Source-linked AI summary
Deep Lattice Networks and Partial Monotonic Functions
Seungil You, David Ding, Kevin Canini, Jan Pfeifer, Maya Gupta
TL;DR
The paper addresses how to learn flexible deep models that are guaranteed monotonic with respect to selected inputs. It combines constrained linear embeddings, calibrators, and lattice ensembles, and reports strong classification and regression performance while retaining monotonicity guarantees.
Problem
The paper focuses on learning flexible functions that remain monotonic with respect to a user-specified subset of inputs.
Method
The method alternates constrained linear embeddings, calibrators, and ensembles of lattices, with adaptive embeddings learning feature interactions for lattice inputs.
Results
Experiments on benchmark and real-world datasets show that DLNs provide better accuracy than min-max networks or crystals in reported classification comparisons.
Takeaways & Limitations
DLNs combine deep-network flexibility with monotonicity-based regularization, interpretability, and debuggability advantages.
Takeaways & Limitations
The user-specified monotonicity formulation assumes monotonic means non-decreasing, with nonincreasing behavior handled by flipping an input’s sign.
Abstract
from arXiv · showhide
We propose learning deep models that are monotonic with respect to a user-specified set of inputs by alternating layers of linear embeddings, ensembles of lattices, and calibrators (piecewise linear functions), with appropriate constraints for monotonicity, and jointly training the resulting network. We implement the layers and projections with new computational graph nodes in TensorFlow and use the ADAM optimizer and batched stochastic gradients. Experiments on benchmark and real-world datasets show that six-layer monotonic deep lattice networks achieve state-of-the art performance for classification and regression with monotonicity guarantees.
1 Introduction
The paper proposes deep lattice networks (DLNs), which combine multiple lattice layers to learn flexible functions that remain monotonic on user-specified inputs. Calibrators and adaptive embeddings extend this constrained modeling approach, with experiments on benchmark and real-world scenarios.
- Motivation and contribution: DLNs stack multiple lattice layers to model flexible partially monotonic functions.Partial monotonicity means the model is guaranteed monotonic with respect to a user-specified subset of inputs.
- Motivation and contribution: Monotonicity acts as a regularizer while improving model interpretability, debuggability, trustworthiness, and test-data generalization.
- Motivation and contribution: The proposed architecture alternates linear embeddings, calibrators, and ensembles of lattices, with each layer trained under structural-risk and monotonicity constraints.The paper illustrates a nine-layer network composed of these layer types.
- Layer extensions: Calibrators nonlinearly transform individual inputs and can be inserted between deep-model layers or used as a final output transformation.This extends their earlier use as preprocessing layers in two-layer models.
- Evaluation: The paper reports experimental results on benchmark and real-world scenarios to demonstrate the potential of DLNs.
2 Deep Lattice Network Layers
DLN layers preserve partial monotonicity by constraining embeddings, calibrators, and lattice outputs, while calibrators and lattice ensembles provide piecewise-linear nonlinear transformations. Linear embeddings also learn how features interact within lattices.
- Layer framework: DLNs treat monotonicity as non-decreasing behavior and allow constraints on any user-specified subset of inputs.Nonincreasing behavior can be obtained by flipping an input’s sign.
- Linear embedding layer: Linear embedding layers separately transform monotonic and non-monotonic inputs, with constraints imposed on the monotonic embedding to preserve monotonicity.The embedding matrices and bias vector are discriminatively trained.
- Calibration layer: Calibration layers apply separate one-dimensional piecewise-linear transforms that map each input to [0, 1].Each calibrator uses fixed keypoint inputs and trained output parameters, with out-of-range inputs clipped to the fixed range.
- Calibration layer: Monotonic calibrators are enforced by constraining successive output parameters to be non-decreasing.The paper also experiments with making all calibrators monotonic for more stable or regularized training.
- Ensemble of lattices layer: Each lattice ensemble contains G lattices that interpolate parameters over an S-dimensional unit hypercube using multilinear interpolation.Each lattice has 2^S vertex parameters, and the ensemble produces one output per lattice.
- Partial monotonicity: A lattice output marked monotonic becomes a monotonic input to the next layer, so composing constrained components guarantees end-to-end partial monotonicity.A linear embedding before the first lattice ensemble adaptively learns feature combinations that determine lattice interactions.
- Hyperparameters: Layer hyperparameters must match adjacent dimensions, such as equating the embedding outputs with the next ensemble’s total lattice inputs.
3 Related Work
Prior work established constrained lattice and neural architectures for partial monotonicity, but expressive demonstrations remained limited in input dimensionality and scale. DLNs address these limitations through deeper layers, adaptive feature interactions, and trainable intermediate calibration.
- Earlier approaches: Earlier lattice models progressed from single calibrated lattices to two-layer calibrator–ensemble architectures for expressive partial monotonic functions.The earlier ensemble approach was restricted to around D ≤ 16 inputs because each lattice requires O(2^S) parameters.
- Deep architectures: DLNs alternate layers to form deeper networks that are potentially more flexible than prior two-layer architectures.This is identified as the first key difference from prior work.
