AlphaSAGE: Structure-Aware Alpha Mining via GFlowNets for Robust Exploration
Authors: Binqi Chen
Hongjun Ding
Ning Shen
Taian Guo
Jinsheng Huang
Luchen Liu
Ming Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Alpha Mining
Generative Flow Networks
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: Alpha mining with Gflownets

Abstract: The automated mining of predictive signals, or alphas, is a central challenge in quantitative finance. While Reinforcement Learning (RL) has emerged as a promising paradigm for generating formulaic alphas, existing frameworks are fundamentally hampered by a triad of interconnected issues. First, they suffer from reward sparsity, where meaningful feedback is only available upon the completion of a full formula, leading to inefficient and unstable exploration. Second, they rely on semantically inadequate sequential representations of mathematical expressions, failing to capture the structure that determine an alpha's behavior. Third, the standard RL objective of maximizing expected returns inherently drives policies towards a single optimal mode, directly contradicting the practical need for a diverse portfolio of non-correlated alphas. To overcome these challenges, we introduce **AlphaSAGE** (**S**tructure-**A**ware Alpha Mining via **G**enerative Flow Networks for Robust **E**xploration), a novel framework is built upon three cornerstone innovations: (1) a structure-aware encoder based on Relational Graph Convolutional Network (RGCN); (2) a new framework with Generative Flow Networks (GFlowNets); and (3) a dense, multi-faceted reward structure. Empirical results demonstrate that AlphaSAGE outperforms existing baselines in mining a more diverse, novel, and highly predictive portfolio of alphas, thereby proposing a new paradigm for automated alpha mining. Our code is available at https://anonymous.4open.science/r/AlphaSAGE-3BA9.

ALPHASAGE: SESCCTRRE-AARRE ALPHA MINING VIA GFLOWNETS FOR ROBUST EXPLORATION
Binqi Chen
Hongjun Ding
Ning Shen
Taian Guo
Jinsheng Huang
Luchen Liu
Ming Zhang
School of Computer Science, National Key Laboratory for Multimedia Information Processing,
PKU-Anker LLM Lab, Peking University, China 7hengren Quant, Beijing. China
Baruch College, City University of New York Statistics, University of British Columbia {cbq, taianguo, his} @ttu.pku.edus hongjun. wang.baruchmfeee @mmil..cmmmmm ning.shen @ sta..ubc.ca, liulc @ zwengrenquant.com com, mzhang cs @ku..edu.cn
ABSTRACT
The automated mining of predictive signals, or alphas, is a central challenge in quantitative finance. While Reinforcement Learning (RL) has emerged as promising paradigm for generating formulaic alphas, existing frameworks are fun- damentally hampered by triad of interconnected issues. First, they suffer from reward sparsity, where meaningful feedback is only available upon the comple- tion of full formula, leading inefficient and unstable exploration. Second, they rely on semantically inadequate sequential representations of mathematical expressions, failing to capture the structure that determine an alpha's 's behavior. Third, the standard RL objective of maximizing expected returns inherently drives policies towards a single optimal mode, directly contradicting the practical need for diverse portfolio of non-correlated alphas. To overcome these challenges, we introduce AlphaSAGE (Structure-A ware Alpha Mining via Generative Flow Networks for Robust Exploration), novel framework built upon three corner- stone innovations: (1) a structure-aware encoder based on Relational Graph Con- volutional Network (RGCN); (2) new framework with Generative Flow Net- works (GFlowNets); and (3) a dense, multi-faceted reward structure. Empiri- cal results demonstrate that AlphaSAGE outperforms existing baselines in min- ing a more diverse, novel, and highly predictive portfolio of alphas, thereby proposing new paradigm for automated alpha mining. Our code is available at
INTRODUCTION
The primary objective in quantitative trading is to identify and exploit market inefficiencies, pur- suit centered on the mining of "alphas". These alphas are predictive signals, typically represented mathematical expressions, that aim to forecast asset returns and thus serve the cornerstone of systematic trading strategies.' Therefore, alpha mining (efficient construction of high-quality al- phas) constitutes the core of quantitative research: high-quality alphas enable more accurate return forecasting, improved risk-adjusted portfolio construction, and ultimately superior excess returns.
Traditionally, alpha mining has been a manual, hypothesis-driven process. Researchers propose financial or economic hypotheses, translate them into candidate alphas, and validate their predictive power through statistical tests or backtesting. While this pipeline has led to influential discoveries such as value, momentum, and quality alphas (Kakushadze, 2016), it suffers from limited scalability and strong reliance on human intuition. With the increasing complexity of financial markets, the bypothesis-drivee paradigm struggles to cope with vast, non-linear interactions in high-dimensional data, making it increasingly challenging to uncover novel and uncorrelated signals.
"Internship at Zhengren Quant. 1 'Corresonding author.
An example of alpha shown in Figure 1.

