Compute-Optimal Grid Search Framework

• Compute-optimal grid search is an algorithmic framework that reduces computational cost by strategically narrowing the search space for hyperparameters.
• It leverages analytical bounds and adaptive techniques, such as hierarchical refinement and grid pooling, to maintain statistical efficiency.
• Practical implementations like SHGS, SSGS, and GEX demonstrate significant improvements in tuning performance and model architecture efficiency.

A compute-optimal grid search is an algorithmic framework that systematically reduces the computational costs of grid-based parameter or hyperparameter search, while targeting a prescribed level of statistical efficiency or optimization accuracy. Its principle is to minimize the number of candidate settings evaluated—whether they be discrete factor levels, regularization parameters, neural architecture dimensions, or acquisition function maximizers—by analytically or heuristically constraining the search space, applying problem-specific reductions, and exploiting theoretical bounds on solution quality versus search granularity.

1. Principles of Compute-Optimality in Grid Search

The notion of compute-optimality in grid search stems from the observation that naïve enumeration of all candidate settings in high-dimensional or continuous domains is infeasible due to prohibitive scaling in time and resources. The objective is to limit the number of evaluations required to achieve near-optimal statistical or optimization performance, by leveraging structure in the parameter space and properties of the objective. This includes:

• Analytical bounds on grid density versus target error, e.g., $O(\epsilon^{-1/d})$ grid points for order-$d$ convex losses (Ndiaye et al., 2018).
• Exploratory algorithms that quickly identify promising subspaces, drastically reducing the candidate set (Zhou et al., 2024, Jiang et al., 2024, Harman et al., 2021).
• Adaptive grid pooling and hierarchical refinement stages, focusing computation on regions empirically or theoretically most likely to yield optimal solutions.
• Utilization of scaling laws to shrink the grid over model shapes or hyperparameter assignments that match compute constraints (Alabdulmohsin et al., 2023).

2. Analytical Frameworks for Grid Density versus Accuracy

Compute-optimal grid search is formalized mathematically by establishing necessary grid resolution required for approximate optimality with respect to a desired precision $\epsilon$. Key results include:

• Uniformly convex losses (order $d$): The optimal grid cardinality is $T_\epsilon = O(\epsilon^{-1/d})$, with each grid point solved only to a safely-calibrated train duality gap (Ndiaye et al., 2018).
• Generalized self-concordant functions: Grid complexity reduces to $O(\epsilon^{-1/2})$ (Ndiaye et al., 2018), applicable to logistic regression and related objectives.
• Validation error targeting: Global convergence bounds for hyperparameter selection via grid search, enabling practitioners to set the search resolution as a direct function of desired risk or validation error gap (Ndiaye et al., 2018).

This analytic theory guides practitioners in setting minimal grid densities required to achieve a prescribed accuracy and supports adaptive construction of the grid through mechanisms such as bilateral grid steps, which halve the effective search space when order-$d$ convexity is available.

3. Adaptive and Heuristic Mechanisms: SHGS, SSGS, GEX

Algorithmic schemes for real-world compute-optimal grid search are characterized by:

• Single Hyperparameter Grid Search (SHGS): Systematic one-dimensional sweeps for each hyperparameter, holding others fixed to random backgrounds, mapping performance response curves and then constraining subsequent multidimensional grid search to high-performing subranges, yielding orders-of-magnitude reduction in total candidate grid points (Zhou et al., 2024).
• Sweet-Spot Grid Search (SSGS) and Randomized Grid Search (RGS): A multi-stage approach composed of broad univariate pruning, a coarse multivariate calibration phase, successive localized refinements around the best currently-known setting, and a final stochastic search to avoid local traps. SSGS strategically narrows the grid in cycles, while RGS offers global coverage in the final phase—shown to produce sizeable improvements in predictive AUC with highly restricted budgets (Jiang et al., 2024).
• Galaxy Exploration Algorithm (GEX): For multifactor experimental design, explore only local neighborhoods of promising designs, rather than exhaustively scanning the entire grid. GEX iteratively expands a small exploration set via hill-climbing and star-neighborhood expansion, and solves the discrete optimal design on this subset—demonstrated to outperform exhaustive and coordinate-exchange methods on high-dimensional problems (Harman et al., 2021).

