Ensuring that there's little information loss during selecting invariant features, we maximize the entropy H (F)mmasked(()) through the following reconstruction objective:
N DT ΣΣ azy
1
Crecon
22)²
(17)
NDT
Finally, the total learning objective of the feature selection module combines these elements:
Le =C + a Σ Wilh + +3Ck -C
(18)
This comprehensive objective reflects a multifaceted approach to feature selection, emphasizing the impor- tance of accurate prediction, effective reconstruction, and alignment of latent variables. By addressing these aspects, InvariantStock'' feature selection module aims ensure robust and reliable stock return predictions.
4.4 Inference Progress
The proposed training framework, not required during the inference phase, ensures that the selected features remain invariant. to environmental changes. During inference in real-world applications, only the mask

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module and the environment-agnostic prediction are retained. In contrast, the environment-aware module and the reconstruction model are utilized exclusively during the training process. As a result, the compu- tational complexity and resource consumption during inference are significantly reduced compared to those during the training phase.
5 Experiments
To address the ensuing research inquiries:
Docs the InvariantStock model demonstrate efficacy on previously unobscrved stocks within fluctu- ating market conditions? (RQ1)
Is the applicability of the InvariantStock model consistent across diverse markets? (RQ2)
What features predominantly contribute to the effectiveness of predictions in the shifting market? (RQ3)
5.1 Dataset
We conducted comprehensive assessments on both the China and the US stock markets, spanning more than 20 years. The details of these datasets summarized in Table T collect extensive range of stock data, striving to accuralely represent the real market conditions pertinent to portfolio selection. This extensive data collection was also aimed at minimizing potential biases in the dataset, thereby ensuring a more reliable and accurate assessment.
China Stocks In assessing the ability to manage distribution shifts, The test set focuses on the period from early 2020 to 10/2022, a timeframe notably impacted by the COVID-19 outbreak. This global health crisis led to the widespread adoption of loose monctary policies, which, turn, drove stock markets new highs. However, by late 2021, the siluation shifted due to mounting concerns about high inflation and expectations of rate hikes by the Federal Reserve. These factors contributed to consistent downturn in the global financial market, reflected in the China stock market well, shown by the Shanghai Composite Index (Figure 5). Here, an initial uptrend before 02/2021 and a subsequent downtrend are evident, marking significant. fluctuations in market conditions. The test set, comprising 1386 new stocks relative to the training set, presents an optimal scenario for evaluating the adaptability of our methods to these distribution shifts.
The approach to the US stock data was similar to that for the China stocks, with a comparable strategy for data segmentation. However, the US stock data characterized by a more limited set of features, specifically six: open, high, low, close, volume, and the ratio of price change.
US Stocks
Set
# Date
#Stocks
# Features
Train 05/1995-12/2016
2662
China
Validation
01/2017-12/2019
3440
26
Test
01/2020-10/2022
4048
Train 01/1990-12/2018
4120
US
Validation
01/2019-12/2020
4831
6
Test 01/2021-01/2024
6314
Table 1: The temporal range, the number of stock entities, and the number of distinct. features employed for the training, validation, and testing datasets for China and US stock markets.
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5.2 Training setup
The experimental setup for our study was conducted using a GTX3090 GPU. The look-back window length is set as 20. Adam with a learning rate of 0.0005 is used as the optimizer and it's scheduled by One cycle scheduler. both a and B and 0 are set to 1. There's no further tuning for these parameters.
5.3 Baselines
In analysis, we benchmarked InvariantStock against both the market standard and the latest machine learning (MI.)-assd methodologies. The comparisons are as follows:
Market Benchmark: For an overarching gauge of market performance, we utilize the CST300 index for the China market and the DJIA for the US market.
Dataset Benchmark: Ackuowledging that our datasets do not encompass stocks in the markets, particularly in terms of historical data, we introduce a 'dataset benchmarki, In this benchmark, each stock is allocated equal percentage of the total asset value in the portfolio. It is crucial to note that this benchmark does not factor in commission fees.
Factor As outlined in Duan et al. (2022) Factor VAE is a method predicated on prior- posterior learning and utilizes a Variational Autoencoder (VAE) framework as detailed by Kingma & &Welling (2013). This VAE framework also serves as the backbone for the prediction module InvariantStock.
Proposed by Zhao et al. 2023 DoubleAdapt employs both data adapter and a model adapter to diminish the impact of domain shifts. Additionally, it leverages meta-learning to progressively capture market evolution.
