Behavior Learning (BL): Learning Hierarchical Optimization Structures from Data

Zhenyao Ma; Yue Liang; Dongxu Li

cs.LGFeb 23, 2026

Behavior Learning (BL): Learning Hierarchical Optimization Structures from Data

Authors: Zhenyao Ma, Yue Liang, Dongxu Li

Organizations: Xiamen University · University of Tübingen · Xi’an Jiaotong University

Abstract

Inspired by behavioral science, we propose Behavior Learning (BL), a novel general-purpose machine learning framework that learns interpretable and identifiable optimization structures from data, ranging from single optimization problems to hierarchical compositions. It unifies predictive performance, intrinsic interpretability, and identifiability, with broad applicability to scientific domains involving optimization. BL parameterizes a compositional utility function built from intrinsically interpretable modular blocks, which induces a data distribution for prediction and generation. Each block represents and can be written in symbolic form as a utility maximization problem (UMP), a foundational paradigm in behavioral science and a universal framework of optimization. BL supports architectures ranging from a single UMP to hierarchical compositions, the latter modeling hierarchical optimization structures. Its smooth and monotone variant (IBL) guarantees identifiability. Theoretically, we establish the universal approximation property of BL, and analyze the M-estimation properties of IBL. Empirically, BL demonstrates strong predictive performance, intrinsic interpretability and scalability to high-dimensional data. Code: https://github.com/MoonYLiang/Behavior-Learning ; install via pip install blnetwork.

Figures & tables

Figure 1: Behavior Learning (BL). (a) Human behavior modeled as a UMP. (b) Learning scheme of BL, where CompU denotes the compositional utility function. (c) BL offers intrinsic interpretability (via symbolic form as a hierarchical optimization structure), identifiability, and inference capability. (d) Three architectural variants of BL, from single UMP to deep compositions.

Figure 2: (a) Visualization and symbolic form of BL(Single) trained on the Boston Housing dataset, modeling the UMP ( $\max U\;\;\text{s.t.}\;\mathcal{C}\leq 0,\;\mathcal{T}=0$ ) of a representative buyer in Boston housing (details in Section 3.2 ). Top: computational graphs of the polynomials inside the three penalty functions— $\tanh$ (preference), $\mathrm{ReLU}$ (budget), and $|\cdot|$ (belief). Each graph is respectively centered on $\tanh^{-1}(U)$ , $\mathcal{C}$ , and $\mathcal{T}$ from left to right, with surrounding nodes representing input features. Directed edges (shown only if coefficient $\geq 0.3$ ) indicate how each feature contributes to the corresponding term. Bottom: approximate symbolic formulation of the trained BL model as a UMP. (b) The BL[2,1] architecture. Layer 1 identifies two key micro-level preference types: the Economic-sensitive Buyer and the Location-sensitive Buyer . Layer 2 aggregates these two components into an effective representative buyer. (c) The BL(Deep) [5,3,1] architecture. Layer 1 recovers five distinct micro-level housing preference types. Layer 2 identifies three macro-level trade-off types capturing different ways these primitive preferences interact. Layer 3 aggregates them into the overall representative buyer. Table 10 provides detailed descriptions of each type. BL(Deep) provides a hierarchical explanation consistent with the coarse-graining principle ( Kadanoff, 1966 ) in statistical physics, reconstructing the full micro-to-macro optimization hierarchy. In addition, the preference and trade-off patterns uncovered by BL(Deep) are well documented in the classical economics literature (see Table 11 ). (d) BL can be applied to a broad class of hierarchical optimization structures in science, including hierarchical need structures, hierarchical social–organizational structures, and renormalization-style coarse-grained structures in physics.

Figure 3: Predictive performance of BL and baselines. Left/Middle: relative AUC and F1-Macro gains over DT, sorted by mean (excluding BL). Right: mean F1-Macro ranks (↓ better). BL achieves first-tier performance in both metrics. Its variants rank second and third in mean F1-Macro rank, with BL(Shallow) showing no statistically significant difference from state-of-the-art models.

