Paper: Reinforcement Symbolic Regression Machine (ICLR-2024)

Reviewer comment (Writing):
The paper has space for improvement in terms of presentation. The paper is not well written and the authors should carefully proofread the paper to improve the presentation quality.

Author response:
**Q1. Clarity and presentation of the paper.**

**Reply:** Thank you for this comment. We have followed the reviewer’s suggestion and thoroughly revised the manuscript. Here are some examples of revision (more details are found in the revised manuscript):

To better illustrate the novelty of our proposed model, an additional paragraph highlighting the motivation and novelty of the Modulated Sub-tree Discovery (MSDB) has been included.

- *We introduced a new modulated sub-tree discovery (MSDB) block, the critical component of the proposed RSRM model, to heuristically learn and define new math operators to improve the representation ability of underlying math expression trees. Here, we would like to emphasize that MSDB addresses a crucial observation that models often struggle to generate complete expressions but excel in capturing certain components. For instance, NGGP may discover an expression like $x^4-x^3+\cos(y)+x-1$, while the ground truth is $x^4-x^3-0.5y^2+x$. Notably, it successfully recovers the simplified expression $x-0.5y^2$ with the same distribution. To this end, MSDB offers a new alternative to simplify expressions by subtracting specific components in the context of a sub-tree, as exemplified by the subtraction of $x^4-x^3$ in the aforementioned case.*

Given the brevity of the description of the workflow in the previous version, we have revised the caption of Figure 1. 

- *RSRM comprises several steps. First, the data is input into the data pre-processing module to determine the parity. Each epoch, the process initiates with the utilization of the MCTS module to generate expressions (Step ①). Subsequently, these expressions undergo the addition of forms and reward calculation (Step ②). The expressions are then refined through GP (Step ③), leading to more accurate expressions (Step ④). Following this, the equation forms are integrated and rewards are calculated (Step ⑤). This iterative cycle, involving MCTS and GP, continues for several rounds, ultimately resulting in the renewal of forms based on the most proficient expressions (Step ⑥).*

In the absence of a comprehensive description of the design of the three sub-tree expression forms, an additional paragraph has been added to explicate the genesis of these forms.

- *In fact, the three forms (namely, ${\mathcal{A} + f(x)}$, $\mathcal{A}\times f(x)$, $\mathcal{A}^{f(x)}$) are general and applicable to express sub-tree structures of any generic formula. Our approach involves the establishment of these forms based on the initial token of the expression tree, because the root of an expression tree serves as a focal point, indicating the primary operation or function in the expression. Thus, we can separate the sub-tree forms based on it.  Specifically, if the first token corresponds to addition ($+$) or subtraction ($-$), the method proceeds to learn the generation of the left and right sides of the respective operators. Similarly, for tokens such as multiplication ($\times$), division ($\div$), or exponentiation (\^{}), a similar procedure is followed. In the case of unary expressions, such as trigonometric functions ($\sin$ and $\cos$), the MCTS and GP models effortlessly derive the complete expression. Therefore, while our method involves a degree of empirical design in identifying the sub-tree expression forms, it possesses a universal nature.*

In addition, here are some examples of the revision of uncleared sentences:

- *Symbolic Regression (SR) can be defined as the task of automatically distilling equations from limited data.*

- *However, there still exist bottlenecks that current methods struggle to break when the expression we need to explore tends toward infinity and especially when the underlying math formula is intricate.*

- *In the early years, polynomial analysis was commonly used to fit expressions.*

- *By incorporating double Q-learning into MCTS, we effectively balance exploration and exploitation of SR tasks*

- *The objective of SR can be transformed into the generation of an optimal expression tree.*

We believe these revisions significantly enhance the clarity of our manuscript, making it easier to follow. We would like to take this opportunity to thank the reviewer for the careful reading and putting forward these detailed and constructive comments.

Reviewer comment (Writing):
(Abstract) "In nature, the behaviors". Consider changing to "In nature, the behavior".

