Compute-Allocation Rule: Finite Budget Strategies

• Compute-allocation rule is the set of mathematical strategies for optimally distributing a finite simulation budget across designs to maximize the probability of correctly selecting the best system.
• Classical OCBA leverages KKT conditions to derive optimal, asymptotic allocation rules that guarantee improved selection performance under idealized simulation scenarios.
• Budget-adaptive methods like FAA and DAA adjust allocations in finite-sample settings, empirically improving selection accuracy by 5–15 percentage points over traditional methods.

The compute-allocation rule refers to the set of mathematical strategies, algorithms, and policies for distributing a finite computational (simulation, sampling, or inference) budget across alternative options to maximize a specific objective, often selection accuracy or utility. In simulation-based ranking and selection (“R&S”), this principle governs how simulation replications are allocated to a set of designs to maximize correct identification of the best system. The optimal computing budget allocation (OCBA) framework, and especially its budget-adaptive extension, offers rigorous foundations for compute allocation rules under finite budgets, yielding practical heuristics with provable guarantees and substantial empirical advantages (Cao et al., 2023).

1. Problem Setting and Motivation

Let $k$ alternatives (“designs” or “systems”) indexed by %%%%1%%%% be tested with outputs %%%%2%%%%, where $\mu_i$ is an unknown performance metric and $\sigma_i^2$ its variance. With a total simulation budget $T$ (replications), the goal is to allocate $N_i$ samples to each design such that the probability of correct selection (PCS)

$$\mathrm{PCS} = \Pr\{ \hat{b} = b \} = \Pr\Bigl( \forall i\ne b: \hat{\mu}_b < \hat{\mu}_i \Bigr)$$

is maximized, where $b$ is the true best ($\mu_b = \min_i \mu_i$) and $\hat{\mu}_i$ is the empirical mean after $N_i$ replications on design $i$. The compute-allocation rule determines the fractions $w_i = N_i/T$ that induce the highest PCS (Cao et al., 2023).

2. Classical Asymptotic OCBA Allocation Rule

As $T\rightarrow\infty$, PCS can be lower-bounded via a Bonferroni argument as

$$\mathrm{APCS} := 1 - \sum_{i\ne b} \Phi\left( -\frac{\mu_i - \mu_b}{\sqrt{\sigma_i^2/(w_iT) + \sigma_b^2/(w_bT)}} \right)$$

Optimizing this surrogate, the Karush-Kuhn-Tucker (KKT) conditions imply, for $i\ne b$,

$$\frac{\sigma_i^2}{w_i} = \frac{\sigma_b^2}{w_b} \left(\frac{\mu_i-\mu_b}{\theta}\right)^2$$

and

$$w_b = \sigma_b\,\sqrt{ \sum_{i\ne b} \frac{w_i^2}{\sigma_i^2} }$$

with normalization $\sum_{i} w_i = 1$. The solution yields the classical OCBA allocation:

• For $$i\ne b: \quad w_i^* \propto \frac{\sigma_i^2}{(\mu_i - \mu_b)^2}$$
• For $$b: \quad w_b^* = \sigma_b\,\sqrt{ \sum_{i\ne b} (w_i^*)^2/\sigma_i^2 }$$

This allocation rule asymptotically maximizes the exponential rate of PCS and thus is optimal in the limit of large $T$ (Cao et al., 2023).

3. Budget-Adaptive Allocation Rule for Finite Budgets

For realistic finite $T$, the asymptotic solution fails to account for finite-sample effects. By retaining $T$-dependence and linearizing the full finite-budget KKT condition,

$$-(\delta_{i,b}^2)/(2(\sigma_i^2/w_i + \sigma_b^2/w_b))\,T + \log\bigg(\frac{\delta_{i,b}\sigma_i^2}{ (\sigma_i^2 / w_i + \sigma_b^2 / w_b)^{3/2}} \bigg) - 2 \log w_i = \lambda$$

(with $\delta_{i,b} = \mu_i - \mu_b$), one obtains

$$\log I_i + \log w_i + T w_i / I_i = \lambda, \quad I_i = \sigma_i^2 / (\mu_i - \mu_b)^2$$

Linearizing $\log w_i$ at $w_i^*$ allows solving for the explicit budget-adaptive allocation rule:

• For $$i \ne b: \quad W_i(T) = w_i^* \cdot \alpha_i(T), \quad \alpha_i(T) = (\lambda - 2\log I_i)/(1 + T/\sum_j I_j)$$
• For $$b: \quad W_b(T) = \sigma_b \sqrt{ \sum_{i \ne b} W_i(T)^2/\sigma_i^2 }$$
• $\lambda$ is chosen to enforce $\sum_i W_i(T) = 1$.

Negative allocations are clamped at the smallest nonnegative value feasible for $T$.

Critically, $\alpha_i(T)$ discounts allocation for hard-to-distinguish designs ($I_i$ large) and boosts easy ones ($I_i$ small). As $T \to \infty$, $\alpha_i(T) \to 1$, retrieving classical OCBA.

4. Theoretical Properties and Asymptotic Optimality

• The finite-budget adaptive allocation exactly solves the full KKT optimality conditions for PCS up to linearization.
• Non-negativity and normalization ($\sum_i W_i = 1$) are always satisfied.
• As $T\to\infty$, classical OCBA is recovered, ensuring asymptotic optimality.
• For finite $T$, the adaptive rule provably and empirically achieves superior PCS compared to classical OCBA, especially in regimes with small budgets.

5. Practical Heuristic Algorithms

Two fully sequential implementations—Final-Budget Anchorage Allocation (FAA) and Dynamic-Anchorage Allocation (DAA)—make the rule practical.

FAA: At every step, estimate means/variances, compute OCBA and T-adaptive proportions for the final budget $T$, and allocate the next sample to the design with greatest $(t+1)W_i^{(t)}(T)-N_i^{(t)}$ (i.e., most under-allocated so far).

DAA: Similar, but at each step use the remaining budget $t+1$ instead of the fixed $T$, improving responsiveness as data accumulates.

Both FAA and DAA require only $O(k)$ arithmetic and are efficient for large $k$ (Cao et al., 2023).

6. Empirical Performance and Implementation

Empirical validation with synthesized Gaussian and real-world facility-simulation benchmarks reveals:

• FAA/DAA outperform Equal Allocation, OCBA, and AOAP by 5–15 percentage points in PCS over all budgets and problem scales.
• FAA is slightly better at small $T$ (below threshold), DAA at larger $T$.
• Runtime per iteration matches OCBA (both $O(k)$), much faster than AOAP ($O(k^2)$).
• In expensive simulation settings, the overhead for FAA/DAA is negligible.

This suggests budget-adaptive allocation achieves substantial efficiency gains by dynamically reweighting exploration towards easier designs when budget is scarce.

7. Context, Implications, and Extensions

Budget-adaptive compute-allocation bridges the gap between classical OCBA and practical, finite-budget allocation needs in R&S. By discounting hard cases and boosting clear winners at small $T$, it preserves asymptotic optimality as $T$ grows but achieves much higher PCS when $T$ is limited. The practical impact is particularly evident for high-throughput simulation workflows and scalable model selection tasks.

Heuristic sequential algorithms (FAA/DAA) implement this logic with minimal overhead, making the rule deployable in real-time, online ranking and selection, optimal resource allocation for cloud services, and experimental design scenarios where measured data is incrementally available and compute remains a principal constraint.

By analytically quantifying the impact of budget on allocation proportions, budget-adaptive compute allocation sets a new standard for simulation-based selection policies under realistic computational constraints (Cao et al., 2023).