Figure 4: Interpreting deeper BL architectures as hierarchical structures of interacting agents. Each block

\mathcal{B}

represents an interpretable agent solving its own UMP, while a layer corresponds to a set of heterogeneous agents operating in parallel. The next layer then aggregates and reallocates the negative energies from the previous layer, thereby performing higher-level coordination across agents. This layered organization provides a natural compositional interpretation of deep BL: bottom-layer modules encode local objectives, while upper layers synthesize these into collective outcomes. Analogous structures arise in biological and social systems—for example, in ant colonies, individual ants (first-layer agents) follow simple local rules, yet their collective behavior is coordinated through higher-level interactions (second-layer aggregation), yielding globally efficient resource allocation and task division.

Model	MNIST		Fashion-MNIST
Model	ECE	NLL	ECE	NLL
E-MLP (depth=1)	0.02 $\;\pm\;$ 0.00	0.20 $\;\pm\;$ 0.02	0.08 $\;\pm\;$ 0.00	0.74 $\;\pm\;$ 0.01
BL (depth=1)	0.02 $\;\pm\;$ 0.00	0.26 $\;\pm\;$ 0.01	0.05 $\;\pm\;$ 0.00	0.36 $\;\pm\;$ 0.01
E-MLP (depth=2)	0.02 $\;\pm\;$ 0.00	0.23 $\;\pm\;$ 0.02	0.09 $\;\pm\;$ 0.00	0.89 $\;\pm\;$ 0.03
BL (depth=2)	0.02 $\;\pm\;$ 0.00	0.16 $\;\pm\;$ 0.01	0.07 $\;\pm\;$ 0.00	0.44 $\;\pm\;$ 0.01
E-MLP (depth=3)	0.02 $\;\pm\;$ 0.00	0.16 $\;\pm\;$ 0.02	0.09 $\;\pm\;$ 0.00	0.85 $\;\pm\;$ 0.04

Table 2: ECE and NLL on image and text datasets. BL and E-MLP are evaluated at depths 1–3 with matched parameter counts. Top-two per column are blue and red .

Model	MNIST	FashionMNIST	AG News	Yelp
E-MLP (depth=1)	100.59 $\;\pm\;$ 0.29	73.57 $\;\pm\;$ 1.20	14.69 $\;\pm\;$ 0.40	179.37 $\;\pm\;$ 0.73
BL (depth=1)	110.63 $\;\pm\;$ 3.34	96.52 $\;\pm\;$ 2.90	17.20 $\;\pm\;$ 0.06	181.07 $\;\pm\;$ 1.80
E-MLP (depth=2)	102.64 $\;\pm\;$ 0.26	78.25 $\;\pm\;$ 0.28	15.76 $\;\pm\;$ 0.06	179.22 $\;\pm\;$ 0.66
BL (depth=2)	122.85 $\;\pm\;$ 3.95	114.43 $\;\pm\;$ 3.72	21.78 $\;\pm\;$ 0.08	180.38 $\;\pm\;$ 1.44
E-MLP (depth=3)	104.52 $\;\pm\;$ 0.30	85.57 $\;\pm\;$ 1.19	16.95 $\;\pm\;$ 0.05	178.99 $\;\pm\;$ 1.42
BL (depth=3)	140.17 $\;\pm\;$ 4.42	130.03 $\;\pm\;$ 4.96	26.29 $\;\pm\;$ 0.24	180.36 $\;\pm\;$ 0.91

Table 3: Training time (seconds) of BL vs. E-MLP on high-dimensional datasets (mean

\pm

std).

Figure 6: Constraint enforcement test of the BL penalty block on an energy-conservation constraint. The figure reports violation statistics

|T(x,y)|

when varying the temperature

\tau

(left side of panel) and the penalty weight

\lambda

(right side of panel).

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Visualization of polynomial feature maps as computation graphs, where nodes represent variables or outputs and edges represent their effects. The left panel illustrates the linear form

\mathcal{F}=ax+b

, in which the single edge

x\!\to\!\mathcal{F}

directly encodes the marginal effect of

x

\mathcal{F}

. The middle panel shows the quadratic form

\mathcal{F}=ax^{2}+bx+c

, where

x

not only has a direct edge

x\!\to\!\mathcal{F}

but also acts on its own edge (“

x\!\to\!\mathcal{F}

”), thereby modifying the strength of its self-effect through a higher-order contribution. The right panel depicts the interaction form

\mathcal{F}=ax+by+cxy+d

, where

y

has an edge

y\!\to\!\mathcal{F}

and, in addition,

x

acts on this edge (“

y\!\to\!\mathcal{F}

”), thereby modulating the strength of

y

’s contribution to

\mathcal{F}

. Symmetrically,

y

may act on the edge (“

x\!\to\!\mathcal{F}

”), so that each variable can reshape the other’s effect through the interaction term.

