Logic-Logit: A Logic-Based Approach to Choice Modeling
Authors: Shuhan Zhang
Wendi Ren
Shuang Li
Venue: ICLR 2025 Poster (2025)
Keywords: Choice Model
Preference Learning
Interpretability
Rule Learning
Primary area: interpretability and explainable AI

Abstract: In this study, we propose a novel rule-based interpretable choice model, {\bf Logic-Logit}, designed to effectively learn and explain human choices. Choice models have been widely applied across various domains—such as commercial demand forecasting, recommendation systems, and consumer behavior analysis—typically categorized as parametric, nonparametric, or deep network-based. While recent innovations have favored neural network approaches for their computational power, these flexible models often involve large parameter sets and lack interpretability, limiting their effectiveness in contexts where transparency is essential.

Previous empirical evidence shows that individuals usually use {\it heuristic decision rules} to form their consideration sets, from which they then choose. These rules are often represented as {\it disjunctions of conjunctions} (i.e., OR-of-ANDs). These rules-driven, {\it consider-then-choose} decision processes enable people to quickly screen numerous alternatives while reducing cognitive and search costs. Motivated by this insight, our approach leverages logic rules to elucidate human choices, providing a fresh perspective on preference modeling. We introduce a unique combination of column generation techniques and the Frank-Wolfe algorithm to facilitate efficient rule extraction for preference modeling—a process recognized as NP-hard. Our empirical evaluation, conducted on both synthetic datasets and real-world data from commercial and healthcare domains, demonstrates that Logic-Logit significantly outperforms baseline models in terms of interpretability and accuracy.

Published as a conference paper at ICLR 2025
LOGIC-LOGIT: A LOGIC-BASED APPROACH TO CHOICE MODELING
Shuhan Zhang, Wendi Ren & Shuang Li
CUHK-Shenzhen
(shuhanzhang, wendiren/@@innk cuhk. edu cn, wwwhuungg@uuhkkccomm edu.
ABSTRACT
In this study, we propose a novel Logit, designed to effectively learn and explain human choices. Choice models have been widely applied across various domains-such as commercial demand forecasting, recommendation systems, and consumer behavior analysis-typicalyy categorized parametric, nonparametrie, or deep network-based. While re- cent innovations have favored neural network approaches for their computational power, these flexible models often involve large parameter pretability, limiting their effectiveness in contexts where transparency is essential.
interpretable choice model, Logic-
and lack inter-
Previous empirical evidence shows that individuals usually use hearistic decision rules to form their consideration sets, from which they then choose. These rules are often represented as disjunctions of conjunctions (i.e., OR-of-ANDs). These rules-driven, Smnsiderthhencchoss decision processes enable people to quickly screen numerous alternatives while reducing cognitive and search costs. Mo- tivated by this insight, our approach leverages logic rules to elucidate human choices, providing a fresh perspective preference modeling. We introduce unique combination of column generation techniques and the Frank-Wolfe al- gorithm facilitate efficient rule extraction for preference modeling-a process recognized as NP-hard. Our empirical evaluation, conducted on both synthetic datasets and real-world data from commercial and healthcare domains, demon- strates that Logic-Logit significantly outperforms baseline models in terms of in- terpretability and accuracy.
INTRODUCTION
In today's complex docision-making environment, individuals usually encounter overwhelming array of options to choose from. In high-stakes areas like healthcare, patients face critical decisions such selecting healthcare providers, hospitals, and insurance plans- choices that directly influ- ence their health outcomes and financial well-being. Similarly, doctors must navigate a multitude of medications and treatment plans determine the best course of action for their patients. In the realm of consumer behavior, homebuyers are tasked with evaluating properties based on various factors, such as location, price, and amenities. all of which contribute to their decision-makie cesses. This abundance of choices not only highlights the daily challenges individuals face but also underscores the need for models that can effectively capture the heterogeneous and intricate patterns inherent in their decision mmaking.
pro-
Empirical research reveals that individuals often simplify decision-making through consider-then- choose process (Hauser 2014). In this cognitive framework, people first establish a consideration using heuristic rules-logeee
guidelines that filter options before selecting from this reduced set. These rules, often informed by past experiences and contextual knowledge, are typically rep- resented as disjunctions of conjunctions (OR-of ofR-of-ANDs) This logical structure enables individuals to efficiently screen alternatives, manage the complexity of decision-making and reduce cognitive effort and search costs. Understanding and modeling this structured decision-making process is critical for developing systems that can predict and explain human choices.
Inspired by these insights, we propose the Logic-Logit model, a novel rule-based choice model designed to capture and explain heterogencous human choice patterns through OR-of-ANDs logic rules. Our approach addresses two fundamental challenges: 1) representing diverse human deci-
"Corresponding author

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sion rationales using OR-of-ANDs logic rules, and 2) efficiently extracting compact rule sets from preference datasets despite the inherent combinatorial complexity.
In our framework, human preferences are modeled through a rule-based utility function, capturing recision-making via disjunctions of conjunctions (OR-of-ANDs, logic rules. These logic-informed features, grounded in data, provide structured and interpretable representation of how individuals evaluate and prioritize options. The utility of each option is modeled as a linear combination of these logic-based features, ensuring that the model remains interpretable while capturing the complexity and heterogeneity of human preferences, where different preference types may utilize various rules with distinct weights. To estimate human preference types and their associated rule sets, we employ a two-tiered optimization framework:
In the outer loop, we utilize the Frank- Wolfe algorithm (conditional gradient) to systematically search for new preference types and estimate their proportions. This algorithm is well-suited for constrained convex optimization, enabling efficient exploration of the model space by iteratively updating preference types based on their respective logic rules (Jagabathula et al. 2020). Continuous refinement through the Frank- Wolfe algorithm ensures the model captures diverse decision-making behaviors while optimizing overall structure.
In the inner loop. the column generation algorithm effectively addresses the combinatorial nature of rule search by iteratively refining the rule set (Wei et al. 2019). This approach alternates between two essential tasks: solving a master problem to update the weights of the existing rules and tackling a subproblem to identify new rules that can enhance model performance. We begin with a small, manageable rule set and progressively introduce new rules as the algorithm iterates. The subproblem employs a risk-seeking search strategy, focusing on rules with the greatest potential for performance gains, which encourages more exploratory approach. This ensures that the rule set remains diverse and prevents premature convergence to suboptimal solutions.
Our contributions can be summarized as follows:
1) We introduce the Logic-Logit model, which effectively combines interpretability with computa- tional efficiency, providing a structured framework for capturing human choice patterns.
2) By utilizing OR-of-ANDs logic rules and integrating the Frank-Wolfe algorithm with column generation, we address the complexities involved in understanding human decision-making logic and rule extraction.
3) Our empirical evaluation, conducted on synthetic datasets and real-world data from commercial and healthcare domains, demonstrates that the Logic-Logit model significantly outperforms baseline models interpretability and accuracy, effectively bridging the gap between identifying heterogene- ity and clarifying underlying decision-making logic.
RELATED WORKS
2.1
CHOICE MODEL AND PREFERENCE LEARNING
Choice models provide structured frameworks for examining how individuals assess various al- ternatives during the recision-making process. This constitutes pivotal element in preference learning. Guided by the Random Utility Theory (RUT) proposed by |McFadden (1981), the primary objective of choice modeling is to establish mapping function that translates option features into utility value, which is a real number.
The Multinomial Logit (MNL) Model, introduced by Hausman & McFadden (1984), serves as the cornerstone for nearly all extant models and has been diversified into numerous extensions to accommodate various scenarios. For instance, the Nested Logit Model, developed by Wen & Kop- pelman (2001) and Hensher & Greene (202)), addresses issues of independence among options. The Mixture MNL, as explored by Dagabathula et al.] (202)). Bhat (1997), and Greene & Hensher (2003). is designed to handle complex distributions within the mapping function. Both our model and the model proposed by Jaagabttulla et al. (2020) are grounded in the Frank-Wolfe framework; however, the utility value of options in his model is computed using a linear approach.
More recently, Neural Network-based MNL models, such as those presented by Aouad & Désir (2022) and Wang et al. (2020), have been introduced to capture intricate and non-linear relationships between utility values and option features. These models leverage the power of neural networks to enhance the predictive accuracy and flexibility of traditional MNL models, addressing the limitations of linear assumptions and enabling more sophisticated modeling of choice behaviors.
