Monotonicity Constraints

Stochastic optimization models sequential decision-making under uncertainty, typically assuming known input distributions. When these distributions are unknown, \citet{agarwal2026semibandit} give an online-to-offline reduction for monotone stochastic optimization under semi-bandit feedback, where only probed variables are observed. Its expected regret relative to the offline guarantee is $\widetilde O(n\sqrt T)$, where $n$ is the number of variables and $T$ the horizon. For optimal stopping and Pandora's box, \citet{liu2025improved} achieve $\widetilde O(\sqrt{nT})$ using problem-specific algorithms. Can a general online-to-offline reduction achieve this improved rate? We give such a reduction for strongly monotone stochastic optimization, answering the corresponding open question. Strong monotonicity is defined relative to the chosen offline algorithm. Our oracle-efficient reduction achieves $\widetilde O(\sqrt{nT})$ expected regret while preserving approximation guarantees for oracles satisfying the same condition. It combines Bernstein-type optimistic distribution estimates with a Hellinger stability bound for adaptive policy transcripts, aggregating squared marginal errors weighted by the executed policy's probing probabilities. The rate is optimal up to logarithmic factors for $T\ge n$, even under full feedback. Under monotonicity and uniformly bounded coordinate-wise variation, the same algorithm achieves $\widetilde O(n^{3/4}\sqrt T)$ expected approximation regret, improving the previous linear dependence on $n$. The exponent $3/4$ is sharp for its ordinary regret over instances and exact oracle choices. Our framework applies to prophet inequality, Pandora's box, best-of-$K$ selection, stochastic knapsack, stochastic orienteering, and stochastic submodular maximization.

Regulated credit scoring requires scores monotone non-decreasing in every exposure input. Deployed pipelines -- hand-crafted monotone aggregates feeding sign-constrained gradient boosting -- already meet this by composition; the open question is what learned temporal aggregation is worth inside one. We answer on five production-scale credit datasets at matched admissibility (one priced baseline convention excepted), with a monotone recurrent architecture whose per-input guarantee we extend, with proofs, to vector-valued inputs and to exogenously macro-conditioned decay gates, severities, thresholds, and peak memory. Two findings result. First, a strictness ladder: the value of learned monotone features rises with governance-frame strictness -- zero on unconstrained engineered panels, maximal in summaries-only frames -- replicated across two datasets and an official temporal-stability metric, though unconditioned features degrade on externally adjudicated later weeks. Second, a conditioning-delivery asymmetry under regime shift. On a train-on-boom, test-on-crisis mortgage design, two public macroeconomic series hurt as input columns, yet conditioning the recurrence on them delivers the paper's only learned-block crisis-cohort uplifts. The confirmed effect: +0.006 to +0.013 AUC on an internally pre-registered Freddie Mac replication, at all five held-out seeds. The discovery estimate: +0.015 to +0.021 on Fannie Mae (three of five seeds post hoc), worth 10-27 basis points of defaulted balance at an 80% approval cutoff, and grows with early-prepaid loans excluded. A state-level test identifies the mechanism: between-cohort calibration transfer. A pandemic-band episode bounds scope: under forbearance-distorted labels the gain generalizes at a quarter to a third of crisis size on Fannie Mae, on Freddie Mac only against the capacity control.

Risk regulation imposes directional constraints on scores; we adopt their strict per-input form -- the score monotone non-decreasing in every exposure input -- as a normative commitment. Deployed pipelines -- monotone hand-crafted aggregates feeding sign-constrained gradient boosting -- already satisfy it by composition, so constrained-versus-unconstrained comparisons price a guarantee the incumbent has for free. We instead hold admissibility fixed on both sides and measure what learning the aggregation is worth. Our instrument is a recurrent network whose state is classical risk statistics (an exponentially weighted moving average and a high-water mark with learned transforms), monotone by construction in every input and per MC-dropout sample. The central finding, by functional regression, is a subsumption boundary: a learned monotone channel reproduces the geometrically weighted separable family of hand-crafted statistics, one channel per member, to Spearman $ρ\ge 0.996$, approximates window statistics with measurable ceilings, and fails at consecutivity ($ρ= 0.924$) and time localization (0.628), both structural, and at the exposure floor (0.829), a learnability boundary. One explicit admissible basis repairs each failure (rank correlation 1.000). In or near the separable family, learned and engineered aggregation are substitutes, and the learned channel is never statistically behind at full sample size and specified capacity. Its advantages are incumbent-specific: a committed grid pays up to 0.019 AUC in decay regions it leaves uncovered (the learned channel stays within 0.004 of the strongest engineered consumer at every swept point); the highest-dimensional comparator degrades fastest with scarce data; and beyond the training support, grid-fed tree-ensemble scores go flat while a strictly increasing head keeps ranking. No single incumbent is dominated on all three axes.