DoubleAdapt:
Diffusion Variational Autoencoder (DVA): Introduced by Koa al. (2023). the DVA predicts multi-step returns using hierarchical VAE combined with diffusion probabilistic model. For evaluation and port backtesting in our study, we consider only the initial element of the predicted return sequence for each stock.
5.4 Investiment Simulation
Based on the predictions generated, we employ the TopK strategy for portfolio construction, wherein cach stock within the portfolio is assigned equal weight. The TopK strategy involves selecting the top stocks with the highest predicted returns for a long position, while simultaneously shorting the stocks with the lowest predicted returns. It's important to note that in the China stock market, due to specific policies, only long positions are executed. In China stock market, there are regulatory mechanisms such 'limit-up" up' 'limitddonn' rules. When a stock hits its limit-up, it becomes hard for further buying orders be filled. similarly, hitting the limit-down prevents selling further orders. Conversely, in the US market, both long and short positions are feasible. Additionally, commission fee is uilaterally charged at rate of 0.0015 in both markets.
5.5 Evaluation Metrics
In assessing the performance of various stock prediction methodologies, several performance metrics employed:
Information Coefficient (IC), IC Information Ratio (ICIR) mentioned in Du et al. (2021) Rank Information Coefficient (RankIC), Rank Information Coefficient Information Ratio (RankICIR), which are defined as:
Information Coefficient (IC): describe the correlation between targets and predictions, and is expressed as:
TC corr Y7)
(19)
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IC Information Ratio (ICIR): measures the consistency and reliability of the prediction and is expressed as:
mean(IC)
ICIR
(20)
std(IC)
Rank Information Coefficient (RankIC): measures the correlation between the ranking of pre- dictions and ranking of targets, and is expressed
RonkIC = corr Rank
(21)
where rank(Y)) is the target ranking of stocks returns and rank(Y;) signifies the predicted ranking of stock returns.
Rank Information Coefficient Information Ratio (RankICIR): measures the consistency and reliability the predicted ranking.
RankICIR= mean (ankkT)) sld( Rankla)
(22)
The backtesting metrics include annualized return rate (ARR), maximun drawdown (MDD), and Sharpe Ratio (SR). Higher ARR, higher SR, and lower MDD It is crucial to note that performance metrics are specifically applicable to prediction-based methods, including FactorVAB, DoubleAdapt, DVA, and Invari- antStock, while Metrics such ARR, MDD, and SR are primarily utilized for evaluating the backtesting effectiveness of the compared methods.
5.6
Performance on China Stock market (RQ1)
The performance results of the compared methods are shown in Table and Figure
Method
IC
ICIR
RankIC
RankICIR
ARR
MDD
SR
CSI300
-0.035
-0.3902
-0.2988
Dataset benchmark
0.1257
-0.2828
0.4625
FactorVAE
0.0331
0.5769
0.0447
0.6627
0.0057
-0.3048
-0.0825
DoubleAdapt
0.0376
0.4911
0.0707
0.9482
0.1701
-0.2668
0.6037
DVA
0.0002
0.0119
0.0005
0.0336
-0. -0.1919
-0.5352
-0.9803
InvariantStock
0.0576
0.6896
0.1005
1.0900
0.8315
-0.1878
3.7198
Table 2: The performance of predictions and backtesting outcomes for the methodologies compared within the Chinese stock market.
InvariantStock distinctly outshines all other methods in the comparative analysis, achieving the highest performance across both performance and backtesting metrics in the China stock market. The superiority of InvrriantStock over FactorVAE is evident through the integration of a Feature Selection Module and diverse learning objectives, effectively serving as an ablation study for the feature selection process. Therefore this comparison not only underlines the model performance but also highlights the critical role of targeted feature selection in improving functionality. Mcanwhile, the return trend of DoubleAdapt aligns more closcly with the dataset benchmark. Notably, the DVA method fails to deliver effective results in this context.
These findings underscore the capability of InvariantStock to adeptly navigate the distribution shifts char- actcristic of China stock market, feat that other bascline methods struggle to accomplish. As evidenced by the backlesting performance after 02/2021, the cumulative return stays increasing (Figure 2 even with a declining trend of SSEC (Figure 5). While most methods face challenges achieving commendable per- formance amidst market shifts and with unseen stocks, InvariantStock demonstrates robust performance both uptrend and downtrend market scenarios by selecting the features that encourage the independence between the environment and stock return.