Author response:
In addition, here are some examples of the revision of uncleared sentences:

- *Symbolic Regression (SR) can be defined as the task of automatically distilling equations from limited data.*

Reviewer comment (Writing):
(Abstract) "Automatically distilling these equations from limited data is cast as a symbolic regression (SR)". This sentence is confusing. Symbolic Regression **is** the task of distilling equations from data. The sentence gives the impression that SR is something else and that the authors are using SR to solve that something else. Please clarify.

Author response:
In addition, here are some examples of the revision of uncleared sentences:

- *Symbolic Regression (SR) can be defined as the task of automatically distilling equations from limited data.*

Reviewer comment (Theory):
(Abstract) "The discrete search space tends toward infinity". This statement is not mathematically rigorous. The search space in SR (with variable length expressions) is always infinite. Do the authors mean the complexity of the search space? How do they measure it? Please clarify.

Author response:
In addition, here are some examples of the revision of uncleared sentences:

- *However, there still exist bottlenecks that current methods struggle to break when the expression we need to explore tends toward infinity and especially when the underlying math formula is intricate.*

Reviewer comment (Writing):
(Method) "we introduce an interpolation method (e.g., data pre-processing)". What do the authors mean by "interpolation method" and how is "data pre-processing" related to it? Please clarify.

Author response:
**Q2. The functionality of the interpolation method and its relation to data pre-processing.**

**Reply:** Thanks for your comment. We employ an interpolation method to generate $f(-x)$ and directly compare it with $f(x)$ for each data point, establishing the presence of odd, even, or neither symmetry within the function. The determination of parity remains consistent throughout and precedes the search step, remaining unchanged thereafter. Hence, this meticulous procedure is a fundamental component of our pre-processing stage.

Reviewer comment (Evaluation):
(Method) "the MSDB examines whether the few expressions that perform well adhere to a specific form. For instance, if both $\exp(x) − x$ and $\exp(x) + x$ yield favorable results, the expression can be confirmed as $\exp(x) − f(x)$, thereby allowing us to focus on finding $f(x)$". How do the authors determine what "perform well" means? The reviewer understands the intuition that if both $\exp(x) − x$ and $\exp(x) + x$ yield favorable results, then the expression must be close to the middle of both expressions, i.e., $\exp(x)$. However, how to determine the threshold in the fitting error to define "perform well" is not clear. Is this threshold the same for all datasets? If not, how is it determined? Please clarify.

Author response:
**Q3. The meaning of “performing well” and details about the selection of expression forms.**

**Reply:** Thank for your suggestion. Firstly, performing well means greater reward.

Next, we give details of selecting expression forms: (1) In the form of multiplication ($\mathcal A \times f(x)$) or exponentiation ($\mathcal A^{f(x)}$), we select the best expression as the chosen form. (2) In the case of addition ($\mathcal A + f(x)$), we opt for expressions with a reward greater than $k_p \times \max\{\text{reward}\}$.Then those expressions are split and count each part. Additionally, we select components whose occurrences satisfy $o > k_s \times \max\{o\}$. The values of $k_s$ and $k_p$ are provided in Appendix A.1. For instance, if $x^5$ occurs 50 times, $-x^4$ occurs 4 times, and $x^3$ occurs 11 times, we finds the form as $x^5 - x^3 + f(x)$. These thresholds remain consistent across all datasets.

Reviewer comment (Writing):
(Method) "This approach effectively reduces the difficulty associated with specific equations." This statement is not clear. Please clarify.

Author response:
**Q4. The unclear part of the form-discovery algorithm.**

**Reply:** Thank for your notice. Please see our explanation as follows.

> "This approach effectively reduces the difficulty associated with specific equations." This statement is not clear. 

For example, obtaining an expression like $x^4 - x^3 - 0.5y^2 + x$ directly can be challenging. However, by identifying its components such as $x^4$ and $-x^3$, we construct the form $x^4 - x^3 + x + f(x)$ to alleviate the difficulty of searching for the term $-0.5y^2$.

Reviewer comment (Writing):
(Expression Tree) "The underlying objective of SR is to transform the given task into the generation of an optimal expression tree". This statement is not clear. Which task is SR transforming into the generation of an optimal expression tree? Please clarify.