We present an oracle-optimal method for finite-sum monotone inclusions under mean-square Lipschitz continuity. Our switching regularization method finds a point $y$ and a certificate $g\in G(y)$ with $(\mathbb{E}\|F(y)+g\|^2)^{1/2}\le\varepsilon$ using $\mathcal{O}(n+\sqrt{n}LR/\varepsilon)$ expected component evaluations and resolvent evaluations. It removes the additive $n\log n$ cost of restarting a variance-reduced solver at every regularization stage by switching to a centered stochastic proximal iteration at regularization strength $L/\sqrt{n}$. Carrying an operator estimate between the remaining stages limits their total cost to $\mathcal{O}(n)$. A matching $Ω(n+\sqrt{n}LR/\varepsilon)$ lower bound holds for randomized linear-span component-oracle algorithms with adaptive stopping and expected query budgets. Thus, for $0<\varepsilon\le LR/2$, our method attains the optimal worst-case expected component complexity in this oracle model, up to universal constants.

Background and Objective: Clinicians expect recurrence risk to climb with cancer severity. In a UK multicentre trial, an unconstrained XGBoost model learnt that higher tumour stage and carcinoma in situ predicted lower recurrence risk, and discrimination, calibration, and SHAP were all blind to it. We developed a counterfactual testing framework to detect this inversion and a monotonic-constraint framework to remove it without hurting performance. Methods: BOXIT enrolled 472 patients with protocol-mandated cystoscopy across 51 UK sites (2007-2012); 435 had at least two years' follow-up (153 recurrences, 35.2%). We developed a counterfactual direction test and a monotonic-constraint correction, with constraint directions drawn from the EORTC and EAU risk systems, and evaluated both against unconstrained XGBoost and logistic regression on 18 predictors (seven directed) over 50 cross-validation folds. The test worsened each patient on one directed feature at a time to check whether risk fell; SHAP direction and calibration were also assessed. Key Findings and Limitations: Tumour stage and carcinoma in situ were associated with lower recurrence, opposite to medical intuition; the unconstrained model reversed carcinoma in situ counterfactuals in 90.2% of cases and stage in 74.3%. Discrimination ($Δ$AUC 0.005, p=0.47), calibration, and SHAP magnitude were all blind to the inversion. Monotonic constraints eliminated every violation at no cost to discrimination (0.723 vs 0.718) and outperformed EORTC (p=8.9e-16). Limitations: single trial, internal-external validation only. Conclusions and Clinical Implications: A model that had learned this inversion passed every conventional check. A counterfactual direction test, run as a single refit with pre-specified monotonic constraints, catches this failure at no cost to performance and should be routine before clinical deployment.

Bakhtiari, Lattimore and Szepesvári (COLT 2025) proved that Thompson sampling (TS) has Bayesian regret $\tilde O(d^{5/2}\sqrt n)$ for bandit convex optimisation with convex monotone ridge losses $f(x)=\ell(\ip{x}θ)$, and asked whether monotonicity of the link is necessary. We give a qualitative negative answer. For every prior on $[0,1]$-valued, $1$-Lipschitz convex ridge losses with an arbitrary convex, possibly non-monotone, link, and for any fixed measurable selection of minimisers, exact-posterior TS has Bayesian regret $O\big((d+1)^4\sqrt{dn}\,\log(e+nd\max\{1,\diam K\})\big)=\tilde O(d^{9/2}\sqrt n)$. The monotone proof relies on a single-removal John-ellipsoid dichotomy; we show by an explicit twelve-point configuration that this dichotomy fails for non-monotone links, and replace it by an $O(d^2)$ cardinality bound for ``uninformative'' configurations. The bound uses a Boolean rounding argument: a $0$-$1$ matrix within $1/(4r)$ in max-norm of a rank-$r$ matrix has rank at most $2r-1$. We construct $d(d+1)$ uninformative losses, showing that the cardinality bound is tight up to constants in the large-diameter-to-gap regime, and give a self-contained information-ratio-to-regret transfer that is uniform over fixed measurable selections. Whether the $d^{5/2}$ dependence of the monotone case can be retained remains open.