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5
CSI300
Dataset
Cumulative Returns
4
FactorvAE
DVA
DoubleAdapt
3
InvarlantStock
2020-01
2020-05
2020-09
2021-01
2021-05
2021-09
2022-01
2022.05
2022.09
Figure Comparative analysis of cumulative returns by different methods in the China stock market highlighting the superior performance of InvariantStock amidst market shifts.
5.7 Performance on US stock market (RQ2)
The performance outcomes of the methods under comparison are presented in Table and Figure Invari- antStock, when evaluated on performance metrics, does not yield results comparable to those observed on the China dataset. This disparity may stem from the limited variety of stock features in the US dataset, which primarily includes only six price-related features. Nevertheless, InvariantStock still demonstrates superior practical performance in terms of RankIC and RankICIR compared to other baseline methods. Moreover, the backtesting results on ARR and SR for InvariantStock remain more commendable than the others. This suggests that InvariantStock is capable of picking profitable stocks even though the overall performance is not good, While DoupleAdapt also shows commendable results, but not as good as InvariantStock's The ARR of FactorVAE slightly better than that of the DJIA. DVA method does not exhibit effective perfor- mance in the US market either. It noteworthy that the MDD of all the proposed methods inferior when compared to the DJIA, which means all the methods cannot handle the risk well in the US market. This discrepancy may be caused by the lack of features in the US market, we make further analysis in Section
5.8
Method
IC
TCTR.
RankIC
RankICIR
ARR
MDD
SR
DJTA
0.0810
-0.2194
0.3657
Dataset
-0.0550
-0.3773
-0.3576
Factor VAE
0.0085
0.0757
-0.0031
-0.0208
0.0953
-0.4461
0.2343
Adapt
0.0114
0.1270
-0.0127
-0.1169
0.3192
-0.3422
1.2788
DVA
-0.0013
-0.0866
-0.0009
-0.0636
-0.3427
-0.7109
-4.7056
InvariantStock
0.0090
0.0840
0.0021
0.01605
0.5810
-0.3107
1.9119
Table 3: The performance of predictions and the backtesting outcomes for the methodologies compared within the US stock market.
5.8 Invariant Features Selection (RQ3)
We computed the mean values of the masks applied to the China test set, depicted in Figure 4 Within this representation, certain features exhibit a light hue, indicating their prominence while the dark hue is opposite. The less important features include the open price [0], high price [1], close price [3], last close price [4], percentage of change [6), total share [19), and next open price [24]. A comprehensive list of all features along with their corresponding indexes is detailed in Table 5 The analysis reveals that most of the lighter features in the dataset are fundamental features. This observation leads to the conclusion that
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4.0
DJIA
3.5
Dataset
FactorvAE
Cumulative Returss
DVA
2.5
DoubleAdapt
InvarlantStock
2.0
1.5-
2021-01
2021.05
2021-09
2022-01
2022-05
2022.09
2023-01
2023-05
2023:09
2024-01
Figure The cumulative returns from various methods in the US Stock Market indicate that with strategically chosen number of stocks in the portfolio, approaches like InvariantStock, DoubleAdapt, and FactorVAE are capable of yielding improved results. However, these methods consistently encounter a higher maximum drawdown, suggesting increased risk factor in their performance.
-08
Average Mask Value
14.
17
18
Figure d: Visualization of the mean value of the mask across all test data, indicating feature significance. Lighter colours represent features of higher prominence. The features specified in Table 5 The selection model demon- strates a preference for most fundamental features, while price-related features are generally disfavored.
fundamental features play a crucial role stabilizing predictions across different market distributions, while price fcatures tend introduce volatility in predictions. The feature selection module of InvariantStock effectively identifies and utilizes these invariant features, contributing stable stock return predictions. Since there only price features in the US dataset, this finding explains why InvariantStock does not yield comparable results as in the China datasot.
Additionally, The lookback window is set to 20 days, An important observation from this setting is the relatively consistent colouration across the time dimension in the feature representation. This consistency suggests that cach fcature contributes equally to the prediction over this 20-day period, indicating a stable influence of each feature within the look-back window. A particularly intriguing finding is the notable significance of the low price in contributing to stable predictions, which is worth further investigation.
