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Decision Trees for Decision-Making under the Predict-then-Optimize Framework

Adam N. Elmachtoub, Jason Cheuk Nam Liang, Ryan McNellis

arXiv:2003.00360v2cs.LGmath.OCstat.ML

TL;DR

The paper addresses the lack of tractable decision-tree methods that directly minimize SPO loss. It develops algorithms for this purpose and reports that SPOTs produce higher-quality decisions with lower model complexity than prediction-error-based tree methods.

  • Problem

    Directly training models with SPO loss is difficult because the loss is nonconvex and discontinuous.

  • Method

    The paper presents tractable methodologies for training decision trees under SPO loss within the predict-then-optimize framework.

  • Results

    SPOTs simultaneously provide higher-quality decisions and lower model complexity than tree-building methods designed to minimize prediction error.

  • Takeaways & Limitations

    SPOTs offer a decision-tree approach that combines SPO-loss training with the comparatively lower complexity reported for prediction-error-based alternatives.

  • Takeaways & Limitations

    With limited training data, near-perfect decisions are no longer possible, while direct SPO-loss optimization is likely infeasible because the loss is nonconvex and discontinuous.

Abstract

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We consider the use of decision trees for decision-making problems under the predict-then-optimize framework. That is, we would like to first use a decision tree to predict unknown input parameters of an optimization problem, and then make decisions by solving the optimization problem using the predicted parameters. A natural loss function in this framework is to measure the suboptimality of the decisions induced by the predicted input parameters, as opposed to measuring loss using input parameter prediction error. This natural loss function is known in the literature as the Smart Predict-then-Optimize (SPO) loss, and we propose a tractable methodology called SPO Trees (SPOTs) for training decision trees under this loss. SPOTs benefit from the interpretability of decision trees, providing an interpretable segmentation of contextual features into groups with distinct optimal solutions to the optimization problem of interest. We conduct several numerical experiments on synthetic and real data including the prediction of travel times for shortest path problems and predicting click probabilities for news article recommendation. We demonstrate on these datasets that SPOTs simultaneously provide higher quality decisions and significantly lower model complexity than other machine learning approaches (e.g., CART) trained to minimize prediction error.

1. Introduction

Many decision-making problems can be framed as optimization with uncertain inputs, but standard prediction-error training does not directly target downstream decision quality. The paper introduces tractable SPO Trees and shows they produce higher-quality decisions with lower complexity than prediction-error-trained tree methods.

  • Framework: The predict-then-optimize framework first predicts uncertain optimization inputs from historical data and then solves the optimization problem using those predictions.Applications mentioned include travel-time prediction for routing and click-probability prediction for advertising or recommendation.
  • Motivation: Training models with SPO loss is difficult because the loss is nonconvex and discontinuous, while surrogate losses are only approximations and lack guaranteed SPO-optimal recovery.Prior work proposed convex or differentiable surrogates because direct SPO-loss training was considered likely infeasible.
  • Contribution: SPO Trees train decision trees to minimize decision error under the predict-then-optimize framework rather than prediction error.The method directly uses SPO loss, which measures the suboptimality of decisions induced by predicted input parameters.
  • Method: The paper exploits decision-tree structure to simplify SPO-loss training and extends greedy and integer-programming approaches to build SPO Trees and SPO Forests.The authors present what they describe as the first tractable methodology for training an ML model with SPO loss for a general class of decision-making problems.
  • Results: Across synthetic and real experiments, SPOTs achieve higher-quality decisions and significantly lower model complexity than trees trained only to minimize prediction error, such as CART.The reported complexity advantage includes fewer leaves, making SPOTs more interpretable while maintaining high-quality decisions.

2. The Predict-then-Optimize Framework

The predict-then-optimize framework predicts unknown cost vectors from features and uses those predictions to make optimization decisions. SPO loss evaluates prediction quality by the excess cost of the induced decision rather than prediction error, motivating decision-tree methods that optimize it directly.

