Budget-Splitting Experimental Design

• Budget-splitting experimental design is a framework for optimally allocating finite resources across multiple experimental arms to enhance statistical power and decision quality.
• It employs techniques such as submodular optimization, convex relaxation, and two-stage adaptive allocation to balance costs with maximal information gain.
• Empirical and theoretical results demonstrate significant improvements in estimation precision and power, with methods that can boost performance by up to 30× in specific settings.

Budget-splitting experimental design refers to a broad class of frameworks and methodologies for optimally allocating a limited experimental budget—monetary, sample count, or resource units—across possible experimental interventions, arms, units, cells, or settings. The central purpose is to maximize inferential power, estimation precision, or decision quality, subject to strict constraints. Budget-splitting enters at multiple levels of the experimental pipeline: selecting which experiments or units to run, deciding treatment probabilities, splitting total samples across phases, or enforcing resource feasibility in marketplace or networked environments. Techniques span submodular optimization, information-theoretic design, randomized rounding, convex relaxation, mechanism design, and empirical variance minimization.

1. Mathematical Formulation and Problem Classes

Budget-splitting designs are unified by their focus on a constrained combinatorial or continuous optimization of experimental allocation under a total budget constraint. The canonical mathematical form is:

$$\max_{S\subseteq V} f(S) \quad\text{s.t.}\quad \sum_{x\in S} c(x) \leq B$$

where $V$ is the set of candidate experiments, $c(x)$ is the (possibly heterogeneous) cost of including $x$, $B$ is the budget, and $f(S)$ measures the expected utility (information gain, variance reduction, regret, etc.) from running $S$.

Specific instantiations include:

• Submodular information gain maximization for model correction, e.g. mutual information in GP design (Shulkind et al., 2017).
• Dose- or cell-based allocation for marginal treatment effect estimation, with polynomial constraints capturing moments of the underlying dose–response curve (Waisman et al., 2023).
• Knapsack- and matroid-constrained selection in classical linear regression experimental design ($D$-optimality, etc.) (Horel et al., 2013, Lau et al., 2024).
• Two-stage adaptive allocation in best-arm identification, splitting samples across arms/phases for minimax or Bayes optimality (Kato, 30 Jun 2025, Zhang et al., 3 Jun 2025).
• Budget splitting across submarkets or units in interference-sensitive online multi-agent markets (Liu et al., 2020, Liao et al., 2024, Guo et al., 2023).
• Rounding fractional allocations into binary assignments that respect hard budget limits (Yamin et al., 15 Jun 2025).

2. Core Methodological Approaches

Submodular and Information-theoretic Design

When $f(S)$ is monotone and submodular (e.g., GP mutual information, trace reduction, entropy), the cost-sensitive greedy algorithm yields a $(1-1/e)$-approximation to the optimal value. One iteratively adds the experiment maximizing marginal gain per unit cost until the budget is exhausted (Shulkind et al., 2017, Agrawal et al., 2019). This approach applies to targeted causal discovery, nonparametric dynamical model correction, and adaptive information collection under constraints.

Convex Relaxation and Rounding

Convex relaxation—often via replacing combinatorial selection with continuous weights—permits tractable surrogates (e.g., log-det relaxations) which are then pipage/polynomially rounded to binary allocations with provable approximation ratios. Deterministic rounding via interlacing polynomials achieves near-optimal guarantees for $D$-, $A$-, $E$-type objectives, even in small-budget regimes (Lau et al., 2024).

Two-Stage, Phase-Split Designs

In adaptive best-arm identification, fixed-budget experimental design leverages a natural two-phase split: an initial (pilot) phase allocates budget to explore/estimate variances, followed by an allocation phase distributing remaining budget proportionally to empirically high-variance arms. This uniform-then-variance (“TS–EBA”) design is asymptotically minimax and Bayes optimal for simple regret (Kato, 30 Jun 2025). Analogous two-phase splits appear for multi-metric treatment identification with a validation phase for high-confidence inference (Zhang et al., 3 Jun 2025).

Mechanism-based and Strategic Allocation

In settings with strategic agents (e.g., subjects reporting private costs), mechanism design frameworks enforce budget feasibility, approximate truthfulness, and individual rationality, while approximating $D$-optimality or other information-based objectives. This is accomplished through monotone relaxation-heuristics with threshold payments and pipage rounding (Horel et al., 2013).

Dependent Randomized Rounding

Dependent (e.g. swap) rounding optimally converts fractional assignment probabilities into binary assignments under exact budget constraints, preserving marginals but introducing negative dependency to reduce the estimator variance. This yields strictly lower variance for IPW estimators and is robust to arbitrary initial probabilistic splits, provided the total treatment allocation is enforced (Yamin et al., 15 Jun 2025).

