Distributionally Robust Regularization of Sparse Integer Programming Trained Learning Models

arXiv:2311.01994 · stat.ML, cs.AI, cs.LG, math.OC · Submitted 2023-11-03 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Distributionally Robust Regularization of Sparse Integer Programming Trained Learning Models".

Jane: The gist Obtaining Explainable Classification Models using Distributionally Robust Optimization proposes a new formulation to learn an ensemble of rule sets that simultaneously addresses the trade-off between rule set sparsity…

Tom: First, who's behind it and why it matters.

Title and authors: Tom: The core idea here is learning an ensemble of rule sets simultaneously, which addresses the tension between how sparse the rules are and how accurate they are >

Jane: So instead of picking one set of rules, they’re building a collection and then using some clever optimization to pick the best combination >

Lu: They use distributionally robust optimization, or DRO, to handle that uncertainty about where the unseen data might come from >

Meng: I always wonder how they manage that uncertainty without making everything impossibly slow to train on our actual production data streams >

Lalam: The paper says they use a robust loss R(h) defined as the maximum of the loss over a probability ball P centered at some distribution P0, where P is within a distance Dϕ(P, P0) of zero >

Tom: It's like they're optimizing for the worst-case scenario within a certain range of possibilities instead of just optimizing for one specific way the data might look >

Jane: That makes sense. They are building this ensemble to be stable across a wider range of potential data distributions, not just one specific training set >

Lu: And they define complexity as the sum of the complexities of all the rule sets in that ensemble, which is a key constraint they’re trying to manage >

Meng: Managing that complexity while keeping things sparse seems like a tough balancing act when you're actually deploying this on something real >

The paper's summary: Tom: The main improvement they propose is using two optimization formulations that work together sequentially to build this ensemble >

Jane: They first solve for an initial good rule set model, and then they use that to update the collection of rules in a second step >

Lu: This second formulation involves selecting an optimal sparse convex combination of those rule sets from the collection they just built >

Meng: That sounds like a lot of work to set up, but if it enforces sparsity through an integer program, maybe it’s manageable for deployment >

Lalam: The paper shows that this approach allows them to improve generalization performance over single rule sets while only incurring a penalty related to the largest complexity of its constituents >

Tom: That's significant. It suggests that combining rules in a specific way can actually give you better results than just using one complex rule set alone >

Jane: It also points out that by setting a low value for C prime, which is related to the complexity bound, they get more flexibility to choose the combination, even beating ensembles with higher bounds >

Lu: They found that this framework can use less computational effort compared to state-of-the-art methods and still match or improve generalization on both sparse rule sets and other complex models >

The paper's improvements: Tom: So, to wrap up, the paper introduces this method using DRO to learn an ensemble of sparse rule sets that balances complexity and accuracy while controlling computational cost >

Jane: It seems like the main implication is that we don't have to rely on expensive cross-validation just to pick a good sparsity level anymore >

Lu: The theoretical guarantee, Proposition two shows that convex combinations of rule sets offer an advantage over a single rule set because the generalization penalty only grows with the largest complexity of its parts > <ref:2311.01994#pg2>

Meng: From an engineering standpoint, the paper suggests that this approach has relatively low complexity when you set a cutoff at C less than or equal to thirty which is good for interpretability and cost >

Lalam: I think it shows that these sparse convex ensembles can be quite competitive with models like CG or CART and in some cases even exceed their generalization performance >

Tom: Exactly. It seems like a solid way to get explainable models that actually perform well without needing massive computational resources for tuning the complexity parameters >

Jane: It’s a promising direction for building robust, interpretable classifiers, and it leaves us wondering how easy it is to implement all these optimization steps in practice >

Lu: We'll have to see how robust this framework is when we move away from the idealized distributionally robust setting into real-world data scenarios >

Conclusion: Tom: So today we’re talking about "Distributionally Robust Regularization of Sparse Integer Programming Trained Learning Models." Basically, they figured out how to build an ensemble of rule sets that is both sparse and accurate by using distributionally robust optimization >

Jane: That’s right. The core idea is solving this trade-off between how few rules you have and how well they predict things, and they use this DRO thing to handle the uncertainty about where your data might come from >