- Adaptive interactions: A linear embedding before each lattice ensemble learns feature interactions discriminatively and adaptively during training.Each lattice input can thereby be a linear combination of the original features rather than a directly assigned feature.
- Calibration: DLNs fix calibrator keypoints uniformly over bounded domains instead of using feature quantiles, which evolve for intermediate layers during training.Earlier methods fixed keypoints a priori from feature quantiles.
- Neural baselines: Positive-weight and min–max-pooling neural networks provide related monotonic constructions, but prior demonstrations covered at most D = 10 features and only a few thousand examples.Earlier neural-network work also reported negative results concerning flexibility for positive-weight models with multiple hidden layers.
4 Function Class of Deep Lattice Networks
The function class of DLNs depends on the interpolation scheme. Multilinear cascades collapse to a single lattice, whereas simplex-interpolated networks remain locally linear and can support universal approximation through cascaded layers.
- 4.1 Cascaded multilinear lookup tables: Multilinear lattice cascades are equivalent to a single multilinear lattice defined on all D input features.The result follows inductively through every layer of the cascaded lookup table.
- 4.1 Cascaded multilinear lookup tables: A lattice whose inputs are multilinear functions on disjoint feature subsets preserves that functional form, changing only coefficients and interpolation weights.The resulting output is another multilinear polynomial over the union of the input feature subsets.
- 4.1 Cascaded multilinear lookup tables: The multilinear-polynomial result follows because each interpolation product contains one of fi or 1 − fi, and the inputs depend on distinct feature subsets.Sums of multilinear polynomials remain multilinear, and any multilinear polynomial can be converted to a multilinear lookup table.
- 4.2 Universal approximation of partial monotone functions: Two cascaded simplex-interpolated lattice layers with a linear embedding can approximate any partial monotone function on the hypercube.This construction uses the fact that simplex-interpolated lattices can represent min and max pooling.
- Interpolation and flexibility: Simplex interpolation makes a DLN locally linear, with a D-input lattice having D! linear pieces; an ensemble cascade is hypothesized to reach O((D!)!) pieces.The factorial-scale count is stated as a hypothesis for the cascaded ensemble construction.
5 Numerical Optimization Details for the DLN
The DLN implementation represents calibrators and lattice interpolation as TensorFlow operators and optimizes parameters with projected stochastic updates. Initialization uses positive-biased embeddings and linear starting functions for lattices and calibrators.
- Implementation and optimization: Calibrators and multilinear lattice interpolation are implemented as new C++ TensorFlow operators, with each layer represented as a computational-graph node.The implementation uses existing TensorFlow operators alongside the new operators.
- Implementation and optimization: ADAM and batched stochastic gradients update the model parameters, followed by projection onto the monotonicity constraints after every gradient step.Projection enforces the required monotonicity constraints during optimization.
- Initialization: Linear embedding components are initialized with IID Gaussian noise N(2, 1) to bias them positive before the first monotonicity projection.Because preceding calibration outputs are expected to have mean 0.5, this initialization also introduces an initial bias.
- Initialization: Lattice parameters start as a linear function spanning [0, 1] with added IID Gaussian noise, while calibrators start as linear maps from [x_min, x_max] to [0, 1].Noise is added to lattice parameters but not to calibrators.
6 Experiments
Experiments evaluate monotonic models on a benchmark and real-world classification and regression tasks, with every model trained under the same monotonicity guarantees. DLNs outperform competing models across the reported case studies and benchmark.
- Experimental setup: The experiments use the Adult benchmark and three internet-services problems with monotonicity constraints specified by product groups.Every model is trained with guarantees on the same set of monotonic inputs.
- Experimental setup: Validation sets tune architecture and optimization hyperparameters, including learning rate, training steps, ensemble size, and lattice inputs.Calibrators use 100 keypoints over the input range [−100, 100].
- User intent case study: Two DLN architectures outperform crystals and min-max net in test accuracy on the user-intent classification case study.The results are summarized in Table 3 and sorted by validation accuracy.
- Adult benchmark: DLN provides better accuracy than the min-max network or crystals on the Adult income-classification benchmark.The tested DLN architecture is Cal-Lin-Cal-EnsLat-Cal-Lin, with monotonicity specified for selected features.
- Rater score prediction: DLN has slightly better validation and test MSE than all other models for rater-score prediction with 10 monotonic features.Rater scores are averaged over 1–5 raters and take 5–25 possible values.
- Usefulness case study: DLN has better validation and test accuracy than other models for predicting whether a candidate result contains useful information.All 9 features in this task are required to be monotonic, using a Cal-Lin-Cal-EnsLat-Cal-Lin architecture.
7 Conclusions
The paper combines calibrators, linear embeddings, and lattices into models that retain deep-network flexibility while supporting monotonicity constraints. This combination also provides the stated regularization, interpretability, and debuggability advantages of constrained models.
- Conclusion: DLNs combine calibrators, linear embeddings, and lattices to learn flexible functions constrained to be monotonic on any subset of inputs.The paper frames this combination as a new class of models.