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Furthermore, examined the most profitable transactions, as illustrated in Table 5 The analysis shows that the mask varies across different dates, with more similarity observed in closer dates. Notably, most profitable transactions occur in the Second-board Market, identifiable by stock codes beginning with "300" This trend is attributed to the more lenient price limitations in the Second-boart Market compared to the Main-board Market.
5.9 Ablation Study
Weighted Mask Vs Binary Mask We test weighted and binary mask methods for the feature selection on the China market, and the results are shown in Table 4 We can see that their performance (IC, ICIR, RankIC and RankICIR) on the test set are close. However, the backtest results (ARR, MDD, and SR) are different, the result. by binary mask better than that by weighted mask. The reason could be the use of the TopK strategy for the portfolio selection. Although the performance of the two methods is close, the performance of the selected stocks is different, which leads to the difference in the backtesting.
Method
IC
ICIR
RankIC
RankICIR
ARR
MDD
SR
Binary mask
0.0576
0.6896
0.1005
1.0900
0.8315
-0.1878
3.7198
Weighted mask
0.0552
0.6492
0.1007
1.0522
0.1953
-0.4752
0.4858
Table 4: The performance of predictions and the backtesting outcomes on China market for the different feature selection techniques, including weighted mask and binary mask.
Portfolio Size In both China market and US market, we examine how the Sharpe Ratio of compared methods fluctuates with the number of stocks included the portfolio, as depicted in Figure g It is observed that Invariant consistently surpasses other methods, regardless of the portfolio size. This consistent outperformance further substantiates the robustness of the InvariantStock approach. Moreover, the portfolio with 100 stocks could achieve the best performance, which means it relatively more stable than the best portfolio with fewer stocks in the US market when facing risks. With the increase in the number of stocks included in the portfolio, the performance of all the methods tested tends to converge towards the dalaset benchmark. That's also why we introduce the dataset benchmark into the comparisons.
6 Limitations and future work
In this study, we focus on lcarning invariant features across different environments, with the model showing a preference for fundamental features over price-related ones. However, the exclusion of certain price features may result in a loss of predictive capability. Consequently, if some confounding variables between the target and features are unobservable, the model may tend to discard these features to maintain stable predictions across varying environments, thereby limiting the model's performance. Therefore, exploring and identifying the latent variables from the observational ones could bridge this gap, which worth studying in the future. Another potential future direction involves the selection of environmental variables. In this work, we utilized the date index as the cnvironmental variable. However, market conditions, asset types, and goographical regions could also be considered as alternative environmental variables for stock prediction and are worth exploring.
7 Conclusion
In this study, we clucidate the phenomenon of market distribution shifts resulting from alterations in policy and economic landscapes. We introduce a learning framework named InvariantStock, designed to master these distribution shifts by learning invariant features. Rigorous experimentation on previously unseen stocks substantiates the resilience and efficacy of InvariantStock. Additionally, backtesting both the Chinese and United States stock markets reveals that InvariantStock attains unparalleled performance, affirming its consistent reliability. Intriguingly, our findings indicate that robust stock prediction hinges more significantly
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fundamental features opposed to those related to stock prices. However, it is noteworthy that the portfolio in this study was arranged in a basic manner, suggesting that future research could potentially enhance this aspect for further refinement.
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A The Trend of SSEC
M4
5500
SSE Composite Index
Mmmch
5000
4500
M ywrr
4000
3500
2020-01
2020-05
2020-09
2021-01
2021-05
2021-09
2022-01
2022-05
2022-09
2023-01
Figure The historical trend SSEC depicting the distribution shifting in China stock market.
B
Ablation Study
Figure
6 and Figure 7 are the sharp ratios from the backtesting with different numbers of stocks on the China stock market and the US stock market.
FactorVAE
DVA
3-
DoubleAdapte InvariantStock
Sharp Ratio
100
150
250
300
Number of stocks in portfolio
Figure 6: The SR results from backlesting various numbers of stocks in the China stock market reveal notable trend. Specifically, when the portfolio comprises 100 stocks, the SR achieved by InvariantStock is the highest among the methods tested. In comparison, the SRs from other baseline methods lag significantly behind InvariantStock's 'ss, underscoring its superior performance this context.
C Features Analysis
Table is the features in the China. dataset and the corresponding index. Figure 8 is the feature mask of best cases during the backtesting.
