Cross-Scenario Expectation Constraints

Updated 23 October 2025

Cross-scenario expectation constraints are defined as average-based limitations that permit rare individual breaches while ensuring overall performance across multiple scenarios.
They utilize methods such as weak dynamic programming and convex duality to reformulate nonconvex or path-dependent control problems into tractable and robust optimization frameworks.
These constraints are applied in diverse areas including robust optimization, financial risk management, and large-scale recommender systems to ensure consistent performance and scalable solutions.

Cross-scenario expectation constraints are constraints in stochastic optimization, control, and learning that require the controlled process, decision variables, or model predictions to satisfy average (expectation-based) limitations across a range of ^{^{^{^{1^{^{^{^,}}}}}}} environmental states, or random outcomes, rather than pointwise or worst-case requirements. This class of constraints arises in stochastic optimal control, robust optimization, machine learning with weak supervision, and large-scale recommender systems, and underpins tractable and flexible approaches for enforcing consistent performance or safety across diverse operating conditions.

1. Formal Definition and Theoretical Foundations

Expectation constraints require that a function of the controlled state or decision variable, evaluated under the relevant probability measure (i.e., scenario), satisfies an inequality or equality of the form: $\mathbb{E}_{P}[g(X(T))] \leq m$ or more generally,

$\mathbb{E}_{P}[g_i(\cdot)] \leq y_i \quad \text{and} \quad \mathbb{E}_{P}[h_j(\cdot)] = z_j$

for all relevant constraints $i, j$ , where $P$ is the probability law conditioned by the control or scenario, and $g, h$ are (possibly path-dependent and functional) observables (Bouchard et al., 2011). Unlike almost-sure constraints (hard constraints that must hold in all realizations), expectation constraints allow for rare constraint violations as long as the average under the scenario law is controlled.

Key theoretical insights:

Expectation constraints are less conservative than robust (worst-case) or almost-sure constraints and often admit convexification of otherwise nonconvex control or optimization problems.
They enable reformulations of dynamic trading or state constraints into expectation-based forms—e.g., state process $X$ remaining in a set $O$ can be encoded via the auxiliary process $Y(s) = 1 \land \inf_{r \in [t,s]} d(X(r))$ and setting $g(x,y) = \mathbf{1}_{(-\infty,0]}(y)$ (Bouchard et al., 2011).
In multi-scenario or multi-stage problems, expectation constraints unify requirements across time and scenarios, allowing for tractable dynamic programming and Lagrangian duality (Pfeiffer et al., 2020, Bayraktar et al., 2023).

2. Weak Dynamic Programming and Viscosity Solutions

Traditional dynamic programming methods face major technical hurdles in the presence of expectation constraints, primarily due to measurability issues and lack of regularity in the value function. To address this, a weak dynamic programming principle is employed, where the value function is characterized via test functions and relaxation:

For any $(t, x, m)$ , the principle asserts: $V(t,x,m + \varepsilon) \geq \mathbb{E}\big[\varphi(T, X(T), M(T))\big]$ where $\varphi$ is a smooth test function and $M$ is an auxiliary martingale tracking the evolution of the constraint (Bouchard et al., 2011).
Relaxation by $\varepsilon>0$ enables local "patching" of admissible controls and circumvents the need for universally measurable selectors, allowing for robustness under perturbations.
The approach leads naturally to the Hamilton–Jacobi–Bellman (HJB) equation in the viscosity solution sense, with state space extended to include the constraint level $m$ : $-\partial_t V(t, x, m) + H(x, D_x V, D^2_x V) = 0$ interpreted with respect to the upper and lower semicontinuous envelopes of $V$ due to its potential discontinuity (Bouchard et al., 2011).

A comparison theorem for the HJB equation ensures uniqueness of the viscosity solution, validating that the relaxed weak principle is equivalent (in the $\varepsilon \to 0$ limit) to strict enforcement of the expectation constraints.

3. Scenario Approach in Convex Optimization

For convex programs with probabilistic or chance constraints,

$\mathbb{P}_{\delta}[f(x,\delta) \leq 0] \geq 1 - \epsilon$

where $\delta$ indexes scenarios, expectation constraints can be replaced by deterministic constraints on samples ("scenarios") of the uncertainty. The scenario approach involves:

Replacing the chance constraint by a finite set of $K$ sampled constraints $f(x, \delta^{(k)}) \leq 0$ , $k=1,...,K$ .
Solution $x^*$ is feasible for the original problem with controllable violation probability, explicitly bounded by the beta function in terms of the so-called support rank $\alpha$ of the constraint: $\mathbb{P}(V_i > \epsilon_i) \leq \mathrm{B}(\epsilon_i; \alpha_i-1, K_i)$ where the support rank measures the effective dimension impacted by each constraint (Schildbach et al., 2012).
When multiple chance/expectation constraints are present, each with limited support rank, the required number of scenarios per constraint is reduced, yielding significant computational savings while maintaining theoretical guarantees.

Sampling-and-discarding schemes further trade off between looser constraints and feasibility, enabling less conservative yet rigorous solutions. This scenario approach avoids the over-conservatism of robust optimization and the intractability of full stochastic programming.