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Recent advances have motivated the shift towards automated alpha mining, where machine learning algorithms systematically search through the enormous combinatorial space of possible formulas. Early efforts often relied on Genetic Algorithm (GA) (Chen et al., 2021; Zhang et al., 2020; Cui et al., 2021), which evolves candidate formulas using mutation and crossover operators. Despite producing interpretable formulas, GA methods can be computationally inefficient and tend converge to local optimum if mutation rate is not carefully designed. More recently, Reinforcement Learning (RL) (Yu al., 2023; Zhu Zhu, 2025; Zhao et al., 2024; Xu et al., 2024) has emerged a powerful alternative, framing alpha construction as sequential decision-making process in which an agent incrementally builds formulas. RL-based methods promise higher efficiency and scalability but also inherit several critical challenges, including reward sparsity, structural underrepresentation and limited diversity generated alphas.
The direct application of RL to alpha mining is gen- erally fraught with significant obstacles that limit its efficacy, First, current methodologies often suf- fer from a severe "cold-start" problem, as the re- alpha's Infor-
(a)
(b)
Log
ward signal-typically based mation Coefficient (IC) is extremely sparse (Zhao al., 2025). Second, most existing approaches rep- resent alpha expressions as simple sequences of to- kens, often processed by sequential models such as LSTMs. Such representations fail to capture the logical and hierarchical structure inherent formu- las. Finally, the traditional RL, which is designed maximize singular reward function, tends to produce relatively uniform path for alpha mining, lacking the diversity essential for constructing robust portfolios (Tang et al., 2025).
Log(TsStd(Slow, 10d))
TsSid
Slow
10d
C
BEG
Slow
10d
TsStd
Log
SEP
Figure Different forms of alpha: (a) For- mulated alpha; (b) Alpha's expression tree;
(c) Reverse Polish Notation (RPN) for alpha.
To overcome these limitations, we propose AlphaSAGE (Structure-Aware Alpha Mining via Generative Flow Networks for Robust Exploration), a comprehensive framework designed to ad- dress the core problems of exploration, semantic understanding, and diversity alpha mining. To evaluate the effectiveness of the model, we conducted extensive experiments based real historical data from both the Chinese and U.S. stock markets. The experimental results demonstrate that the model outperforms existing models across different markets.
In summary, our contributions are as follows:
We introduce a structure-aware encoder based Relational Graph Convolutional Net- work (RGCN) (Schlichtkrull et al., 2018) that operates on Abstract Syntax Tree (AST) representations of alphas to capture their semantic and compositional nature.
We propose generative framework using Generative Flow Networks (GFlowNets) (Ben- gio et al., 2021; Malkin et al., 2022; Bengio et al., 2023) that learns to sample diverse set of candidates, directly addressing the need for varied alpha portfolio.
We present dense, multi-faceted reward function that combines terminal perfor- mance with intrinsic rewards for structural integrity and novelty to effectively guide the GFlowNet's exploration.
BACKGROUND AND RELATED WORK
2.1 ALPHAS IN QUANTITATIVE TRADING
In quantitative equity trading, an alpha is function that maps past information about each asset, such as historical prices, volumes, or fundamental ratios, to real valued score for every stock on given date. At each rebalancing time the alpha values St,a across stocks are typically standardized in the cross section and then transformed into a portfolio signal, for example by going long stocks with high scores and short on stocks with low scores under risk and leverage budget.
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The next period portfolio return is
R₂ = Σ Whnn L,t
where Wy,n are the portfolio weights derived from the alpha signal and 1, are realized returns. The predictive quality of an alpha is often summarized by the information coefficient, defined as the cross sectional correlation between st, and rt+1,n over time. In practice, production systems maintain a library of many such alphas and combine a subset of them into a diversified trading signal, which motivates automated methods for discovering alphas with strong predictive power and low mutual correlation.
2.2 ALPHA MINING AND COMBINATION
In quantitative finance, an alpha is a deterministic transformation of historical market data into signal that aims to forecast future returns. When expressed as a symbolic program (c.g., an abstract syntax tree), alpha remains interpretable and auditable. Alpha quality is commonly summarized by correlation brased metrics (e.g., IC) computed between alpha outputs and subsequent returns.
manual and hypothesis- driven. More recent automation most no- tably genetic algorithm (GA) (Chen et al., 2021; Zhang et al., 2020; Cui et al., 2021) and reinforce- ment learning (RI.) (Yu et al., 2023; Zhu & Zhu, 2025; Zhao et al., 2024; Xu et al., expa)-expanded the search space but introduced three recurring challenges: sparse and delayed rewards, weak encod- ing of alpha structure, and mode collapse toward a few similar solutions. Because single alphas are typically unstable across time and markets, practitioners assemble a library of alphas and combine them into a portfolio-level signal. Simple linear combinations are prevalent in practice, yet high correlations among alphas can make coefficient estimates unreliable and reduce both robustness and interpretability.
Early discovery pipelines
These observations motivate frameworks that jointly optimize for predictive power and diversity. Our design follows this principle: it encourages structurally distinct alphas during generation and combines them with transparent, adaptively weighted scheme that emphasizes low cross-alpha dependence. Additional details appear in Appendix B.I.
2.3 GRAPH NEURAL NETWORKS
Graph neural networks (GNNs) (Scarselli et al., 2008; Yao et al., 2019; Schlichtkrull et al., 2018) update each node by gathering information from its neighbors and then refining the node's repre- sentation with that context. Stacking layers allows information to propagate over multiple hops, so nodes capture both local attributes and broader structural relations. When factor candidates are rep- resented as graphs-such structural constraints more naturally than sequence models. This makes them attractive for learning embeddings of factors, guiding search over symbolic expressions, and measuring diversity at the representation level. Additional details appear in Appendix B.2.
trees for formulaic alphas-GNNs GNNs can encode semantic similarity and
2.4 GENERATIVE FLOW NETWORKS
Generative Flow Networks (GFlowNets) (Bengio et al., 2021; Malkin et al., 2022; Bengio et al., 2023) are generative learners that construct objects step by step and aim sample diverse set of high-reward solutions rather than collapsing to single optimum. They treat generation moving through a directed acyclic state space from an initial empty state to a terminal, valid object. By learn- ing complementary forward and backward policies and matching "flow" through states, GFlowNets approximate a sampling distribution that is shaped by the downstream reward. Practically, this yields exploration that is both reward-awar and diversity seekiigg, producing portfolio of candidates with varied structures and competitive quality-propertic properties that are well aligned with the needs of alpha discovery and combination. Additional details appear in Appendix B.3.
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3 METHODOLOGY
3.1 FRAMEWORK OVERVIEW AND PROBLEM FORMULATION
The primary objective of automated alpha discovery is navigate vast, combinatorial search space X' of potential mathematical expressions, or "alphas". Each alpha a E X' is a function that maps historical market data for a universe of N assets with A features at day d. denoted as Xa Rx to vector of predictive signals zd a(Xa) E R. The quality of these signals is evaluated against future asset returns Yd E R...
Existing RL frameworks model this as sequential decision making problem to construct a syner- gistic portfolio of alphas. In this paradigm, an agent iteratively generates new alphas to add to an evolving pool, F. The reward for generating new alpha, Onew, is its marginal contribution to the performance of a combination model a(+) trained on the updated pool. The objective each step to find alpha that maximizes this improvement:
new arg max E [R(a[FF),
(2)
OE
where the reward is defined as R(a(FF) IC(c(X;FU fa))) IF) This formulation creates a non-stationary Markov Decision Process, as the reward for any given alpha changes when- ever the pool F, is updated. While this approach encourages synergy within the single, greedily constructed portfolio, it does not learn a global distribution over all high-quality alphas.
We reformulate alpha discovery models the distribution of high-quality alphas over the entire space X. The policy is trained such that the probability of sampling any alpha is proportional to a carefully designed reward function R(a), which reflects its intrinsic quality and novelty:
problem of learning generative policy Po(a) that directly
Pa(a) x R(a), E
(3)
By sampling from this learned global distribution, rather than following single construction path, we can generate a more diverse and robust portfolio of candidate alphas.
3.2 ALPHA GENERATION VIA GENERATIVE FLOW NETWORKS
To address the need for diverse portfolio, we propose new framework with GelowNets. A
GFlowNet is a probabilistic generative model designed to learn a stochastic policy for sampling objects from a space X' with probability P(a) proportional a given reward function R(a).
The construction of an alpha is modeled as trajectory т = (so - s1 + ... + a) in a state space represented as directed acyelic graph (DAG).
States (s E S): Partially constructed ASTs. The initial state so is an empty tree. Terminal states are complete and valid ASTs, forming the space X' C.
Actions (a E A): Adding new token (operator feature) partial AST. According to the state 8, invalid actions are masked and the next token is sampled from the masked distribution.
open leaf node of
*
Complete Trajectories: A full trajectory corresponds to constructing valid expression tree. Only such trajectories are considered terminal and eligible for evaluation.
To prevent expressions from growing excessively long or from being forcefully terminated into invalid states after exceeding the maximum token length, we incorporate an early stop mechanism. Specifically, when the current stack already forms valid expression, the generation process may stop with a probability:
Len(s:)
p MaxLen
where Len(s;) is the number of nodes in st, and MaxLen is the maximum allowed length. This mechanism balances exploration of longer expressions with the efficiency of producing syntactically valid formulas.
A GFlowNet learns a forward policy (p ss; 8) for constructing objects and a backward pol- icy PB 1:41; 0) for deconstruction. The training objective enforces flow-mate mat-hinggngg condition
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Embedding Space
Behavior
Emnbedding
BEG
St+1
Log
Distance
Div
Rsa
Slow
Shigh
St
KNN
Div
Quality Check
Slow
Shigh
Log
SEP
co
An
Cn
Rg
Early Stop
SEP
Slow
Shigh
GNN Encoder
Alpha Pool
Sample
GFN
Masked
History Data
Ryc
Logits
Masks
Sub Log Div
SEP
Curve
Softmax
Token
Action Logils
Logits
Buffer
(Tr (s))
LENT
Tree Node
Figure 2: An overview of AlphaSAGE. (a) AlphaGenerator. Starting from an empty state, we iteratively construct partial AST for the formula prefix. An RGCN encoder produces node and pooled graph embeddings, which condition the GFlowNet forward policy to sample the next to- ken from syntactically valid actions only. Rollouts terminate by emitting SEP once the minimal length met or MaxLen reached. The resulting formula a, and its embedding 6, are stored for reward evaluation and training. (b) AlphaEvaluator. For each sampled On,, compute reward components on historical cross-sectional data: RSA (structure alignment), RNov (novelty via low correlation with reference set), and Ric (predictive IC). We additionally use an entropy regularizer CENT to encourage exploration. The combined reward R(a,) updates the GFlowNet and encoder via the trajectory balance objective LTB augmented with LENT-
throughout the state space, ensuring that the probability of generating complete alpha matches target distribution:
R(a) Z
P(a)=
Σ PF(T)
(5)
T:Sn=a
where Z -- R(a') is learnable parameter representing the total flow or partition function. While several loss functions exist, we use the Trajectory Balance (TB) loss, which is particularly suitable for our scenario as it focuses full trajectories. The TB loss for given trajectory is:
n
a
(TR(T)= log Zg Σ log Pr(ss log R(s,)
Σ log Pnn Pn(st_1/ss::)
where Ze a learnable scalar approximating total flow. Minimizing this loss over sampled trajecto- ries trains the policy, resulting in a model that produces a diverse set of high-reward alphas.
3.3 GNN EMBEDDING AND STRUCTURE-AWANT REWARD
A fundamental limitation of existing methods is their reliance sequential encoders (e.g.. LSTMs) operating on flattened representations like Reverse Polish Notation. Such an approach fails to cap- ture the hierarchical structure of mathematical expressions, treating logically equivalent formulas (e.g., close + open and open + close) as different sequences. To overcome this, we first parse every formulaic alpha into its corresponding AST, denoted as T. (Va Ea), where Va is the set of nodes (operators and features) and E, the set of edges representing the computational hierarchy. This representation is invariant semantically inconsequential syntactic variations.
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To capture the heterogeneity of relations between different types of operators and features in the Ta we adopt RGCN as the encoder. Unlike standard GNNs that treat all edges uniformly, RGCNs explicitly model multiple relation types, which is crucial for distinguishing. for example, the edge between a temporal operator and a feature versus the edge between a temporal operator and its window length.
Each node v E Va is initialized with a feature vector (1) At layer I, the hidden representation of node is updated as:
1 w I
h ReLU
Σ
wo'l h(
Cu.r
FER (e)
Ca MaxPooling (he (1) leev.).
where is the set relation types, (T( denotes the neighbors of node o connected via relation Corr is a normalization constant (e.g., (v)),, w: is the trainable weight matrix specific to relation r, and W is self-loop transformation matrix. This embedding provides a Glation-awaan and structure-aware representation of the alpha.
To ensure that the learned embedding not just structurally aware but also predictive of the alpha's actual behavior, introduce Structure- (SA) reward. The goal is to further ensure that alphas with similar structural embeddings exhibit similar behavioral patterns.
Let Zj R DxN be the time-series vector of cross-sectionally normalized outputs for on and (d) RN N is the output at day d. We define a behavioral distance based on the outputs of az and ay:
dechav (a;;; a1))
D Σ (z(d) -
1
exp(-Ileo, - Co, 113)
Wij EkENK((a)) exp(-Ilea, Coi [2] jENx(a)).
(10)
=exp Rsa(a;) - exp
Σ
Wij dochar (at a4))
(11)
jENic(as)
where (a;) is K -earrrttt neighbors of as.
3.4 MULTI-FACETED REWARD FUNCTION AND TRAINING OBJECTIVE
The effectiveness of the GFlowNet is critically dependent the design of the reward function R(a). To address reward sparsity and guide exploration effectively, we design a dense, multi-faceted reward function that dynamically combines several components.
The total reward for a completed a at training step a weighted sum of three components:
Terminal Performance Reward (Ric): The primary measure of an alpha's predictive power, defined as its Information Coefficient:
Cov(a(Xa).uu va)
(ac IC(a, v)
Eg /Var(a(Xa)) Var(ya)
(12)
2. Structure-Aware Reward (Rsa): As defined in Eq. 11, this reward provides a dense signal for aligning the alpha's structural embedding with its behavior.
3. Novelty Reward (RNov): To encourage the discovery of novel alphas, we introduce elty reward. It penalizes similarity to a dynamically updated library Fanowa of previously discovered high-quality alphas. The definition is:
Rxov(a)
(IC(a,a)))
(13)

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These reward components combined using time-dependent weighting scheme to balance dif- ferent objectives throughout the training process. The final reward function is:
R(a,T) = Ric(a) + (sa) + (TT)))
(14)
where A(T) = (1 Tamed ) Amax is scheduling function that gradually decreases the weight of the structure-aware reward, and (T()
T
Tameal
Mmax a weight for the novelty reward.
Furthermore, prevent premature convergence and encourage fine-grained exploration at the action level, we add a policy entropy bonus to our final training objective. The objective is to minimize the expected Trajectory Balance loss regularized by the entropy of the forward policy:
CENT - PPr
Σ (T(o(((ss))
(15)
Limal = (F..P)(r.)) [TTB(T)] + LENT,
(16)
where (TTo(118x) is the entropy of the action selection policy at state 8 and ,3 is a hyperparametee controlling the strength of the entropy regularization. This comprehensive objective guides Al- phaSAGE to learn generative policy that produces a diverse, novel, and highly predictive portfolio of alpha alphas.
Compatibility of reward components. By construction, the three reward components share the same sign for desirable behaviours. Higher predictive performance, higher structure behaviour alignment, and higher novelty all increase the overall reward, so there is no explicit negative cou- pling between terms. In the alpha-mining context, Ric drives predictive quality, RSA shapes the embedding space so that structurally similar alphas behave similarly, and Rvov discourages redun- dancy with respect to the evolving factor library rather than competing with Ric. The components are therefore designed to be compatible and to act in a complementary way during training, rather than to optimise conflicting objectives.
3.5 ALPHA COMBINATION
For the combination stage, we follow the approach proposed in AlphaForge (Shi et al., 2025a). Specifically, instead fixing a static set of alphas, the framework performs a dynamic re-selection and linear combination of mined alphas. At each period, recently effective alphas are filtered and re-weighted through simple linear regression, yielding a time-varying "Mega-Alpha."
This design is advantageous because it adapts quickly to regime shifts while maintaining inter- pretability: alpha contributions remain transparent, and the portfolio avoids overfitting by discard- ing stale redundant signals. Compared with complex non-linear combiners, this method offers balance between robustness, efficiency, and explanatory clarity.
EXPERIMENTS AND RESULTS
4.1 EXPERIMENT SETTING
Based on prior work (Yu et al., 2023; Tang et al., 2025) and real-world trad- ing scenarios, we employed two types of metrics for model evaluation.(I): Correlation Metries, including Information Coefficient (IC), IC Information Ratio (ICIR), Rank Information Coefficient (RIC), RIC Information Ratio (RICIR); (2): Portfolio Metrics, including Annualized Return (AR), Maximum Drawdown (MDD), Sharpe Ratio (SR). All metrics are better when higher. Detailed definitions and backtest settings are provided the Appendix D.I.
Evaluation Metrics.
Datasets. We selected three important subsets from two major markets (Yang et al., 2020): the CSI300, CS1500 and CSI1000 in the Chinese market, and the S&P500 in the U.S. market. The split of the training set/validation set/test set for the Chinese market is defined as follows: 2010- 01-01 to 2020-12-31, 2021-01-01 to 2021-12-31 2022-01-01 to 2024-12-31, For the US market: 2010-01-01 to 2016-12-31 / 2017701-01 to 2017-12-31 / 2018-01-01 to 2020-12-31. Detailed hy-
perparameter settings are provided in Appendix D.2.
Due limitations in the data source, the US market data used in this study concludes on 2020-12-31.