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2.5
Sharp Ratio
-255
-5.0
-7.5
FactorVAE
-10.0
DVA
DoubleAdapte
-2255
InvariantStock
150
200
250
360
Number of stocks in portfolio
Figure 7: The Sharpe Ratio (SR) results from backtesting with varying numbers of stocks in the US stock market show that InvariantStock continues to outperform other methods, albeit the margin of superiority not as pronounced observed in the China market. This suggests that while InvariantSlock maintains lead in performance, the competitive gap between it and other methods is narrower in the US market context.
Index Feature
2
3 Close
4 Previous close
5 Price change
6 Percentage
Open
High
Low
of change
Index Feature
8 Amount
9 Turnover rate
10 Turnover rate (CMC)
11 Volume ra- tio
12
13
Volume
PE (TTM)
Index
14
15
16 PS (TTM)
17 Dividend yield
18 Dividend yield (TTM)
19 Total share
20 Circulated share
Feature
PB
PS
Index
21
22 Market
23 Total Capi- talization
24 Next open
25
Feature
Free ciren- lated share
Overnight
Capitaliza- tion
price
change
Table 5: Features and its correspouding index in China datasel, Circulated Market Capitalization(MMCC Trailing Twelve Months(TTM) Price-to-sales Ratio(PS) Prock/book value ratio(PB) Price-earning ratio(PE). Among those, (0-11,2 2,,,24,25 are the price features or technique indicators, which are decided by the trading between investors, while 12-23 are the fundamental features that reflect the decision of the company or the financial situation of the company.
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Figure 8: The most profitable transactions made by InvariantStock in China stock market. the position filled in black means the masked features.
19


Reviews:
[{"review_id": "W1Dyp7pvaj", "reviewer": "Reviewer_vRBt", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Pros\n\n1. The introduction of invariant feature learning to address distribution shifts is a significant advancement in stock return prediction methodologies.\n\n2. The combination of environment-aware and agnostic modules provides a robust mechanism for feature selection and prediction.\n   \n3. Empirical performance is strong to show evidence of the framework's effectiveness.\n   \n4. The feature selection module ensures that only the most relevant and consistent features are used. \n\n## Cons\n\n1. The framework's performance in the US market was notably less impressive than in the China market, indicating potential issues with generalizability across different datasets.\n   \n2. The higher MMD compared to market benchmarks suggests the framework may not effectively manage risk in volatile / OOD conditions.\n   \n3. The multi-step training process and complex modules might lead to significant computational overhead, limiting real-time applicability.\n   \n4. Heavy reliance on historical data could be a limitation in highly volatile or unprecedented market conditions, where past data may not be indicative of future trends.\n   \n5. The binary mask and straight-through technique used in feature selection is possible with iterative alignments.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work introduces InvariantStock, a learning-based framework that addresses distribution shifts in stock markets by focusing on invariant feature learning across various environments.\n   \nA feature selection module is integrated, designed to identify features that remain consistent across different market distributions, thereby enhancing predictive stability and accuracy.\n   \nOne interesting idea from this work is to use maximizing mutual information between invariant features and the target variable while ensuring robustness against environmental variability.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "No requested changes, some optional adjustments. \n\n- Conduct additional experiments across different market conditions, asset types, and geographical regions to further validate the framework's robustness and generalizability.\n\n\n- To improve performance in markets with limited feature diversity, consider integrating additional sources of data (e.g., macroeconomic indicators, sentiment analysis from news articles, etc.) to enrich the feature set.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "BvrlQ1DL30", "reviewer": "Reviewer_dtfG", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n1. The problem is both practical and interesting.\n2. The approach is simple and easy to implement \n3. The experimental results demonstrate the effectiveness of the proposed approach\n\nWeaknesses: \n1. The overall technical strength is low; most of the technical details are simply adopted from other papers or rather standard techniques\n    For instance, most of the contents in page 4, 5, 6. \n2. The technical highlight is not clear, and it doesn't delve into too much on the invariant feature discovery and what it takes to get them. \n3. For this type of research, I am a bit worried about its practical and theoretical contribution: for the practical impacts, the industry likely adopt much more sophisticated approaches; for the theoretical side, it lacks of deep technical merits.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a new approach to learn invariant features for dealing with stock market shifting. They use an information theoretical criterion in helping select the feature sets which are invariant to environmental factors. The experimental results show the effectiveness of the proposed approach.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. I would like to see some in-depth discussions on discovering the invariant features, what technical challenges are there and how you solve it?\n    What makes this problem unique and challenge if any?