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In the above work, all alternatives are evaluated simultaneously. In contrast, Consider-thers Choose models adopt a two-stage decision process: first, consumers form a consideration set using screening rules or heuristics; second, they select the option that maximizes utility within the consideration set. Liu & Arora (201T) highlight the conjunctive screening rule, where a product is considered only if all its features meet specific criteria. AAuad et al. (202T) develop dynamic programming framework to analyze the computational aspects of assortment optimization under this paradigm. [Akchen & Mitrofanov (2023) emphasize the importance of consideration sets, defining nonparametric choice model characterized solely by distribution over these sets.
2.2 LINEAR RULE LEARNING
Logic Rule Rule learning has long been important technique in supervised learning, partic- ularly in fields that demand model interpretability. As indicated by [Rives|] (1987) and WWi et al.] 2019), Logic Rules conceptualized as the process of thresholding original option features into binary features. These rules can take the form of conjunctions represented by the logical operator A.
Rule Learning The number of potential rules that can be derived from thresholding and conjunc- tions virtually infinite, making the identification of uppropriate rule sets and their corresponding weights in influencing choice behavior an NP-hard problem. Various methods have been proposed to tackle this challenge. Ruczinski et al. (2003) employed multiple logic trees fitted to the data, utilizing regression on the weights of these trees; however, the adjustment of logic trees proved to be time-consuming.
To address the inefficiencies associated with traditional approaches, Wei et al. 2019 introduced Generalized Linear Rule Models and implemented a column generation algorithm an advanced technique for optimizing problems afflicted by dimensionality issues (Desaulniers et al. 2006). This approach involves an iterative process of searching for and evaluating potential rules, thereby en- abling the refinement of the rule set through reweighting. Chen & Misić (2022) proposed decision tree-based choice model that incorporates column generation to capture the mutual effects among options. However, the applicability of this model is limited by its inability to leverage aption-speciffe features and the interpretability deficit inherent in random forests, whereas rules can be readily un- derstood and explicated.
BACKGROUND: DISCRETE CHOICE MODEL
Discrete choice models describe individuals' preferences among a set of alternatives. Let S be a set of alternatives, each characterized by a feature vector a, E S. Let [a.s)ees- Given choice set, predicts the probability that an individual select one of the alternatives. The choice probability of alternative s based on its utility, which is a function of the feature vector. For example, one can consider a linear utility function, f(x.) ) = w Vg. where w is the taste vector to be estimated.
Multinomial Logit This is the most classic choice model traced back to Plackett (1975) Luce (159). With a linear utility, it models the choice probability as a softmax function
exp (w" xx)
8 8 S.
Eves exp (w )
This model be explained through Random Utility Maximization theory (McFadden 1981) In this framework, the utility of item is random, expressed as U, = w" x, + <s, where 6, is the Gumbel noise. When the decision maker chooses the item that maximizes the random utility, i.e., arg max,es U.. the resulting choice probability is given by the softmax expression above.
Mixed Logit To account for heterogeneity in the taste vector wz, the mixed logit model assumes that w drawn from latent distribution A, referred to as the mixture distribution. Each user behaves according multinomial logit, but their specific taste vector is unobservable and treated as a random sample from the population distribution p. Consequently, the choice probability is
exp (w (x))))))))) See exp (w axs)
Eng ~
1 8 8 S.
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4 OUR PROPOSED MODEL: LOGIC-LOGIT
4.1 MOTIVATING EXAMPLE
example. Different buyer types have unique house- pouse-purchasing pref- erences, represented by human-readable OR-of-ANDs logic rules for filtering unsuitable options. Middle-class families with children prioritize homes meeting criteria like A (price below $500, 000), B (within 5 miles of a good school), or C (within 10 miles of work). Their choice rule is (A B) V A A C), preferring affordable homes near good school work. Single professionals focus on criteria such as D (modern amenities), E' (within miles entertainment), or F' (within miles of work). Their rule is (D A E) V (D F), favoring homes with modern features and close to entertainment or work. Wealthy buyers look for G (luxury features), H (exclusive neighborhood), or / (large lot size). Their choice logic is (G H) I)V (H /), preferring luxury properties in an exclusive area or with a large lot.
Let's take bouse-buyerr
These logic- based preferences provide structured way to capture diverse customer needs. In our model, each customer type has its own set of rules.
4.2 LOGIC-INFORMED UTILITY AND LOGIC-LOGIT
For each customer type, e.g., middle-class families with children single professionals, each has its own distinct OR-of-ANDs logic rule to determine their preferences. We define a set of predicates
P = fpm 1m where each predicate Pm X' + {0, 1} maps feature vectors, x E X to Boolean values X' refers to the set of all choice features. A predicate evaluates to 1 if the condition it specifies is satisfied and 0 otherwise. These mappings, from real-valued features to predicates, are predefined and fixed. A possible way of predefining the pm and P is shown in Appendix A which is also used in our real data experiments.
Given these predicates, OR-of-ANDs logic rule can be expressed as
R R=
V A
Pm
Pm P...A.
where R represents the disjunction of U conjunction rules specific to a customer type. This is a general form that can express any logic rules. Each conjunction consists of predicates in the set A... corresponding to the w-th conjunction for that particular type. Given M predicates, there are 2 - possible different conjunctions. each of which corresponds
non-empty subset of P. the set of all predicates. The evaluation of the rule R for instance is R (x) = -V (Ap..... Pm (1t)
For our proposed Logic-Logh model, the utility function for the customer is modeled as a weighted combination of these conjunctions, with each conjunction acting Boolean feature:
Σ Wn (Apm.. Pm (a))
P...A.
where Wn is the weight associated with the u-th conjunction. Under the multinomial logit model, the probability of selecting item s from the offer set S can then be expressed as
exp (Eun W.. (Api..A Pmn (mn())))
Pw (s S,x) Evesese exp Eum Wa (ApmEA, Pm( pm(
S S S.
(1)
We remark that the dimension of w is 2M - 1, which corresponds to the number of all possible Boolean features generated from conjunction rules derived from the predicate set P. On the other hand, it is sparse vector with only U nonzero elements {w., Jueuo.
4.3 MIXED LOGIC-LOGI MODEL
To account for the diversity in user preferences, we extend the above-introduced Logic-Logit model Mixed Logic-Logit model. In this case, the rule weight vector w is modeled as a sample drawn from a latent mixture distribution p(w). The choice probability of selecting item j from the offer
S, under this mixture distribution is
(s S, a)
Exarp (Pw [S,=)]]
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where Pw (s S.x) is computed in Eq. I We denote the mixed logit choice probability vector from a mixture distribution p as
Po(S, x) PPp x)]e.s.
LEARNING ALGORITHM
Let D be the data distribution, with each data point (S, 2..)) y) records the choice set S, their feature vectors æ and the corresponding choice frequency which is either probability vector or a one-hot vector. To learn the mixture distribution p. consider minimizing the negative log-likelihood
min PEP
L(p) == -B(..a.))=2 [v] log (S)
(3)
This problem is infinite-dimensional functional optimization problem over the mixture distribu- tion p. addition, from the discussion in Section 4.3 the taste vector ur potentially of high dimensions due to exponentially many potential rules. On the other hand, in many practical situ- ations, only small number of simple rules serve as the main driving force of a decision-making process. To facilitate interpretable way to generate sparse mixture distributions, we adopt a func- tional conditional gradient algorithm to solve (3). The algorithm's 's road map is illustrated in Fig,
5.1 FUNCTIONAL CONDITIONAL. GRADIENT ALGORITHM FOR ESTIMATING THE MIXING DISTRIBUTION
We perform functional conditional gradient (a.k.a. Frank- Frank-Wolfe) algorithm (Bach 2017) to solve the problem Eq. (3). To this end, we first compute the functional gradient of L at p. which is itself function, as
(')))w)) = (s)
Us Pa(s S, LESS
Σ
Ps Sxxx
(4)
The core of the conditional gradient descent algorithm is the Frank-Wolfe step. At each iteration k, given feasible solution Pk. this step computes the optimal solution Ps by solving the following optimization problem based on the gradient L'(px):
Pk arg min (I'(p)),P).
PEP
where the inner product is between a function and a distribution, i.e., (I'(p),,)) = (p())(px))] Using the functional gradient (4). the above minimization problem can be rewritten
min PEP
ESS...
Pp LeES
Σ
Vs 5, E, (Puoss S.
Thanks to the linearity of the objective in p. suffices to restrict to Dirac measures, i.e., p= 6np for
some taste vector w. This leads to the following equivalent form of the Frank-Wolfs step:
min ((S.
Σ
Pw
S.
(FW)
LOES
Therefore, the Frank- step finds new customer type, denoted as wi- Given dwn, the next iteration Pk+1 is defined as
Pk+ (1 7)k 1)kppx + m
where 7e is a suitable step size parameter, often chosen as F4I to achieve a sublinear conver- gence rate (11/K), or determined by line search.