Modeling the Martian nightside thermosphere remains challenging due to sparse in situ sampling and strong coupling among transport, magnetic, and seasonal processes. Purely data-driven models can produce non-physical artifacts, such as density inversions, in poorly sampled altitude regimes. We present a multi-task physics-informed neural network that simultaneously predicts the base-10 logarithmic densities of four neutral species (O, CO$_2$, N$_2$, and Ar) using more than a decade of MAVEN/NGIMS observations (MY 32-38, 2014-2025). A shared backbone learns a common representation of the nightside thermospheric state and branches into species-specific output heads. A weak monotonicity prior is incorporated via automatic differentiation by penalizing positive vertical gradients in logarithmic density. Experiments using an orbit-disjoint train/validation/test split show that physics-informed regularization substantially reduces non-physical inversions while preserving predictive skill and slightly improving it in the best-performing configuration, as measured by RMSE, MAE, and $R^2$. The resulting model provides a computationally efficient surrogate for nightside thermospheric reconstruction with improved vertical consistency.

Shape-constrained and physics-informed learning reports an accuracy cost of enforcing a prior and treats it as a property of the prior. We show it is mostly a property of the free features and the validation split. Let P be the excess risk of restricting a hypothesis class to functions with a shape constraint on features S, and D the excess risk of the ablated model that ignores S. Because a function constant in x_j is both non-decreasing and non-increasing in x_j, the ablated class is contained in the constrained class, so 0 <= P <= D for every risk functional, with no convexity, smoothness, or realizability assumption. Empirically the bound is a sign test: a constrained model must never be beaten by its own ablation. We instantiate it on an ordinal wildfire-severity task (N = 26,681, K = 3) with hard monotone constraints on four meteorological drivers, coordinates left free, and a validation ladder from i.i.d. resampling to 2-degree spatial blocking. Coordinates act as a shield: alone they recover 92.9% of the full model's macro-F1 under spatial blocking, collapsing D from 0.1288 to 0.0427; the same prior costs 0.0473 shielded and 0.3470 unshielded, a ratio of 7.3 with identical physics. Because D is protocol-dependent it does not transfer: coarsening blocks from 1 to 10 degrees drives D from 0.0942 to 0.0050, leaving two configurations unidentifiable a priori. Inversions of the certified nesting bound the pipeline's additive resolution: over 318 comparisons they give a self-calibrating floor of 0.0220 macro-F1, below which no reported price is interpretable, including four cells in our own headline grid. Cost and compliance are independent: the unconstrained model violates the prior at rate 0.48-0.49 while enforcing it costs 0.0473. We give a two-fit screen that rejects unidentifiable experiments before a constrained model is trained.

Tabular foundation models achieve strong accuracy on choice prediction tasks, but their predictions often violate the economic logic those tasks require: raising a price can increase predicted demand, implied willingness-to-pay estimates are frequently negative or implausible, and unavailable alternatives receive nonzero probability. We propose a two-stage adapter that takes a foundation model's predicted choice probabilities as a precomputed feature and embeds them inside a multinomial logit's utility. In Stage 1, we fit the multinomial logit's structural coefficients by maximum likelihood with sign constraints; in Stage 2, we freeze those coefficients and fit a small neural correction operating on the foundation model's predictions. We prove that this composition exactly preserves the multinomial logit's marginal rate of substitution, so analytically computable value-of-time becomes a mathematical guarantee rather than an empirical accident. Across three datasets and two foundation models, the adapter gains 6.4 percentage points (pp) of test accuracy on average over the multinomial logit and up to 12.8 pp, maintains 100% cost monotonicity, and produces values of time within the published transportation-economics range on the transportation datasets. Performance degrades gracefully under foundation-model context restriction, retaining at least 6 pp of accuracy gain even at 10% of the original foundation-model context.

Monotonicity has been a long-running architectural inductive bias for neural networks, motivated by tabular, scientific, and economic settings where outputs are known to respond monotonically to certain inputs. Existing approaches are MLP- or flow-based and lack per-edge functional transparency; the only Kolmogorov--Arnold Network (KAN) variant with monotonicity, MonoKAN, enforces the constraint only on a restricted parameter subset and requires a projection-style training procedure. We close this gap with MKAN, a KAN with hard monotonicity guaranteed for all parameter values via exponential reparameterization of B-spline coefficients, positive edge weights, and a monotone base activation. Training reduces to standard unconstrained gradient descent. Our headline theoretical contribution is a representation-cost theorem: any $C^K, K >0$ feature extractor inducing a ball-shaped semantic-neighborhood partition admits a monotone realization of the equivalent neighborhood structure at $N' = N^* + k \le 2N^*$, where $k$ is the number of non-monotone coordinates of the original. The bound is architecture-agnostic and gives a principled sizing rule for monotone encoders. Empirically, MKAN is competitive with state-of-the-art monotone NNs on the SMM/ICML-2024 benchmark while being the only method that combines hard unconstrained monotonicity with KAN's per-edge functional transparency; the $2N^*$ prediction is validated in a self-supervised feature-size sweep on four real datasets, and on a controlled monotone-generative dataset MKAN recovers ground-truth factors with substantially higher Spearman alignment than KAN, MLP, and linear baselines.