Author response:
In addition, here are some examples of the revision of uncleared sentences:

- *The objective of SR can be transformed into the generation of an optimal expression tree.*

Reviewer comment (Writing):
(Modulated sub-tree discovery) "This search form focuses on identifying expressions of the form like ex − x and ex + x." What do the authors mean by "search form"? Please clarify.

Author response:
**Q4. The unclear part of the form-discovery algorithm.**

**Reply:** Thank for your notice. Please see our explanation as follows.

> The concept of "search form" and the use of "..in this search form" is not clear. Is search form a synonym of "search space"? Please clarify.

The term “search form” denotes the continuous exploration based on a predefined expression structure. For instance, in the form $e^x + f(x)$, we regard $e^x$ as an integral part of the overall expression. Therefore, if the outcome of the MCTS is $x + x^2$, the resultant expression would be $e^x + x + x^2$. Our calculations for RMSE and reward are based on this resultant expression.

The form ($\mathcal A + f(x)$) considers that the entire expression is constructed by the concatenation of terms through sign addition ($+$) or subtraction ($-$). Consequently, if expressions such as $e^x - x$ and $e^x + x$ yield favorable results, it suggests that the overall expression may be of the form $e^x + f(x)$. 

Similarly, when the form($\mathcal A \times f(x)$) involves concatenation through multiplication ($\times$) or division ($\div$), the evaluation of expressions like $1.57e^x$ and $1.56e^x + x$ helps identify that the expression could be of the form $e^x \times f(x)$.

Reviewer comment (Writing):
(Modulated sub-tree discovery) "In this search form, we obtain good expressions such as 1.57ex and 1.56ex + x,". The concept of "search form" and the use of "..in this search form" is not clear. Is search form a synonym of "search space"? Please clarify.

Author response:
**Q4. The unclear part of the form-discovery algorithm.**

**Reply:** Thank for your notice. Please see our explanation as follows.

> The concept of "search form" and the use of "..in this search form" is not clear. Is search form a synonym of "search space"? Please clarify.

The term “search form” denotes the continuous exploration based on a predefined expression structure. For instance, in the form $e^x + f(x)$, we regard $e^x$ as an integral part of the overall expression. Therefore, if the outcome of the MCTS is $x + x^2$, the resultant expression would be $e^x + x + x^2$. Our calculations for RMSE and reward are based on this resultant expression.

The form ($\mathcal A + f(x)$) considers that the entire expression is constructed by the concatenation of terms through sign addition ($+$) or subtraction ($-$). Consequently, if expressions such as $e^x - x$ and $e^x + x$ yield favorable results, it suggests that the overall expression may be of the form $e^x + f(x)$. 

Similarly, when the form($\mathcal A \times f(x)$) involves concatenation through multiplication ($\times$) or division ($\div$), the evaluation of expressions like $1.57e^x$ and $1.56e^x + x$ helps identify that the expression could be of the form $e^x \times f(x)$.

Reviewer comment (Evaluation):
(Results) How many expressions were evaluated in each algorithm? Is this the same number for all algorithms? This is very important to assess the computational complexity of the algorithm proposed. Please clarify.

Author response:
**Q5. Number of expressions evaluated.** 

**Reply:** This is an excellent comment. In fact, there is a tradeoff between the recovery rate (accuracy) and the computational complexity of the model. We have performed a trade-off experiment in the manuscript illustrating the recovery rate in terms of the number of evaluations. Please see Appendix Section C.7 and Fig. S2. Here is the table of this experiment result for your reference.