Practically, one start with single customer type Po Swn. where wo is initial parameter vector, such as the best multinomial logit weights. The algorithm then iteratively adds new customer types the mixture distribution by solving the Frank-Won step (FW). To achieve better empirical performance, one can also perform a fully corrective step, which updates the mixture distribution by re-optimizing the weights on all customer types:
pk+1 +- arg min [(p): : p= Σ a, }
jm0
This is convex optimization problem over a and can be solved efficiently.
On a high level, this algorithm shares similar aspects as the mnite-dimensional conditional gradient algorithms in Dagabathula et al. (2020); Hu et al. (2022). which estimate mixed logit model by re- parameterizing it as constrained convex optimization over choice probability vectors. Different
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from theirs, we develop algorithm using the functional conditional gradient. Moreover, it is important to note that in our logic-rule-basee problem, the Frank- Wolfe step (Fw) has different computational challenge than they have, ours involves searching over space with combinatorial dimension 2 1. We devote the next section to discussing how to solve this problem efficiently.
Column Gemeration
No
Frank-Wolfe Stop Initialization
Stopping Criterion
New Preference Type
Searching
Update Rule Weights
Yes
No
Stopping
Criterion
Main Problem Initialization
Support Finding
Proportion Update
Preference Type List
Target Update
Output
Figure I: The framework of our Algorithm.
5.2 SOLVING FRANK-WOLFE STEP USING COLUMN GENERATION
Plugging in our logic-logit model o(w (Aw)
( to the Frank-Wolfe step
E W
yields the following objective
(5)
exp EE Wu (ApmEs. Pm (2x))
Σ
Us
(6)
Pou (. SES
EVEs exp E Wu (ApmEA. Pm (2)g)
When the number of predicates M is large, it results an exponentially large number of possible conjunctions. To address this challenge, devise column generation algorithm. This approach begins with a smaller, more manageable problem using a limited search space, and then gradually expands the space by adding new rules iteratively. We start with an empty rule set: k(O)) = O and initialize rule weights (0) = [- We will repeat the following procedures until the convergence or the stopping criterion is met.
Master Problem: Update Rule Weights Suppose this current rule set is R(U) = [.A)) the master problem optimizes the weights by minimizing the Frank- Wolfe objective
(( +
arg min (w [A,)
This step adjusts the weights for the current set of rules to best fit the data by minimizing the Frank- Wolfe objective Φb Note that this optimization is on the U-dimensional space and thus more tractable.
Subproblem: Search for New Rule With the rule weights (!) fixed, the subproblem focuses on identifying vector Ve(((()) (U)) E TR2 not zero vector, then there exists a descent direction that reduces the objective (5). To identify non-zero component of this high-dimensional gradient vector, we conduct a rule search as follows.
rule Ar+1 that can further enhance the objective. Specifically, if the gradient
The conjunction rule is identified as path in a tree, where each node represents a predicate. We employ two search strategies: Breadth-First Search (BFS) and Depth-First Search (DFS). BFS first examines all rules with the same length and then gradually increases rule length; whereas DFS focuses on generating new rules by performing conjunctions on the rules that have already been identified. To streamline the process, we impose a limit on rule length to ensure that we focus on rules that are both computationally feasible and manageable. Moreover, we take a risk-seeking approach by prioritizing the maximization of potential gains over average outcomes. Namely, when comparing the performance of different rules, we focus on specific quantiles of the performance distribution rather than the average. This allows to identify rules that could yield higher rewards or provide valuable insights. Such an exploratory strategy is especially useful empirically, where certain rules might produce substantial gains in specific scenarios but could be overlooked when

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Conjunction Node
Potental Node
Non Poceriial Node
Figure 2: DFS Visualization In the figure, each node corresponds to a predicate. When the search rule conjunction length increases from n + 1, the depth-first search (DFS) algorithm initiates from the previously detected n-conjunction rules and exhaustively searches through all extensions with other predicates order to discover a potential augmentation of the existing r--conjunction.
considering only average performance. Once the new rule Ar+1 is found, we add it to the rule set RU+1= <- R(a) U (Au+1).
The algorithm stops until new rule can significantly improve the objective after predefined
number of iterations. The algorithm outputs the final rule set along with their weights. After song the master problem, we may choose eliminate rules with weights below specified threshold, ensuring that only the most impactful rules remain in the set, thereby improving the algorithm's s efficiency and effectiveness. The pseudo-code and additional details are provided in Appendix
EXPERIMENT
To validate the effectiveness of our proposed method, this experimental section begins with evalu- ations of synthetic datasets with clearly defined predicates, ground truth consumer types, and deci- sion rules. We then assess distribution learning and event prediction accuracy using two complex real-world datasets. The detailed ways we predefine the predicates for the real dataset is shown in appendix A Our method is systematically compared against several baseline approaches provide a comprehensive performance evaluation.
6.1 EXPERIMENTAL SETUP
Baselines We employ MNL and Mix-MNL Jagabathula et al. 2020) representatives of the clas- sic models, while Vanilla NN, TasteNet (Han et al. (2020). DeepMNL. WWag et al., 2020 (Sifringer jet al. 2020), Mix-DeepMNI. (Jagabathula et al. 2020), and RUMNet Aouad Désir (2022) serve neural network-based alternatives. A comprehensive description and comparison of models on utility value calculation, complexity, and interpretability is provided in Table on Appendix B
Metric We report the training and testing losses as negative log-likelihoods (Fquation 3) and the top-1 accuracy for both datasets, where top-1 accuracy is the proportion of instances where the most selected product is predicted to have the highest choice probability. All results are reported as the average over 10 runs.
6.2 SYNTHETIC DATA
Generation of Offersets and Choice Data A total of 5,000 offer sets are generated, each com- prising 20 products. Each product is characterized by 50 synthetic binary features sampled from Bernoulli distribution, representing the grounded features of 50 product predicates. The complete set of predicates associated with products can be denoted as p {pr, 1 Additionally, each offer set is associated with 15 binary features induced by offerset predicates, denoted as p {pr, ) J These types of predicates are frequently encountered in practical scenarios and universal for all choice in offerset, such as search criteria in a query or the age of a patient requiring medication. The total sales volume N; for each offer set is fixed as 100 units. Three distinct preference types are predefined. Each type follows three decision rules, i.c. conjunctions predicates. Only the first 10 product predicates and offer set predicates are utilized within these preference types, adhering to the sparsity of effective predicates and demonstrating the model's selectivity when faced with redundant predicates. The choice result is computed as 100 mixed choice probability.
Rule Comparison Metric To facilitate the comparison of rules, use Jaccard index between the rules" predicate sets, defined as follows.
IAn B
J(A B)
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I- refers the set size, A and B refer to the predicate sets of the two rules. For example, the Jaccard index the following two rules p1 Aps and PiApp/Ap is 0.5, with AnB = (p2, Ps} [AnB]
and AU B {pi, P2. P4} | U B| 4. This metric directly shows the difference between the
rule contents.
Rule Comparison
0 GI Rules
GT Typcl
0.8
Gt 1pe3 GT typ2
0.6
0.4
8 7
0.2
Learned Type
Learned Type Learned Rules
Type
Figure 3: Rule Comparison. The figure shows the Jaccard index between the ground truth (GT) rules
(Y-axis) and the rules learned by our risk seeking model (X-axis), denoted by the color intensity.
Ground Truth Type
Ground Truth
Learned Rules
Jaceard Index
Rules
pi
Type
P5 A Ps
D2
A ps
P1 PS A Pd
P4 A P5
h pb
Type
P A P5 A pio
P4 A p. A pio
Pi1 A Api
Pi1 Api
Pia
Pi2
Type
Pia APii
Pia Pia
Pis A.2
Pis A Ap2
Table 1: Ground truth rules and closest learned rules on the synthetic dataset.
Results and Discussion Figure illustrates the comparison of learned rules with respect to the ground truth rules. Rule associated with the top-3 preference types, regarding their proportions, are considered as learned rules. A comprehensive comparative analysis between the ground truth rules and the identified rules provided Table [ Remarkably, eight out of nine ground truth rules are perfectly learned. The results demonstrate that Logic- effectively and selectively captures the majority of the rules without being encumbered by other irrelevant predicates included within the data. Provides more intuitive result visualization.
