Final Budget Anchorage Allocation (FAA)

• Final-Budget Anchorage Allocation (FAA) is a strategy that optimally assigns a fixed terminal budget to maximize system-level objectives such as accuracy, fairness, or utility.
• It employs adaptive, sequential, and clustering methodologies to address strategic uncertainty and nonlinear dynamics in domains like simulation R&S, federated learning, and multi-task scheduling.
• FAA's rigorous frameworks and numerical experiments demonstrate superior performance over traditional methods, providing actionable insights for practical resource management.

Final-Budget Anchorage Allocation (FAA) is a resource allocation paradigm that seeks optimal assignment of a fixed final budget—whether monetary, computational, annotation, or incentive—to competing agents, tasks, or systems under uncertainty, strategic behavior, and nonlinear dynamics. This allocation is often anchored to maximize a specific system-wide objective, with sequential, adaptive, or batch methodologies formalized for simulation optimization, federated learning, multi-task model selection, or fair divisible resource allocation.

1. General Definition and Frameworks

FAA refers to the allocation of a fixed budget at the terminal stage of a process or across a finite horizon, with the goal of optimizing an aggregate objective such as utility, social welfare, accuracy, probability of correct selection (PCS), or fairness. This is distinguished from incremental or online allocation mechanisms by its focus on global budget exhaustion and judicious anchoring—making allocation decisions with respect to the final available resources.

Mathematical frameworks for FAA typically specify:

• Decision variables: the vector of allocations $\{b_i\}$ to agents/tasks/alternatives.
• Objective function: a system-level performance measure (accuracy, welfare, PCS).
• Constraints: exhaustion ($\sum_i b_i \leq B$), agent/task-specific capacity or effectiveness. FAA is instantiated in multiple domains:

Domain	FAA Objective	Allocation Variables
Simulation R&S	Maximizing PCS	Replication counts
Federated Learning	Maximizing accuracy	Incentive for clients
Multi-task Learning	Joint performance	Labeling efforts
Resource Mechanisms	Nash welfare	Resource fractions/pay

2. Simulation-Based Ranking and Selection

In simulation-based ranking and selection (R&S), FAA addresses the allocation of simulations to competing designs under a finite budget, maximizing PCS—the probability that the selected design is truly best. The budget-adaptive allocation rule (Cao et al., 2023) modifies the classical OCBA (Optimal Computing Budget Allocation) procedure:

• OCBA (asymptotically):

$$w^*_i = \frac{\sigma^2_i}{(\mu_i - \mu_b)^2} / \sum_{j \neq b} \frac{\sigma^2_j}{(\mu_j - \mu_b)^2}$$

• FAA (finite budget):

$$W_i(T) = w^*_i \cdot \alpha_i(T), \quad \alpha_i(T) = \frac{\lambda - 2 \log I_i}{1 + T/S}$$

with $\lambda$ chosen to normalize allocations.

• FAA algorithm: Allocates budget up-front using final target ratios, distributing replications to the most "starving" (under-allocated) design.

FAA thus discounts competitive designs under small budgets and boosts allocations to easy-to-distinguish cases, adapting as budget increases. Numerical experiments demonstrate FAA and its dynamic anchoring variant outperform OCBA and other baselines, especially with limited resources.

3. Federated Learning: BARA Algorithm and Incentive Structures

FAA in federated learning appears in the BARA framework (Yang et al., 2023), which optimizes allocation of a fixed reward/incentive budget over multiple communication rounds to improve final global model accuracy. The allocation problem is: $$\max_{n:1\leq n<N} a_0 + \sum_{t=1}^{T(n)} \Delta a_{t,n}, \quad \text{s.t.} \quad \sum_{t=1}^{T(n)} B_t(n) \leq B_{total}$$ where $n$ is the number of clients per round and $B_t$ is roundwise budget.

BARA deploys Bayesian optimization with a Gaussian Process (GP) to learn the round-client-budget-to-accuracy mapping:

• Interpolates missing accuracy improvements using Newton’s polynomial.
• Maximizes Upper Confidence Bound (UCB) at each round:

$$n_{t+1} = \arg\max_n \mu_t(n) + \sqrt{\beta_{t+1} \sigma_t(n)}$$

FAA is implicit here: allocation decisions are made relative to the total remaining reward, with BARA dynamically anchoring choices for optimal accuracy. Experiments show substantial accuracy improvement (1.3%–10%) over even and monotonic allocation strategies.

