Learning Samples Importance: Parameterizing Dual Variables in Everywhere Learning

Ignacio Boero; Jonathan Nixon; Alejandro Ribeiro

cs.LGSep 28, 2026

Learning Samples Importance: Parameterizing Dual Variables in Everywhere Learning

Authors: Ignacio Boero, Jonathan Nixon, Alejandro Ribeiro

Organizations: University of Pennsylvania

Abstract

Everywhere learning provides a principled framework for training AI models under constraints that must hold throughout the data distribution. In the dual domain, these pointwise constraints give rise to functional dual variables. In this work, we propose to learn these dual variables, motivated by the fact that their values encode useful information about the underlying constrained problem. By representing the dual variable as a parametric function of each sample, we enable the learned multiplier to be evaluated on new, unseen samples. This contrasts with standard empirical dual formulations, which assign an independent multiplier to each training sample. We characterize the error in the recovered primal solution induced by restricting the dual variable to a parametric function class and show that it is controlled by how well this class approximates the optimal statistical multiplier. Moreover, we show that the learned parametric multiplier retains the sensitivity interpretation of the optimal statistical multiplier, yielding approximate sensitivity guarantees that extend beyond the samples used for training. We empirically validate our theory across a variety of everywhere learning tasks, showing that the resulting constrained problems can be solved efficiently and that the learned dual variables provide meaningful representations of sample-level sensitivity.

Figures & tables

Figure 1: Primal recovery on TOFU. Left: distribution of forgetting-constraint values, where negative values indicate constraint satisfaction. Center: KL divergence on the retain set, measuring deviation from the original model. Right: aggregate unlearning performance across memorization, privacy, and utility metrics. The parameterized dual closely matches the pointwise in constraint satisfaciton, objective value and downstream performance.

	Optimality Gap (%)		Mean Violation (%)		Max Violation (%)
	Mean	Std	Mean	Std	Mean	Std	P95	Max
Param	$-3.63$	1.15	2.70	1.26	6.26	2.61	8.04	15.80
Dual-P	0.18	1.41	0.49	0.62	1.83	2.15	5.65	19.94
Dual-H	$-0.08$	0.55	1.05	0.30	5.61	0.27	6.03	6.65
Dual-S	2.18	1.25	1.34	0.30	8.22	1.98	11.61	13.47

Table 1: Primal recovery on the IEEE 30-bus benchmark. Param replaces the independent per-sample multipliers of Dual-P with a learned dual network conditioned on primal embeddings. Metrics are computed per test operating point and aggregated across the test set. The remaining baselines are reproduced from Owerko et al. (2025) , Fig. 10.

	Active-set base rate	Dual recovery on unseen samples
Dataset	Active-set base rate	Tight-set AUC $\uparrow$	NDCG $\uparrow$	Spearman $\rho$ $\uparrow$
MNIST	$0.47\%$	$0.923\pm 0.013$	$0.836\pm 0.036$	$0.277\pm 0.026$
OPF	$5.13\%$	$0.877\pm 0.063$	$0.902\pm 0.008$	$0.324\pm 0.035$
TOFU	$80.00\%$	$0.713\pm 0.078$	$0.386\pm 0.040$	$-0.048\pm 0.051$
FLORA-Bench	$54.08\%$	${0.935\pm 0.001}$	${0.914\pm 0.001}$	$0.211\pm 0.008$

Table 2: Generalization of learned dual multipliers to unseen samples. Predicted multipliers are evaluated against an independently computed pointwise dual reference. Tight-set AUC measures identification of active constraints, while NDCG and Spearman’s

\rho

measure recovery of the multiplier ranking. Results are reported as mean

\pm

standard deviation.

Figure 2: Sensitivity estimation on MNIST. Left: objective increase after imposing held-out constraints versus their mean predicted multiplier. Larger $\lambda_{\phi}$ values induce larger increases than random draws. Right: representative low- and high-sensitivity samples with their classwise multipliers.

