Recall-Constrained Optimization

• Recall-Constrained Optimization is a framework that maximizes a primary utility while imposing lower bounds on recall metrics to ensure adequate true positive rates.
• It employs surrogate loss functions, implicit function techniques, and proxy-Lagrangian approaches to handle non-convex and non-decomposable recall constraints.
• This approach is widely applied in machine learning, resource allocation, and quantum optimization to balance precision and recall in complex systems.

Recall-constrained optimization is a family of constrained optimization methodologies in which the goal is to optimize a primary objective (often precision, reward, or global utility) subject to explicit lower bounds or targets on recall-type rates—such as true positive rates, coverage levels, or cumulative identification fractions. This paradigm is prevalent in machine learning for model selection, ranking, retrieval, continual learning, resource allocation, portfolio selection, and quantum memory recall. The essential challenge stems from the non-decomposability, potential non-convexity, and threshold-dependent nature of recall metrics, requiring surrogate modeling, specialized constraint-handling schemes, and careful algorithmic design.

1. Mathematical Formulation of Recall Constraints

Recall-constrained optimization problems generally formalize the constraint on recall (or a recall-like rate) as a thresholded expectation or count, e.g.

$$\text{maximize}_{\theta} \quad \mathcal{U}(\theta) \quad \text{subject to} \quad \text{Recall}(\theta, t) \geq \beta$$

where $\mathcal{U}$ is the primary utility (precision, F-score, reward, or other), $\theta$ are model parameters, and $\text{Recall}(\theta, t)$ is recall at threshold $t$, often defined as $\frac{\text{tp}(\theta, t)}{|Y^+|}$ with $\text{tp}$ the true positive count among positives $Y^+$. The constraint can be absolute, per-subgroup, dynamic, or smoothed. The threshold $t$ itself may be a hyperparameter or an implicit function of the model parameters.

Specialized rate constraints are adopted for fairness (e.g., equal opportunity demands group recall rates $p^+(D_k[y=1]) \geq \gamma p^+(D[y=1])$), continual learning (requiring recall of past information post-update), portfolio selection (limiting risk or ensuring recall of profitable instances), bandit formulations (identifying feasible recall of arms), and combinatorial settings (ensuring coverage or solution recall constraints).

2. Computational Approaches and Surrogate Losses

Classical constrained optimization frameworks—Lagrangian duality, projected gradient, or iterative feasibility correction—face challenges due to non-convex, non-differentiable recall constraints (indicator functions, combinatorial dependencies). To mitigate these, surrogate convex relaxations—especially hinge loss for classification,

$\ell_h(f_b, x, y) = \max(0, 1 - y(f(x) - b))$

are used. Lower and upper bounds on recall-related counts are derived, such as: $$\text{tp}(f_b) \geq \sum_{i \in Y^+}(1 - \ell_h(f_b, x_i, y_i)), \quad \text{fp}(f_b) \leq \sum_{i \in Y^-}\ell_h(f_b, x_i, y_i)$$ Rewriting the original recall constraint, e.g.

$$\text{tp}(f) \geq \alpha [\text{tp}(f) + \text{fp}(f)]$$

enables surrogate-constrained saddle-point optimization via stochastic gradient descent. This is effective for large-scale systems, as demonstrated for AUCPR, F-measure, and recall-precision trade-offs (Eban et al., 2016).

Alternative frameworks "bake in" recall rate constraints using threshold quantile estimation, where the operational threshold $q$ is adaptively set so that a desired proportion of positives is predicted (Mackey et al., 2018). This transforms the problem into an unconstrained smooth minimization over a surrogate quantile-based loss, compatible with SGD.

3. Implicit Function and Non-Decomposable Objectives

Recent work directly models the recall constraint as an implicit function of model parameters, such as setting $$g(\theta, t) = \text{Recall}(\theta, t) - \beta = 0$$ and then, by the Implicit Function Theorem, expressing the threshold as $t = h(\theta)$ (Kumar et al., 2021). This enables unconstrained optimization of $f(\theta, h(\theta))$ using chain-rule differentiation: $$\nabla_\theta f(\theta, h(\theta)) = \nabla_\theta f(\theta, t) + \frac{\partial f}{\partial t} \cdot \nabla_\theta h(\theta)$$ where $\nabla_\theta h(\theta)$ is obtained by differentiating the constraint.

This technique avoids dual variables, maintains strict constraint satisfaction at each iteration, and empirically yields superior results for high-recall scenarios compared to Lagrangian relaxations.