4. Multi-Task Learning and Label Budget Allocation

FAA principles are crucial in multi-task label budget allocation (Sun et al., 2023), where the problem is to distribute a total annotation/labeling budget $B$ across diverse tasks $\{t_k\}$: $$\max_{b_1, ..., b_K} \mathcal{P}(b_1, ..., b_K), \quad \text{s.t.} \sum_k b_k \leq B$$ The Task-Adaptive Budget Allocation algorithm quantifies the marginal "extent of new information" per unit annotation, guiding dynamic reallocation for maximum joint performance.

FAA, in this context, involves anchoring the remaining budget in the final labeling stage specifically to the tasks most likely to boost overall information content. The adaptive strategy outperforms heuristic baselines on PASCAL VOC and Taskonomy, especially where transfer between tasks and annotation cost disparities are pronounced.

5. Lipschitz Value Functions and Clustering-Based Allocation

FAA is formalized for model selection and hyperparameter optimization in (Bateni et al., 12 Oct 2025) as the Unknown Value Probing (UVP) problem over a Lipschitz space: $$\max_{b_1,\ldots,b_n} \left\{ \max_{x_i \in X} A(x_i, b_i) \right\}, \quad \sum_{i=1}^n b_i \leq B$$ with performance functions $A(x, b)$ satisfying monotonicity and Lipschitz continuity: $$|A(x_i, b) - A(x_j, b)| \leq \epsilon \|x_i - x_j\|_2$$ Cluster-based algorithms (FullCent,\eFullCent) allocate budget to diverse, distant candidate configurations, anchoring the final budget on promising regions as guided by feedback and performance-aware metrics.

Provable tight bounds (gap proportional to $\epsilon r_k^\star$) ensure near-optimality, and practical implementations show empirical superiority over random search, Hyperband, and Bayesian optimization baselines in hyperparameter problems.

6. Fair Division, Strategic Mechanisms, and Welfare Optimization

FAA is integral to divisible resource mechanisms under budget constraints (Caragiannis et al., 2017). Here, users submit signals for a share of a unit resource, subject to payment capped by budget $c_i$. The efficiency benchmark is liquid welfare: $LW(d) = \sum_{i=1}^n \min\{ v_i(d_i), c_i \}$ Kelly’s mechanism, allocating by signal proportion, achieves a liquid price of anarchy (LPoA) of exactly 2—the best possible for any number of users with a general mechanism. Two-player FAA-inspired mechanisms (E2-PYS, E2-SR) achieve lower LPoA via differential equation-optimized allocation rules.

Strategic equilibrium analysis reveals that worst-case welfare gaps are highly structured, often with all but one user capped at budget and equilibria computable via mathematical programs. FAA emerges as a tool for “anchoring” allocation rules to worst-case efficiency in budget-constrained environments.

7. Sequential Fair Allocation: MDP Formulation and Concentration Bounds

FAA is also modeled in sequential fair resource allocation via Markov Decision Processes (MDPs) (Hassanzadeh et al., 2023). The SAFFE(-D) algorithms allocate a finite resource budget to agents arriving over time, with uncertainty in future demand:

• State variable includes cumulative allocations and revealed demands.
• Bellman recursion solves for Nash Social Welfare-optimal policies. Regularization via SAFFE-D anchors allocation in the face of uncertainty by penalizing expected future demand estimates with their standard deviation: $$Y_i^t = X_i^t + \sum_{\tau=t+1}^T [\mathbb{E}[X_i^\tau] - \lambda\,\operatorname{std}(X_i^\tau)]^+$$ Rigorous bounds on envy, efficiency, and total allocation deviations are established by concentration inequalities, ensuring FAA approaches hindsight optimality.

Summary Table: FAA Instantiations

Domain/Algorithm	FAA Principle	Objective/Metric
OCBA (Simulation R&S)	Final budget ratios	Probability of correct selection (PCS)
BARA (FedLearn)	Dynamic anchoring	Final model accuracy
Task-Adaptive (MTL)	Marginal info gain	Joint test performance
FullCent/AdaCent (HPO)	Cluster/prune effort	Best observed model value
Kelly/E2-PYS (Mech.)	Signal/pay caps	Liquid welfare (LPoA)
SAFFE(-D) (MDP/Seq.)	Regularized expectation	Nash Social Welfare, utilization

Conclusion

Final-Budget Anchorage Allocation encompasses a family of resource allocation strategies that optimize the deployment of scarce, terminal resources under uncertainty and competition. FAA is mathematically rigorous and empirically validated across domains, with adaptive, clustering, and regularization techniques ensuring improved system-level outcomes over classical approaches that ignore budget anchorage. Its theory and practice span simulation optimization, federated learning, multi-task scheduling, hyperparameter exploration, fair division, and sequential MDP-based allocation, each with domain-specific objectives, constraints, and solution guarantees.