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Baselines. We compare AlphaSAGE with several baseline approaches: (1) Traditional machine learning methods include MLP (Murtagh, 1991). LightGBM (Ke et al., 2017). and XGBoost (Chen & Guestrin, 2016); (2) GA-based methods include GP (Chen et al., 2021); (3) RL-based methods include AlphaGen (uu et al., 2023) and AlphaQCM (Zhu & 7hu, 2025); (4) Generative adversarial setworks-based methods include AlphaForge (Shi et al., 2025a); (5) Large Language Model(L..M-
based methods include AlphaAgent (Tang et al., 2025). The details of baseline are available at Appendix D.3.
4.2 OVERALL PERFORMANCE
Table summarizes results across CSI300/500/1000 and S&P500: AlphaSAGE ranks first all correlation metrics, with notably higher ICIR/RICIR, and these gains translate into the best portfolio outcomes (highest annualized return, lowest drawdown, highest Sharpe).
we examined robustness with respect to random initialization. On CSI300, running multiple seeds for all methods yields consistent improvements of AlphaSAGBi over strong baselines in both cor- relation and portfolio metrics, with the same performance ranking preserved across runs (see Ap- pendix 12). This suggests that the empirical gains of AlphaSAGE are stable and not attributable to a particular random seed.
Figure 3 further shows that CSI300 cumulative returns, with smoother drawdowns, faster recoveries, and stronger rebound capture; the CSI300 index lags throughout, underscoring the value of active factor discovery and combination.
AlphaSAGE maintains persistent lead in
1.3
AlphaSAGE
MLP
AlphaForge
XGBoost
AlphaAgent
AlphaQCM
LightGBM
CSI300 Index
AlphaGen
Cumulative Return
1.1
1.0
0.9
0.8
0.7
2022-01
2022-05
2022-09
2023-01
2023-05
2023-09
2024-01
2024-05
2024-09
2025-01
Date
Figure 3: Cumulative return on CSI300 (2022-2024). Comparison among AlphaSAGE (ours), all baselines, and CS1300 Index benchmark.
4.3 ABLATION STUDY
Table 2 shows that the plain GFlowNet baseline is weakest; adding only early stopping (ES) further hurts, implying ES needs a stronger encoder. Replacing the sequence encoder with a GNN provides the largest single lift across correlation and risk metries, underscoring the value of structure-and aware representations. Adding the structure-aware reward (SA) improves ranking stability (ICIR/RICIR) and tightens drawdowns. Introducing the novelty reward (NOV) raises both signal quality and trad- ability by reducing redundancy among factors. Finally, the entropy regularizer (ENT) yields the best overall results-higher IC/RIC, AR, and Sharpe with controlled MDD-iindccatng improved exploration without brittleness and supporting the method's 's robustness component choices.
4.4 SENSITIVITY ANALYSTS
We vary the weights of novelty reward (Rxov) and structure-awand reward (Rsa) on CSI300 (Fig. 4). For RNOv, correlation and portfolio metries improve small-moderate levels and remain
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Table 1: Performance Comparison of Different Methods CSI300, CSI500, CST1000 (China) and S&P500 (U.S.). Bold and underlined numbers represent the best and second-best performance across all compared approaches, respectively.
Correlation Metrics
Portfolio Metrics
Dataset
Method
IC
ICIR
RIC
RICIR
AR
MDD
SR
MLP
0.020
0.158
0.019
0.142
3,54%
-00.9%
0.68
Ligh
0.011
0.124
0.006
0.064
2.61%
-18.5%
0.53
XGBoost
0,031
0.243
0.033
0.248
5,40%
-17.5%
1.26
GP
0,026
0.215
0.028
0.216
6.80%
-17.6%
1.55
CS1300
AlphaGen
0,058
0,414
0.057
0.360
4.00%
-22.6%
0.76
AlphaQCM
0,043
0.262
0.042
0.246
1.95%
-24.8%
0.36
AlphaForge
0.041
0.259
0.052
0.306
3.90%
-21.9%
0.88
AlphaAgent
0,051
0.325
0.056
0.329
2.16%
-26.9%
0.65
AlphaSAGE(ours)
0.079
0.496
0.094
0.583
7.62 %
17.3%
1.71
MLP
0.017
0.185
0.020
0.233
1.56%
-24.3%
0.27
LightGBM
0.024
0.305
0.021
0.264
4.61%
-17.5%
0.89
XGBoost
0.039
0.365
0.052
0.528
5.50%
-17.1%
1.15
GP
0.014
0.238
0.022
0.233
3.04%
-19.4%
0.56
CSI500
AlphaGen
0.032
0.270
0.031
0.230
1.15%
-32.4%
0.19
AlphaQCM
0.048
0.378
0.073
0.546
4.06%
-24.0%
0.75
AlphaForge
0.053
0.345
0.083
0.600
4.18%
-16.7%
0.93
AlphaAgent
0.053
0.396
0.065
0.495
1.82%
22.4%
0.36
AlphaSAGE(ours)
0.054
0.379
0.084
0.637
5.53%
16.0%
1.20
MLP
0.048
0.384
0.069
0.621
3.22%
25.7%
0.47
LighIGBM
0.067
0.501
0.083
0.656
4.98%
-2..7%
0.98
XGBoost
0,062
0,498
0.086
0.695
4.72%
-3.55
0.91
GP
0.058
0,474
0.079
0.657
4.32%
-24.7%
0.67
CST1000
AlphaGen
0.071
0.540
0.092
0.713
5.27%
-24.0%
0.92
AlphaQCM
0.065
0.453
0.107
0.682
7.12%
-20.6%
1.31
AlphaForge
0.071
0.537
0.095
0.742
6.07%
-1111%
1.06
AlphaAgent
0.072
0.579
0.089
0.712
5.51%
-0.5%
1.01
AlphaSAGE(ours)
0.076
0.582
0.096
0.736
7.62%
20.3%
1.33
MI.P
0.035
0.287
0.020
0.143
12.85%
-5.6%
3.35
LightGBM
0.023
0.196
0.018
0.165
11.11%
-5.1%
4.22
XGBoost
0.016
0.159
0.026
0.168
13.25%
-8.3%
3.61
GP
0.032
0.308
0.002
0.016
13.39%
-13.0%
3.15
S&P500
AlphaGen
0.044
0.396
0.013
0.127
10.31%
-5.5%
3.96
AlphaQCM
0.038
0.262
0.010
0.071
13.86%
-13.0%
3.30
AlphaForge
0.039
0.422
0.031
0.324
17.24%
-5.0%
6.30
AlphaAgent
0.048
0.479
0.033
0.315
18.66%
-5.7%
6.27
AlphaSAGE(ours 0.052
0.493
0.038
0.382
19.47
-4.2%
6.32
broad plateau before tapering when novelty dominates. For RSA, improvements are largely mono- tonic across correlation and portfolio metrics with stable drawdowns. Overall, AlphaSAGE exhibits smooth responses without abrupt performance drops, indicating robustness wide range of hy- perparameter choices and low sensitivity around the operating region.
CONCLUSION
We introduced AlphaSAGE, a structure-aware. diversity-seeking framework for formulaic alpha discovery and combination. The approach unifies GNN encoder for symbolic expressions,

---

Table 2: Ablation study on CS1300. The base model GftowNets, where ES denotes Early Stop- ping, GNN indicates whether a GNN or LSTM is used as the encoder, SA stands for Structure-Aware Reward, NOV represents Novelty Reward, and ENT denotes Entropy Loss.
Included Components
Correlation Metrics
Portfolio Metrics
ES
GNN
SA
NOV
ENT
IC
ICIR
RIC
RICIR
AR
MDD
SR
X
X
X
X
X
0.048
0.393
0.057
0,437
3.63%
-22.9%
0.72
X
X
X
X
0.046
0.313
0.060
0.397
-0.47%
-24.8%
-0.11
V
X
X
X
0.070
0.495
0.088
0.554
5.58%
-19.4%
1.25
V
X
X
0.071
0.453
0.088
0,566
4.68%
-17.6%
1.14
V
V
x
0.075
0.494
0.092
0.614
6.77%
-17.8%
1.53
V
V
0.079
0.496
0.094
0.583
7.62%
-17.3%
1.71
Correlation Metries
Correlation Metries
0.7
0.7
0.15
0.15
0.6
0.6
ICIR RRRCIR
IC/ RIC
0.10
0.5
0.10
0.5
0.4
0.4
0.05
0.05
0.3
0.3
0.00
0.2
0.00
0.2
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
Protfolio Metrics
Protfolio Metries
0.0
AR. MMD
2
2
SR
-0.1-
-0.1
-0.2
-0.2
0.2
0.4
0.6
0.8
1.0
0.4
0.6
0.8
Weight of Novel Reward AR
Weight of Self-Supervised Reward ICIR
IC
RIC
MMD
RICIR
SR
Figure 4: Sensitivity analysis of the weights for RNOV and Re on CS1300. For the y-axis. IC, RIC, AR, and MDD refer the axis on the left; ICIR, RICIR, and SR refer the axis on the right.
GFlowNet generator that explores multiple high-reward modes, and a multi-signal training objective coupling predictive quality, representation behaavio alignment, novelty pressure, and entropy- based regularization. A transparent, dynamic linear combiner then translates candidate alphas into trad- able portfolio signal while maintaining interpretability.
Empirically, AlphaSAGE delivers first-rank correlation metrics across CSI300/500 and S&P500 and consistently converts these gains into superior portfolio outcomes. On CSI300 (2022-2024) its cumulative return curve maintains persistent lead with smoother drawdowns, faster recoveries, and stronger rebound capture, underscoring robust generalization across market regimes. Ablations attribute the largest single lift to structure-aware encoding (GNN), with self-supervised alignment improving rank stability and risk control, novelty rewarding useful diversity that lifts both signal quality and tradability, and entropy regularization sharpening exploration without brittleness. Sen- sitivity studies show smooth responses over broad ranges of the novelty and alignment weights, indicating low tuning burden and practical robustness.
Together, these results demonstrate that coupling structure- aware representation, iiversity-seeking generation, and principled multi-signal supervision yields reliable improvements in both signal qual- ity and its conversion to realized returns, while preserving transparency how factors are generated and combined.
10

---

ACKNOWLEDGMENTS
This paper is partially supported by grants from the National Key Research and Development Pro- gram of China with Grant No. 2023YFC3341203 and the National Natural Science Foundation of China (NSFC Grant Number 62276002).
REFERENCES
Emmanuel Bengio, Moksh Jain, Maksym Korablyov, Doina Precup, and Yoshua Bengio. Flow network based generative models for non-iterative diverse candidate generation. Advances in neural information processing systems, 34:27381-27388 --7399,,,,, 2021.
Yoshua Bengio, Salem Lahlou, Tristan Deleu, Edward J Hu, Mo Tiwari, and Emmanuel Bengio. Gflownet foundations. Journal of Machine Learning Research, 24(210):1-55, 2023.
Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv preprint arXiv:2104.13478, 2021.
Lang Cao, Zekun Xi, Long Liao, Ziwei Yang, and Zheng Cao. Chain-of-alpha: Unleashing the power of large language models for alpha mining in quantitative trading. arXiv preprint arXiv:2508,066 06312, 2025.
Dangxing Chen. Can trust the explanations? investigating explainable machine learning methods for monotonic models. arXiv preprint arXiv:2309. 2023.
Tianqi Chen and Carlos Guestrin. XGBoost: A scalable tree boosting system. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 785-794. ACM, 2016. ISBN 978- 778-1-4503-4223--2 doi: 10.1145/99997722299955 URL https://dlacmm.or//d////
Tianxiang Chen, Wei Chen, and Luyao Du An empirical study of financial factor mining based gene expression programming. In 2021 4th International Conference Advanced Electronic Materials, Computers and Software Engineering (AEMCSE), pp. 1113-1117. IEEE, 2021.
Wei-Lin Chiang, Xuanqing Liu, Si Si, Yang Li, Samy Bengio, and Cho-Jui Hsich. Cluster-gen: An efficient algorithm for training deep and large graph convolutional networks. In Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining, pp- 257-266, 2019.
Can Cui, Wei Wang, Meihui Zhang, Gang Chen, Zhaojing Luo, and Beng Chin Ooi. AlphaEvolve: A learning framework discover novel alphas in quantitative investment. In Proceedings the 2021 International Conference on Management of Data, pp. 2208-2216. ACM, 2021. ISBN 978- wwt..o////o/01..115555... 1-4503-8343-1. doi: 10.1 10.114557444800133553444 URL https::/dllacm.or///ooo 10. 10513//444445/ 3457324.
Justin Gilmer, Samuel Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. In International conference on machine learning, pp. 1263-1272 Pmlr, 2017.
Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. Advances in neural information processing systems, 30, 2017.
Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. Advances in neural information processing systems, 33:221 18-22133, 2020.
Moksh Jain, Emmanuel Bengio, Alex Hernandez-Gardd Jarrid Rector Rector-Brooks, Bonaventure FP Dossou, Chanakya Ajit Ekbote, Jie Fu, Tianyu Zhang, Michael Kilgour, Dinghuai Zhang, et al. Biological sequence design with gflownets. In International Conference on Machine Learning. pp. 9786-9801. PMLR, 2022.
Zura Kakushadze. 101 formulaic alphas, 2016. URL URL.hhgg.///axxrrrra//1110000000