3. Applications and Instantiations

Budget-splitting experimental design permeates multiple domains:

Domain	Budget unit	Optimization/Design Approach
GP correction for dynamical models	Experiment cost	Submodular greedy, MI maximization
Digital ads—MTE estimation	User exposure $B$	Multicell, polynomial moment system
Survey/sampling with heterogeneous cost	Sampling cost	Two-stage, Neyman-proportional, stratified design
Marketplace A/B testing (auctions)	Buyer/arm budgets	Parallel submarkets, debiasing
Treatment arm selection	Sampling budget	Two-phase (uniform–variance), minimax optimal
Targeted causal mechanism discovery	Intervention/sample	DR–submodular greedy, batch splitting
Online controlled experiments	Sample size, cost	Convex program, randomized rounding

Notable insights:

• In two-sided marketplaces, splitting buyer budgets and creating disjoint submarkets restores SUTVA, eliminates interference/cannibalization bias, and vastly increases statistical power—up to $30\times$ relative to conventional campaign-level randomization (Liu et al., 2020, Liao et al., 2024, Guo et al., 2023).
• Multicell dose-response experiments enable full recovery of marginal treatment effect curves with no increase in budget, enabling intensive margin optimization (Waisman et al., 2023).

4. Statistical Properties and Theoretical Guarantees

Approximation and Optimality Bounds

Across submodular, convex, and information-gain objectives, greedy and convex relaxation–rounding designs achieve provable approximation ratios:

• $(1-1/e)$-factor for monotone submodular maximization under general costs (Shulkind et al., 2017, Agrawal et al., 2019)
• $(B/(B-d+1))$-factor for $D$-/A-/E-designs under deterministic interlacing rounding (Lau et al., 2024)
• Constant-factor (e.g. $\leq 12.98$) approximations for budget-feasible linear regression design under truthfulness (Horel et al., 2013)
• Asymptotically exact minimax and Bayes risk for two-phase best-arm allocation (Kato, 30 Jun 2025)

Variance and Unbiasedness

Dependent rounding ensures unbiased estimation for sample average treatment effects and strictly reduces (or does not increase) estimator variance under typical estimators (IPW, linear) (Yamin et al., 15 Jun 2025). Strong submodularity or mutual-information criteria directly encode diminishing returns, which dictates that marginal information gain per budget dollar declines as budget increases.

Practical Power: Comparison with Baselines

Budget-splitting designs yield dramatic power increases in resource-constrained online experiments, especially as the number of experimental units (e.g. users, sellers) far exceeds the strategic agents (e.g. buyers, advertisers). Power boosts of $20$–$50\%$ (or severalfold in practical settings) are observed both theoretically and empirically (Liu et al., 2020, Shulkind et al., 2017).

5. Implementation Protocols and Empirical Findings

Implementation follows a sequence of:

• Problem-specific relaxation or mutual-information modeling to generate optimal assignment probabilities or candidate sets;
• Rounding (deterministic, randomized, swap) to produce feasible assignments or splits;
• Batched or phase-wise allocation for multi-round settings;
• Plug-in, bootstrap, or debiased plug-in inference for valid uncertainty quantification even under budget-induced dependencies (Liu et al., 2020, Liao et al., 2024, Yamin et al., 15 Jun 2025).

Empirical studies confirm the asymptotic efficiency, robustness to heterogeneity, and realized power gains predicted by the theory. For instance, swap rounding based on pilot variance estimates outperforms independent IPW assignment in both variance and mean squared error across a variety of datasets and simulation architectures (Yamin et al., 15 Jun 2025).

6. Limitations, Extensions, and Future Directions

Several open questions and limitations remain:

• Approximation ratios for truthful, budget-feasible mechanisms are bounded below by 2, with tightening of these bounds an active area (Horel et al., 2013).
• Multicell and submarket splitting require sufficient sample size in each cell to maintain estimator stability; small budgets may degrade precision or require careful adaptive allocation (Waisman et al., 2023, Liu et al., 2020).
• Current dependent rounding techniques handle only hard partition budget constraints; more general polytopal or network flow constraints suggest further combinatorial and algorithmic development (Yamin et al., 15 Jun 2025).
• Extensions to non-linear and doubly-robust estimators, multi-arm or multi-dose interventions, sequential (online) adaptation, and heavy-tailed or adversarial noise models are being developed in response to limitations outlined in empirical studies (Yamin et al., 15 Jun 2025, Lau et al., 2024).

7. Synthesis and Operational Guidance

In summary, budget-splitting experimental design formalizes and operationalizes the statistical and algorithmic allocation of finite experimental resources for maximal inferential or decision-theoretic impact. Techniques leverage submodular optimization, convex relaxations, strategic mechanism design, and advanced rounding or sampling to navigate feasibility constraints and heterogeneity, achieving state-of-the-art precision, unbiasedness, and practical implementability in a wide array of experimental settings (Shulkind et al., 2017, Yamin et al., 15 Jun 2025, Waisman et al., 2023, Liu et al., 2020). Best practices include leveraging problem-specific information (variance, cost, potential effect heterogeneity) in the design phase, employing greedy or relaxation-based allocation, and using variance-reducing dependent rounding wherever feasible. These approaches enable optimal or near-optimal learning under strict budget constraints in both classical and modern, high-dimensional experimental paradigms.