Type
Type
Sh Avd
Type
Leamed Type
Type
Leamed Type 3
Ground Preference Types
Learned Types
Figure 4: Rule Coverage
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PageRank Effect Capture Comparison
Mix VAL (Ponk dott)
Pako Shapelark-
3000
PegeRank Preforence Jipe
503
Pegetonk
Suck
Rask
Reck
Prize c Hask
Volue-for-monyy Preference Type
3188 MussriedPier 9133
Parfor
Figure 5: Preference Type Analysis of Expedia Hotel Dataset
6.3 EXPEDIA HOTEL DATASET EXPERIMENT
Dataset Introduction The Expedia Hotel Datase iis released in 2013 Kaggle competition for improving the recommendation system of Expedia. The dataset is characterized by structured for- mat comprising both hotel search queries and corresponding search outcomes. Each search instance in the dataset consists primarily of two key components: the search criteria and the characteristics of the retrieved hotel results. The set of hotels retrieved through search can be conceptualized as an 'offerset', wherein each hotel represents an option. The hotel that the customer ultimately books can be considered as the selected choice within this set. The detailed feature of each instance and experiment setting is shown in AppendixD
Results and Discussions Figure 5 illustrates the two primary preference types identified in the Expedia Hotel Dataset: the PageRank Preference Type (85%) and the Value-for-Money Preference Type (15%). It is important to note that this does not imply that 85% of customers strietly follow PageRank when selecting hotels; rather, individuals typically exhibit characteristies of both types, with PageRank serving as a significant factor influencing choice behavior.
The ability of MiX-MNL and Logic-Logit to capture the PageRank effect is shown by comparing the aggregated choice probabilities of options at different ranking positions with actual sales data. Logic-Logit fits rules to model sales increases from lower to higher ranks. For example, the rule Rank 3rd is derived from the substantial sales difference between hotels ranked 1 and 2nd versus those ranked 3rd to 10th In scenarios where sales remain consistent across ranks, such between 4th to 6th and 7th to 9th,, no rules are detected. Overall, Logic-Logit segments the PageRank into distinct stages with defined rules and models the non-linear effect by assigning varying weights to these rules, whereas Mix-MNL primarily captures linear effect.
Furthermore, the analysis of the Value-for-Money preference type indicates that customers asso- ciate price with multiple factors, including PageRank, star rating, location score, and historical price, through conjunction rules. Unlike neural network-based approaches that encode features into complex latent layers, Logic-Logit explicitly captures the correlations between features in an in- terpretable manner. A comprehensive performance comparison between Logic-Logit and baseline methods using the Expedia hotel dataset is included in Appendix C
6.4 MIMIC-IV DATASET EXPERIMENTS
an electronic health record dataset of patients admitted the intensive care unit (CCU) JJhnson al. 2023). We considered patients diagnosed with sepsis (Sarial 2018), one of the major causes of mortality in ICU due to septic shock. Septic shocks are medical emergencies and early recognition and treatment would improve survival. We aim uncover the treatment choice
MIMIC- IV
behavior of doctors.
https://www.kaqgh/ com /otaaeaseco/tttttt/s//i/iii / hotel https://mimic.. mi

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Treatments and Vital Signs Suggested by (Komorowski et al. 2018). we extracted 14 treatments categorized as vasopressors, antibiotics, and auxiliary treatments associated with sepsis. We then select 16 vital signs that are highly related to sepsis. A detailed introduction to extraction and data processing can be found in Appendix
Drug Name
Detected Rules
Norepinephrine
PT 12.6 A ABPd 59.0
PTT > 31.6 A age 5 ABPd 51.0
Vasopressin
PCO2[Arteria))
41.0
INR > 1.1 A anchorage S 66.0
Packed Red Blood Cells
ABPd S 57.0 A ABPm S 56.0
PT > 12.6 A INNR > 14 RR & 32.0
PlateletCount s 95.0 PTT > 31.6
Table 2: MIMIC Drug Preference Rules Examples.
Results and Discussion In the context of drug prediction tasks, the feature set of the data is pre- dominantly centered on patient characteristics rather than the attributes of the drugs being consid- ered. Consequently, classical linear models such as Multinomial Logit (MNL) and Mixed Multino- mial Logit (Mix-MNL) are ill-equipped to address this task effectively. Table 3 shows NN-based models also perform badly due to the lack of product features. Furthermore, the medical domain ne- cessitates pacision-mmakin that is interpretable and traceable. Our model is capable of extracting the underlying logical rules and their associated weights that govern physicians' medication selection processes.
Table I1 presents exemplar rules that support drug preference learning, consistent with established medical knowledge. Taking packed red blood cells as an illustrative example, the rule ABPd < 57.0 mmHg PB 56.0 mmHg indicates that the patient's 's blood pressure is below the normal threshold. Additionally, elevated levels of PTT, PT, and INR, along with a deficiency in platelet count, suggest that the patient unable to achieve hemostasis. These rules collectively indicate that the patient is severely injured and experiencing significant blood loss, which provides interpretable evidence for the doctor's preference for packed red blood cells.
Train accu (Topl)
Test accu
Category
Method
Train loss
Test loss
(Top1)
NN
2.5215(=0.033) 2.3118(=0.137)
0.1261(=0.043)
0,0727(=0.031
TasteNet
2.422((=0.000)
2.4323(=0.000)
0.1079(=0.008)
0.0727(#0.06:
DeepNN
DeepMNL
2.3696(+0.000) 2.4331(=0.000)
0.1102(=0.008)
0.0818(20.063)
miXDeepMNL
2.3394(=0.000)
2.0000(=0.000
0.1060(=0.008)
0.0848(-0.065)
RUMNet
+..000(+0.000)
2.4426( 2.442((=0.000)
0. 0.1000(=000022)
0.0000(+00064)
Ours-NM-50
2.098( F0.0165)
2.2047(
10.020)
0.2961(+00010)
0.264(+0..02)
Ours*
Ours-RS 50
(.0.91(10.011
2. 1.13 L0.005))
0.26 (00003
Table 3: Treatment choice prediction on MIMIC-IV dataset. We report Negative log likelihood (NLL) loss and Top-1 choice prediction accuracy of the training and testing set. NM refers to the normal mode of our model, while RS indicates the risk- risk-seeking mode. We extract 50 rules for both types. Our Logit-Logit model performs much better than all the baselines.
7 CONCLUSION
In this study, we have proposed the Logic-Logit model, which incorporates logic rules into choice modeling as a pivotal strategy for enhancing both interpretability and accuracy. Our approach uti- lizes column generation for effective rule learning and leverages the Trank-Wolft algorithm to facil- itate the learning of mixture distributions for preference types. The empirical results obtained from real-world datasets demonstrate that our model significantly outperforms traditional neural network models in terms of both interpretability and predictive accuracy. This finding underscores the po- tential of integrating logical reasoning into choice models to produce more transparent and reliable decision-make tools.
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ACKNOWLEDGMENT
Shuang Li's research was in part supported by the NSFC under grant No. 62206236, Shenzhen Stability Science Program 2023, Shenzhen Science and Technology Program ZDSYS20230620091002222 Longgang District Key Laboratory of Intelligent Digital Economy Security, and SRIBD Innovation Fund SIF20240010.
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APPENDIX OVERVIEW
In the following, we will provide supplementary materials better illustrate our methods and ex- periments.
Section
presents more details of our model and implementation.
Section
introduces the baseline methods we considered in our paper.
Section
provides more additional experiment results about synthetic datasets.
Section
provides more additional experiment results about Expedia Hotel datasets.
Section
provides more details about MIMIC-IV dataset experiments.
A IMPLEMENTATION DETAILS
A.1
PREDEFINING THE PREDICATES
The specification of predicates can be quite flexible. Any method generate binary label from the original choice feature can be used as predicates, such as traditional classification model. Here introduce the most direct way to achieve this, which is also used the two real dataset experiments.
Discrete Features Discrete features inherently possess finite conditions due to the limited set of possible values that the feature can assume, such as if number of rooms = in a house. house containing one to ten rooms appear in the dataset, then there will naturally be ten predicates indicating how many rooms are in a house.
