Optimal Design Algorithms for PSI

• Optimal design-based algorithms for PSI are principled strategies that combine statistical optimal design, simulation-based inference, and modern optimization to efficiently identify informative experimental configurations.
• They leverage techniques such as Monte Carlo integration, likelihood-free methods, randomized relaxations, and adaptive discretization to overcome the challenges of high-dimensional and complex design spaces.
• These approaches reduce estimation variance and sample complexity, enabling efficient solutions in applications like sensor selection, multi-objective bandits, and inverse problem solving.

Optimal Design-Based Algorithms for PSI

Optimal design-based algorithms for PSI (Pareto Set Identification or Posterior Sample Inclusion) comprise a family of principled strategies grounded in statistical optimal design, simulation-based inference, and modern optimization. These methods aim to identify informative experimental configurations or adaptively allocate observations to maximize information gain, minimize estimation variance, or efficiently solve multivariate multi-objective identification tasks. Across both classical and contemporary research, approaches span Monte Carlo and MCMC integration, likelihood-free simulation (ABC), convex and randomized relaxations, adaptive discretization, and scalable approximate algorithms for high-dimensional or structure-exploiting PSI settings. This article surveys leading methodologies and theoretical insights underlying optimal design-based algorithms applied to PSI, with attention to their mathematical underpinnings, computational techniques, and practical performance.

1. Simulation-Based and Likelihood-Free Approaches

Simulation-based optimal design starts with the specification of an expected utility criterion. Given a (possibly unknown) probability model $p(y \mid \theta, d)$ for outcome $y$ under design $d$ and parameters $\theta$, and a utility $u(y, d, \theta)$, the objective is to maximize

$$U(d) = \iint u(y, d, \theta)\, p(y \mid \theta, d)\, p(\theta)\, d\theta\, dy.$$

As analytical integration is usually infeasible, Monte Carlo approximations are used: sample $\theta^{(t)} \sim p(\theta)$, $y^{(t)} \sim p(y \mid \theta^{(t)}, d)$, and estimate

$$U(d) \approx \frac{1}{T} \sum_{t=1}^T u(y^{(t)}, d, \theta^{(t)}).$$

Optimization over $d$ can either be conducted by searching for the mode of the estimated $U(d)$ through stochastic optimization (simulated annealing, exchange algorithms, SPSA) or through integrated approaches. In particular, MCMC schemes that sample $(d, y, \theta)$ jointly—such as the algorithm of Müller (1999)—allow design inference and utility integration in a single stochastic process. The acceptance probability for a proposed state update is constructed so that the marginal law of $d$ is proportional to $U(d)$, ensuring concentration of the chain around the optimal design as the chain progresses or as an annealing parameter $J$ is increased (1305.4273).

In "likelihood-free" scenarios, where $p(y \mid \theta, d)$ is intractable but data can be simulated, Approximate Bayesian Computation (ABC) methods are used. ABC replaces the unknown likelihood with a kernel $p_\epsilon(y, y^*)$, which quantifies closeness between simulated and observed data summaries, often via a sufficient statistic $T(\cdot)$. ABC can be embedded both in MCMC-based utility sampling (by redefining the acceptance ratios) and in two-stage schemes that first sample an approximate posterior and then perform standard Monte Carlo design search.

Key advantages of simulation-based, likelihood-free methods for PSI include:

• Flexibility to handle intractable or unspecified likelihoods, provided there is a model-based simulator.
• Seamless incorporation of prior observations and contexts where exact likelihood evaluation is unavailable or computationally prohibitive.
• Generality to optimize expected information gain or other design objectives by fully simulating the joint model.

Limitations stem from computational cost, especially as the accuracy of the ABC kernel $\epsilon$ is increased (leading to high rejection rates and slow mixing), and from the need to calibrate proposal distributions or annealing parameters to ensure efficient exploration.

2. Approximations, Randomization, and Relaxations

Modern optimal design algorithms for PSI often rely on approximate or randomized solutions to statistically or computationally hard design problems.