  • Optimization problem: The underlying decision problem minimizes c^T w over a feasible region S, with tractable optimization assumed for any cost vector c.The framework permits general decision-making problems with known constraints and unknown linear objective coefficients.
  • Predict-then-optimize: The framework minimizes an empirical loss for models that map feature vectors x to predicted cost vectors ĉ, then uses w*(ĉ) as the proposed decision.Training data consist of feature-cost samples, and the predicted cost vector is supplied to the optimization problem for a new feature vector.
  • SPO loss: SPO loss measures the excess true cost c^T w*(ĉ) − z*(c) caused by using the decision induced by predicted costs.Unlike MSE, it scores the quality of the resulting decision rather than the prediction error itself.
  • SPO loss: SPO loss uses the worst-case optimal decision associated with a predicted cost vector when multiple predicted-cost optima exist.The set W*(ĉ) may contain multiple optimal solutions, so the loss is defined with respect to the worst-case decision from that set.
  • Motivation: Direct SPO-loss training is difficult because the loss is nonconvex, discontinuous, and nondifferentiable in the prediction.Prior work therefore developed surrogate approaches, whereas this paper proposes multiple strategies for training decision trees directly under SPO loss.

3. Decision Trees for Decision-Making

Decision trees partition feature space into leaves that assign predicted costs and corresponding optimization decisions. In the illustrative shortest-path problem, SPOT identifies the decision boundary directly and achieves high-quality decisions with lower depth than CART.

  • Illustrative problem: The illustrative shortest-path problem selects between two edges with unknown travel times using weekday, hour, and snowfall features.The goal is to choose the path with the smallest cost given the observed feature vector.
  • Decision-tree structure: A decision tree repeatedly splits feature space, and each leaf assigns one predicted cost vector and its associated optimal decision.Leaf depth is the number of splits needed to reach it, while tree depth is the maximum leaf depth.
  • 3.1. An Illustrative Example: With depth up to 4, CART nearly recovers the optimal boundary because combined splits finely partition feature space into regions with accurate local predictions.This illustrates how greater tree complexity can compensate for splits that individually have less decision-making value.
  • Small-data setting: Limited training data restrict tree depth because each leaf needs enough observations to estimate its cost prediction accurately.This makes maximizing each split’s contribution to decision quality especially important in small-data settings, where shallower interpretable trees are also preferred.
  • 3.1. An Illustrative Example: At depth 1, SPOT identifies the true boundary x < 0.28 and achieves zero decision error, whereas CART reaches near-zero error only at depth 4.CART has comparatively high decision error at depths 1–3, while SPOT maintains zero error across the evaluated depths.
  • 3.1. An Illustrative Example: SPOT provides high-quality decisions with significantly lower complexity than the CART tree required for comparable decision quality.The comparison uses normalized extra travel time, defined as cumulative SPO loss normalized by cumulative optimal decision cost.

4. Methodology

The methodology trains decision trees directly against SPO loss, using leaf-level cost averages to simplify optimization and recursive or integer-programming procedures to construct trees.

  • SPO Trees are trained by partitioning observations into leaves whose assignments follow the structure of a decision tree.The resulting objective aggregates SPO losses induced by the decisions associated with each leaf.
  • Theorem 1 shows that each leaf’s average cost vector minimizes its within-leaf SPO loss, simplifying the tree-training objective.This result assumes the underlying decision problem has a unique optimal solution for the relevant average cost vector.
  • 4.1. SPOT: Recursive Partitioning Approach: Recursive partitioning greedily selects the split with the lowest training SPO loss by evaluating leaf averages, induced decisions, and combined losses.The procedure recursively repeats split selection until practitioner-specified stopping criteria are met.
  • 4.1. SPOT: Recursive Partitioning Approach: The recursive procedure can be computationally prohibitive when all continuous-feature split points are considered, motivating quantile-based candidate split heuristics.The method also supports pruning with SPO loss to reduce model complexity.
  • 4.2. SPOT: Integer Programming Approach: The integer-programming formulation expresses SPO Tree training as a mixed integer linear program that can be solved to optimality for modestly sized instances.A feasible greedy solution provides a bound, and the MILP solution is at least as optimal as that greedy solution if solved appropriately.
  • 4.3. SPO Forests: SPO Forests extend the approach by combining greedy SPO Trees into an ensemble.

Experimental Results

Across shortest-path and news-recommendation experiments, SPO Trees generally deliver better decisions than prediction-error-trained trees while using simpler models. Their advantage is strongest for shallow or moderately restricted trees, whereas CART can become comparable at greater depth.

  • Noisy Shortest Path: SPO Trees outperform CART in all small-data shortest-path settings and can achieve lower SPO loss at depth 1 than unrestricted CART.The authors attribute CART’s weaker decision performance to its prediction-error objective and limited training data.
  • Noisy Shortest Path: SPO Tree forests outperform CART forests by 20.5% on average and show lower performance variance across the four parameter settings.The MILP SPO Tree approach further improves decision quality over the greedy approach and random forests.
  • Noisy Shortest Path: At n = 10000, SPOTs perform better with shallow or modest depths, but CART becomes comparable or slightly better at depth six or more.The results show that deeper CART trees can finely partition features even when individual splits have little decision value.
  • Noisy Shortest Path: Large or unrestricted SPOTs contain less than half as many leaves as CART while achieving comparable accuracy and greater interpretability.CART random forests have a very slight edge over SPO Forests in normalized extra travel time on the test set.