Continuous Features The possible value of continuous feature is infinite, thus we cannot use the equivalence condition like discrete feature. We have continued with the approach of Wei et al. (2019) to get predicates by thresholding the feature with the percentiles within the dataset, such as if house price ranks top 20% and if house area fail rank in top 60%. The number of these percentile thresholds can be determined accuracy for the rule content, but also lead to larger searching space and difficulty for the rule searching step. This methodology is based the distribution of the feature, and eliminate the influence outlier features and feature scales, which may cause trouble in linear models.
hyperparameter. Dense thresholds bring about
A.2 SUBPROBLEM SOLVING: RULE SEARCHING WITH COLUMN GENERATION
The pseudo-code of the rule searching step shown below:
Algorithm Solving Frank-Wolfe Step using Column Generation
1: Input: max conjuntion length K
2: Initialize: k: = ruleset == O), rule weight w
3: while stopping condition 1 not met do
4: while & < K do
b BFS
5:
0
6: while stopping condition 2 not met do
7:
AR Generale Candidale Rules R, q) Compute (WWne)) for each new rule in A R
8:
9:
AR" Selecc Top-N potential Rules from A with Volwnewl information
10:
R + RU AR
w arg min (w R)
12:
+ q +1 end while k - k +1 end while
14:
15:
16:
end while
17:
return ruleset R, rule weight w
The algorithm 1 shows the entire road map of Solving Frank-Wolfe Step using Column Genera- tion, including the rule searching and potential evaluation with gradient. The stopping condition controls the number of total preference types to be found. It can be set predefined number or depends on the convergence of NLL loss. The stopping condition 2 controls how many rules with searching length will be searched in preference type. It can also be set as a predefined number or
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depends the convergence of Frank- Wolfe step loss (as show in Eq. set is R(()) the detailed formulation for computation of the V Wnewr) is shown below:
5 Suppose the current rule
EE
Nge (k-1)
exp (zee)
[e(Wnew)
Ineew (20je)
(3ex))))))
EES. Eves, exp (zpre)
t=
gjt
(k-1) refers the choice probability of product where Ett- Σ Wh. (A... Pm (2x)) and Ojt
in offer set St mixed by the previous found h preference types. With weights of rules that have already been in rule set fixed, we initialize the candidate rules weights Wnew to be zeros, and calculate the gradient of loss upon this weights. In other words, we use the gradient to determine if changing the weight of the candidate rule from zero can bring about improvement in Frank-Woll step solving. The absolute value of the gradient directly implics the potential of candidate rule. Therefore we can select top-n most potential rules and put them R.
Algorithm Generate Candidate Rules
Input: R, k, q
2: Initialize: AR = {
3: if q then
4: for rule R do
5: if rule conjunction length = then
6:
Extend rule conjunction with each predicate respectively Put all extended rules in AR
DFS
7:
8:
end if
9: end for
10: else
11: Randomly Sample k predicates and form conjunction
Repeat line 11 for multiple times and put generated rules in AR
13:
end if
14:
return AR
From Algorithm and Algorithm 2, we can find out that the our model searches rules following the order of rule conjunction length, i.e. BFS. Each time the model finishes length-k rule searching then start length-k+1 rule searching. Moreover, to improve the rule searching efficiency, we also introduce DFS when the searching change from length-k to length-k+1. For example, when start to search for length-2 rule we first go through all the length-1 rule in the current rule set and form conjunctions with other predicates get length-2 rules induced by length-1 rules in current rule set. This comes from the simple intuition that choices satisfy condition A A B must satisfy condition A and condition simultaneously. Although single predicate A doesn't depict the A A B perfectly, increasing the weight of predicate A can still increase the utility of choices satisfy A A B and bring about improvement.
A.3 OPTION FEATURE AND OFFER SET FEATURE
Our model distinguishes between option-specifi features and offerset-specific features, treating them differently. The offerset features exclusively considered within the context of conjunction rule searching. Any rule composed solely offerset features is skipped during the rule searching process due the inherent structure of the logit. Irrespective of the weights assigned to these rules, they are inevitably discarded during the softmax process since offerset features are common across all options within the same offerset. Consequently, rules involving offerset features are only fune- tional when they are conjunct with option-specing features. The rules pertaining to option features elucidate the attractiveness of individual options, whereas the rules associated with offerset fea- tures delineate the scenarios (or offersets) in which various option-specific rules should be applied. This nuanced approach enables Logic-L.ogit to successfully incorporate offerset features into choice modeling. thereby enhancing its explanatory power and predictive accuracy.
A.4 RULE SET EXPANSION, RE- WEIGHTING, AND PRUNING
The rule space to be explored is extensive, necessitating careful control of the rule set size for efficient subproblem resolution. An overly large rule set can lead to time-consuming optimization and compromise interpretability.
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Rule Set Expansion During single subproblem iteration, substantial number of rules are identi- fied, many of which exhibit promise. However, only the top-k most promising rules are incorporated into the rule set. This strategy curbs the expansion rate of the rule set, maintaining its sparsity and interpretability.
Rule Set Pruning Our algorithm employs two phases of pruning. The first occurs during subprob- lem resolution, while the second is executed post-resolution.
During subproblem resolution, to ensure the problem remains tractable, we impose a maximum rule set size, typically generously. When the size exceeds this threshold, rules with low weights discarded. Post-discard, the remaining rules undergo re-weighting.
Following subproblem resolution, to uphold interpretabilityy and sparsity, an ultimate maximum size limit is enforced, significantly smaller than during the subproblem phase. Similarly, rules are pruned, and the retained rules are re-weigh
Re-weighting For each preference type, the number of rules must be manageable to preserve interpretability. Selecting the most sigmificant rules is critical during the re-weighting process. We apply relatively high /1 penalty when optimizing rule weights ensure sparsity. Notably, this penalty is not applied post the second pruning phase, as the rule set size is already constrained by the ultimate maximum limit.
A.5 COMPUTING INFRASTRUCTURE
Our experiments applicable to both CPU and GPU. For this paper, GPU NVIDIA GeForce
RTX 3090.
A.6
OPTIMIZER
We choose the Adam optimizer do stochastic gradient descent in both rule weights learning and preference type proportion update.
B BASELINES
Classic Choice Models
MNL (McFadden] 1981) Multinomial logit model
Mix-MNL (Jagabathula et al. 2020) MNL with feature weights belonging to finite discrete support learned with Frank Wolfe algorithm framework.
Deep Learning-Drit Driven Choice Models
Vanilla NN is simple and standard feed-forward neural network that takes a concate- nated vector of all product and customer attributes as input, and outputs the utility of each choice alternative. These utility values are then passed through a softmax layer to calculate the choice probabilities for each alternative.
TesteNet (Han et al.) 2020) uses deep learning to model individual tastes/preferences directly in choice modeling context.
DeepMNL. (Wang et al. 2020) combines the traditional MNL model with deep learn- ing architectures to capture more complex choice behavior.
Mix (Jagabathula et al. 2020) is mixture of DeepMNL. models that allow for heterogeneity in preferences across individuals.
RUMNet (Aouad & Désir) 2022) is deep learning-based model that combines ran- dom utility theory with neural networks. It provides efficient approximation of random utility maximization (RUM) discrete choice models.
Hyperparameters setting of Deep learning driven choice models We apply similar hyper- parameters for all neural network-based models followed the setting RUMNet (Aouad & Désir 2022). We select the standard ADAM optimizer with a learning rate of 0.001 and batch sizes of 32. We set label smoothing as 0.0001, a norm- based regularization method on the neural network's outputs. We tune the neural network structure and report the best one in all tables. In particular, for each model. we vary the parameters (I, w) E (3, 10), (5, 20), (10, 30), where I denotes the depth of the network and denotes its width.
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Model
Utility Value
Parameters
Grounds for Interpretability
Logic-Loggi (Ours)
Rule-Based
O(Nn)
Rules Content & Weights
MNL
O(d)
Linear
Feature Weights
Mix-MNL
[(Nd)
Vanilla NN
TasteNet
Deep- MNL
NN-Based
Oddd, + Idft)
Mix- Mix-DeepMMN
RUMNet
Table 4: Model Comparison, In the # parameter column, .V represents the consumer type number, n denotes the maximum rule number, and d indicates the dimensionality of the product feature vector. For NN-based models, de are dimensions of latent features and L is the number of layers. As the inherent complexity of NN-based models, their interpretability is often considered negligible.
C DETAILED EXPERIMENTS RESULTS
In this section, we present the detailed experimental results by comparing our model with several baseline approaches. In ours, NM refers to the normal mode of our model, while RS indicates the risk-seeking mode. The number at the end denotes the count of rules for each preference type.
Train (Topl)
Test
Category
Method
Train loss
Test loss
(Topl)
MNL
2.7503 (000003)
2.7751(+0.0055
0.16(+0.000)
4.332((00.005)
Classic
MiX-MNL
(Trank-Wolle)
2.0.00
2.7679 2.7679(10.004)
(.0.00(
0.0.81(10.011]
NN
2.9671(=0.016
3.0075 (0.007)
0.0739(+0.008)
1.0553(+0000)
TasteNet
(...01(+0.054)
0.1759(+0.012)
8.1620(=0.02))
DeepNN
DeepMNL
2.1603(+..024
2.6001
0.2561((0.0001
0.2099 0.2099(10.006
miXDeeMMNN
2.4538 (.0.00
2.5798(
1.0.55)
H.2401((000001)
0.2253( 0.2253(10.011) 10.019)
RUMNet
(0.000
2.5749 +0.071)
02442(40.011)
0.2186(10.011) F0.048)
0.21867
Ours-NM-5 NM-5
2.6032 2.6032(10.017 E0.017)
(66397(10002)
(.0051(
10.011)
Ours-NM-20
2.5242(110.006)
2.5871((0.00885.