Proportional volume sampling, as developed for $A$-optimal and $D$-optimal criteria, picks design subsets $S$ with probability proportional to

$\mu(S) \cdot \det(V_S V_S^T)$

where $V_S$ is the matrix of design vectors in $S$ and $\mu(S)$ is a weight derived from an underlying measure (hard-core distributions with parameters $\lambda$). For $A$-optimal design, the average variance is linked to symmetric polynomials of the information matrix, enabling the design of approximation algorithms with guarantees: a $d$-approximation for small sample size ($k=d$), and a $(1+\varepsilon)$-approximation for large sample regimes. Tightness results and NP-hardness bounds delineate the boundary of tractable approximations, especially for $E$-optimality, where similar methods do not yield good ratios (1802.08318).

Other work reformulates optimal design in terms of modern convex and sparse optimization. The Bayes $A$-optimal design criterion can be recast as minimizing a quadratic plus a squared group lasso penalty, enabling solution by fast splitting (proximal gradient, FISTA) or block coordinate methods, which benefit from guaranteed convergence rates and direct control over sparsity properties, leading to efficient, scalable algorithms that yield sparse, interpretable designs well suited for scenarios such as sparse sensor selection in PSI (1809.01931).

Randomization is also leveraged in the context of high-dimensional PDE-based inverse problems. Here, A-optimal design—minimizing the posterior covariance trace—is rendered tractable by randomized subspace iteration and reweighted $\ell_1$ minimization. This matrix-free, parallelizable approach exploits rapid eigenvalue decay, making it feasible to select informative sensor subsets in large-scale settings. Error analysis guarantees estimator reliability and guides parameter choices (1906.03791).

3. Adaptive, High-Dimensional, and Continuous Design Spaces

For continuous or very high-dimensional design spaces, adaptive discretization and function approximation form the backbone of recent algorithmic advancements for PSI.

Adaptive discretization algorithms proceed by initially considering a finite candidate set and iteratively refining this discretization via sensitivity analysis:

1. Solve a convex optimization over the current discretization for the design measure $\xi_k$.
2. Evaluate the sensitivity function $v(M(\xi_k),x) = D \Phi(M(\xi_k)) (m(x) - M(\xi_k))$ across the space.
3. If the minimum of $v$ over $x$ is below a threshold $-\epsilon$, add the worst-violating $x$ to the candidate set.

Termination and convergence are guaranteed under mild regularity conditions; notably, if the design criterion is additionally strongly convex and the design space is finite, a linear rate of convergence is obtained. Applications to chemical engineering demonstrate the practical efficiency of these methods (2406.01541).

Another avenue employs adaptive sampling and high-dimensional function approximation (ADA-GPR). Here, Gaussian process regression is used to model the directional derivative of the design criterion, allowing the algorithm to prioritize evaluations of the expensive model only in regions of uncertainty or likely improvement. Acquisition functions combining predicted mean and variance guide the search efficiently, resulting in substantial reductions (orders of magnitude) in the number of required simulations, validated via chemical process engineering tasks (2101.06214).

Such adaptive methods are particularly impactful for PSI-type problems, where the size and complexity of the candidate design space would otherwise make exhaustive search or uniform grid evaluation computationally infeasible.

4. G-Optimal Design and Meta-Heuristic Approaches

Meta-heuristic algorithms, such as particle swarm optimization (PSO), have been adapted to compute exact G-optimal designs—those minimizing the maximum prediction variance across a design region. In this context, each particle encodes a full design matrix, and the standard velocity–position update rules are applied in the matrix domain, possibly with "reflecting wall" confinement to maintain feasibility.

Empirical and theoretical studies demonstrate that PSO finds equal or superior G-optimal designs compared to coordinate exchange or genetic algorithms, while reducing computational cost, especially for designs involving many factors (dimensions). The maximization of scaled prediction variance (SPV) over a fine grid is used as the objective for each candidate design. These approaches are particularly useful for multi-factor industrial experiments, where the design space is large but global optimization techniques can efficiently identify near-optimal configurations (2206.06498).

5. Application to Structured Pareto Set Identification

Recent advances connect optimal design-based strategies directly to PSI in multi-objective, multi-arm or multi-output bandit models. In a structured multi-output linear model, each arm is defined by a feature $x_i \in \mathbb{R}^h$ and $d$-dimensional mean response $\mu_i = \Theta^T x_i$, with unknown common parameters $\Theta \in \mathbb{R}^{h \times d}$.