Figure 3: Effect of dual parameterization capacity on FLORA-Bench. Left: ranking recovery on the training and test sets as the number of dual-network parameters increases. Right: decomposition into approximation and generalization errors. Increasing capacity substantially reduces approximation error and improves sensitivity recovery, while introducing a growing generalization error, illustrating the tradeoff between dual expressivity and statistical complexity.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Primal recovery on MNIST. Comparison of the parameterized dual formulation with independent pointwise multipliers and the corresponding unconstrained or partially constrained baselines. The parameterized formulation closely matches the pointwise solution in constraint satisfaction and primal objective while preserving classification performance.

Figure 5: Primal recovery on FLORA-Bench. Comparison of ERM, feasibility-only training, independent pointwise multipliers, and parameterized dual multipliers. Pointwise and parameterized duals achieve near-zero constraint violations and comparable objective values while maintaining essentially identical test accuracy.

	$\mathrm{Re}(\mathbf{s}^{g})$ (%)		$\mathrm{Im}(\mathbf{s}^{g})$ (%)		$\|V\|$ (%)		$\|\mathbf{f}_{f}\|$ (%)		$\|\mathbf{f}_{t}\|$ (%)
	Mean	Max	Mean	Max	Mean	Max	Mean	Max	Mean	Max
Param	0.32	0.53	5.68	5.68	0.12	0.12	4.90	4.90	2.49	2.49
Dual-P	0.20	0.25	0.69	0.70	0.74	0.85	0.68	0.68	0.15	0.15
Dual-H	0.35	0.72	0.90	0.92	3.76	5.61	0.25	0.25	0.00	0.00
Dual-S	0.55	0.86	0.14	0.14	6.00	8.21	0.00	0.00	0.00	0.00

Appendix

Figure 6: Constraint violations on IEEE 30-bus, disaggregated by constraint type. For each test operating point, the mean and maximum violation are computed separately for active generation

\mathrm{Re}(\mathbf{s}^{g})

, reactive generation

\mathrm{Im}(\mathbf{s}^{g})

, voltage magnitude

|V|

, and branch-flow magnitudes

|\mathbf{f}_{f}|

and

|\mathbf{f}_{t}|

, and are then averaged across the test set. Violations are expressed as percentages of the corresponding feasible range. Voltage-angle constraints are omitted because no violations are observed. Baseline values are reproduced from Owerko et al. (2025) , Fig. 11.

Figure 7: Effect of dual parameterization capacity on MNIST. Ranking recovery and error decomposition as the number of parameters in the dual network increases. Increasing capacity primarily reduces approximation error and improves recovery of the pointwise multiplier ranking, while the generalization error remains comparatively stable across model sizes.

Figure 8: Sensitivity estimation on FLORA-Bench. Change in the optimized objective after imposing constraints on unseen samples, grouped according to the magnitude of their predicted multipliers. Sets with larger predicted multiplier values induce larger objective changes, providing an operational validation of the sensitivity information encoded by the learned dual.

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Everywhere Learning: Artificial Intelligence with Pointwise Constraints

Ignacio Boero, Ignacio Hounie, Luiz Chamon, Alejandro Ribeiro

Department of Electrical and Systems Engineering, University of Pennsylvania · École polytechnique, Institut Polytechnique de Paris

Constrained Optimization Statistical Learning Theory

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Everywhere learning is a new paradigm whereby Artificial Intelligence (AI) systems are trained to satisfy loss constraints with probability one over the data distribution. This is in contrast to the standard paradigm of training AI systems to minimize average losses. We develop an approximate duality theory to substantiate a generalization analysis that establishes the proximity between solutions of empirical and statistical everywhere learning problems. Our results show that dual variables reweigh the data distribution towards points in which loss constraints are more difficult to satisfy and that generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration of mass on points where constraints are more difficult to satisfy. We further show that we can control generalization with a sparse L1 penalty on constraint relaxations. We illustrate the merits of everywhere learning with an experiment in agentic classification for language model tasks.

Abstract

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