---

Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang. Wei Chen, Weidong Ma, Qiwei Ye, and Tie- Yan Liu. LightGBM: A highly efficient gradient boosting decision tree. In Advances in Neural Infor- mation Processing Systems, volume 30. Curran Associates, Inc., 2017. URL https///rroceeding. thex/ppper2011//as//644449999999 httaa//ahh4444444444440000660000 (02dee888866hhhh4996666676666 www.://66676a---atttttt.o.o
neurips.cc/paper files/paper/201 7/has//64499444../.
Salem Lahlou, Tristan Deleu, Pablo Lemos, Dinghuai Zhang, Alcxandra Volokhova, Alex Hernández-Garen, Léna Néhale Ezzine, Yoshua Bengio, and Nikolay Malkin. A theory of con- tinuous generative flow networks. In International Conference on Machine Learning, pp. 18269- 18300. PMLR, 2023.
Yann LeCun, Sumit Chopra, Raia Hadsell, M Ranzato, Fujie Huang, et al. A tutorial on energy- based learning. Predicting structured data, 1(0), 2006.
Pan Li, Yanbang Wang, Hongwei Wang, and Jure Leskovec. Distance encoding: Design provably more powerful neural networks for graph representation learning. Advances in Neural Information Processing Systems, 33:4465-4477 2020.
Zhiwei Li, Ran Song, Caihong Sun, Wei Xu, Zhengtao Yu, and Ji-Rong Wen. Can large language models mine interpretable financial factors more effectively? a neural-symbolic factor mining agent model. In Lun-Wei Ku, Andre Martins, and Vivek Srikumar (eds.). Findings of the Asso- ciation for Computational Linguistics: ACL 2024, pp. 3891-3902, Bangkok, Thailand, August 2024. Association for Computational Linguistics. doi: 10.18653/v1/2024 100.://///22244nnnnnnn----a andings-acll233. URL https://aclanthology http:://acaamhoooooogog02222222 fiw.iiinn.s.niss-----.aaa ac1.233/.
Kanika Madan, Jarrid Rector-Brooks, Maksym Korablyov, Emmanuel Bengio, Moksh Jain, An- drei Cristian Nica, Tom Bose, Yoshua Bengio, and Nikolay Malkin. Learning gflownets from partial episodes for improved convergence and stability. In International Conference on Machine Learning, pp. 23467-23483. PMLR, 2023.
Nikolay Malkin, Moksh Jain, Emmanuel Bengio, Chen Sun, and Yoshua Bengio. Trajectory balance: Improved credit assignment in gflownets. Advances in Neural Information Processing Systems, 35:5955-5967, 2022.
Fionn Murtagh. Multilayer perceptrons for classification and regression. Neurocomputing, 2(5-6): 183-197, 1991.
Junji Ren, Junjie Zhao, Shengcai Liu, and Peng Yang. From Linear to Hierarchical: Evolving Tree- structured Thoughts for Efficient Alpha Mining, August 2025. URL http://arriv.or//bb//550.. 16334. arXiv:2508. 16334 [es].
Tim Salimans, Diederik Kingma, and Max Welling. Markov chain monte carlo and variational inference: Bridging the gap. In International conference on machine learning, pp. 1218-1226. PMLR, 2015.
Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model. IEEE transactions on neural networks, 20(1):61-80, 2008
Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne Van Den Berg, Ivan Titov, and Max Welling. Modeling relational data with graph convolutional networks. In European semantic web conference, pp. 593-607. Springer, 2018.
Hao Shi, Weili Song, Xinting Zhang. Jiahe Shi, Cuicui Luo, Xiang Ao, Hamid Arian, and Luis Angel Seco. Alphaforge: A framework to mine and dynamically combine formulaic alpha factors. In Toby Walsh, Julie Shah, and Zico Kolter (eds.), AAAI-25, Sponsored by the Association for the Advancement of Artificial Intelligence, February 25 March 4, 2025, Philadelphia, PA, USA, pp. 12524-12532. AAAI Press, 2025a. doi: 10.1609/AAAI 33912.3335.... URL https://doi.org// org//10. 1609/aaaixx9111.33333
Yu Shi, Yitong Duan, and Jian Li. Navigating the alpha jungle: An I1m- tln-powered mets framework for formulaic factor mining. arXiv preprint arXiv:2505.1 /22, 2025b.
12

---

Ziyi Tang, Zechuan Chen, Jiarui Yang. Jiayao Mai, Yongsen Zheng, Keze Wang, Jinrui Chen, and Liang Lin. Alphaagent: Lim-driven alpha mining with regularized exploration to counteract alpha decay. In Proceedings of the 31st ACM SIGKDD Conference Knowledge Discovery and Data Mining V. 2, pp. 2813-2822, 2025.
Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. arXiv preprint arXiv:1710.10903 2017.
Feng Xu, Yan Yin, Xinyu Zhang, Tianyuan Liu, Shengyi Jiang, and Zongzhang Zhang- {/text(Aipha) Discovering Logical Formulaic Alphas using Deep Reinforcement Learning, June 2024. URL http://arxiv.orr/abb///006 16505, arXiv:2406. 16505 [cs, q-fin].
Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference Learning Representations.
Xiao Yang. Weiqing Liu, Dong Zhou, Jiang Bian, and Tie- Yan Liu. Qlib: An ai-oriented quantitative 11199.
investment platform, 2020. URL
Liang Yao, Chengsheng Mao, and Yuan Luo. Graph convolutional networks for text classification. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pp. 7370-7377, 2019.
Shuo Yu, Hongyan Xue, Xiang Ao, Feiyang Pan, Jia He, Dandan Tu, and Qing He. Generating synergistic formulaic alpha collections via reinforcement learning. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 2023. doi: 10.1145, 3580305.3599831.
Dinghuai Zhang, Hanjun Dai, Nikolay Malkin, Aaron C Courville, Yoshua Bengio, and Ling Pan. Let the flows tell: Solving graph combinatorial problems with gflownets. Advances in neural information processing systems, 36:1 1952-1199999 2023.
Tianping Zhang, Yuanqi Li, Yifei Jin, and Jian Li. AutoAlpha: an efficient hierarchical evolutionary algorithm for mining alpha factors in quantitative investment, 2020. URL http://arxiv.org/abb. 2002.08245.
Junjie Zhao, Chengxi Zhang, Min Qin, and Peng Yang. QuantFactor REINFORCE: Mining Steady Formulaic Alpha Factors with Variance-bounded REINFORCE, October 2024. URL http://arxiv.
org/abs/2409.05144 144. arXiv:2409.05144.
Junjie Zhao, Chengxi Zhang, Chenkai Wang, and Peng Yang. Learning from expert factors: Trajectory-level reward shaping for formulaic alpha mining, 2025. URL https//arxxv.orr/abb// 2026..
Da Zheng, Chao Ma, Minjie Wang. Jinjing Zhou, Qidong Su, Xiang Song. Quan Gan, Zheng Zhang, and George Karypis. Distdgl: Distributed graph neural network training for billion-scale graphs. In 2020 IEEE/ACM 10th Workshop on Irregular Applications: Architectures and Algorithms (IA3), pp. 36-44. IEEE, 2020.
Zhoufan Zhu and Ke Zhu. Alphaqem: Alpha discovery in finance with distributional reinforcement learning. Forty-second International Conference on Machine Learning. 2025.
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A THE USE OF LARGE LANGUAGE MODELS (LLMS)
We declare that the use of large language models (LLMs) during the drafting of this manuscript was confined to language-related assistance, such sentence refinement and grammatical corrections. All substantive content was independently authored by the authors and underwent rigorous review and verification following any modifications based on LLM assistance. This research did not involve any other processes reliant upon large language models.
B SUPPLEMENTARY BACKGROUND ON RELATED WORK
B. ALPHA MINING AND COMBINATION
Search paradigms. Early work emphasized manual, hypothesis-driven construction of formulaic alphas; automated search later expanded with genetic algorithms (mutation/croossoeer over expres- sion trees) (Chen et al., 2021; Zhang et al., 2020; Cui et al., 2021) and reinforcement learning (sequential decision-me Becision-mmaing over token spaces) (Yu et al., 2023; Zhu & Zhu, 2025; Zhao et al., 2024; Xu et al., 2024). In addition, there also approaches based Large Language Models (LLMs) (Cao et al., 2025; Shi et al., 2025b; Tang et al., 2025; Li et al., 2024; Chen, 2023; Ren et al., 2025) that generate alphas using LLMs refine existing alphas
Combination and multicollinearity Given library {zi, 1k=1 linear combination remains prevalent for transparency:
(k) K
8i.
wk
(k) Zi,
WE WER
(17)
1=1
However, high cross-alpha correlation inflates estimator variance. mitigate this:
Regularization and constraints
min Σ |l
y++a - Z.w/ + A2 A2/ + A1/lw/1 s.t. w = [ lo S 8;
(18)
(K) stacks alpha columns, and optional constraints control turnover or
where Ze exposure. Diagnostics such condition number or VIF help monitor collinearity. Beyond static weights, practice often uses rolling or regime-condee condie-coddtiooeed reweighting.
B.2 GRAPHI NEURAL NETWORKS (GNNS)
Message passing view.
A broad class of GNNs can be written as
(8)
h euv),
(8)
Lue m g-+1
(8)
h
(++1)
Pupa (h
a
(19)
where is a permutation-invant -nvvaiiiainnn aggregator (sum/mean/mmax or attention). Classical instances include GCN (Yao et al., 2019), GraphSAGE (Hamilton et al., 2017), GAT (Veličković al., 2017),
GIN (Xu et al.), the MPNN family (Gilmer et al., 2017), and relational/heterognnooss variants (R- GCN) (Schlichtkrull et al., 2018).
Spectral perspective (GCN). Let Å = A+ and D = diag E, Asj). The layerwise propagation
is
((++1) = -/b
(20)
interpretable as low-pass filter on the graph. Repeated smoothing risks over-smoothing, where node embeddings become indistigguisaable; residual connections, normalization, and careful depth mitigate this (Bronstein et al., 2021).
Attention and heterogeneity. exp(LeaayRel.U (Wh, Wh,))) h' of Σ
GAT computes attention weights Que over neighbors:
auo W hn
E EweN(w) (eppTeeakRRe.Je [Wh,, (hh,)))
uEA((o)
(21)
14

