Bayesian Optimization of Function Networks with Partial Evaluations

Bayesian Optimization is a powerful technique for optimizing black-box functions, especially when these functions are expensive to evaluate. It is widely used in machine learning, engineering, and hyperparameter tuning due to its ability to handle noisy, costly, or unknown objective functions. When it comes to function networks, which consist of multiple interconnected functions that work together, optimization can become more complex. Adding another layer of complexity is the idea of partial evaluations, where only a subset of the network’s functions can be evaluated at a given time.

In this article, we’ll explore the concept of Bayesian optimization of function networks with a focus on partial evaluations, diving deep into how this method works, its advantages, and its applications in various fields.

What is Bayesian Optimization?

Bayesian Optimization (BO) is a strategy for optimizing functions that are expensive or time-consuming to evaluate. Instead of relying on traditional gradient-based methods, which require many function evaluations, BO uses probabilistic models (like Gaussian Processes) to model the objective function. This model is then used to guide the search for the function’s optimum by focusing on areas where the model predicts high potential gains.

The main steps involved in Bayesian Optimization include:

1. Surrogate Model Creation The core of Bayesian Optimization is a surrogate model (commonly a Gaussian Process or Random Forest) that approximates the objective function. The surrogate model provides a probabilistic estimate of the function’s behavior, predicting both the expected value of the function and the uncertainty in that prediction.
2. Acquisition Function An acquisition function is used to balance the exploration (searching unknown regions) and exploitation (focusing on promising regions) of the surrogate model. Popular acquisition functions include:
   • Expected Improvement (EI)
   • Upper Confidence Bound (UCB)
   • Probability of Improvement (PI)
3. Iterative Optimization The process iteratively selects new points to evaluate based on the acquisition function, updates the surrogate model with the new data, and repeats the cycle until convergence or budget constraints are met.

What are Function Networks?

Function Networks refer to a group of functions that are interrelated or dependent on each other. These can be used to model systems or processes where multiple objectives or constraints need to be optimized together. Each node in the network represents a function, and the edges between nodes signify dependencies or relationships between these functions.

For example, in machine learning models, each layer in a deep neural network could be thought of as a “function” within the broader network of the model. Similarly, in complex engineering systems, various subsystems (each modeled by a function) may interact to produce a final output.

Partial Evaluations in Function Networks

Partial evaluations occur when not all functions in the network can be fully evaluated due to various constraints—such as time, resources, or computational limits. This creates a challenge because evaluating only parts of a system may provide incomplete information, yet we still need to make optimization decisions based on this partial knowledge.

In a function network with partial evaluations, only some of the nodes (functions) in the network are evaluated at each iteration. The goal is to leverage the information from these partial evaluations to make educated guesses about the rest of the network and guide the optimization process effectively.

Bayesian Optimization of Function Networks with Partial Evaluations

Combining Bayesian Optimization with function networks and handling partial evaluations is a complex task that requires balancing between exploring different parts of the network and exploiting the known relationships between functions.

Here’s how this method works:

1. Surrogate Modeling for Function Networks The surrogate model in this case doesn’t just model a single function, but rather a network of interdependent functions. It incorporates both the partial evaluations from the current iteration and the correlations between functions across the network. For example, Gaussian Processes with multi-output capabilities can be used to model such networks, allowing predictions for nodes that haven’t been fully evaluated.
2. Incorporating Partial Evaluations When only a subset of functions in the network is evaluated, the model uses the available information to predict the behavior of the remaining functions. This allows the optimization process to continue even when complete information is not available. The uncertainty in the surrogate model increases for unevaluated nodes, but Bayesian Optimization can handle this uncertainty using the acquisition function.
3. Acquisition Function in Networks The acquisition function used in this setting takes into account both the evaluated and unevaluated nodes in the network. It must balance the trade-off between exploring parts of the network with high uncertainty (i.e., where few or no evaluations have been made) and exploiting parts of the network that appear to offer promising results based on previous evaluations.
4. Iterative Process Like traditional Bayesian Optimization, this method operates iteratively. At each step, it decides which parts of the network to evaluate next based on the acquisition function and the current state of the surrogate model. This ensures that computational resources are allocated efficiently, focusing on the most informative parts of the network.

Advantages of Bayesian Optimization with Partial Evaluations

There are several benefits to using Bayesian Optimization for function networks, particularly in settings where only partial evaluations are possible:

1. Efficiency Partial evaluations allow for more cost-effective optimization. Instead of evaluating the entire network—which may be expensive or time-consuming—the optimizer can make progress by evaluating only a subset of the functions.
2. Handling Complexity Function networks often represent complex systems with interdependencies. Bayesian Optimization’s probabilistic modeling is well-suited to handling this complexity by modeling the relationships between functions, even when full evaluations are not available.
3. Flexibility This approach is highly adaptable, making it useful for a variety of applications, from hyperparameter tuning in machine learning models to optimizing engineering systems with multiple components.
4. Uncertainty Quantification Bayesian methods naturally provide a measure of uncertainty in their predictions, which is particularly valuable when working with partial evaluations. The optimizer can focus on areas with high uncertainty, leading to better exploration of the function space.

Applications of Bayesian Optimization in Function Networks

Bayesian Optimization of function networks with partial evaluations has a broad range of applications across various domains:

1. Hyperparameter Tuning in Machine Learning In deep learning, for example, different layers or architectures of a neural network can be viewed as part of a function network. Bayesian Optimization can be used to optimize hyperparameters by evaluating only some aspects of the model (such as subsets of layers) at each iteration, leading to faster convergence.
2. Engineering Design Optimization In fields like aerospace, automotive design, or electronics, complex systems are often broken down into interdependent subsystems. Partial evaluations in such systems allow for efficient exploration of the design space without having to simulate or test every component in full detail.
3. Drug Discovery In pharmaceutical research, multiple processes or biological pathways (modeled as functions) interact to determine the effectiveness of a drug. Bayesian Optimization can help navigate the large search space of potential drug formulations with partial data, speeding up the discovery process.
4. Financial Modeling In finance, complex models often incorporate several interrelated functions (e.g., market conditions, risk factors). Using partial evaluations allows for optimizing investment strategies without fully evaluating all possible market conditions, saving both time and computational power.

Challenges and Future Directions

While Bayesian Optimization of function networks with partial evaluations offers many advantages, it also presents certain challenges:

• Scalability: As the size of the function network grows, the computational complexity of the surrogate model and acquisition function increases. Efficient implementations are needed to ensure scalability.
• Modeling Complexity: Accurately modeling the relationships between functions, especially when only partial data is available, can be challenging. Advanced models like deep Gaussian processes may offer a solution, but they introduce additional computational overhead.
• Exploration vs. Exploitation: Striking the right balance between exploring unknown regions of the network and exploiting known regions is key to achieving optimal results.

Conclusion

Bayesian Optimization of function networks with partial evaluations is a sophisticated and efficient method for optimizing complex systems, especially when full evaluations are costly or impractical. By leveraging surrogate models, acquisition functions, and probabilistic reasoning, this technique allows for intelligent exploration of function spaces while making the best use of limited evaluation budgets.

This approach has broad applications across fields like machine learning, engineering, and finance, where systems are often composed of interdependent components and resources for evaluation are limited. By embracing the principles of Bayesian Optimization, organizations can achieve faster, more efficient solutions to their optimization problems.