4. Proxy-Lagrangian, Rate Constraints, and Stochastic Solution Mixtures

For non-differentiable constraints, the proxy-Lagrangian approach decouples optimization over smoothed surrogates (model update) and original hard constraint evaluation (dual/Lagrange multipliers) (Cotter et al., 2018). Specifically,

• The "w-player" minimizes a smooth proxy-Lagrangian using surrogate constraints.
• The "λ-player" maximizes over the original constraint functions.
• The algorithm returns a stochastic mixture over iterates, which can always be reduced to a sparse support of at most $m+1$ deterministic solutions (where $m$ is the number of constraints).

This ensures approximate optimality and feasibility, theoretically and empirically achieving desired recall and fairness guarantees in practical systems.

5. Applications in Bandits, Resource Allocation, and Model Ensembling

In stochastic multi-armed bandits, recall-constrained optimization corresponds to maximizing a primary attribute (e.g., recovery rate) while satisfying constraints on secondary attributes (risk, cost) (Kagrecha et al., 2020). The Con-LCB algorithm maintains confidence bounds for feasibility on recall-type metrics, and selects arms so that suboptimal or infeasible arms are rarely played (logarithmic in horizon), while correctly identifying feasible recall levels.

For combinatorial RL, constraints—including recall on solution sets or operational coverage—are encoded either by masking (for maskable constraints) or by penalty inclusion in the CMDP reward function (for post hoc feasible constraints). Lagrangian relaxation adapts the policy to optimize reward while penalizing recall violations (Solozabal et al., 2020).

In ensembling for constrained optimization, multicalibration ensures models are unbiased on slices of data relevant to recall constraints. White-box and black-box ensemble methods combine predictions or policies to align self-assessed utility (e.g., recall rates) closely with the true performance, and are provably efficient and convergent (Globus-Harris et al., 27 May 2024).

6. Rate-Constrained Online and Long-Term Optimization

Online learning with recall-type constraints includes bounded-recall algorithms, where only recent history is accessible. Asymmetric weighting of recent rewards enables regret that scales as $O(1/\sqrt{M})$ with window size, which is tight (Schneider et al., 2022). In meta-frameworks with long-term constraints, e.g., cumulative recall over many rounds, primal-dual approaches guarantee sublinear regret and sublinear violation, with guarantees calibrated by a feasibility parameter $\rho$ that quantifies the strictness of the constraint (Castiglioni et al., 2022).

7. Extensions: Quantum Recall, Generative Models, and Hybrid Systems

In adiabatic quantum optimization, the recall task corresponds to energy minimization in a Hopfield network represented by an Ising Hamiltonian. The recall-constrained problem is determined by learning rule selection, input noise, and energy landscape separation (Seddiqi et al., 2014). Heuristic and projection rules yield well-separated minima, improving recall probability at the cost of annealing times.

Generative models such as GANs and normalizing flows require explicit balancing between precision (sample quality) and recall (sample diversity). A family of PR-divergences is introduced, each corresponding to a particular precision-recall trade-off (Verine et al., 2023). Minimizing a specific PR-divergence yields a unique operating point on the precision-recall curve, and any f-divergence is a weighted sum of PR-divergences. The training approach improves either precision or recall beyond previous heuristics.

Hybrid systems for recall-constrained extraction combine learning-based (high d-recall) and pattern-based (high e-recall) methods, merging their outputs and optimizing a combined metric $$R_\text{total} = \alpha \, d\text{-recall} + (1-\alpha)\,e\text{-recall}$$ (Goldberg, 2023).

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Summary Table: Recall Constraint Methods Across Key Domains

Technique	Constraint Handling	Scalability/Guarantee
Surrogate loss (hinge, quantile)	Smooth bounds on recall metrics	Efficient SGD, often per minibatch
Implicit function theorem	Threshold as a function of params	Exact constraint, compatible w/ SGD
Proxy-Lagrangian, stochastic mixtures	Surrogate + exact constraint eval	Sparse support, approximate optimality
Con-LCB (bandits)	Confidence bounds / feasibility	Logarithmic regret, feasibility flag
RL with CMDP + penalty signals	Penalty for constraint violation	Real-time, maskable/non-maskable constraints
Model ensembling with multicalibration	Local calibration on subpopulations	Consistent utility, extensions to recall
Quantum optimization (Ising/Hopfield)	Energy landscape, bias tuning	Recall varies by rule, landscape separation
PR-divergence (GANs, flows)	f-divergence minimization	Unique precision-recall trade-off per λ
Bounded-recall online optimization	Windowed history; asymmetry	Optimal regret scaling in window size
Long-term regret minimization (meta-alg.)	Primal-dual, cumulative constraint	Sublinear regret and constraint violation

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Recall-constrained optimization encompasses a broad spectrum of algorithmic techniques and theoretical guarantees for satisfying recall or related rate constraints in high-dimensional, large-scale, and sequential decision-making environments. These methods have become central in domains requiring careful trade-offs between coverage (recall), selectivity (precision), risk management, and long-term performance guarantees.