---

R-GCN introduces relation-specife parameters:
h (++1)
Σ E PE (EN.(e)
1 w + n.)
(22)
Corr
Expressivity and readout. GIN links message passing to Weisfeiler-Lehmar tests and uses a sum-aggregation MLP to approach maximal discriminative power in the I-WL regime (Xu al.). Graph-level outputs use readouts
hg READOUT (th!")
e.g., sum sum/mcan//ma mmax.
(23)
Positional structural encodings (e.g., Laplacian eigenvectors, distance encodings) can further en- hance global awareness (Li et al., 2020). Practical training relies on sampling and partitioning for scale (Chiang et al., 2019; Zheng et al., 2020), with OGB benchmarks standardizing evaluation (Hu et al., 2020).
Over-smoothing and over-squashing. Deep stacks may over-smooth; curvature-inspired rewiring, residuals, and normalization layers are common responses. Over-squashing the com- pression of exponentially many distant signals into fixed-size messages-can be alleviated by atten- tion/edge weighting, graph rewiring, and subgraph-based encoders (Bronstein al., 2021).
B.3 GENERATIVE FLOW NETWORKS (GFLOWNETS)
Objective: sampling proportional to reward. Given a set of terminal objects X' and non- negative reward R. X R2o. GFlowNets seek policy that samples e X' with
Po(x) & R(x), Po(x)
Σ TET(#)
Po(T),
(24)
3 is trajectory in DAG of states and T(x) the set of trajectories ending at (Bengio et al., 2021; 2023).
where 7 (50 - ......... +x)
Detailed-balance (DB) and trajectory-balance (TB). Let Fo(s) > 0 denote a learnable flow through state s. DB enforces local conservation:
Fo(s) Po(s') s) Fo(s') Po(s) s') for edges stss'.
(25)
TB provides path-wise condition linking forward/backward policies and a scalar Zg (partition function):
CTB = E,
[ [log Po(r) + log Za - llo R(x))²]
(26)
encouraging (z(x)) x R(x) when minimized (Malkin al., 2022). Subtrajectory balance (SubTB) generalizes TB to partial paths for credit assignment (Madan et al., 2023).
Forward/backward policies and partition function. A typical parameterization factors Po(T) = TT--1 Po(st+1 st), Po(s: 8t41) learned for DB/SubTB, and treats Zo as a learn- able scalar (or function) estimating Σ, R(x). Estimation stability can be improved via baselines, variance reduction, and regularization.
Mode coverage vs. RL/EBM/MMMC. Unlike standard RL objectives that often favor a single high-return mode under sparse rewards, GFlowNets learn a distribution covering multiple modes. Compared to energy-based models (EBMs) and MCMC, GFlowNets amortize sampling via learned policies, reducing the need for long chains while retaining a reward-shaped target (LeCun et al., 2006: Salimans et al., 2015; Bengio al., 2023). Empirical applications span molecular design, program synthesis, and discrete structure generation (Zhang et al., 2023; Jain et al., 2022), with ongoing work offline training, replay buffers, and credit assignment (Lahlou al., 2023).
15

---

C IMPLEMENTATION DETAILS
C.I PSEUDO CODE
The pseudo code of AlphaSAGE (core of mining framework) is shown Algorithm 1. And the code is available at http:://giilbbbcmo////cccc/cc/ts
Algorithm 1: AlphaSAGE
Input: Stock features X, stock trend labels y, action set A
Output: Final alpha pool FF
Initialize: GFN parameters 0; probability buffer Bpnsh + 2; embedding buffer Bemb +- Ø; alpha pool F
for step t = ..., Tmax do
Parse current state - astt;
Compute state embedding e <- faNN(astz):
11 Eq.
Output action distribution Tp(- astt); append to Bpoosi
rand()2 (St) then
O2 t SEP;
else
L Sample or select E A from Te(: Jaste);
10
if a2 =
SEP then
Build expression BuildExpr(s)):
Compute alpha + ComputeAlpha(a, X);
12
13
Compute correlation reward Ric t IC(
// Eq.
14
Compute novelty to pool members RNov + (2,
// Eq. 13
15
Run KNN on terminal embedding e, against pool embeddings:
N + (NN(C, Emb(F), k);
16
Build distance weight matrix W (iistt2eiggh(N) and performance similarity a sim -amm-Ppe Perrf W);
17
SA reward RSA f- SAReward(sim);
// Eq. 11 Eq. 14
Total reward R + Combine (Ric RNOv, RSA.t).
19
PasssTTreshooddd then
F FU-UU(A (a): Append e, to Bemb:
21
22
Trajectory Balance loss Lm <- Tra jetoo (Bprbb R); Entropy regularizer Lent Ent (Bpob)) clear Bprob
// Eq.
/ Eq. 15
24
Total loss Linnal f- LTB + AenCCani
/ Eq. 16
Update 0 t- VoLniinll
26
Reset S4.1 + InitState(); clear Bposi
27
else
State transition S4+1 Transition(S,.a))
return F:
C.2 RELATION TYPE OF RGCN
To denote combinations between different operators and features, we have defined the follow- ing relationships: Unary operator with operand; Commutative operator with operands; 3 Non-commutative operator with left operand; 4 Non-commutative operator with right operand; $ Rolling operator with feature operand; @ Rolling operator with time operand.
C.3 FEATURES AND OPERATORS
All operators and features available during alpha mining listed in the Table 3.
16

---

Table 3: Raw features and operators. F: base market features; U/B: unary/binary operators; CS: cross-sectional operation (within-day across assets); TS: time-series operation (rolling window). The lookback length denotes the past d trading days, and the is used only for numerical stability. Description
Name
Type
Open
Opening price
Close
Closing price
High
Daily highest price
Low
Daily lowest price
Vwap
Daily average price, weighted by the volume trades each price Trading volume (number of shares)
Volume
Abs
Absolute value of the input
SlogIp
Signed log transform: sign(input) times log plus the absolute value)
Inv
Reciprocal of the input; add avoid division by zero
Sign
Sign of the input, returning 0, or Natural logarithm of the input; add for numerical stability
Log
Rank
U-CS
Cross-sectional rank normalization within day, mapped the range
Add
Sraennt-wise addition two inputs Element wise subtraction: first minus second Element-wiss multiplication
Sub
Mul
Div
Flement-wise division; add small constant the denominator for stability
Pow
Element-wise power: raise the first input the power of the second
Greater
Element-wise comparison: if first input is greater than second, else 0
Less
Element-wiss comparison: if first input is less than second, else 0
Ref
U-TS
Lag operator: the value from d days ago Rolling mean the past days Rolling over the past d days Rolling standard deviation over the past d days
TsMean
U-TS
TsSum
U-TS
TsStd
U-TS
Tslr
U-TS
Rolling information ratio the past days
MiMinfaxDiff
U-TS
Rolling range over the past d days (rolling max minus rolling min)
TsMaxDiff
U-TS
Current value minus the rolling max over the past d days
TsMinDiff'
U-TS
Current value minus the rolling min over the past d days
TsVar
U-TS
Rolling variance over the past d days Rolling skewness over the past d days Rolling kurtosis over the past d days Rolling maximum over the past days Rolling minimum over the past d days Rolling median over the past d days Rolling median absolute deviation over the past d days Rolling rank of the current value within the past days, mapped to [0, 1J
TsSkew
U-TS
TsKurt
U-TS
TsMax
U-TS
TsMin
U-TS
TsMed
U-TS
TsMad
U-TS
TsRank
U-TS
TsDelta
U-TS
Change over days: current value minus the value d days ago
TsDiv
U-TS
Ratio over days: current value divided by the value days ago Percentage change over the past days
TsPeiChange
U-TS
TsWMA
U-TS
Linearly decaying weighted moving average over the past d days
TsEMA
U-TS
Exponential moving average with decay over the past days
TsCov
B-TS
Rolling covariance between two inputs over the past days
TsCorr
B-TS
Rolling Peanson correlation between two inputs over the past days
17

---

D EXPERIMENT DETAILS
D.1 METRIC DETAILS
For all evaluation metrics, provide definitions and brief interpretations. Let Cov(a(Xa), ya)
Pa V Var((((X)) Var(ya)
(cross-scctional correlation day d),
(27)
and let Ra denote the portfolio return constructed from a on day d, K the number of periods per year (e.g., K-252 for daily), ry the risk-free rate, and
W. = Σ(1+ R.) u.t.
(cumulative wealth).
(28)
Information Coeflicient (IC): See Eq. 12. Interpretation. Cross-sectional predictive power the factor-how well a(Xa) aligns with next-period outcomes yd- Using the absolute value isolates magnitude rather than sign (long/short direction can be flipped). Higher IC indicates more informative date-wise rankings and is a prerequisite for con- structing profitable long-short portfolios.
Information Ratio of IC (ICIR):
Ealpa
ICIR
(29)
Varra(pa)
Interpretation. Time-series consistency of cross-sectional predictability: mean IC relative its volatility. Under weak dependence, ICIR approximates signal-to-noise measure
(akin to i-statistic for E[(pa), favoring factors that work consistently rather than sporadi- cally.
Rank Information Coefficient (RankIC):
Cov(rank(a rank(ya))
RankIC = Ea poak
rank
(30)
Var(rank (axaa))(((aa))) (Xx)) Var(rank(ua))
Interpretation. Spearman-style counterpart IC that evaluates whether higher-ranked sig- nals correspond to higher-ranked outcomes. RankIC is robust to outliers and monotone transforms of a, aligning with rank-based portfolio constructions.
Information Ratio of RankIC (RankICIR):
Eg Pank
RankICIR
(31)
Var (prank)
Interpretation. Time-series stability of rank-based predictive power, prioritizing factors whose cross-sectional ordering remains reliable across time.
Annualized Return (AR):
AR Interpretation. Economic value produced by the portfolio rule induced by a. When com- pounding is material, geometric annualization via W, is preferred.
K Eg(Ra)
(32)
Maximum Drawdown (MDD):
MDD max (1
W.
(33)
max W.
Worst peak-to-trough loss of the wealth process; trajectory- and tail-risk metric not captured by variance alone. It is critical for leverage, risk limits, and investor experience.
Interpretation.
Sharpe Ratio (SR) (anmualized, excess over risk-free);
VK [. Tyal
SR
(34)
/aar RRa rsaa)
Interpretation. Risk-adjusted return per unit of volatility for the a-induced portfolio, en- abling fair comparison across methods, universes, and rehalancing frequencies.
18