4. Expectation Constraints in Stochastic Control and Stopping

In advanced stochastic control and stopping problems (possibly non-Markovian), expectation constraints are imposed either at the terminal time or at all intermediate times: $P(t, w, m) = \left\{ P \in \mathcal{P}(t, w) : \mathbb{E}_P[g(s)] \leq m\ \forall s \in [t,T]\right\}$ (Chow et al., 2018, Bayraktar et al., 2023).

Main developments:

Such constraints accommodate path-dependent restrictions (e.g., drawdown, floor, quantile hedging) by suitable choice of $g$ (possibly indicator-type functionals of the path).
Measurable selection techniques are used to construct universally measurable families of controls or probability kernels, as admissibility for one initial condition may not extend to nearby ones.
In weak formulations, the enlarged canonical space includes the Brownian motion, controlled state process, diffusion controls, and stopping times, with admissible probability measures characterized by a countable family of martingale conditions (Bayraktar et al., 2020, Bayraktar et al., 2023).
The dynamic programming principle persists via the introduction of auxiliary supermartingales and by interpreting the conditional expected cost as an additional state variable during recursion.

These strategies ensure that cross-scenario expectation constraints are handled consistently in path-dependent and non-Markovian settings, critical for financial applications where time-consistent risk management is essential.

5. Duality and Optimization Algorithms for Expectation Constraints

Convex duality frameworks extend naturally to expectation-constrained problems:

Lagrange multipliers $\lambda$ are associated with the equality or inequality expectation constraints, leading to saddle-point formulations of the form

$D(0) = \sup_{\lambda}\inf_{\gamma} \mathbb{E}[ \Phi(X^\gamma) + \lambda \cdot \Psi(X^\gamma) ]$

with strong duality shown to hold under appropriate constraint qualifications (Pfeiffer et al., 2020).

For stochastic optimization problems with multiple expectation constraints, primal algorithms such as Cooperative Stochastic Approximation (CSA) and its multi-constraint extensions operate without projections onto complicated feasible sets or inner dual iterations. Instead, they adaptively update decision variables based on unbiased estimators of constraint satisfaction, conditioning objective and constraint descent on feasibility (Lan et al., 2016, Basu et al., 2019).
Augmented Lagrangian and linearized proximal algorithms are used to handle convex stochastic programs with expectation constraints, achieving $\mathcal{O}(1/\sqrt{K})$ convergence for both objective and constraint violations under standard conditions (Zhang et al., 2021).
In the context of conservative stochastic optimization, constraints are "tightened" by a positive margin $\upsilon = O(T^{-1/2})$ , ensuring zero average constraint violation with optimal convergence rate for the objective (Akhtar et al., 2020).

Collectively, these algorithms enable scalable, efficient, and theoretically sound optimization subject to expectation constraints even when only noisy, sample-based evaluations are possible.

6. Expectation Constraints in Bayesian Inference and Learning

Expectation constraints are not limited to control and optimization—they also serve as a foundation for model updates in the presence of side information or structural constraints:

In Bayesian inference, updating a prior density by imposing expectation constraints (rather than data samples) leads directly to exponential family posteriors: $P(u|I) = \frac{1}{Z(\lambda)} \exp[-\lambda f(u)] P(u|I_0)$ with $\lambda$ determined by matching the required posterior expectation. This result is derived solely from consistency and independence assumptions and is equivalent to the outcome of the maximum entropy principle, but conceptually independent of entropy maximization (Davis, 2015).
In semi-supervised machine learning and weakly supervised structured prediction, auxiliary expectation constraints guide learning. Techniques such as alternating projections (AP) optimize a composite objective combining supervised log-likelihood and constraints on expectations with respect to auxiliary distributions, iteratively projecting back and forth between distributions that are close to the model and those that satisfy the constraints (Bellare et al., 2012). This enables the inclusion of domain knowledge and expressive structural properties that would otherwise be intractable.

Expectation constraints thus function as a unifying principle across inference, learning, and control, providing a flexible mechanism for the integration of prior information or desired outcome structure in a broad range of mathematical models.

7. Relevance in Modern Large-Scale and Multi-Scenario Systems

Recent advances apply cross-scenario expectation constraint principles at scale for recommendation and ad ranking systems:

Cross-scenario recommender systems (e.g., RED-Rec) aggregate and synthesize behavioral signals from heterogeneous scenarios (search, feed, ads) by unifying expectation-based user interest representations. This is enabled by LLM-powered two-tower frameworks and scenario-aware dense mixing policies that fuse diverse modalities and scenarios, ensuring balanced and holistic user modeling at billion scale (Xu et al., 16 Oct 2025).
In multi-scenario ad ranking, hybrid contrastive losses (generalized and scenario-specific) enforce cross-scenario consistency and distinctiveness via expectation constraints on the latent space representations of samples, improving generalization and representational power across scenarios (Mu et al., 2023).

Such approaches ensure that systems optimize not for isolated scenario-level objectives, but for aggregate constraints and metrics spanning platform-wide user behaviors, which is essential for coherent personalization and robust performance in complex environments.

Cross-scenario expectation constraints represent a robust theoretical and algorithmic foundation for enforcing averaged limitations or risk controls across random, adversarial, or complex operating environments. Through the development of weak dynamic programming principles, duality and convex analysis, scenario-based sampling, efficient stochastic optimization, and hierarchical representation architectures, the field has seen sustained progress in both mathematical theory and practical system design. This paradigm enables robust, interpretable, and scalable solutions across finance, engineering, learning, and user modeling domains.