0.2204
(0000034)
1.186(110.00
Ours*
Ours-RS-3
2.5026(10.008)
2.6305(10.011)
0.2208(+00.005)
0.2058(10.011)
Ours-RS-24
2.5215(1000041)
2.5836(10.004)
1.2231(10.0013
0.1908(H1) 10.004))
Table 5: Choice prediction results on the synthetic dataset. We report Negative log likelihood (NLL) loss and Top-1 choice prediction accuracy the training and testing set. NM refers to the normal mode of our model, while RS indicates the risk risk-seeking mode. We extract 5 / 20 rules for both types.
The synthetic data is generated with single sales, implying that within every offerset, only one option selected. The Logic-Logit model demonstrates comparable performance in terms of both Negative Log-Likelihood (NLL) Loss and Top1 accuracy. Notably, the risk-seeking mode model yields improved outcomes compared to the normal mode, as it aids the model fitting to uncovered samples, maintaining an equilibrium between exploration and exploitation.
In both the Expedia Hotel Dataset and the MIMIC Dataset, our model demonstrates superior perfor- mance over neural network- based models and classic models in terms of loss and accuracy.
Within the hotel dataset, it is evident that increasing the maximum rule number enhances the expres- siveness of the Logic-Logit model and improves its fit to the training data, although this may lead over-fitting. Notably, the risk-secking heuristic proves ineffective in this context due the dom- inance of preference-type sales and the PageRank effect. Enhancing rule diversity does not yield significant performance benefits.
The MIMIC dataset predominantly comprises patient features, specifically offerset-speeeeee features shared among all drug options. Consequently, primarily focused models capable of learning both drug and patient features. Classic linear models and Vanilla Neural Networks are not suitable to this scenario.
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Train accu
Test accu
Category
Method
Train loss
Test loss
(Top1)
(Top1)
MNI.
3.1482(+0.002)))
3.1423( +0.032)
0.177(+0.000)
0. 0.1596(+0.005)
Classic
Mix-MNI. (Frank- Wolfe)
2.9654((+0.00))
2.9576 +0.025)
++0.088)
0.1657 0- (DD))))
NN
3.099I(10.025)
0.2168(10.000 3.0863(H0.015)
(1.669(-0.023)
0.2120 011)
TasteNet
3.0909(=0:017)
0.1503(+0.007)
0.1431(+0.013)
DeepNN
DeepMNIL
3.0494(=0.011)
(.069)((0.022)
0.1566(+0.002)
0.1586(+00007)
DeepMNN
3.0283(+0.020)
3.0755(+0.0255
O. 0.1600(00.0TO)
0.1507(+0.010)
RUMNet
2.0182(1.000)
L.736611.022
(.1.31(10.0044
0.1489(
0.1489(10.0005
Ours-NM-30
2.9505 2.9505(10.019)
2.9754 10.013)
(.0000()
0.1698
10.009)
Ours*
Ours-RS-30
2.9473(10.017)
2.9675(10.016) 0.0000(00.002)
0.1738 (10000))
Table 6: Choice prediction on the Expedia Hotel dataset. We report Negative log likelihood (NLL) loss and Top-1 choice prediction accuracy of the training and testing set. Our Logit-Logii model performs best. NM refers to the normal mode of our model, while RS indicates the risk-seeking mode. We extract 30 rules for both types.
D ADDITIONAL EXPEDIA HOTEL EXPERIMENTS
Feature Introduction The Expedia Hotel Datase is publicly available dataset released on Kag- gle in 2013. It records customers' garch-and-bookingg interaction with Expedia platform. The fea- tures listed in table 7 are used in the experiment.
Feature Type
Feature Name
Variable Type
Description
PageRank
Discrete
The recommendation position the hotel
Star Rating
Continuous
The historical star rating
Location Score
Continuous
The desirability of a hotel's 's location
Historical Price
Continuous
Mcan price of the hotel over the last period
Hotel Feature
Branded
Binary
II the hotel is part of major hotel chain
Promotion
Binary
Ii the hotel a sale price promotion
Price
Continuous
Displayed price of the hotel
Booking
Binary
If customer book the hotel
Booking Window
Discrete
Number of days in the future from search date
Length Stay
Discrete
Number nights stay
Adults Count
Discrete
The number of adults
Search Criterion
Children Count
Discrete
The number of children
Room Count
Discrete
The number of rooms
Saturday Night
Binary
If the stay includes Saturday night the displayed sort is random
Random
Binary
Table 7: Descrption of feature description on the Expedia Hotel dataset.
Experiment Setting To prepare dataset, we divided it into training and testing subsets using a 3:1 ratio based on search timestamps. We selected offersets with 28 to 32 search results, whose results is not randomly displayed, i.e. sorted by Expedia Recommendation System. Searches with no bookings, hotels priced over $1000, or booking windows longer than one year were excluded. Finally, we randomly sampled 10,000 instances for the training set and 1,000 instances for the test set.
The hyper parameters of our model is set as below. This setting is shared for Expedia Hotel experiments in both Appendix and Appendix [
Model Setting
Overall Convergence Criterion: The NLL loss decreases less than 0.001 after a new prefer-
ence type is found and proportion is updated.
The ultimate output rule number for each preference type is set to be 30 for Table 6 30 and 100 for Table g
The maximum rule number of the rule set (Rule Prune Threshold) R during column gener- ation is set to be 100.
https://www/ e htt.
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The candidate rule set size generated during the column generation iteration is set to be 100 * searching rule length.
The number of percentile thresholds we put on the continuous features to get predicates is set be 30, i.e. a threshold per 3.3%.
We search rules up to 3 conjunctions. For each length-1 1 to length-3 conjunctions we search for 10, 50, 100 iterations.
Additional Large-Scale Expedia Hotel Dataset Experiments Moreover, in order to show our model's scalability on bigger dataset, also evaluate model with 50,000 training instances and 5000 testing instances, containing 15-32 search results
Train accu (Top1)
Test accu
Category
Method
Train loss
Test loss
(Topl)
MNL
(05525(10.1.))
2.9578(10.131)
0.1633
(..042)
0.1607(L0.011)
Classic
MNL (Frank- )
2.8267(+0.029)
(..021
0.1883 :0.014)
0.1875(H0.013)
NN
3.0.11(+0.015)
3.2699( +-0.11))
0.1138( (.0.033
0.0863(+-0.003
TasteNet
2.8278(10.06)
2.83841 +0.025)
0.1877(F00000))
U. 0..660000000)
DeepMNL
2.0022
(.0.33
(0.000))00006)
0.2020((00000)
DeepNN
miXDeepMNL
2.7453(10.025
2.7017(10.015)
0.2036
:0.004)
0.2035(+0.008)
RUMNet
2.7244(10.0099
2.7532(10.016)
0.2105(+0.0033
0.2072(+0.002)
Ours-NM-30
2.8258(110001)
2.8310
:0.002)
0.2015 (0.000)
0. (.0080(100001)
Ours-NM-In TOO
2.8014(+0.008S
2.8166
+0.005)
0.2058 +0.002)
U. (0.7(
Ours
Ours-RS-30
2.0.81(+0.003)
2.8328(+0.008 +0.003)
0.2013 #0.000)
O. 0.1952(+0.001
Ours-RS RS-
2.7983(+0.002,
2.8165(+0.003)
0,2207
O. (.1992(+0.002)
Table 8: Choice prediction on larger scale Expedia Hotel dataset. We report Negative log likelihood (NLL) loss and Top-1 choice prediction accuracy the training and testing set. We extract 50/100 rules for both types.
In large-scale datasets, our model performs comparably to NN-based models, excelling in- sampee accuracy the majority of NN-based models. Logic-Logit, with significantly fewer parameters compared to neural network models, simultaneously demonstrates superior performance and rule- based interpretabilityy
E ADDITIONAL MIMIC-IV EXPERIMENTS
MIMIC-IVI is publicly available database sourced from the electronic health record of the Beth Israel Deaconess Medical Center (Johnson et al. 2023). Available information includes patient measurements, orders, diagnoses, procedures, treatments, and deidentified free-text clinical notes. Sepsis is leading cause of mortality in the ICU, particularly when it progresses to septic shock. Septic shocks are critical medical emergencies, and timely recognition and treatment are crucial for improving survival rates. In the real-world healthcare data experiments on the MIMIC-IV dataset, we aim to understand how doctors choose the treatments diagnose.