Key points in algorithmic design include:

• Sample allocation is dictated by a $G$-optimal experimental design over current arms, so that after sampling, least-squares estimation of $\Theta$ yields uniformly accurate mean estimates for all arms.
• Arm elimination is guided by empirical estimates of sub-optimality gaps (defined via min-max differences across objectives), and elimination thresholds are adjusted adaptively per round.
• The complexity (sample budget or confidence level) is shown to depend only on the "h" smallest gaps (arms), rather than all $K$ arms, reflecting the advantage of exploiting model structure (2507.04255).

The G-optimal Empirical Gap Elimination (GEGE) algorithm, for both fixed-budget and fixed-confidence settings, demonstrates this principle concretely. It batches sampling based on the optimal design, applies conservative elimination based on gap estimation error, and iterates until the Pareto set is identified or the budget is exhausted.

Empirical results validate the theory: the number of samples, or the error rate, scales mainly with the feature dimension $h$ and not the total arm count $K$. In practice, GEGE matches or outperforms unstructured PSI solvers (e.g., EGE-SH/SR, APE, PAL) when $K \gg h$, on both synthetic and real-world network optimization problems.

6. Connections to Broader Methodological Themes and Applications

Optimal design-based algorithms for PSI feature deep ties to statistical decision theory, experimental design, inference under computational or model structure constraints, and multi-objective optimization. The development of scalable, structure-exploiting, and likelihood-free algorithms enables their application in domains as diverse as sensor network planning, high-dimensional inverse problems (PDEs), sparse regression, feature selection, phase retrieval, and adaptive process monitoring.

Common themes and practical lessons include:

• The importance of model structure (e.g., linear, transductive, or grouped-arm models) in reducing sample complexity or computational cost.
• The ability of randomized and meta-heuristic algorithms to overcome intractabilities present in combinatorial or high-dimensional design spaces.
• The role of adaptive and sequential methods, including active learning and function approximation, in handling otherwise prohibitive continuous or high-dimensional settings.
• The emergence of rigorous links (e.g., between group lasso and A-optimality) that bridge experimental design with machine learning and optimization literature.
• Hard barriers for approximation in some design settings (e.g., NP-hardness of A-optimality with minimum sample size; impossibility of good performance for E-optimal design within proportional volume sampling frameworks).

A summary table relates core methodologies to practical settings:

Methodology	Key Setting / Structure	Primary Benefit
Simulation-based + ABC	Intractable likelihood, Bayesian design	Flexibility, likelihood-free integration
Proportional volume sampling	Linear model, A-/D-optimality	Efficient approximation, guarantees
Convex + group-lasso reform.	Bayesian A-optimal design	Sparse, scalable, convergence rates
Adaptive discretization/GPR	Nonlinear/continuous high-dim.	Drastically reduced simulation cost
PSO/meta-heuristics	G-optimal, complex design region	Robust optimization, scalability
GEGE structure-exploiting	Multi-output linear bandit PSI	Sample complexity in h not K

7. Future Perspectives and Limitations

Ongoing research in optimal design for PSI is focused on:

• Extending current algorithms to richer, possibly nonparametric or non-linear settings while maintaining sample efficiency and tractability.
• Incorporating robustness to model mis-specification or adversarial perturbations, especially pertinent in applications such as security or privacy-preserving design (e.g., homomorphic PSI protocols (2204.11334)).
• Clarifying limits: for example, the impossibility of improving E-optimality guarantees with volume sampling, and hardness results for constant-factor approximation in A-optimal design when $k = d$.
• Developing hybrid methods that combine adaptive discretization, randomized relaxations, and Bayesian updating to handle real-world PSI problems with non-standard constraints or hierarchical/multistage decision-making.

A central message is that optimal design-based algorithms for PSI, by combining probabilistic modeling with principled search and structure exploitation, yield robust and efficient solutions across a range of statistically and computationally challenging PSIs, with documented impact from signal processing to sensor networks and multi-objective optimization.