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Reporting conventions. (i We report IC/RankIC in absolute value (cf. Eq. 12, Eq. 30) because factor signs are arbitrary up to inversion. (ii) For SR, we use excess returns Ra ry.d To ensure fair comparison while simplifying the process, explicitly. (iii) Due to differing rules between the CS1300/500 indices and the S&P500, when back- testing the CSI300/500, we purchase the top 20% of stocks each trading day and sell them after 20 days (long positions only); For the S&P500, we purchase the top 10% of stocks each trading day and sell them after 20 days, while simultaneously selling the bottom 10% of stocks and repurchasing them after 20 days (long-short short combination).
set f. for comparability and state this choice
D.2 HYPERPARAMETER SETTING
The hyperparameter settings of AlphaSAGE are listed in Table 4.
Table 4: The hyperparameter settings of AlphaSAGE.
Name
Description
Value
Max Length
The maximum number of tokens in the st
20
Hidden Dim
The dimension of hidden state The number of layers in RGCN encoder
128
Encoder Layer
2
Entropy Coef
The weight of CENT The learning rate to optimize
0.01
Learning Rate
0.0001
SA Weight
The weight of RSA
1.0
NOV Weight
The weight of RNOV
0.3
Pool Capacity
The maximum number of alphas in alpha pool
50
Episodes
The number trajectory sampling instances
10000/20000
D.3 BASELINE DETAILS
We selected seven methods as the baseline:
MLP (Murtagh, 1991): feedforward neural network that maps tabular features return targets, capturing nonlinear interactions. It is strong generic baseline but can overfit without careful regularization and offers limited interpretability.
LightGBM (Ke et al., 2017): A gradient-boosted decision tree learner with histogram- based splits and leaf-wise growth, well suited to large, sparse, heterogeneous financial features. It trains fast and handles missing values natively, though leaf-wise growth can overfit small samples without constraints.
XGBoost (Chen & Guestrin, 2016): Boosted trees optimized with second-order informa- tion, shrinkage, column subsampling, and explicit regularization. Reliable on tabular alpha features, but the ensemble remains hard interpret structurally and be sensitive to label leakage or distribution shift.
GP (Chen et al., 2021): Genetic programming performs symbolic regression by evolving expression trees via mutation and crossover, yielding human-readable formulas. It explores large search spaces but is prone to bloat.
AlphaGen (Yu et al., 2023): Proposes mining synergistic sets of formulaic alphas by directly optimizing the downstream combination model's performance. Uses reinforce- ment learning to explore the expression search space, assigning the improvement in portfo- lio/combiner performance as the RL return so the generator preferentially discovers alphas that work well together.
AIphaQCM (Zhu & Zhu, 2025): Frames synergistic alpha discovery non-stationary, MDP and adopts a distributional RL approach. Learns both Q-function and quantiles, then applies quantiled conditional moment method to obtain unbiased variance estimate; the learned value and variance jointly guide exploration under non- stationarity, improving search efficiency on large universes.
19

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AlphaForge (Shi et al., 2025a): Introduces a two-stage framework that couples a genera- tive-preet pre-predictive neural module for factor proposal (encouraging broad, diverse exploration) with a dynamic combination stage. The combiner selects by recent performance and adapts weights over time, addressing inconsistency and rigidity of fixed-weight ensembles and yielding stronger portfolio results in empirical tests.
AlphaAgent (Tang et al., 2025): multi-agent factor discovery framework proposed. First, the Ideal Agent generates alpha idcals based market observations and financial knowledge derived from inputs. Subsequently, the Factor Agent generates computable alpha expressions. Finally, the Eval Agent assesses the alphas, feeding the feedback back into the Ideal Agent for refinement.
For AlphaGinn', AlphaQCM AlphaForge" and AlphaAgent', follow the default setting in their open-source projects. And for other baselines, we follow the setting in http://githuh.com/Dllyhtt AlphaForge.
E ADDITIONAL RESULTS
E.1 BACKTESTING RESULTS
Figure shows that CSI500 (2022-2024) 2024) AlphaSAGE delivers the strongest end-period wealth and sustains a clear lead for most of the horizon. It experiences smoother drawdowns around mid- 2023, recovers earlier from late-2024 stress, and retains more of the subsequent rally; all baselines trail, while the CS1500 index lags markedly throughout.
AlphaSAGE
AlphaGen
LightGBM
1.2
AlphaForge
GP
MLP
AlphaQCM
XGBoost
CSI500 Index
Cumulative Retur
1.1
1.0
0.9
0.8
0.7
2022-01
2022-05
2022-09
2023-01
2023-05
2023-09
2024-01
2024-05
2024-09
2025-01
Date
Figure 5: Cumulative return on CS1500 2024). Comparison among AlphaSAGE (ours), all baselines, and CSI500 Index benchmark.
Figure 6 shows that S&0000 (2018-2021) AlphaSAGE tracks near the top during calm phases, then recovers faster and compounds higher after the 2020 drawdown, finishing with the best cu- mulative return. Several baselines (c.g., AlphaForge) are competitive early but fail to match the late-period acceleration; the market index remains below AlphaSAGE by the end.
5 https///gghhibbooomm/////haaaaan
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1.6
AlphaSAGE
AlphaGen
LightGBM
AlphaForge
MLP
1.5
AlphaQCM
XGBoost
SP500 Index
Cumulative Retur
1.4
1.3
1.2
1.1
1.0
2018-01
2018-05
2018-09
2019-01
2019-05
2019-09
2020-01
2020-05
2020-09
2021-01
Date
Figure 6: Cumulative return on S&P500 (2018-2020). Comparison among AlphaSAGE. (ours), all baselines, and S&P500 Index benchmark.
E.2 PARAMETER ANALYSIS
This appendix reports five practical knobs: training steps, candidate-pool size, Mux Length, Hidden Dim and Encoder Layer. We track correlation metrics (IC/RIC, ICIR/RICIR) and portfolio metrics (AR, MDD, SR).
Training steps. As shown in Fig. 7, GelowNets converge faster and train more efficiently than PPO in the alpha-mining setting. IC/RIC and ICIR/RICIR rise sharply in early iterations and reach high, stable plateau with lower variance; PPO improves more slowly and exhibits larger oscillations throughout.
IC / RIC Comparison
ICIR / RICIR Comparison
0.10
ICIR /RICIR
0.06
IC/
0.3
0.04
0.2
0.02
0.1
0.00
0.0
50000
100000
150000
200000
50000
100000
150000
200000
Training Steps
Training Steps
PPO IC
PPO RIC
PPO ICTR
PPO RICTR RICIR
GEN
GEN
Figure 7: Learning dynamics vs. training steps. GFlowNets (GFN) achieve higher plateaus earlier and with less volatility than PPO across IC/RIC and ICIR/RICIR.
21

---

Candidate-pool size. Figure shows that increasing the factor pool yields rapid gains fol- lowed by saturation. All correlation metrics (IC/RIC, ICIR/RICIR) and portfolio metrics (AR, SR) improve markedly when moving from very small to moderate pool sizes, then flatten into broad plateaus; MDD improves monotonically with no instability.
0.10
2.0
0.6
1.5
0.5
0.00
0.08
IC CRCC
0.4
JCB /IICR AR MDD
0.05
1.0
0.06
-0.10
0.3
0.5
0.15
0.04
0.2
00.00
0.0
0.02
0.1
0.25
0.00
0.0
-0.30
-0.5
20
30
40
20
30
Pool Size
Pool Size
RIC
MDD
ICIR
RICIR
SR
Figure 8: Effect of candidate-pool size. Metrics increase quickly at small-moderate pool sizes and then stabilize, indicating diminishing returns beyond modest pool.
MaxLen (maximum AST length). We further study the effect of MaxLen in Eq. which con- trols the maximum size of generated abstract syntax trees and thus the early-stop behaviour of the sampler. Table 5 reports mean AST depth and performance on CSI300 for different values of MaxLen. We observe clear trade-off: very small MaxLen restricts expressiveness and yields weak IC and portfolio metrics, performance improves steadily up to MaxLen 20, and larger values hurt performance despite deeper trees. This pattern suggests that overly long expressions introduce noise and make optimization harder, while a moderate cap (around 20 in our setting) balances expressiveness and stability.
Table 5: Sensitivity to MaxLen on CSI300. MaxLen controls the maximum AST length (and thus early-stop threshold). Mean AST depth is measured at convergence.
MaxLen
Mean depth
IC
ICIR
RIC
RICIR
AR
MDD
SR
5
2.02
0.031
0.209
0.041
0.269
-0.66%
-27.3%
0.14
10
3.36
0.037
0.253
0.040
0.264
-1.95%
-27.6%
0.43
15
3.68
0.059
0.391
0.072
0.493
1.31%
-23.6%
0.31
20
4.06
0.079
0.496
0.094
0.583
7.62%
-17.3%
1.71
30
5.18
0.043
0.286
0.056
0.369
-1.65%
-25.6%
0.41
40
5.66
0.031
0.245
0.035
0.270
-0.92%
-28.9%
0.21
50
5.78
0.042
0.288
0.048
0.322
1.22%
-25.4%
0.27
Encoder Layer. We next vary the number of RGCN layers in the structure-awar encoder on CSI300. As summarized in Table 6, performance is stable across depths, with shallow 2-layer encoder performing best. Increasing the depth to three or four layers yields slightly weaker IC and portfolio metrics, which is consistent with over-smoothing and harder optimization under limited structural supervision. This suggests that AlphaSAGE does not require finely tuned encoder depth and is robust to reasonable choices.
Hidden Dim. We also probe the effect of the GFlowNet backbone size by varying the hidden dimension CSI300. Table 7 shows that moderate hidden size of 128 achieves the best trade-
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Table 6: Sensitivity Encoder Layer CSI300.
Layers
IC
ICIR
RIC
RICIR
AR
MDD
SR
2
0.079
0.496
0.094
0.583
7.62%
-17.3%
1.71
3
0.073
0.473
0.088
0.572
7.08%
-18.5%
1.50
4
0.068
0.451
0.083
0.556
6.58%
-19.4%
1.34
between expressiveness and generalization. smaller size 64 underfits and degrades both correlation and portfolio metrics, while enlarging to 256 offers further gains and slightly hurts out-of-sample performance. The qualitative conclusions of the paper remain unchanged across this range, indicating that AlphaSAGE is not overly sensitive the specific hidden dimension choice.
Table 7: Sensitivity to GFlowNet hidden dimension on CSI300.
Hidden Dim
IC
ICIR
RIC
RICIR
AR
MDD
SR
64
0.059
0.368
0.078
0.464
5,42%
-19.6%
1.31
128
0.079
0.496
0.094
0.583
7.62%
-17.3%
1.71
256
0.073
0.448
0.089
0.561
6.88%
-18.1%
1.59
E.3 LEWARD-WEIGHT SCHEDULES
Our multi-dimensional reward combines predictive performance, structure-behaviour alignment, and novelty via time-dependent weights the structure-aware term ASa and the novelty term Rxov (see Eq. (xx)). The design follows a pragmatic curriculum intuition: early in training we encourage exploration and representation learning in the alpha space, while later place more emphasis on realized predictive performance.
To assess the impact of the weight schedules, compare three simple choices for the decay func- tion g(T) on CS1300: (i) a constant schedule (no decay), (ii) a linear decay schedule, and (iii) an exponential decay schedule. In all cases, the initial coefficients (Ap, 7o) kept fixed, and only the functional form of g(T)) is changed.
Table reports IC/RIC, ICIR/RICIR, and portfolio-level metrics (AR, MDD, SR). Linear decay yields the best overall performance, exponential decay performs similarly, and constant weights underperform. These results indicate that AlphaSAGE not overly sensitive to the exact sched- ule shape within a reasonable range, and that annealing the structure- and novelty terms is beneficial in practice.
Table 8: Sensitivity to reward-weight schedules on CSI300. Comparison of constant, linear-decay, and exponential-decat schedules for the time-dependent weights on RSA and RNov.
Schedule type
IC
ICIR
RIC
RICIR
AR
MDD
SR
Constant
0.043
0.317
0.051
0.368
3.26%
-0.3%%
0.69
Linear decay
0.079
0.496
0.094
0.583
7.62%
-17.3%
1.71
Exponential decay
0.075
0.482
0.095
0.578
7.56%
-17.6%
1.69
E.4 ENCODER ARCHITECTURES
Beyond the main results, we also compare structure aware encoder with sequence based al- ternative. In our framework, the default encoder is an RGCN that operates on abstract syntax trees (ASTs) and explicitly exploits the operator feature graph structure. As reference, we build Trans- former encoder that consumes a lincarized expression sequence (for example, prefix notation) under similar parameter budget.
23