Algorithm	Domain	Complexity Reduction
SHGS	Hyperparameter tuning	$O(d\,n) \rightarrow O(m^d)$
SSGS+RGS	Deep learning	$O(n^d)\rightarrow$ adaptive cycles
GEX	Design of experiments	Adaptive local expansion

4. Scaling Laws and Model Shape Optimization

In model architecture search, compute-optimality can be achieved through analytic scaling laws. For vision transformers:

• Scaling exponents for width, depth, and MLP size specify how to set model shape as a function of budgeted compute $C$. The exponents $(s_w, s_d, s_m)$ encapsulate empirical relationships and are fitted by star sweeps at small compute budgets (Alabdulmohsin et al., 2023).
• Shape-prediction formulas:

$$w^\star \propto C^{s_w}, \quad d^\star \propto C^{s_d}, \quad m^\star \propto C^{s_m}$$

enable grid search to focus only on a small set of locally-optimal architecture candidates, as opposed to exhaustive exploration (Alabdulmohsin et al., 2023).

Empirical validation demonstrates that shape-optimized models outperform much larger baselines under the same compute constraints, both in training and inference efficiency.

5. Random Grid Search in Bayesian Optimization

In Bayesian black-box optimization, the practical maximization of acquisition functions (e.g., UCB, TS) can be replaced by randomized grid search with provable regret bounds:

• Additive inaccuracy management: As long as the accumulated suboptimality $\sum_{t=1}^T \epsilon_t$ grows sublinearly with $T$, regret remains sublinear, matching the scaling of algorithms with exact acquisition maximization (Kim et al., 13 Jun 2025).
• Grid density versus regret: For Lipschitz acquisition, using $m_t = \Theta(t)$ candidate grid points at each round ensures sublinear regret, and more aggressive growth $m_t = \Theta(t^d)$ recovers nearly exact behavior (Kim et al., 13 Jun 2025).
• Computational advantages: Linearly growing grids drastically reduce computational cost compared to gradient-based maximization, demonstrated on synthetic benchmarks and real-world AutoML pipelines (Kim et al., 13 Jun 2025).

6. Practical Considerations and Implementation Guidelines

Compute-optimal grid search requires several tactics for maximum efficacy:

• Range reduction: Preliminary one-dimensional sweeps or validation loss bounds to constrict continuous hyperparameter ranges (Zhou et al., 2024, Jiang et al., 2024).
• Staged allocation: Division of compute budget among exploratory, calibration, and refinement phases (Jiang et al., 2024).
• Adaptive expansion or refinement: Iteratively focus search on neighborhoods near promising solutions (e.g., star neighborhoods, cycle shrinkage) (Harman et al., 2021, Jiang et al., 2024).
• Safe stopping: Use theory-driven criteria to terminate inner solver early, matching train gap to the minimal guarantee needed for target validation accuracy (Ndiaye et al., 2018).

The combination of these strategies enables grid search to scale tractably to high-dimensional, multi-factor, or continuous domains while providing principled guarantees on performance loss and computational cost.

7. Domains of Application and Empirical Effectiveness

Compute-optimal grid search frameworks show systemic efficacy across a wide set of domains:

• Deep learning hyperparameter tuning: SHGS and three-stage mechanisms yield up to $10^8$ fold reductions in search space cardinality over naïve grid, and 16–18$\%$ improvements in predictive AUC over random search (Jiang et al., 2024, Zhou et al., 2024).
• Multifactor experimental design: GEX attains $>99.9\%$ optimality, even in 10-factor logistic models, with computational demands substantially lower than exhaustive alternatives (Harman et al., 2021).
• Regularization path selection: Uniformly convex and GSC loss frameworks enable globally guaranteed risk bounds with optimal complexity (Ndiaye et al., 2018).
• Bayesian optimization acquisition maximization: Random grid search achieves regret matching that of exact maximizers but with far less computation (Kim et al., 13 Jun 2025).
• Transformer architecture optimization: Scaling law–guided grid search yields SoViT architectures that match much larger baselines at half the inference cost (Alabdulmohsin et al., 2023).

The consistent finding is that compute-optimal grid search, underpinned by analytic principles and adaptive search strategies, delivers scalable hyperparameter, architecture, and acquisition maximization across diverse statistical and machine learning contexts.