Mutual Information-Aware Optimization

• Mutual information-aware optimization is a strategy that uses expected information gain to balance exploration and exploitation in Gaussian process-based global optimization.
• The GP-MI algorithm adapts its acquisition function by decaying the exploration bonus based on cumulative mutual information, leading to sharper regret bounds and faster convergence.
• This approach demonstrates robust performance across synthetic and real-world tasks by effectively managing the trade-off between discovering unknown regions and exploiting promising candidates.

Mutual information-aware optimization is a principled strategy for global optimization that leverages information-theoretic concepts to control the trade-off between exploring unknown regions and exploiting promising candidates. In the context of Gaussian process (GP) optimization, mutual information-aware algorithms—such as the Gaussian Process Mutual Information (GP-MI) algorithm—use the expected information gain about the function as a primary determinant for query selection. This approach enables more adaptive and theoretically robust optimization, resulting in sharper theoretical regret bounds and demonstrably better empirical performance than traditional methods that use static exploration bonuses.

1. Principles of Mutual Information-Aware Optimization

Mutual information (MI) quantifies, for a given candidate query $x$, the expected reduction in uncertainty about an unknown objective function $f$ following observation at $x$. In Bayesian optimization with GPs, MI is used to adaptively balance:

• Exploration (sampling points that reveal new information about $f$)
• Exploitation (sampling points expected to be near the optimum)

Formally, for a GP prior ($f \sim \text{GP}(0,k)$) and noise variance $\sigma^2$, the information gain from observing at points $X_t = \{x_1,\dots,x_t\}$ is

$$I_t(X_t) = I(f; Y_t) = \frac{1}{2} \log\det\left(I + \sigma^{-2}K_t\right),$$

where $K_t$ is the $t \times t$ kernel matrix at points $X_t$.

MI-aware optimization algorithms select points to maximize an acquisition function that is directly calibrated by MI and the information already gathered, ensuring exploration is automatically reduced as knowledge about $f$ accumulates.

2. The GP-MI Algorithm: Workflow and Mathematical Formulation

The GP-MI algorithm introduces a mutual information-adaptive acquisition rule for sequential Bayesian optimization:

1. GP Posterior Update: After $t-1$ queries, compute posterior mean $\mu_t(x)$ and variance $\sigma_t^2(x)$ over $\mathcal{X}$.
2. Adaptive Acquisition Function: Select $x_t$ as

$$x_t = \arg\max_{x \in \mathcal{X}} \quad \mu_t(x) + \sqrt{\alpha \, \sigma_t^2(x)}$$

where exploration bonus (the "uncertainty term") is directly tied to the mutual information accumulated so far, via tuning of the parameter $\alpha$.

1. Observation: Observe $y_t = f(x_t) + \epsilon_t$ and augment data.
2. Iteration: Repeat steps 1–3.

The core innovation is that $\alpha$ is not a static or deterministically growing (as in GP-UCB), but adapts as a function of the total mutual information already acquired. When little is known about $f$, exploration is emphasized. As MI grows, the algorithm automatically focuses more on exploitation.

Regret-Based Analysis

A key theoretical result relates cumulative regret $R_T$ to the MI $$\gamma_T = \max_{A \subset \mathcal{X}, |A|=T} I(f; Y_A)$$ as

$R_T = O(\sqrt{C_1 \gamma_T}),$

where $C_1$ depends on noise variance. For RBF kernels, this yields

$R_T = O\left((\log T)^{d+1}\right)$

in $d$ dimensions, representing an exponential improvement over previous regret bounds.

Step	Description
1	GP posterior computation
2	Exploration bonus via cumulative mutual info
3	Mutual info-aware maximization of acquisition fn.
4	Update and iterate

3. Comparison to GP-UCB and Other Approaches

The classic GP-UCB algorithm uses a static or monotonically increasing exploration coefficient (typically $\beta_t = O(\log t)$), potentially causing "over-exploration" as $t$ increases: $$x_t = \arg\max_x \mu_t(x) + \sqrt{\beta_t} \sigma_t(x).$$ In contrast, GP-MI's exploration term explicitly decays as more information is acquired, focusing on the actual growth of information rather than iteration count, avoiding wasteful exploration.

Aspect	GP-MI (MI-aware)	GP-UCB
Exploration term	Adaptive, based on MI gathered	$O(\log t)$ (grows)
Regret bound (RBF)	$O((\log T)^{d+1})$ (exponential gain)	$O(T^{1/2} (\log T)^{d+1})$
Empirical performance	Lower regret, rapid convergence	More exploration/slower
Calibration sensitivity	Robust	Needs careful tuning

GP-MI has been experimentally shown to outperform both GP-UCB and the Expected Improvement (EI) heuristic across synthetic and real, high-dimensional, and multimodal optimization tasks, rapidly zeroing in on regions of interest without excessive exploration or premature exploitation.

4. Application Domains

Mutual information-aware optimization via GP-MI has demonstrated significant advantages on:

• Synthetic Benchmarks: Mixtures of Gaussians, Matern kernel GPs, Himmelblau, Branin, and Goldstein-Price functions.
• Engineering/Scientific Simulations: Tsunami run-up (parameter tuning for maximum effect), Mackey-Glass equations (chaotic, delayed systems), and other high-cost simulations.

In all tasks, GP-MI's information-driven exploration-exploitation tradeoff enables faster convergence, lower regret, and robust performance even as problem complexity grows.

5. Theoretical Impact and Future Directions

The paper is the first to demonstrate that cumulative regret in Bayesian optimization can be directly bounded in terms of the mutual information acquired, rather than more abstract measures or only as a function of time. This formal connection opens new avenues for:

• Broad application to any GP-based optimization setting (batches, parallel, universal GP algorithms).
• Scalable extensions, including sparse GPs, stochastic approximations, or non-GP surrogates.
• Generalization of MI-aware strategies to non-parametric and more complex optimization frameworks.

Future research is likely to focus on more scalable mutual information computation, richer observation models, and deployment to domains where judicious use of expensive evaluations is critical.

6. Implementation Considerations

• Scalability: Most computational cost arises from kernel matrix computations ($O(t^3)$ at iteration $t$), but since MI is only computed for observed points, sparsity and batch strategies can be leveraged.
• Parameter calibration: The MI-aware exploration schedule is robust to hyperparameter choices, simplifying deployment compared to methods needing careful tuning of $\beta_t$.
• Extension: The MI-aware exploration formulation portends advantages for batch, asynchronous, or non-GP Bayesian optimization with suitable surrogates.

Summary Table: Key Features

Aspect	GP-MI	GP-UCB
Exploration decay	via mutual information (adaptive)	grows with $\log t$
Regret bound	$O((\log T)^{d+1})$	$O(T^{1/2}(\log T)^{d+1})$
Empirical regret	Lower, faster convergence	Slower
Applications	Broad (simulation, science, engineering)	Same, but less optimal

The GP-MI algorithm illustrates the power and practical impact of mutual information-aware optimization: by adapting exploration to actual knowledge gained, it enables demonstrably more efficient, theoretically justified, and robust global optimization in expensive and challenging environments.