**Table.** Trade-off experiment: Average Number of Evaluations / Average Recovery Rate (%) of the Nyugen Benchmark over 100 parallel runs.

| Name      | 5 epochs       | 15 epochs      | 25 epochs       | 35 epochs       | 45 epochs       
|-----------|----------------|----------------|-----------------|-----------------|------------------|
| ALL       | ${15654}/{53}$ | ${27543}/{77}$ | ${41525}/{87}$  | ${55395}/{98}$  | ${58240}/{100}$ 
| Nyugen-4  | ${28389}/{5}$  | ${41638}/{43}$ | ${54151}/{75}$  | ${58483}/{100}$ | ${76216}/{100}$ 
| Nyugen-5  | ${25084}/{0}$  | ${57033}/{33}$ | ${77075}/{80}$  | ${118661}/{83}$ | ${149868}/{100}$
| Nyugen-11 | ${20932}/{41}$ | ${36487}/{62}$ | ${67237}/{88}$  | ${77757}/{100}$ | ${84902}/{100}$ 
| Nyugen-12 | ${26345}/{0}$  | ${65122}/{13}$ | ${123571}/{41}$ | ${175970}/{89}$ | ${207889}/{100}$

Please do not hesitate to let us know if you have any further questions. Thank you very much.

Reviewer comment (Theory):
There are no theoretical results to verify the proposed approach. However, the method does obtain strong empirical performance.

Author response:
**Q1. Theoretical Analysis.**

**Reply:** Thanks for this comment. Since primary part of the search method we employed in our proposed RL model is the UCT-based MCTS, the theoretical analysis (e.g., convergence, guarantees) of this algorithm can be found in literature such as Shah et al. (2019). We have clarified this in our revised manuscript.

Reviewer comment (Writing):
Page 3, top. $c$ is undefined. It could perhaps be helpful to state what $c$ is in the UCT algorithm or provide a reference to either another paper or a forward reference to where it is defined.

Author response:
**Q2. Definition of $c$ in the UCT algorithm.**

**Reply:** Thanks for the reviewer’s comment. $c$ typically represents the exploration-exploitation tradeoff parameter. The exploration-exploitation tradeoff is a fundamental challenge in decision-making under uncertainty. In the context of UCT, it refers to the balance between exploring new possibilities (potentially discovering a better move or solution) and exploiting known information (favoring moves or solutions that seem promising based on existing knowledge).

Choosing an appropriate value for $c$ is crucial, as it influences the algorithm's behavior. A higher $c$ encourages more exploration, while a lower $c$ emphasizes exploitation. The optimal value for $c$ often depends on the specific problem or environment being addressed. It may need to be tuned through experimentation for different scenarios. In this paper , we take $c = \sqrt{2}$.

Reviewer comment (Experiments):
Page 5, "we find that Gaussian random numbers with a unit mean and variance provide more effective initial values for optimization.". There seems to be no empirical evidence in the paper supporting this. Although you may have results for this, it could be helpful for the reader if you included an additional Appendix experiment and linked it to this statement as the empirical evidence for this.

Author response:
**Q3. Experiment on initial value of constant optimization.**

**Reply:**  This is an excellent remark. We conducted an experiment considering different initialization approaches for constant optimization. Specifically, we explored seven methods shown as follows (Case 1 – Case 7).

Case 1: Initializing constants with a vector of ones (the initialization method in former models).
Case 2: Initializing constants with a vector of random uniform values between 0 and 1.
Case 3: Initializing constants with a vector of random uniform values between 0.5 and 1.5.
Case 4: Initializing constants with a vector of random Gaussian values with a mean of 0 and standard deviation of 1.
Case 5: Initializing constants with a vector of random Gaussian values with a mean of 1 and standard deviation of 1.
Case 6: Initializing constants with a vector of random Gaussian values with a mean of 1 and standard deviation of 0.5.
Case 7: Initializing constants with a vector of random Gaussian values with a mean of 0 and standard deviation of $\frac{1}{3}$.

We evaluated the recovery rate of each expression using different constant initialization methods across diverse benchmarks, employing ranges of input values such as [0,1], [-1,1], [0,10], [-10,10], [0,50], [-50,50], and data sizes of 20 or 500. Notably, Table 1 demonstrates that the average recovery rates across all initializing methods are remarkably close.

However, examining the distribution of expression recovery rates below 50%, 10%, and 0%, it becomes apparent that the method employing a vector of ones exhibits the highest percentage from Table 2. This observation indicates that the Case 1 initialization method (with the vector of all ones) has the highest number of expressions unable to converge once across 100 parallel runs.