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Table 9 reports the results CSI300. The RGCN encoder achieves higher correlation metrics (IC, RIC, ICIR, RICIR) and stronger portfolio performance (AR, SR) with a slightly smaller maximum drawdown. This comparison supports the view that the structural inductive bias provided by graph based encoding ASTs is beneficial for formulaic alpha mining, and that the gains of AlphaSAGE cannot be attributed solely increased model capacity.
Table Encoder comparison on CSI300. RGCN operates on ASTs, while the Transformer encoder consumes linearized expression sequences under similar parameter budget.
Encoder type
IC
ICIR
RIC
RICIR
AR
MDD
SR
RGCN 0.079
0.496
0.094
0.583
7.62%
-1.33
1.71
Transformer
0.065
0.479
0.085
0.537
4.97%
-18.8%
1.22
E.5 RL VERSUS GFLOWNET AND THE EFFECT OF STRUCTURE AWARE COMPONENTS
To disentangle the effect of the generative framework from that of the proposed encoder and reward design, we consider four configurations on CS1300: (i) an RL baseline with the original sequential encoding and IC only reward: (ii) the same RL baseline augmented with the RGCN encoder and the composite reward; (iii) a GFlowNet with the original sequential encoding and IC only reward; and (iv) a GFlowNet with the RGCN encoder and the composite reward, which corresponds to Alp.aSAGE.
Table 10 summarizes the results. Two observations are consistent with the main text. First, adding the structure aware components improves both RL and GFlowNet, increasing IC/RIC as well as AR and SR while reducing maximum drawdown. Second, without these components the GFlowNet configuration struggles under sparse IC only feedback and underperforms the RL baseline, whereas with the RGCN encoder and composite reward achieves the best overall performance. This sup- ports our claim that both the choice of GFlowNet and the structure aware design are important, and that the gains of AlphaSAGE come from their combination.
Table 10: RL versus GelowNet with and without structure aware components on CSI300. "Proposed components" denotes the of the RGCN encoder together with the composite reward.
Framework Components
IC
ICIR
RIC
RICIR
AR
MDD
SR
RL
x
0.058
0.414
0.057
0.360
4.00%
-22.6%
0.76
RL
/
0.067
0.459
0.079
0.502
5.53%
-20.7%
1.03
GelowNet
X
0.046
0.313
0.060
0.397
-0.47%
-24.8%
-0.11
GelowNet
V 0.079
0.496
0.094
0.583
7.62%
-17.3%
1.71
E.6 RUNTIME AND SCALABILITY
In our implementation, the dominant computational cost of AlphaSAGE comes from evaluating candidate alphas on historical cross-sectional panels. This cost is shared with existing learning- based alpha mining systems such as AlphaGen, AlphaQCM, and AlphaForge, since all of them must compute panel-wise values for large collections of formulaic alphas. The additional components introduced by AlphaSAGE are relatively lightweight.
First, the GFlowNet sampler operates on partial abstract syntax trees (ASTs) with bounded maxi- mum length MaxLen. The number of sampling steps per alpha is therefore proportional to MaxLe and comparable in magnitude to the sequential works. Second, the structure-aware reward only requires computing k-nearest neighbors in a low- dimensional graph-embedding space. These embeddings and neighbor relationships are precom- puted and cached during training, so the incremental overhead is modest. Third, the carly-stopping mechanism controlled by MaxLen stochastically truncates trajectories once valid expression of sufficient length formed, which avoids wasteful exploration of overly long expression trees and empirically accelerates convergence.
steps used in RL-based frame-
24

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In typical quantitative trading workflows, alpha mining is offline relatively low frequency (for example weekly or monthly) to update the alpha library. During portfolio construction and execution, the system evaluates only the fixed set of discovered alphas, so the latency of the mining model does not directly affect real-time trading decisions.
Table 11 reports training time for the main methods several universes. AlphaSAGE is efficient in practice and suitable for large-scale alpha mining under realistic resource constraints.
Table 11: Runtime comparison (hours) on different universes.
Method
CS1300
CS1500
S&P 500
CS11000
AlphaGen
1.34
1.68
1.48
2.11
AlphaQCM
1.72
1.98
1.82
2.32
AlphaForge
1.56
2.16
2.02
2.74
AlphaSAGE
0.49
0.76
0.71
1.12
E.7 STABILITY ACROSS RANDOM SEEDS
To assess robustness with respect to random initialization, we ran multiple independent trials on CSI300 for all RL-based methods using different random seeds and report mean and standard deviation for each metric. As shown in Table 12, AlphaSAGE consistently outperforms the baselines across correlation metries (IC/RIC, ICIR/RICIR) and portfolio metrics (AR, SR) while maintaining a smaller maximum drawdown. The ranking between methods is stable across runs, indicating that our main conclusions are not driven by a particular seed.
Table 12: Multiple runs on CSI300. Mean (standard deviation) over different random seeds.
Method
IC
ICIR
RIC
RICIR
AR
MDD
SR
0.063
0.454
0,061
0.389
4.18%
-22.3%
0.84
AlphaGen
(0.005)
(0.024)
(0.006)
(0.021)
(0.004)
(0.017)
(0.126)
0.045
0.268
0.044
0.259
2.23%
-23.8%
0.52
AlphaQCM
(0.002)
(0.015)
(0.006)
(0.019)
(0.003)
(0.019)
(0.155)
0.040
0.246
0.049
0.291
4.01%
-21.3%
0.96
AlphaForge
(0.003)
(0.017)
(0.005)
(0.022)
(0.003)
(0.020)
(0.112)
0.073
0.473
0.088
0.572
7.08%
-18.5%
1.50
AlphaSAGE (0.004)
(0.022)
(0.008)
(0.026)
(0.005)
(0.022)
(0.143)
E.8 SUBPERIOD PERFORMANCE UNDER REGIME SHIFTS
In the Appendix E. we report that AlphaSAGE delivers consistently strong IC- and ICIR-based per- formance across all markets and horizons, while cumulative return curves on CSI500 (2022-2024) and S&P500 (2018-2020) are less pronounced and in some cases slightly below certain baselines. This section provides additional context for these findings.
First, AlphaSAGE is trained to discover alphas by maximizing a reward in which Ric is the primary component. Information coefficient measures the average cross- sectional correlation between an alpha signal and next-period returns and is a useful proxy for ranking quality, but does not directly optimize path-dependent portfolio outcomes. Over short or turbulent subperiods, an alpha library with stable morizoorion positive IC can still produce flatter cumulative returns if the window ex- hibits unusually high volatility, sharp factor- anctor-premia reversals, elevated turnover costs, or compressed cross- cross-sectional dispersion. In such regimes the mapping from cross- cross-sectional predictability real- ized P&L can weaken temporarily, so or-eeeete discovery does not guarantee dominance in every short subwindow.
Second, the CSI500 interval from 2022 to 2024 coincides with several well-documented regime events that disproportionately affected mid- and small-cap names and induced repeated style rota-
25

---

tions. In 2022, strict zero-COVID policies and the Shanghai lockdown weighed heavily economic activity and risk appetite; the ongoing property-sector stress and mortgage-related incidents further pushed the market into risk-off stance. After the late-2022 policy pivot and reopening, equity leadership rotated quickly across sectors, and in 2023-2024 policy support increasingly emphasized high-dividend and buyback themes that favored larger, more stable firms over the CS1500 universe. Against this backdrop, an IC-optimized alpha library can exhibit relatively muted cumulative returns on CSI500 over this specific window even its average predictive edge remains positive.
Third, the S&P500 window from 2018 to 2020 likewise spans several abrupt regime shifts. The es- calation of the U.S.-China trade conflict and tighter monetary policy in 2018 contributed to a sharp Q4 sell-off and rapid factor rotations; in 2019, persistent trade uncertainty and growth concerns continued to destabilize style leadership. In early 2020, the COVID-19 shock produced an unprece- dented crash followed by policy-driven rebound with extreme cross-sectional dispersion. Such environments are known to distort the short-horizon relationship between historical IC and realized portfolio performance.
Overall, these subperiods illustrate that AlphaSAGE's objective-maximizinng an IC-dominated re- ward over long histories- robust cross-sectional predictability, while realized cumulative turns in short, highly stressed windows can be shaped by regime-specifie shocks and style dynamics that lie beyond the scope of the discovery objective.
F PROOF
Proposition E1 (Alpha Diversity Stabilizes Estimation and Prediction). Let F = (o.....,ay) e T XN collect N standardized alphas (each column has mean and variance 1), and let y E R." be the target return. Define
- FT
FERNNN
1 T FF
(35)
T
The OLS estimator is B = E-'g. g. Suppose y - FD satisfies Bic] = 0 and Var(e) = 0²Ir. Then:
1. (Estimator variance decomposition) Writing the eigendecomppsittion Σ QAQT with eigenvalues 1 2 A 2 ... AN > 0, we have
0² Σ
N
Var(š)
tr Var(8)
(36)
T
T
Consequently, pairwise correlations increase and the spectrum becomes more ill- conditioned (small Amin), the total estimation variance inflates.
2. (Prediction risk amplification) The in-sample prediction variance satisfies
a²
tn( (3r FT) T (((FE-1FT) o² t((EE1)= = o² 3-11 B[IFF(B-B81113 (
E
(37)
but the out-of-sample risk for new design with the same second moments equals
o² Repee o² + E[(a" (8 --))))) = o² + -f ((EE-1) = ) (38)
while the uncertainty allocation across coordinates is governed by Σ-1; higher multi- collinearity (smaller Amia) yields larger coordinate-wise dispersion of B and hence less interpretability.
3. (Sensitivity to perturbations) For perturbations (AE, Ag), the linear system Σ, = obeys the classical bound
| 12
||Agll2
| 12
x2(E)
(39)
1 ||
||l||2
|
26

