Technical Summary: Dynamic Priors in Bayesian Optimization for Hyperparameter Optimization

Problem Statement

Bayesian Optimization (BO) is a standard approach for Hyperparameter Optimization (HPO), yet existing methods typically restrict user influence to the initialization phase. While recent work has introduced interfaces for specifying initial priors (e.g., $\pi$BO) or rejecting candidate evaluations, these approaches lack the ability for continuous, dynamic user steering during the optimization process. In iterative machine learning workflows, practitioners often gain new insights or change priorities as models evolve, but current BO frameworks cannot incorporate these evolving beliefs without restarting or significantly altering the process. Furthermore, existing methods for incorporating user priors often lack robustness against misleading or "deceptive" user inputs, which can degrade performance.

Methodology: DynaBO

The authors propose DynaBO, a framework designed to enable continuous user control over the BO process by integrating dynamic priors at runtime. The methodology consists of three core components:

1. Dynamic Prior-Weighted Acquisition Function

Unlike static priors applied only at initialization, DynaBO generalizes the acquisition function to accept a sequence of user-specified priors $\{\pi^{(m)}\}_{m=1}^M$ provided at different time steps $\{t^{(m)}\}$.

• Mechanism: The standard acquisition function $\alpha_{\hat{f}}(\lambda)$ is augmented by a sum of decaying prior weights:
  $$\alpha_{\hat{f}}^{dyna}(\lambda) := \alpha_{\hat{f}}(\lambda) \cdot \sum_{m=1}^{M} \pi^{(m)}(\lambda)^{\beta/(t - t^{(m)})}$$
• Decay: Each prior's influence decays over time ($t \to \infty$), ensuring that the prior's weight converges to 1. This preserves the asymptotic convergence guarantees of vanilla BO while allowing temporary bias toward user-specified regions.
• Aggregation: The framework uses summation rather than multiplication to combine multiple priors. The authors argue that multiplication acts as an intersection of beliefs (potentially canceling out valid regions if priors disagree), whereas summation acts as a union, preserving multi-modal structures induced by diverse user inputs.
• Candidate Selection: To optimize the modified acquisition function, DynaBO adapts the candidate sampling strategy. It draws starting points for local search (hill climbing) not only from the incumbent's vicinity but also from distributions centered on the provided priors, ensuring user-suggested regions are explicitly explored.

2. Prior Rejection Safeguard

To address the risk of misleading priors slowing down optimization, DynaBO introduces a data-driven detection mechanism.

• Evaluation: Upon receiving a new prior $\pi^{(m)}$, the system samples configurations from the prior distribution ($F_{\pi^{(m)}}$) and from a normal distribution around the current incumbent ($N_{\hat{\lambda}}$).
• Comparison: It compares the expected quality of these regions using a secondary acquisition function $\xi$ (e.g., Lower Confidence Bound).
• Decision: The prior is accepted only if the expected quality of the prior region exceeds that of the incumbent region by a threshold $\tau$:
  $$\mathbb{E}_{\lambda \sim F_{\pi^{(m)}}}[\xi_{\hat{f}}(\lambda)] - \mathbb{E}_{\lambda \sim N_{\hat{\lambda}}}[\xi_{\hat{f}}(\lambda)] \geq \tau$$
• Flexibility: If a prior is rejected, the system can warn the user, allowing for a "human-in-the-loop" override, thus fostering collaborative optimization.

3. Theoretical Guarantees

The authors provide formal proofs for the following properties:

• Almost Sure Convergence: Even with dynamic and potentially misleading priors, DynaBO converges to the global optimum with probability one as $T \to \infty$, as the influence of priors vanishes asymptotically.
• Robustness: The asymptotic simple regret of DynaBO matches that of vanilla BO, even when priors are deceptive.
• Acceleration: If informative priors are provided that concentrate sampling in a region $U_\epsilon$ containing the optimum, DynaBO achieves accelerated convergence in finite-time settings by reducing the effective search space and information gain requirements.

Experimental Results

The authors evaluated DynaBO across diverse HPO benchmarks (including PD1, LCBench, and XGBoost scenarios) using various surrogate models (Random Forests and Gaussian Processes) and acquisition functions (EI and LCB).

• Comparison Baselines: DynaBO was compared against Vanilla BO, $\pi$BO (static priors), and Probabilistic Circuits (PCs) for interactive HPO.
• Performance:
  • Informative Priors: DynaBO consistently outperformed all competitors across Expert, Advanced, and Local prior types, demonstrating significant speedups in finding optimal configurations.
  • Deceptive Priors: While deceptive priors degraded the performance of $\pi$BO and PCs, DynaBO's rejection mechanism successfully filtered out most misleading inputs. In scenarios where rejection occurred, DynaBO recovered performance levels comparable to or better than Vanilla BO.
  • Robustness: The framework maintained Vanilla BO's optimization qualities even when deceptive priors were injected multiple times.
• Overhead: The computational overhead for the rejection mechanism was minimal (approx. 12 seconds per run in a 26-hour context), making it practical for real-world workflows.

Significance and Claims

The paper claims to advance the paradigm of collaborative, human-centered HPO. Its primary contributions are:

1. Continuous Control: It is the first BO framework to support the injection of priors at any time during optimization, bridging the gap between fully automated and manually controlled model development.
2. Safety and Synergy: By combining a safeguard against misleading inputs with a mechanism that synergizes user beliefs and surrogate model beliefs, DynaBO offers a reliable way to incorporate expert intuition without compromising the theoretical guarantees of BO.
3. Theoretical Rigor: The work provides the first proof of finite-time acceleration for dynamic priors while maintaining asymptotic convergence and robustness to deception.

The authors conclude that DynaBO enables reliable and efficient collaborative BO, allowing practitioners to steer optimization processes dynamically as new insights emerge, thereby democratizing access to advanced HPO tools while mitigating the risks of human error.