Name	Size	#Features	Task type	Field
German Credit	1,000	20	Binary cls.	Finance
Adult Income	48,842	14	Binary cls.	Economics
COMPAS (two-years)	5,278	13	Binary cls.	Law & Society
Bank Marketing	45,211	16	Binary cls.	Marketing
Planning Relax	182	12	Binary cls.	Psychology
EEG Eye State	14,980	14	Binary cls.	Neuroscience

Appendix

Table 4: Standard OpenML datasets used in our task. #Features denotes the number of input variables (excluding the target and ID).

Methodological Family	Model Name
Neural networks	Standard MLP
	Neural Additive Model (NAM)
	TabNet
Linear regressors	ElasticNet
	Logistic Regression
	Polynomial Logistic Regression

Appendix

Table 5: Overview of baseline models in the standard prediction task

Model	Parameter	Search space
BL(Single)	learning_rate	{1e $-3$ , 1e $-1$ }
	batch_size	{64, 128, 256, 512}
	max_grad_norm	{1.0, 2.0, 5.0}

Appendix

Table 6: Hyperparameter tuning space for BL(Single)

Model	Parameter	Search space
BL(Shallow)	learning_rate	LogUniform{5e $-5$ , 5e $-3$ }
	batch_size	{64, 128, 256, 512}
	n_layers	UniformInt{1, 3}
	n_first_layer	{24, 30, 36, 40}
	n_middle_layer	{8, 6, 4}
	n_last_layer	{2, 4, 6}

Appendix

Table 7: Hyperparameter tuning space for BL(Shallow)

Model	Parameter	Search space
MLP	learning_rate	LogUniform{1e $-5$ , 1e $-1$ }
	batch_size	{32, 64, 128, 256}
	n_layers	UniformInt{2, 4}
	hidden_size	UniformInt{32, 256}
	weight_decay	LogUniform{1e $-6$ , 1e $-2$ }
NAM	learning_rate	LogUniform{1e $-3$ , 1e $-1$ }

Appendix

Table 8: The hyperparameter tuning space for baseline models used in the standard prediction tasks

Variable	Description
CRIM	Per-capita crime rate by town
ZN	Proportion of residential land zoned for lots over 25,000 sq.ft.
INDUS	Proportion of non-retail business acres per town
CHAS	Charles River dummy variable (=1 if tract bounds river)
NOX	Nitric oxide concentration (parts per 10 million)
RM	Average number of rooms per dwelling

Appendix

Table 9: Boston Housing dataset variables and descriptions.

Layer	Block	Representative preference
Layer 1	Location-Sensitive Buyer	Values river access, transport accessibility, and neighborhood amenities.
	Risk-Sensitive Buyer	Averse to local disamenities such as pollution and environmental risk.
	Economic-Sensitive Buyer	Sensitive to school quality and neighborhood socio-economic composition.
	Zoning-Contrast Buyer	Responds to zoning and land-use patterns that shape local housing supply.
	Affordability-Preferring Buyer	Strongly prefers more affordable housing and dislikes high prices.
Layer 2	Integrated Location–Economic Buyer	Jointly evaluates location and socio-economic attributes in an integrated way.

Appendix

Table 10: Semantic roles of blocks in the deep BL architecture.

Layer / Block	Representative reference
Layer 1: Location-Sensitive Buyer	Gibbons & Machin (2005)
Layer 1: Risk-Sensitive Buyer	Chay & Greenstone (2005)
Layer 1: Economic-Sensitive Buyer	Black (1999)
Layer 1: Zoning-Contrast Buyer	Glaeser & Gyourko (2002)
Layer 1: Affordability-Preferring Buyer	McFadden (1977)
Layer 2: Integrated Location–Economic Buyer	Bayer et al. (2007)

Appendix

Table 11: Each block in the deep BL architecture is aligned with a classic preference mechanism documented in the economics literature.