Lu: They are looking at sparse ensembles of rule sets as binary classifiers, and they define complexity by summing up the complexities of all those rule sets in the ensemble >

Meng: I see. So you’re trying to keep the number of rules small for interpretability, but you want it to still be good on real traffic data >

Lalam: And they use this robust loss R(h), which is basically the maximum loss over a probability ball centered at some distribution P0, where P can be within a certain distance Dphi from that center >

Tom: That means they aren't just optimizing for one specific way the data looks, but for the worst-case scenario within a range of possibilities >

Jane: So they use this DRO to select a combination that is minimax optimal under this robust formulation while also making sure it stays sparse >

Lu: The ensemble construction involves two optimization formulations working together sequentially, starting with finding a good decision tree model and then selecting the optimal sparse convex combination from that collection >

Meng: That sounds like a multi-step process. What about the complexity of those steps? Does it get too slow to run on production data?

Tom: The paper shows they use column generation for the first step to search through all possible rule sets efficiently, and then a smaller integer program for the second step to enforce sparsity >

Jane: And their experimental results show that this method uses less computational effort than some competing state-of-the-art methods >

Lu: They found that these sparse convex ensembles use less computational effort and still match or even improve generalization performance over other sparse rule sets, like the ones from Dash et al. twenty eighteen >

Meng: It’s good to hear it doesn't require massive compute resources for tuning the complexity parameters either >

Tom: And they have a theoretical guarantee in Proposition two showing that convex combinations of rule sets do better than a single rule set because the penalty only grows with the biggest constituent rule set >

Jane: They also found that setting a low value for C prime, which is related to the complexity bound, gives more flexibility and beats ensembles assembled with higher bounds in six out of eight cases >

Lu: The authors found that setting alpha to zero point zero five for the DRO parameter ρ is better than a larger value because it makes the robust formulation less conservative >

Meng: So they settled on a cutoff C less than or equal to thirty for the aggregate complexity of sparse ensembles, which seems like a sweet spot for both performance and low cost >

Lalam: They also pointed out that basing performance on binary loss can cause cyclical behavior because it only gives two values, which is something you have to watch out for >

Tom: So the paper is titled "Distributionally Robust Regularization of Sparse Integer Programming Trained Learning Models," and it shows a solid way to get explainable models that actually perform well without needing massive tuning on the complexity parameters >

Jane: It’s interesting how they manage to keep the theoretical guarantees while still keeping things relatively simple for interpretation >

Lu: It makes me wonder how this approach extends if we move beyond binary classification and into more complex decision problems where those rule sets aren't just binary choices anymore >

Meng: If you can find a way to make this sequential optimization pipeline faster on massive datasets, that would be really interesting for us >

IBM Research

stat.ML, cs.AI, cs.LG, math.OC

Submitted: 2023-11-03

Updated: 2026-10-15

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 87/100

The gist: The gist Obtaining Explainable Classification Models using Distributionally Robust Optimization proposes a new formulation to learn an ensemble of rule sets that simultaneously addresses the

Key concepts

Distributionally Robust Optimization (DRO): A technique used to find a solution that performs well under the worst-case probability distribution within a certain range. It helps select a convex combination of models that is robust against uncertainty in the data distribution, avoiding costly validation steps.
Sparse Ensembles of Rule Sets: This approach creates classifiers by combining many simple rule sets, where complexity is limited to the sum of individual rule set complexities. This limits model size for better interpretability while aiming for higher accuracy than single complex models.
Robust Loss R(h): The robust loss measures how well a model performs across all possible probability distributions within a defined distance from the true distribution. It ensures that the chosen combination of rule sets is reliable even if the data comes from an unexpected, yet plausible, distribution.

Terminology

Explainability — 2× in this paper · explained in 4 episodes
Generalization — 2× in this paper · explained in 13 episodes
Sparsity — 2× in this paper · explained in 4 episodes
Accuracy — 1× in this paper · explained in 1 episode
Regularization — 1× in this paper · explained in 3 episodes

Summary

The gist Obtaining Explainable Classification Models using Distributionally Robust Optimization proposes a new formulation to learn an ensemble of rule sets that simultaneously addresses the trade-off between rule set sparsity and prediction accuracy by utilizing distributionally robust optimization.