---

where (²(
Dmaxx/Dmin Thus, neal-collinyariyy (large R2)) makes B highly unstable under small data noise or distributional drift.
|ا|| ||
=
4. (Two-alpha closed form) For two standardized alphas with correlation p.
1 p
1 p²
1
P
(40)
p 1
-
1
-p
so Var(3)
and K2 (2.) = L+e I-p As p + 1, both the variance and the
T(1-p²1
1
condition number blow up.
5. (Equicorrelated family) If E has equicorrelation p off-diagonal, then
A =1+ (N 1)p,
X = AN = =1 1 P.
(41)
and hence
0² 1 T
1 (N - 1+ (N 1)p + N NN1) ) (2
tr (Br
(42)
1-p P
Even modest p > causes variance inflation linear in N through the (N 1)/(1 p) term; promoting diversity (smaller p) sharply reduces this inflation.
Proof. (1) Since B = (FT FT E- and y FB + E with Var(e) = 0² IT, we have
1
0² Iー1
Var(3) = E-1 72
FT
Var(y) F Σ-1
F
F
F
E-1
(43)
7
T
o² Using Σ = QAQ yields tr Var(š)
T Eitp'.
(2) For sampl variance,
BI|F(B B²) = tr(F Var(3) (ar(3)F)) FT) tr(FE-1FT)
(44)
T
Since FE-1FT and ΣΣ- share the same trace (tr( (r(B = tr BA).. this equals o² tr(IN) = d²N, For new draw with the same second moments, Blaā = Σ, the added generalization o² variance is T tr(EE-1)= = T giving Ryred = 0² + N/T); however the distribution of this uncertainty over coordinates is governed by Σ-1, worsening with ill-conditioning, which harms interpretability of individual B4.
e²N
(3) The stated perturbation bound follows from standard linear system sensitivity: for Σ first-order analysis (or the Bauer-Fike type arguments) gives A S || +
|| normalizing by ||,3|22 and noting ||~ ||E|| = X2 Σ) yields the claim.
(5)-(5) The two-alpha and equicorrelation calculations follow from direct inversion and the known eigenstructure; for equicorrelation, the all-ones vector the top eigenvector with eigenvalue + (N 1)p and the orthogonal complement has eigenvalue 1 p. Plugging these into part (1) gives the trace formula and condition number.
Collectively, (1)-() show that reducing off- diagonal correlations increases the eigenvalues of Σ, de- creases K2 (E), shrinks Var(3), and improves stability and eecepretablliy--frmmaa substantiating the need for diverse, weakly correlated alphas.
Corollary F2 (Regularization as proxy for diversity). For ridge with penalty X > 0, 3xx >T)-1g g and
= +
x² Var(3,) T (E+
tr Var(
0² N
Σ
1
(45)
T X +
Either increasing diversity (raising the Xx) or increasing X reduces variance; explicit diversity con- trol targets the spectrum directly, often achieving lower variance without the shrinkage bias inherent ridge.
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Reviews:
[{"review_id": "ipnW0V7bgv", "reviewer": "Reviewer_Md3w", "summary": "This paper proposes AlphaSAGE, a framework for automated alpha mining in quantitative finance. By identifying three challenges of reward sparsity, inadequate structural representation of mathematical expressions, and lack of diversity in generated alphas, AlphaSAGE integrates a structure-aware encoder based on Relational Graph Convolutional Networks (RGCN) to capture the hierarchical nature of alpha expressions, a Generative Flow Network (GFlowNet) for diverse sampling of high-reward alphas, and a multi-faceted reward function that combines predictive performance, structural alignment, and novelty. Experiments on real-world market datasets demonstrate that AlphaSAGE outperforms baselines in both correlation metrics and portfolio outcomes.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "**S1:** The integration of GFlowNets with structure-aware RGCN encoding is sound. It tackles the diversity issue in alpha generation by sampling from a reward-proportional distribution. The multi-faceted reward design provides dense guidance, mitigating reward sparsity.\n\n**S2:** The experiments across multiple markets with multiple metrics show the superiority of the proposed method over several baselines. The ablation study validates the contribution of each component.\n\n**S3:** The framework provides interpretability through symbolic alpha expressions and a transparent combination scheme, aligning with real-world trading needs.", "weaknesses": "**W1:** The framework may incur high computational costs, especially for large-scale alpha spaces. The paper does not discuss runtime comparisons or scalability limitations, which could affect practicality in resource-constrained settings.\n\n**W2:** The U.S. market data ends in 2020 due to \"limitations in the data source,\" which may reduce the generalizability of results to recent market regimes. Experiments could benefit from including more diverse markets or longer time periods.\n\n**W3:** The paper does not explore the impact of several hyperparameters (e.g., GNN depth, GFlowNet architecture).\n\n**W4:** More recent methods (e.g., LLM-based alpha mining) are not discussed and compared in the related work and experiments.\n\nMinor Comments\n\n**D1:** In the abstract, \"a novel framework is built upon three cornerstone innovations\" --> \"a novel framework built upon three cornerstone innovations\"\n\n**D2:** The reference format should be double-checked. Please use \"\\citep\" by default and \"\\citet\" only when the authors serve as a component of the sentence.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "LhwCPFpoiK", "reviewer": "Reviewer_Yaqu", "summary": "This paper proposes AlphaSAGE, a structure-aware generative framework for automated alpha mining in quantitative finance. The method replaces reinforcement learning–based sequential generation with a Generative Flow Network (GFlowNet) to alleviate reward sparsity and promote diversity. A Relational Graph Convolutional Network (RGCN) encoder operates on abstract syntax trees to capture the structural semantics of formulaic alphas, while a multi-faceted reward combines predictive performance (IC), structure–behavior alignment (RSA), and novelty (RNOV). Experiments on Chinese (CSI300/500) and U.S. (S&P 500) markets show consistent gains in ICIR, Sharpe ratio, and drawdown metrics compared to GA- and RL-based baselines such as AlphaGen, AlphaQCM, and AlphaForge. Ablations indicate that structure-aware encoding and reward densification contribute most to performance.", "questions": "Can the authors formalize how the GFlowNet objective encourages low correlation among generated alphas, perhaps via a diversity regularizer or entropy argument? \nHow does the framework scale with a larger alpha pool, and what is its computational cost relative to RL-based methods? \nIt would be helpful to include a robustness test on extreme market periods (e.g., 2020–2022) to verify generalization under stress. A comparison with a Transformer-based encoder could clarify whether improvements stem from graph reasoning or simply from greater model capacity. \nFinally, reporting statistical confidence intervals for ICIR and Sharpe ratios would substantially strengthen the empirical credibility.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper tackles a meaningful and challenging problem that lies at the intersection of quantitative finance, graph learning, and generative modeling. The idea of using GFlowNets to sample a diverse distribution of high-reward alphas is original within this domain and fits the practical need for low-correlation factors. The RGCN-based encoder provides a principled way to respect formula structure, improving over purely sequential representations. The dense reward design is intuitive and empirically effective, addressing the sparse-feedback issue that limits existing reinforcement-learning frameworks. Experiments span multiple markets and metrics, with ablation and sensitivity analyses that help clarify component contributions. The writing is generally clear, and the motivation for diversity-seeking generation is well articulated.", "weaknesses": "The main concern lies in theoretical and methodological depth. The link between GFlowNet flow conservation and portfolio diversity is described only intuitively; no analytical justification or formal insight is provided. The reward composition—particularly the time-dependent weights λ(T) and η(T)—appears heuristic, and the sensitivity analysis, while helpful, does not demonstrate robustness across market regimes or noisy IC estimation. The experimental setup lacks transparency: complex baselines such as AlphaForge and AlphaQCM are not described in enough detail to ensure a fair comparison, and no significance testing or robustness checks (e.g., bootstrapping, stress periods) are presented. The novelty is somewhat incremental since the components (GFlowNets, RGCNs, multi-objective reward) are all known; the contribution lies mainly in system integration rather than a conceptual advance. Finally, presentation can be improved—several mathematical sections omit key definitions, and Figure 2 is overly dense and difficult to interpret.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "ZUl2MZCPFW", "reviewer": "Reviewer_fD8y", "summary": "The paper proposes AlphaSAGE, a method for automated mining of  alphas (interpretable signals expressed as symbolic expressions) in quantitative finance. It is a direct application of GFlowNets to quantitative-finance tasks via a new encoder for AST representation and curated rewards. Experiments on CSI300/CSI500 (China) and S&P500 (US) show consistent gains over GA, RL, and GAN-style baselines, supported by ablations and sensitivity analyses.", "questions": "1. What is the combination model $c(\\cdot)$? Please give detail description.\n\n2.  The 'MaxLen' in eq.(3) seems to be a important hyper-parameter to avoid overly deep ASTs. Please provide ablation study on how this hyper-parameter affect the early stop mechanism and the GFlowNet performance.\n\n3. Why do all the reported results appear to be based on single trials rather than multiple independent runs?\n\n4. What's the cumulative returns in the plotted curves?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* It is new to use a Relational Graph Convolutional Network and  Abstract Syntax Trees (AST) to encode the information within alphas for GFlowNet training.\n\n* The author designs diverse rewards, $R_{IC}$, $R_{SA}$ and $R_{Nov}$ to better guided the GFlowNet sampling.", "weaknesses": "1. The introduced components (RGCN encoder, composite reward) are primarily engineering improvements tailored to specific applications. These designs are orthogonal plug-ins to the GFlowNet pipeline and, while they improve empirical results, they do not substantively advance GFlowNet methodology or theory—particularly regarding robust exploration.\n\n2. The paper attributes improvements in exploration and reward sparsity primarily to reward design, yet these issues originate from the inherently vast trajectory and state space in GFlowNet training. While the proposed dense and multi-faceted reward function provides a smoother optimization landscape, it only mitigates rather than resolves the intrinsic exploration problem. Effective traversal of such large spaces fundamentally requires exploration mechanisms and policy optimization strategies, which are not the focus of this work.\n\n3. The reward-reshaping component is incremental: although the design here differs, the underlying idea is well established by many prior works. Contrary to the author's claim, the work does not present a new generative framework. The approach remains largely adheres to the standard GFlowNet formulation, incorporating incremental modifications for specific applications.\n\n4. As discussed in the introduction, reinforcement learning (RL) methods are key baselines for comparison. Since the proposed encoder and reward components are orthogonal to both RL and GFlowNet frameworks, it is important to examine performance differences between them both with and without these components. Such comparisons would clarify whether the reported performance gains stem from the GFlowNet framework itself or primarily from the proposed encoder and reward designs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "tbQHnSogkY", "reviewer": "Reviewer_BXEc", "summary": "This paper proposes AlphaSAGE, a structure-aware framework for automated alpha mining in quantitative finance that integrates relational graph convolutional networks (RGCN) with generative flow networks (GFlowNets) to improve exploration efficiency, structural understanding, and diversity of discovered alphas. By representing alphas as abstract syntax trees (AST) and introducing a dense multi-faceted reward, AlphaSAGE effectively addresses reward sparsity and mode collapse issues in reinforcement learning-based approaches. Experiments on both Chinese and U.S. stock markets show that AlphaSAGE consistently outperforms baselines in terms of correlation and portfolio metrics.", "questions": "1. The paper lacks a discussion on the reproducibility of experimental results, as it does not provide information about the variance or stability of the reported outcomes. Were the results obtained from a single training run, or were they averaged across multiple runs to ensure statistical reliability?\n2. The paper does not sufficiently elaborate on how the GNN embedding specifically influences inter-alpha correlations. In a highly collinear factor space, is there any empirical or theoretical evidence demonstrating that the proposed representation effectively mitigates redundancy or correlation among alphas?\n3. The cumulative return results on CSI500 (2022–2024) and S&P500 (2018–2020) are not as pronounced, and in some cases even underperform compared to certain baselines. Could the authors provide further analysis or explanations for these weaker results, such as data regime shifts, market noise, or model sensitivity?\n4. Although the multi-faceted reward design effectively alleviates the reward sparsity problem, the paper does not clearly demonstrate that different reward components are non-conflicting. It remains unclear how the objectives are balanced during training to avoid mutual interference.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper is well-structured and logically coherent, with publicly available reproduction code that ensures strong reproducibility.\n2. The authors introduce Generative Flow Networks (GFlowNets) into quantitative finance, which, compared with conventional reinforcement learning methods, can generate diverse and high-quality solutions while maintaining reward sensitivity, effectively alleviating the mode collapse problem.\n3. The paper employs a Relational Graph Convolutional Network (RGCN) to model the abstract syntax tree (AST) of formulaic alphas, effectively capturing their hierarchical structure and semantic relationships among operators.\n4. The authors design a multi-level reward mechanism that integrates multiple concerns into a dense and stable optimization signal, thereby enhancing both exploration efficiency and convergence stability.\n5. The experimental section covers both Chinese and U.S. markets, and AlphaSAGE consistently outperforms recent baseline methods across various metrics, demonstrating comprehensive and up-to-date experimental comparisons.", "weaknesses": "1. Although the paper proposes a multi-dimensional reward function and provides ablation studies on the weighting parameters, the selection of these weights relies heavily on empirical tuning. The lack of a theoretically grounded or adaptive weighting mechanism may limit the model’s stability and robustness across different market regimes.\n2. While each module in the framework is meaningful, the overall architecture is relatively complex and computationally intensive during training, which may hinder its direct application to high-frequency or real-time trading scenarios. Moreover, the paper lacks a detailed analysis or comparison of computational cost.\n3. The background section remains somewhat vague and may not be sufficiently accessible for readers who are not familiar with quantitative finance, which slightly limits the paper’s readability for a broader audience.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)