Model	MNIST		Fashion-MNIST
Model	AUPR	FPR@95	AUPR	FPR@95
E-MLP (depth=1)	89.37 $\;\pm\;$ 1.52	35.57 $\;\pm\;$ 5.87	91.35 $\;\pm\;$ 1.25	28.24 $\;\pm\;$ 4.37
BL (depth=1)	91.57 $\;\pm\;$ 2.39	47.81 $\;\pm\;$ 11.29	91.79 $\;\pm\;$ 0.90	38.86 $\;\pm\;$ 2.57
E-MLP (depth=2)	91.52 $\;\pm\;$ 1.27	28.89 $\;\pm\;$ 2.85	86.19 $\;\pm\;$ 2.27	47.72 $\;\pm\;$ 4.79
BL (depth=2)	91.20 $\;\pm\;$ 1.22	52.71 $\;\pm\;$ 18.66	89.30 $\;\pm\;$ 2.47	42.65 $\;\pm\;$ 9.53
E-MLP (depth=3)	90.04 $\;\pm\;$ 1.89	31.92 $\;\pm\;$ 5.76	84.30 $\;\pm\;$ 1.50	54.49 $\;\pm\;$ 2.74

Appendix

Table 12: OOD AUPR and FPR@95 (%) on image and text datasets. BL and E-MLP are evaluated at depths 1–3 with matched parameter counts, both without skip connections. Top-two per column are blue and red .

Dataset	Model	# Parameters
MNIST & FashionMNIST	E-MLP (depth=1)	203,530
	BL (depth=1)	208,384
	E-MLP (depth=2)	235,146
	BL (depth=2)	219,264
	E-MLP (depth=3)	238,314
	BL (depth=3)	221,684

Appendix

Table 13: Number of trainable parameters for E-MLP and BL models across high-dimension datasets.

	Block 11			Block 12
Variable	$U_{11}$	$C_{11}$	$T_{11}$	$U_{12}$	$C_{12}$	$T_{12}$
$\lambda$	1.003	0.997	0.999	0.997	1.003	1.000
per capita crime rate (CRIM)	0.21	0.14	0.03	0.12	0.09	0.25
residential land proportion (ZN)	0.23	-0.04	-0.27	0.25	0.00	0.09
non-retail business acreage (INDUS)	-0.06	0.21	0.25	0.16	0.22	0.27
Charles River dummy (CHAS)	0.25	0.04	-0.24	-0.12	-0.20	-0.23

Appendix

Table 14: Estimated UMP block parameters learned by the BL model (layer = [2, 1]) on the Boston Housing dataset. For each block,

U

denotes the Utility component,

C

the Inequality-Constraint component, and

T

the Equality-Constraint component.

Variable	$U_{11}$	$U_{12}$	$U_{13}$	$U_{14}$	$U_{15}$
$\lambda$	1.000	0.998	1.003	1.002	1.000
per capita crime rate (CRIM)	0.21	0.12	0.17	-0.09	0.06
residential land proportion (ZN)	0.23	0.25	-0.07	-0.22	-0.16
non-retail business acreage (INDUS)	-0.06	0.16	0.16	0.23	-0.14
Charles River dummy (CHAS)	0.25	-0.12	-0.22	-0.05	-0.01
nitric oxide concentration (NOX)	-0.06	0.16	-0.14	0.24	0.16

Appendix

Table 15: Estimated UMP parameters for the Layer 1 blocks of the BL model (layer = [5, 3, 1]) trained on the Boston Housing dataset. Here,

U

denotes the Utility component,

C

the Inequality-Constraint component, and

T

the Equality-Constraint component.

	Block 21			Block 22			Block 23
Variable	$U_{21}$	$C_{21}$	$T_{21}$	$U_{22}$	$C_{22}$	$T_{22}$	$U_{23}$	$C_{23}$	$T_{23}$
$\lambda$	1.000	1.000	1.000	0.999	1.003	1.002	1.001	1.002	0.999
Block 11 output ( $b_{1,1}$ )	0.28	0.06	-0.20	-0.31	0.24	0.18	-0.29	-0.08	0.22
Block 12 output ( $b_{1,2}$ )	0.21	-0.11	-0.09	-0.44	0.12	-0.22	0.15	-0.22	0.20
Block 13 output ( $b_{1,3}$ )	-0.40	0.18	-0.44	-0.36	-0.01	-0.09	-0.13	-0.14	0.32
Block 14 output ( $b_{1,4}$ )	-0.27	-0.17	0.30	0.33	-0.34	-0.26	0.28	-0.42	-0.34

Appendix

Table 16: Layer 2 and Layer 3 UMP parameters (

U

C

T

) for Blocks in the BL model (layer = [5, 3, 1]).

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