Enforcing functional inequality constraints such as monotonicity and convexity in neural networks is a fundamental challenge in many industrial and scientific applications. Classical one-sided penalty methods, along with primal-dual methods gated by complementary slackness, provide constraint gradients only at violated locations, resulting in fragile satisfaction. Architectures that guarantee feasibility by construction, on the other hand, remain largely limited to elementary cases and impose additional inductive biases. We introduce neural slack variables, a deep learning native primal-side approach that converts constraint enforcement into a regression problem by coupling the primary network with a jointly learned auxiliary network. The auxiliary network serves as a valid target for the primary network's constraint quantities, inducing feasibility and regularity. Neural slack variables achieve zero measured violations on dense-grid monotonicity and convexity test cases, where penalty and primal-dual baselines leave residual violations, and enable arbitrage-free learning of volatility surfaces, an open industrial challenge in quantitative finance.

When a learner faces a new task with few samples, it must leverage any available side information. In practice, this often comes in the form of model evaluations on related tasks in public benchmarks. A key question then is how to model task relatedness such that it is both realistic and the benchmark evaluations lead to provable gains. Empirically, we observe that weak monotonicity is often approximately satisfied: if a model dominates another on many benchmarks, it also tends to outperform on the new task. We explore the statistical complexity of learning under (approximate) weak monotonicity, leveraging it within two learning paradigms: transfer learning and model selection aggregation. We show that not only can we prune the model class based on monotonicity, but we can also further adapt to the geometry of the available trade-offs by hedging on the frontier.

Tabular foundation models achieve strong accuracy on choice prediction tasks, but their predictions often violate the economic logic those tasks require: raising a price sometimes increases predicted demand, and implied willingness-to-pay estimates are frequently negative or implausible. We propose a two-stage adapter that embeds foundation model predictions within a utility-maximization framework. In the first stage, we estimate a standard choice model whose parameters are constrained to obey economic theory. In the second stage, we freeze those parameters and train a correction term that incorporates the foundation model's predictions as additional information. The result is a model that inherits the foundation model's accuracy gains while guaranteeing monotonic price-demand relationships under policy perturbation and producing analytically computable trade-off measures. On two transportation datasets, the adapter recovers up to 13 percentage points of accuracy over a standard logit model while maintaining perfect economic consistency, something neither the raw foundation models nor conventional distillation achieve.

Accident anticipation aims to predict impending collisions from dashcam videos and trigger early alerts. Existing methods rely on binary supervision with manually annotated "anomaly onset" frames, which are subjective and inconsistent, leading to inaccurate risk estimation. In contrast, we propose RiskProp, a novel collision-anchored self-supervised risk propagation paradigm for early accident anticipation, which removes the need for anomaly onset annotations and leverages only the reliably annotated collision frame. RiskProp models temporal risk evolution through two observation-driven losses: first, since future frames contain more definitive evidence of an impending accident, we introduce a future-frame regularization loss that uses the model's next-frame prediction as a soft target to supervise the current frame, enabling backward propagation of risk signals; second, inspired by the empirical trend of rising risk before accidents, we design an adaptive monotonic constraint to encourage a non-decreasing progression over time. Experiments on CAP and Nexar demonstrate that RiskProp achieves state-of-the-art performance and produces smoother, more discriminative risk curves, improving both early anticipation and interpretability.

Isotonic regression (IR) is shape-constrained regression to maintain a univariate fitting curve non-decreasing, which has numerous applications including single-index models and probability calibration. When it comes to multi-output regression, the classical IR is no longer applicable because the monotonicity is not readily extendable. We consider a novel multi-output regression problem where a regression function is cyclically monotone. Roughly speaking, a cyclically monotone function is the gradient of some convex potential. Whereas enforcing cyclic monotonicity is apparently challenging, we leverage the fact that Kantorovich's optimal transport (OT) always yields a cyclically monotone coupling as an optimal solution. This perspective naturally allows us to interpret a regression function and the convex potential as a link function in generalized linear models and Brenier's potential in OT, respectively, and hence we call this IR extension Brenier isotonic regression. We demonstrate experiments with probability calibration and generalized linear models. In particular, IR outperforms many famous baselines in probability calibration robustly.

Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in settings with discrete treatment and discrete ordinal outcome. We propose new families of monotonicity assumptions and formulate the bounding problem as a linear programming problem. We further introduce a new monotonicity assumption specifically to achieve identification. Finally, we present numerical experiments to validate our methods and demonstrate their application using real-world datasets.