\n\n2. Can this idea be generalized to other settings, the invariant features should have applications beyond stock market. Can you find any?", "broader_impact_concerns": "The approach has the potential to be adopted and further studied by industry and academics related to finance and AI.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "EiEe2m4f8D", "reviewer": "Reviewer_ecVD", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- Important and Impactful Research Problem: The paper addresses a significant and impactful research problem with a well-focused research scope, specifically the challenge of predicting stock returns amidst distribution shifts.\n- Invariant Feature Learning: The approach effectively learns invariant features that can handle distribution shifts in the stock market, enhancing robustness and prediction accuracy.\n- Framework Design: The framework design comprises an environment-agnostic prediction module and an environment-aware prediction module. This dual-module design facilitates the learning of invariant features across different environments.\n- Empirical Validation and Strong Performance: The framework’s effectiveness is validated through experiments in real-world market scenarios, demonstrating the strong performance of InvariantStock compared to state-of-the-art methods.\n\nWeaknesses:\n- Insufficient Technical Justifications: The paper lacks comprehensive technical justifications for all the design choices made in the proposed framework.\n- Weak Evaluation: The evaluation of the proposed method is not robust enough to convincingly demonstrate its effectiveness.\n- Insufficient Discussion of Related Work: There is a lack of thorough discussion of related work, which would provide better context and highlight the contributions of this study.\n- Hard-to-Follow Presentation: The presentation workflow is difficult to follow in some sections, which affects the clarity and readability of the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the significant and impactful challenge of predicting stock returns amidst distribution shifts, presenting a well-focused research scope. The proposed approach effectively learns invariant features that enhance robustness and prediction accuracy in the stock market. The innovative framework design comprises an environment-agnostic prediction module and an environment-aware prediction module, facilitating the learning of invariant features across different environments. The effectiveness of the framework is empirically validated through experiments in real-world market scenarios, demonstrating that InvariantStock outperforms state-of-the-art methods in various evaluation metrics.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Add Needed Technical Justifications: Include thorough technical justifications for design choices such as the use of binary masks instead of weighted ones, and the selection of hinge loss. These justifications should be supported in the ablation study section to demonstrate their impact on the model’s performance.\n- For experiments, there are a couple of changes needed to be made.\n(a) Data Processing and Parameter Choices: Provide a detailed discussion on how the training data is processed and the rationale behind parameter choices.\n(b) Expanded Testing Cases: Include more diverse cases for testing to demonstrate the model’s robustness across different scenarios.\n(c) Impact of Original Number of Features: Test and analyze how the original number of features affects the results, providing insights into the model’s sensitivity to feature selection.\n(d) Explanation of Experimental Results: Improve the explanation of experimental results by discussing what the metric improvements mean in practical terms, rather than merely presenting a table of results.\n- Reflect Novelty: Extend the discussion of related work to better reflect the novelty of the current research. Clearly articulate the fundamental research contributions of this work compared with previous studies. Expand the discussion on feature selection and invariant learning. Clarify the types of features referred to in this work, specifying whether statistical features like moving averages or other economic indicators such as GDP and EPS are considered.\n- Improve Presentation\n(a) Problem Formulation: Enhance the presentation of the paper by clearly formulating the problem in Section 3. Move beyond basic background definitions to a detailed discussion of the specific problem being addressed.\n(b) Equation Explanations: Provide a thorough explanation of Equation 1, including the concept of mutual information between the invariant features F and the target variable Y. Clearly define and explain the H function and its significance in the context of the paper.\n(c) Explain how E is constructed.", "broader_impact_concerns": "The technique may be limited to scenarios where a large number of stock data features are available and may not be applicable to scenarios with only a limited number of features.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The general view of the reviewers is that, although the algorithmic contribution is modest, the work is technically sound and will be of interest to some in the TMLR audience, primarily because stock market prediction is such a challenging real-world application; progress here could have positive repercussions in other critical time series domains.

In their final version, the authors should be sure to incorporate new results and perspectives that were produced during the review process. I would also like to encourage the authors to add a dedicated Limitations section (e.g., directly before the Conclusion) collecting all the limitations mentioned throughout the paper and in the reviewer discussion into a single place. This would be very useful in this kind of paper, since the reader immediately wants to know why they should or shouldn't rush to implement this method in stock trading today.

---AE
Decision: Accept (poster)