We have added a section in the Appendix to discuss this (see Appendix Section C.6).

Reviewer comment (Evaluation):
There are no error bars for the main results tables, Table 1, Table 2, and Table 3.

Author response:
**Q5. No error bars on the main results.**

**Reply:** Thanks for your suggestion. In fact, since the recovery rates of individual equations within the same benchmark dataset do not follow the Gaussian distribution, the application of an error bar denoting the confidence range may not be suitable in this context. This is generally admitted in the symbolic regression community. However, we provide a table (see Table 3 below) outlining the 95% confidence range for each method within each benchmark for your reference.

**Table 3.** 95% confidence range of recovery rate (%) for each method within each benchmark

| Benchmark | OURS | SPL | uDSR | NGGP | DSR | GP |
|:----:|:----:|:----:|:----:|:----:|:----:|:----:|
| Nguyen | **100.00$\pm$0.0** | 93.50$\pm$11.7 | 87.58$\pm$13.6 | 91.75$\pm$13.0 | 83.58$\pm$18.5 | 44.42$\pm$22.1 |
| Nguyen$^c$ | **100.00$\pm$0.0** | 69.57$\pm$35.2 | 60.71$\pm$33.2 | 99.14$\pm$1.2 | 70.43$\pm$35.7 | 8.43$\pm$15.6 |
| Livermore | **97.95$\pm$4.0** | 51.00$\pm$16.9 | 81.05$\pm$14.6 | 80.68$\pm$14.0 | 50.41$\pm$15.7 | 50.77$\pm$20.4 |
| R | **68.83$\pm$28.9** | 0.00$\pm$0.0 | 16.50$\pm$26.2 | 26.67$\pm$31.1 | 0.00$\pm$0.0 | 0.00$\pm$0.0 |
| AIFeynman | **73.50$\pm$24.1** | 66.42$\pm$27.8 | 63.50$\pm$26.0 | 67.25$\pm$27.4 | 66.42$\pm$27.8 | 60.58$\pm$25.7 |

Please do not hesitate to let us know if you have any further questions. Thank you very much.

Reviewer comment (Novelty):
The novelty of the proposed method is unclear. The proposed method seems to be a simple combination of existing methods, including the monte carlo tree search (MCTS) algorithm, double q-learning method, and genetic programming method.

Author response:
Thanks for this question. Firstly, we would like to mention that existing symbolic regression (SR) methods generally struggle with generating lengthy and complex equations, and are faced with issues related to overfitting, e.g., poor generalizability. We aimed to develop a new model (aka., RSRM) to overcome these critical challenges, which has shown SOTA performance as demonstrated over extensive examples on over 200 symbolic test formulas. 

Secondly, we do not agree with the reviewer’s comment on “*a simple combination of existing methods*”. The coordination of different methods or blocks requires a **rational** design (see the workflow in Figure 1) which largely affects the balance between exploration and exploitation. A simple combination would not work. In addition, we introduced a **new** modulated sub-tree discovery (MSDB) block, the critical component of the proposed RSRM model, to heuristically learn and define new math operators to improve the representation ability of underlying math expression trees. Here, we would like to emphasize that MSDB addresses a crucial observation that models often struggle to generate complete expressions but excel in capturing certain components. For instance, NGGP may discover an expression like $x^4-x^3+\cos(y)+x-1$, while the ground truth is $x^4-x^3-0.5y^2+x$. Notably, it successfully recovers the simplified expression $x-0.5y^2$ with the same distribution. To this end, MSDB offers a new alternative to simplify expressions by subtracting specific components in the context of a sub-tree, as exemplified by the subtraction of $x^4-x^3$ in the aforementioned case. 

Therefore, the **novelty** of the proposed model is three-fold: *(1)* Using double Q-learning in junction with MCTS effectively reduces redundant search thus improving the exploration efficiency and efficacy. *(2)* The proposed MSDB block for discovering new sub-tree expression forms takes the divide-and-conquer concept and could significantly improve the overall search performance of the RSRM model. In particular, MSDB can handle equations with symmetry (reducing the complexity), and assist in dealing with long equations by identifying common patterns and defining new math operators on the fly. *(3)* The proposed RSRM model demonstrates clear superiority over many representative baseline models and yields the SOTA SR results, which offers a new approach to tackle very complex SR challenges.