6. Conclusion

The paper presents tractable methods for training decision trees under SPO loss. Across its experiments, SPOTs provide higher-quality decisions and lower model complexity than tree-building methods designed to minimize prediction error.

  • 6. Conclusion: The paper proposes tractable methodologies for training decision trees under SPO loss within the predict-then-optimize framework.This conclusion states the central methodological contribution.
  • 6. Conclusion: SPOTs provide higher-quality decisions and lower model complexity than tree-building methods designed to minimize prediction error.The conclusion summarizes the paper’s reported results across its experimental settings.

Appendix A: Encoding Decision Trees using Integer and Linear Constraints

The appendix encodes decision-tree structure and routing through integer and linear constraints for the SPO Tree MILP. It also documents a correction to inactive-branch logic.

  • Tree Structure: A complete tree of depth H has L = 2^H leaves and B = 2^H − 1 branch nodes, with observations assigned to indexed leaves.The formulation distinguishes leaves from splitting nodes and permits inactive components in the complete-tree representation.
  • Inactive Nodes: Inactive branch nodes can behave as leaves, with descendant splits constrained inactive and observations routed consistently through the tree.The appendix explicitly tracks active leaves and active splits using indicator variables.
  • Split Encoding: Each split uses a binary feature-selection vector a_t and threshold b_t, encoding a one-feature rule such as x_2 < 0.4.Only one feature component may be nonzero in a split vector.
  • Routing Constraints: Ancestor sets and linear constraints encode whether observations follow left or right paths from the root to each leaf.The formulation uses left- and right-ancestor sets to express tree-routing logic.
  • Inactive Nodes: The authors correct prior inactive-branch logic because setting a and b to zero leaves both branch constraints feasible.They set b to one for an inactive branch so the intended routing constraint becomes infeasible and observations are forced left.

Appendix B: SPOT Integer Programming Approach: Additional Implementation Details

The implementation appendix describes regularization, solver-starting, feature-rounding, and relaxation-bound strategies for the SPO Tree MILP. It also notes that cross-validating the regularization parameter may be computationally impractical.

  • Regularization: The regularization parameter α balances concise trees against training error and can be tuned by cross-validation.The experiments instead use no regularization during training and apply CART-style post-pruning with SPO loss.
  • Regularization: Cross-validation over α may be infeasible when repeatedly solving the optimization problem is computationally expensive.This is identified as a practical limitation of tuning the regularization parameter across multiple folds.
  • Computational Strategies: Warm-starting the MILP with the greedy solution is recommended to improve computational time.The appendix lists this as the first suggested strategy for speeding up MILP solution.
  • Computational Strategies: Rounding features, when it does not affect tree quality, can reduce the precision of ε and improve computational performance.The recommendation is illustrated with rounding to a precision such as 1e−2.
  • Computational Strategies: Adding a lower objective bound of zero can help MILP solvers whose LP relaxations otherwise contain large negative solutions.The appendix connects this bound to branch-and-bound performance.

Appendix C: Additional Experimental Details: News Article Recommendation

This appendix details preprocessing the Yahoo! Front Page Today Module interaction data into predict-then-optimize feature-cost pairs and reports SPOT training runtimes. The resulting data comprise six article types and thousands of retained feature-cost pairs across training, validation, and test sets.

  • Dataset: 45,811,883 interaction records between users and news articles form the dataset, with each record containing five-dimensional user and article feature vectors and a binary click outcome.The features were constructed using a conjoint analysis with a bilinear model.
  • Preprocessing: The preprocessing samples half of the May 1–5, 2009 interactions for training, assigns the remaining records to validation, and uses all May 6–10 records for testing.Interaction records are grouped by user type separately within the training, validation, and test sets.
  • Preprocessing: Users are clustered into 10,000 user types and displayed articles into seven article types using K-means before constructing feature-cost pairs.User features are represented by cluster means, while costs use average click probabilities for article types within each user cluster.
  • Preprocessing: After requiring at least 50 interactions per user–article-type average, the data retain six article types and 5,130, 5,105, and 8,768 feature-cost pairs for training, validation, and test.One article type and additional feature-cost pairs were dropped to satisfy the interaction-count threshold.
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