We select 3000 patients that satisfied the following criteria from the dataset: (1) The pa- tients are diagnosed with sepsis (Saria (018). (2) Patients, if diagnosed with sepsis, the timestamps of any clinical testing and timestamps of medication administration and corresponding dosage were not missing.
Patients
Vital signs Referred to (Komorowski et al. 2018 we collected 16 vital signs that are highly related to sepsis from chart events recorded in the ICU. Table 10 shows these vitals with names and descriptions.
Treatment Suggested by (Komorowski et al. 2018). we extracted treatments associated with sepsis which are consistent with expert consensus. Based on the distinct clinical characteristics of these treatments, they can be categorized into the following three groups, which are shown in Table hypotension aims
9 Vasopressor therapy is fundamental treatment of septic-shock-indus at correcting the vascular tone depression and then improving organ perfusion pressure; Antibiotics
https::/mimis mit.
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Published as a conference paper at ICLR 2025
also should be given within few hours of the diagnosis of sepsis; Some auxiliary treatments such as packed red blood cells and invasive ventilation are also necessary in ICU.
Data Preprocessing For each patient, we extract the treatments belonging to the above three cate- gories, as well as the corresponding treatment start and end times. Then, we select the patient's 's vital features during the treatment. We only record the first vital value. During this period, some vital values will be lost, so we only keep the most complete records as the processed dataset. Another problem is lahel imbalance, number of some treatment types is too large, we will delete some records. Finally, we randomly selected 3000 records the training set and 300 records as the test
we only keep 14 treatment types with enough records, and if the
Experiment Settings
The hyper parameters of our model is set below:
Overall Convergence Criterion: The NLL loss decreases less than 0.005 after prefer- ence type is found and proportion is updated.
The ultimate output rule number for each preference type is set to be 50.
The maximum rule number of the rule set (Rule Prune Threshold) R during column gener- ation is set to be 100.
The candidate rule set size generated during the column generation iteration is set to be 100 * searching rule length.
The number of percentile thresholds we put on the continuous features to get predicates is set to be 20, i.e. a threshold per 5%.
We search rules up to 5 conjunctions. For each length-2 to length-5 conjunctions we search for 50, 100, 150, 200 iterations.
Sensitive analysis We conduct sensitive analysis for three significant rule number for each preference type, the maximum conjunction length and rule prune thresh- old we search up to in the rule searching with MIMIC dataset.
ultimate
0.30
Train Top-1 AcC
2.35
Train NLL
Test Top-1 Acc
Test NLL
2.30
0.28
Accuracy
2.20
0.24
2.15
0.22
2.10
10
30
40
10
20
30
40
50
Number
Figure 6: Max Rule Number
0.30
2.40
Test NLL
2.30
Accuracy
2.25
0.20
2.20
0.15
Train Top-1 Acc
Test Top-1 Acc
2.10
Max Rule Length
Max Rule Length
Figure 7: Max Rule Length
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ruleset will Published as a conference paper at ICLR 2025
2.26
0.28
2.24
Tast NLL
0.27
2.22
Accuracy
0.26
2.20
0.25
0.24
2.16
0.23
Train Top-1 Acc
2.14
0.22
Test Top-1 Acc
2.12
150
Rule Prune Threshold
Rule Prune Threshold
Figure 8: Rule Prune Threshold
Fig/oand Fig 7 shows that the model performance improves significantly with the max rule number and max rule length. Both of these two hyperparameters increase the complexity of each preference type's 's rule set. For algorithmic point of view, increasing the maximum complexity enlarger feasible space P for the functional Frank-Wolfe step solving and bring about improvement for the whole model. Meanwhile increasing the rule set complexity will also bring about additional computational cost, trade-off should be made when tuning these hyperparameters. Setting the ultimate rule number to be 50, Fig shows the relationship between the rule prune threshold and model performance. Setting a larger rule prune threshold than the maximum rule number allows for a wider tolerance range for newly searched rules, making it easier for promising rules to be learned. However, if this value too high, excessive deletion of redundant rules during the final pruning phase can result in significant changes in the ruleset, leading to decline in model performance. Therefore, generally recommended the rule prune threshold to 1.5-2 times the maximum rule number.
Category
Treatment
Epinephrine
Phenylephrine
Norepinephrine
Vasoconstrictor
Dobutamine
Dopamine
Vasopressin
Angiotensin (Giapreza)
Vancomycin
Caspofungin
Cctepime
Ceftriaxone
Antibiotic
Gentamicin
Micafungin
Tobramycin
Piperacillin/ Tazobactam
Furosemide (Lasix)
Heparin Sodium
Invasive Ventilation
Auxiliary Treatment
Packed Red Blood Cells
Immune Globulin (IVIG)
Acetaminophen-IV
Table 9: Description of the treatment extracted from MIMIC-IV dataset.
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Published as a conference paper at ICLR 2025
Item
Abbreviation
Description
Normal Range
220546
WBC
White blood count
4,000 10, 000cells/mma
225690
Total Bilirubin
Total Bilirubin
0.1 1.2 mg/dL
220235
PCO2 (Arterial)
Partial pressure carhon dioxide (Arterial)
45 mmHg
226063
PO2 (Venous)
Partial pressure of oxygen (Venous)
30 40 mmHg
229761
Creatinine (whole blood)
Creatinine (whole blood)
0.5 1.2 mg/dL
226253
SpO2 Desat Limit
Oxygen saturation (SpO2)
95% 100%
220210
RR
Respiratory rate
12 20 breaths per minute
220052
ABPm
Arterial Blood Pressure
70 100 mmHg
220051
ABPd
Arterial Blood Pressure diastolic
60 80 mmHg
223762
Temperature C
Temperature Celsius
36.1 37.2
220045
HR
Heart rate
60 100 beats per minute
220050
ABPs
Arterial Blood Pressure systolic
90 120 mmHg
227465
PT
Prothrombin time
11 13.5 seconds
227467
INR
International normalized ratio
0.8 1.2
227466
PTT
Partial thromboplastin time
25 35 seconds
227457
Platelet Count
Platelet Count
1.50 450 x10" platelets/I.
Table 10: Description of the vital signs extracted from MIMIC-IV dataset. Item id is the ID extracted from MIMIC-IV tables.