Model Formulation and Complexity

The paper focuses on sparse ensembles of rule sets as binary classifiers for binary data. The complexity of these ensembles is defined as the sum of the complexities of the rule sets used in the ensemble. The authors aim to limit model complexity to improve interpretability while trying to obtain more accurate classifiers than rule sets of similar complexity levels. They construct their sparse ensemble using two formulations that are solved in coordination with each other.

Distributionally Robust Optimization (DRO)

The method employs a distributionally robust optimization (DRO) approach to avoid the expensive cross-validation step required by existing methods for choosing model complexity. DRO is used to select a convex combination that is minimax optimal for a DRO formulation while satisfying a sparsity constraint. The robust loss R(h) is defined as max P ∈P Eξ∼P l(h(x), y) where the probability ball P is centered at P0 and admits all probability mass functions P = (Pi) that are a distance Dϕ(P, P0) of ρ > 0 from P0.

Ensembling Algorithm

The ensemble construction follows a sequential approach involving two optimization formulations. The first formulation involves solving for hn:= arg min h∈H(C′) EP n−1 l(h(x), y) to identify a good DNF model. This is followed by adding hn to the collection Fn(x) ← n-1/n Fn-1(x) + 1/n hn(x). The second formulation then selects an optimal sparse convex combination of DNF models from the identified collection. This is solved by minimizing min vk EP l Xn k=1 vkhk(x), y s.t. X xk vk>0 c(hk) ≤ C.

Column Generation and Sparsity Control

The first formulation utilizes column generation to efficiently search through the space of all feasible DNF models H(C′). The pricing problem solves a formulation that chooses a set of rules (given by nonzero values of wk) that does not minimize the 0-1 loss but instead a weighted Hamming loss. This process is extended to the general case where the data weights are any arbitrary pmf Pn. The second formulation is solved as a smaller integer program (IP) which can be easily solved with standard tools and enforces sparsity by limiting the number of DNF models considered.

Experimental Results

Extensive numerical experiments establish that the sparse convex ensembles of rule sets constructed by their method use less computational effort than competing state-of-the-art methods. The results demonstrate that the sparse convex ensembles of rule sets we construct (a) use less computational effort than competing state-of-the-art methods; (b) while matching, and in some cases improving, the generalization performance over both the sparse rule sets of Dash et al. (2018) and the best state-of-the-art methods; and (c) are of relatively low complexity leading to easy interpretability. The results show that DR comes close to and in half of the eight cases even exceeds the generalization performance of the competing explainable models CG and CART. The total model complexity C produced for C′ = 5 ensembles are significantly smaller in all instances.

Generalization Guarantees

Proposition 2 provides a bound on the generalization performance of convex rule set combinations F = Pk vkhk. This result shows that convex combinations of rule sets enjoy an important advantage over a single rule set; the generalization penalty (the second term) grows only with the largest complexity of its constituents, while allowing for extra continuous parameters (the vk) to minimize the training error. The analysis shows that setting a low value for C′ = 5 grants more generalization flexibility to the sparse ensembling formulation (10), beating the best ensembles assembled from C′ = 10 in six of the eight cases.

Parameter Sensitivity

Table 6 illustrates that ρ = 0.05 is on balance better than the larger ρ = 0.5 which may make the DRO formulation too conservative. The value of 20, chosen for the results presented in the main body of the paper, seems to attain a good balance over these three competing factors. The aggregate complexity of sparse ensemble determined by (11) with a cutoff at C ≤ 30 appears to be preferable from the view of all three factors of good generalization performance at a low computational cost while producing models with good interpretability.

Implementation Details

The implementation uses Java and version 22.1 of the commercial solver IBM ILOG CPLEX to solve the linear programs and mixed-integer linear programs. The DRO parameter ρ is set to 0.05 as measured using χ2-divergence for the distance function Dϕ(P, P0). The implementation of the sparse convex combination IP formulation (11) – (14) limits the time spent in the solver to 600 seconds. The results presented are based on splitting each input dataset uniformly at random into a training dataset, containing 70% of the data, and a test dataset with the remaining 30% of the data. The implementation then uses the training dataset to execute the sequence of our two formulations and obtain the set of final rules. The total number of iterations n are stopped when no improvements are noticed in 20 consecutive iterations.