Reviewer comment (Novelty):
The motivation and advantages of the proposed method over previous work are unclear.

Author response:
Thanks for this question. Firstly, we would like to mention that existing symbolic regression (SR) methods generally struggle with generating lengthy and complex equations, and are faced with issues related to overfitting, e.g., poor generalizability. We aimed to develop a new model (aka., RSRM) to overcome these critical challenges, which has shown SOTA performance as demonstrated over extensive examples on over 200 symbolic test formulas. 

Secondly, we do not agree with the reviewer’s comment on “*a simple combination of existing methods*”. The coordination of different methods or blocks requires a **rational** design (see the workflow in Figure 1) which largely affects the balance between exploration and exploitation. A simple combination would not work. In addition, we introduced a **new** modulated sub-tree discovery (MSDB) block, the critical component of the proposed RSRM model, to heuristically learn and define new math operators to improve the representation ability of underlying math expression trees. Here, we would like to emphasize that MSDB addresses a crucial observation that models often struggle to generate complete expressions but excel in capturing certain components. For instance, NGGP may discover an expression like $x^4-x^3+\cos(y)+x-1$, while the ground truth is $x^4-x^3-0.5y^2+x$. Notably, it successfully recovers the simplified expression $x-0.5y^2$ with the same distribution. To this end, MSDB offers a new alternative to simplify expressions by subtracting specific components in the context of a sub-tree, as exemplified by the subtraction of $x^4-x^3$ in the aforementioned case. 

Therefore, the **novelty** of the proposed model is three-fold: *(1)* Using double Q-learning in junction with MCTS effectively reduces redundant search thus improving the exploration efficiency and efficacy. *(2)* The proposed MSDB block for discovering new sub-tree expression forms takes the divide-and-conquer concept and could significantly improve the overall search performance of the RSRM model. In particular, MSDB can handle equations with symmetry (reducing the complexity), and assist in dealing with long equations by identifying common patterns and defining new math operators on the fly. *(3)* The proposed RSRM model demonstrates clear superiority over many representative baseline models and yields the SOTA SR results, which offers a new approach to tackle very complex SR challenges.

Reviewer comment (Experiments):
The authors claim that they use a MCTS agent for exploration. However, I found that they use greedy selection rather than exploration based on the upper confidence bound, which is a key component for efficient exploration in MCTS. Thus, whether the MCTS agent can explore the environment efficiently is unconvincing.

Author response:
This is a good question. In fact, the UCT is employed during the MCTS simulation while the greedy selection of the maximum reward is only applied to choose the optimal expression tree (as shown in Algorithm 1). We have tested, when we designed the model, that greedy selection of the maximum reward performs better that the case of the highest UCT score in the context of SR tasks. 

Note that the maximum UCT score-based greedy selection method often selects nodes with a lower possibility to generate non-meaningful equations. For instance, it may prefer selecting $\sin$ nodes across all instances range over $\log$ nodes, because the output of $\sin$ is  within the [-1,1] range, and the latter might yield NaN for negative values. However, this preference doesn't align with our objective, as the $\log$ token could potentially be the optimal choice. Hence, we opt to only consider the best expression (with the maximum reward) to determine which node to use in the selection step.

Reviewer comment (Experiments):
The proposed modulated sub-tree discovery incorporate three specific search forms into the algorithm. However, the three forms may be too specific to be generally applicable to complex real-world problems.

Author response:
Thanks for the reviewer’s excellent comment. In fact, the three forms (namely, ${\mathcal{A} + f(x)}$, $\mathcal{A}\times f(x)$, $\mathcal{A}^{f(x)}$) are general and applicable to express sub-tree structures of any generic formula. Our approach involves the establishment of these forms based on the initial token of the expression tree, because the root of an expression tree serves as a focal point, indicating the primary operation or function in the expression. Thus, we can separate the sub-tree forms based on it.  Specifically, if the first token corresponds to addition ($+$) or subtraction ($-$), the method proceeds to learn the generation of the left and right sides of the respective operators. Similarly, for tokens such as multiplication ($\times$), division ($\div$), or exponentiation (\^{}), a similar procedure is followed. In the case of unary expressions, such as trigonometric functions ($\sin$ and $\cos$), the MCTS and GP models effortlessly derive the complete expression. Therefore, while our method involves a degree of empirical design in identifying the sub-tree expression forms, it possesses a universal nature.