Drug Name
Detected Rules
Norepinephrine
PT > 12.6 ABPd < 59.0
PTT > 31.6 A age & 76.0 ABPd
Vasopressin
(CO2(Arteria)) > 41.0
INR > 1.1 anchorage < 66.0
Epinephrine
RR> 20.0 WBC <= 6.8 6.8 3
ABPs < 77.0
Furosemide (Lasix)
ABPs > 93.0 WBA 14.55
ABPs > 107.0
Dobutamine
INR <= 1.4A PCO2(Arteridd 36.0
PCO2(Arterial) <= 51.0 A ABPs <= 89.0
Dopamine
PTT > 65
PlatelelCoontl <= A PTT 48.8
Packed Red Blood Cells
ABPd < 57.0 A ABPm S 56.0
PT > 12.6 INR > 1.4 RR < 32.0
PlateletCount < 95.0 A PTT 31.6
Antibiotic
age < 66.0 A RR < 30.0 A WBC 2.66
PCO2(Arterial) <= 42.0
Table 11: MIMIC Drug Preference Rules Examples
21


Reviews:
[{"review_id": "6xdnoZ79Wf", "reviewer": "Reviewer_Xxs5", "summary": "This paper presents Logic-Logit, a rule-based interpretable choice model that utilizes logical rules to predict human choices in contexts like healthcare and commercial domains. The authors aim to address limitations in interpretability associated with existing neural network-based models by proposing a model that represents choices through OR-of-ANDs logic rules. These rules enable compact and interpretable representation of human decision-making. The paper introduces an optimization framework using the Frank-Wolfe algorithm combined with column generation to efficiently extract rules, showcasing empirical success in interpretability and accuracy across synthetic, commercial (Expedia Hotel), and healthcare (MIMIC-IV) datasets.", "questions": "- Can the authors provide more insight into how the model scales with an increase in the number of features or the size of the dataset? Would any adjustments be necessary in the Frank-Wolfe and column generation steps?\n- How does the model handle cases where decision criteria overlap significantly between customer types? For instance, if preferences are highly correlated between types, does the model tend to overfit to certain rules or ignore relevant variation?\n- Would incorporating neural components in conjunction with logic-based rules (e.g., neural embeddings for complex feature spaces) enhance performance without compromising interpretability? This hybrid approach could be of interest if straightforward rule-based methods struggle with nuanced distinctions in large feature sets.\n- How can this method be applied for RLHF, can authors provide experimental results to demonstrate this? i.e., Finetuing an LLM for a Diffusion models.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The model’s combination of interpretable rule-based choice modeling with optimization algorithms is innovative. The approach’s focus on interpretable, structured rule extraction addresses a significant gap in choice modeling literature, especially relevant for high-stakes domains.\n- The experimental setup is comprehensive, covering synthetic and real-world datasets. Benchmarks with traditional models and neural networks underscore the model’s effectiveness in balancing accuracy and interpretability. The rule extraction and optimization process is detailed and thoughtfully developed, providing clarity on algorithmic decisions.\n- The paper is generally well-organized. The presentation of the OR-of-ANDs rule structure, Frank-Wolfe algorithm, and column generation steps is clear, supporting reproducibility. The inclusion of rule explanations on real-world datasets aids in understanding practical implications.", "weaknesses": "- While the column generation and rule pruning strategies manage computational demands, further discussion on the model’s scalability with significantly larger datasets would enhance the paper. For instance, scalability tests on larger commercial datasets would demonstrate practical feasibility in data-intensive domains.\n- The approach involves selecting parameters like the number of rules, rule lengths, and pruning thresholds. More empirical insights into the sensitivity of these parameters on model performance, especially in healthcare contexts, could strengthen robustness claims.\n- While the model shows good accuracy on average, there is less discussion about edge cases, where rule-based logic might oversimplify complex decision boundaries. Addressing potential limitations in handling such cases would provide a more balanced view of the model’s capabilities.", "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": "KOtsEJykI0", "reviewer": "Reviewer_59ua", "summary": "This paper introduces an approach to modeling human choice using OR-of-ANDs logic rules. The authors illustrate how any preference can be transformed into a Boolean logic formula, with rules mapped to Boolean space and interconnected through AND and OR operators. This formula is then incorporated into a mixed logit choice model, enabling preference learning. However, this approach results in an infinite-dimensional optimization problem, a major computational challenge. To address this, the authors apply the functional conditional gradient method (Frank-Wolfe) to reduce the optimization’s complexity. Additionally, due to the exponential size of the search space ($2^M - 1$, where $M$ is the number of rules), they use a column generation technique that incrementally expands the search space by adding new rules in each step. Empirical experiments on both synthetic and real-world datasets (from commercial and healthcare domains) highlight the effectiveness and versatility of this approach.", "questions": "1. Could you estimate the computational cost of the proposed methods, particularly for real-world datasets? \n2. Are there specific conditions or data characteristics under which this model is computationally more efficient or challenging?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-organized and motivated. Its focus on interpretable human preferences is valuable for understanding human decision-making, particularly in high-stakes areas like healthcare and autonomous driving, where trust and transparency are essential.\n2. The approach to modeling human choice could also inspire advancements in fields like automated reasoning (the task authors have verified), where a clear understanding of decision-making processes is crucial.", "weaknesses": "1. While innovative, the current solution appears complex, particularly due to the combined use of the functional conditional gradient method and column generation. This complexity may limit its applicability or make implementation challenging for practitioners. A more streamlined or efficient approach could enhance the method’s usability across a wider range of real-world applications.\n2. The search space of combinatorial rules is increased exponentially number of rules $M$, which needs to be further reduced to accelerate the learning process.", "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": "3RWj5KAtOE", "reviewer": "Reviewer_7vMh", "summary": "This paper addresses the development of choice models by constructing a rule-based approach using logical statements that dissect conjunctions. The proposed method employs a dual-optimization process: an outer optimization for determining preference weights, and an inner optimization to identify new rules. The model iteratively refines the rule set by incorporating each newly discovered rule. Assuming convex optimization, the authors apply the Frank-Wolfe algorithm to update preference weights across all rule types. Experiments on a synthetic dataset and the Expedia Hotel Dataset demonstrate that the proposed method consistently outperforms all baseline models.", "questions": "- Item $j$ and $S_t$ are introduced but not used in Equation (1), which creates some confusion.\n- If $p_m$ returns 1 when a condition is met and $x_s$ is a single item, does that mean the resulting conjunctions reduce to just $x_s$?\n- The implementation details of BFS and DFS in the algorithm are unclear, as no specifics or pseudocode are provided. Pseudocode of the algorithm would be helpful with all conditions.\n- The notation $\\mathcal{X}$ is used but not defined.\n- Some figures (1,2) are included but not referenced within the text.\n- Could you clarify the rationale behind the rule distance metric? How would the distance between two unrelated features be interpreted?\n- Why do the neural network (NN) models lack product features?\n- Why aren’t there results for NN-based models on the synthetic dataset?\n- The paper does not provide details on how the baseline methods were trained. What are their hyperparameters and how were they trained?\n- Other discrete choice models, such as graph-based approaches [1], mixed logit models, and network formation models [2], are not included for comparison, which might help comparing tradeoffs between methods.\n\n[1] Tomlinson, Kiran, and Austin R. Benson. \"Graph-based methods for discrete choice.\" Network Science 12.1 (2024): 21-40.\n\n[2] Gupta, Harsh, and Mason A. Porter. \"Mixed logit models and network formation.\" Journal of Complex Networks 10.6 (2022): cnac045.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "- The authors tackle a crucial question in designing interpretable and explainable models.\n- They present an interpretable algorithm tailored for choice modeling.\n- The method is intuitive and demonstrates strong performance.\n- The proposed approach outperforms baseline methods across 2 datasets.", "weaknesses": "- The method does not appear scalable to a large number of features (predicates). What is the computational complexity of this approach?\n  \n- The experimental details, including hyperparameters (e.g., exact convergence condition used) for the proposed algorithm, are missing and were not found in the Appendix. Significant methodological details are lacking.\n\n- Although logic rules enhance interpretability, they may not be applicable for designing all types of features.", "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": "dW1giTYniA", "reviewer": "Reviewer_brV8", "summary": "The paper introduces the Logic-Logit, an interpretable, rule-based choice model, along with a learning pipeline built on column generation techniques and the Frank-Wolfe algorithm. The authors also provide numerical results demonstrating the model's interpretability and predictive accuracy.", "questions": "See weaknesses [a] [b].", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper is generally well-written and presents a new explainable choice model with a learning framework. Numerical results indicate strong interpretability and predictive performance, tested on both synthetic and real-world datasets.", "weaknesses": "[a]There is no discussion regarding the model's capacity. It is unclear whether this model falls under the mixed-logit model or if it is equivalent to a mixed-logit model. If so, why not use the approach in [1] to learn a mixed-logit, given its stronger analytical properties, such as provable convergence guarantees? (Also, the mixed-logit's identified customer segments and the parameters in each segment could also be viewed as the interpretation of the choice?)\n\n[b]I'm confused about if the learning requires prior knowledge of $M$ and $\\{p_m\\}$. For instance, Line 177 mentions, “These mappings, from real-valued features to predicates, are predefined and fixed,” and the learning problem (3) does not appear to learn $\\{p_m\\}$. Yet, in the experiments, the algorithm identifies these rules. This raises questions about the rule search strategies outlined in lines 308-323, and a pseudo-code would help clarify this process.\n\n[c]The literature review lacks a discussion of consider-then-choose models. Since the proposed model follows this approach, it would be beneficial to compare it with other models in this category, such as [2] and [3].\n\nMinor comments:\n\n[a]Given that interpretability is a key advantage of the proposed model, it would be helpful to compare it with other interpretable choice models, such as [4].\n\n[b]Including the variance/std in the experimental results would provide additional insight.\n\n[c]Some potential typos: line 192 should be “item $s$ ” instead of “item $j$”, and also “offer set $S$” instead of “offer set $S_t$”.\n\n\nReferences\n\n[1]Hu, Yiqun, David Simchi-Levi, and Zhenzhen Yan. \"Learning mixed multinomial logits with provable guarantees.\" Advances in Neural Information Processing Systems 35 (2022): 9447-9459.\n\n[2]Liu, Qing, and Neeraj Arora. \"Efficient choice designs for a consider-then-choose model.\" Marketing Science 30.2 (2011): 321-338.\n\n[3]Akchen, Yi-Chun, and Dmitry Mitrofanov. \"Consider or Choose? The Role and Power of Consideration Sets.\" arXiv e-prints (2023): arXiv-2302.\n\n[4]Tomlinson, Kiran, and Austin R. Benson. \"Learning interpretable feature context effects in discrete choice.\" Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining. 2021.", "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)