Cyclical Behavior

The analysis notes that basing the performance on the misclassification losses l(h(x), y) = h(x)−y can lead to undesirable cyclical behavior. This is because l takes only two values and thus the solution P∗(h) = arg maxP ∈P EP l(h(x), y) is correspondingly degenerate, partitioning the dataset S into two subsets of the correctly classified and the misclassified. The histograms show identical histograms in every other iteration, which indicates that the chosen DNF iterates hn flip the instantaneous loss values of a handful of datapoints and leaves the vast majority unchanged. This leads to a roughly equal split of the dataset into three subsets. The results illustrate why the DRO philosophy of training continuously valued parameter models is of limited direct use when the models can produce only binary loss information.

Column Generation for Rule Sets

The formulation (6) searches over the exponentially large space H(C′) of rule sets using column generation. The pricing problem identifies the rule k missing from RMLP that has the lowest reduced cost. This approach is motivated by the fact that there is a solution to the LP in which the number of nonzero variable values is bounded above by the number of constraints. The pricing problem identifies the rule k missing from RMLP that has the lowest reduced cost.

Solving DRO Formulation

The algorithm adapted from Ghosh et al. (2021) solves the DRO formulation (5) for the worst-case pmf. The Lagrangian objective of (9) can be expressed as L(α, λ, P) = X i∈S Pizi + λ 1 − X i∈S Pi! + α N ρ − X i∈S Φ(NPi)! This leads to the optimal objective value of (9) given by R˜∗(Fn) = minα≥0,λ maxPi≥0 L(α, λ, P). The algorithm is adapted from Ghosh et al. (2021), and the details are provided in Section C of the supplement.

Data Preprocessing

The datasets were pre-processed as described by Dash et al. (2018). The standard “dummy”/“onehot” coding to transform categorical features into multiple binary features is used. The non-binary numerical features are transformed into a sequence of binary features where each feature represents the satisfaction or not of a certain threshold and its negation.

Improvements for AI systems

Distributionally Robust Optimization (DRO) for Rule Set Selection: The system can select an ensemble of rule sets that simultaneously addresses these competing factors of rule set sparsity and prediction accuracy, ensuring Good generalization is ensured while keeping computational costs low.
Efficient Rule Set Search via Column Generation: The system utilizes a distributionally robust combinatorial optimization problem solved with column generation to efficiently search the space of DNF classifiers, which allows for the construction of a sparse ensemble of rule sets in contrast to techniques like random forests or boosting.
Sparsity-Constrained Convex Combination Selection: The system employs a second integer program (IP) to select an optimal sparse convex combination from the dense set generated by the first formulation, thereby enforcing sparsity while improving generalization, as described by solving min vk EP l(Xn k=1 vkhk(x), y s.t. X xk ck ≤ C.
Adaptive Complexity Budgeting: The system can adaptively determine complexity parameters by solving two sequential formulations where the second step solves the IP for increasing values of C, stopping when the value of the identified optimal solution does not decrease beyond a given threshold.
Improved Generalization Guarantees for Ensembles: The resulting sparse convex ensembles are theoretically supported by Proposition 2, which shows that they can significantly improve training performance over single rule sets while displaying the generalization performance of its largest constituent rule set.

Abstract

Building explainable machine learning models is crucial for human users to be able to interpret the proposed statistical relationship obtained from the training data. Mixed-integer optimization formulations are often used to train such models with explicit sparsity constraints, aiming to strike the right trade-off between sparsity and prediction accuracy. Existing methods to find the right choice of sparsity -- e.g., via cross-validation -- are computationally expensive. For convex model training formulations, recent advances in distributionally robust optimization (DRO) provide strong generalization while sidestepping this computational burden. We describe an extension of such regularization via DRO to mixed-integer sparse programs, providing statistical guarantees as a function of an associated sparsity parameter of the formulation. We illustrate the use of this approach in the case of building explainable binary classification models using sets of feature value rules.