Reviewer comment (Experiments):
The authors evaluate the generalization ability of their method in a toy environment. It would be more convincing if the authors could conduct the generalization experiments on more complex real-world problems.

Author response:
Thanks for your comment and suggestion. The reason why we tested our model by discovering a surrogate formula to approximate the cumulative density function (CDF) of a normal distribution is because this equation lacks an explicit elementary expression. Hence, it's impossible to obtain an exact solution through SR. Learning this function requires balancing between training error and generalization error, demanding a stronger ability to generalize. Moreover, only half of the curve data is provided, necessitating strong generalization skills from the model to extrapolate and accurately fit the missing portion of the curve. This calls for the proficiency of the model in capturing the distribution of the entire curve. In summary, this is in fact not a simple SR case, in which many existing SR methods fail.

Indeed, we acknowledge the importance of assessing the generalization ability of our model in more complex real-world scenarios. To address this, we conducted tests on the SRBench datasets, a benchmark comprising complex real-world problems with added noise. The results (see Figure 3b and Appendix Figure S3) demonstrate the superior performance of our model compared to other models, including UDSR, which has been recognized as the leading model in the preceding year.

Reviewer comment (Theory):
There is a limited set of hand-designed expression forms, which restricts the applicability of the method to other domains.

Author response:
**Reply:** Thanks for the reviewer’s excellent comment. In fact, the three forms (namely, ${\mathcal{A} + f(x)}$, $\mathcal{A}\times f(x)$, $\mathcal{A}^{f(x)}$) are general and applicable to express sub-tree structures of any generic formula. Our approach involves the establishment of these forms based on the initial token of the expression tree, because the root of an expression tree serves as a focal point, indicating the primary operation or function in the expression. Thus, we can separate the sub-tree forms based on it.  Specifically, if the first token corresponds to addition ($+$) or subtraction ($-$), the method proceeds to learn the generation of the left and right sides of the respective operators. Similarly, for tokens such as multiplication ($\times$), division ($\div$), or exponentiation (\^{}), a similar procedure is followed. In the case of unary expressions, such as trigonometric functions ($\sin$ and $\cos$), the MCTS and GP models effortlessly derive the complete expression. Therefore, while our method involves a degree of empirical design in identifying the sub-tree expression forms, it possesses a universal nature.

Reviewer comment (Experiments):
It is a bit illusive how SR can be applied to practical domains.

Author response:
**Reply:** Thanks for this question. Symbolic regression (SR) has found broad applications across diverse fields, including discovery of fundamental physical laws [1, 2] or governing equations [3-5], modeling material constitutive relations [6], and TCP congestion control [7], among many others. For example, in the case of TCP congestion control, SR can be used to learn the control policy with better interpretability and generalization ability, which also exhibit advantage of computational speed in comparison with neural network regression models. In addition, when the training data is very limited, the neural network model tends to have the over-fitting issue, whereas SR can still produce a model with good generalizability.

Reviewer comment (Experiments):
What are some practical domains that SR is useful in? In particular, what are some scenarios where SR outperforms neural network regression?

Author response:
**Reply:** Thanks for this question. Symbolic regression (SR) has found broad applications across diverse fields, including discovery of fundamental physical laws [1, 2] or governing equations [3-5], modeling material constitutive relations [6], and TCP congestion control [7], among many others. For example, in the case of TCP congestion control, SR can be used to learn the control policy with better interpretability and generalization ability, which also exhibit advantage of computational speed in comparison with neural network regression models. In addition, when the training data is very limited, the neural network model tends to have the over-fitting issue, whereas SR can still produce a model with good generalizability.