Multi Fidelity Optimization - Learning Module

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BOHB: Bayesian Optimization and Hyperband

Intelligent Sampling Meets Efficient Evaluation

Hyperband's power lies in efficient multi-fidelity evaluation, but it samples configurations uniformly at random. This ignores valuable information from completed evaluations. A configuration that performed well might suggest nearby configurations are also promising—yet Hyperband never exploits this structure.

BOHB (Bayesian Optimization and Hyperband), introduced by Falkner et al. (2018), combines Bayesian optimization's intelligent sampling with Hyperband's efficient evaluation. Rather than sampling randomly, BOHB builds a probabilistic model of the objective function and samples configurations likely to perform well.

This combination achieves the best of both worlds: the sample efficiency of Bayesian optimization and the computational efficiency of early stopping.

What You Will Learn

By the end of this page, you will understand: • How BOHB combines Bayesian optimization with Hyperband • Tree-structured Parzen Estimators (TPE) for configuration sampling • Multi-fidelity surrogate modeling strategies • Complete BOHB implementation • When BOHB outperforms vanilla Hyperband

BOHB Algorithm Overview

BOHB modifies Hyperband at a single point: configuration sampling. Instead of sampling uniformly from the search space, BOHB uses a Tree-structured Parzen Estimator (TPE) to sample configurations likely to perform well based on previous observations.

Algorithm Structure:

Run Hyperband's bracket/round structure exactly as before
When a new configuration is needed, use TPE to sample it
Update TPE model after each evaluation completes
Balance exploration (random sampling) with exploitation (TPE sampling)

Hyperband vs BOHB
Aspect	Hyperband	BOHB
Configuration sampling	Uniform random	TPE-guided
Model building	None	Kernel density estimation
Observations used	None	All completed evaluations
Overhead per sample	O(1)	O(n log n) for n observations
Best for	Unknown structure	Exploitable structure

The TPE Sampling Strategy:

TPE models the configuration space by fitting two density estimators:

(\ell(\lambda)): Density of configurations with good performance (top percentile)
(g(\lambda)): Density of configurations with bad performance (remaining)

New configurations are sampled to maximize the ratio (\ell(\lambda) / g(\lambda)), which approximates the Expected Improvement acquisition function.

The "good" threshold is typically the top 15-25% of observed performances, adapting as more observations accumulate.

Tree-Structured Parzen Estimators

TPE is a model-based optimization technique that differs fundamentally from Gaussian Process-based Bayesian optimization. Rather than modeling (p(y|\lambda)) (performance given configuration), TPE models (p(\lambda|y)) (configuration given performance).

Density Estimation:

Given (n) observations ({(\lambda_i, y_i)}_{i=1}^n), TPE:

Sorts by performance (y)
Splits at threshold (y^*) (e.g., 25th percentile)
Fits density (\ell(\lambda)) to configurations where (y < y^*)
Fits density (g(\lambda)) to configurations where (y \geq y^*)

For continuous hyperparameters, densities are modeled as truncated Gaussian mixtures. For discrete/categorical, densities use categorical distributions with smoothing.

tpe_sampler.py
Python
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import numpy as np
from scipy.stats import norm, truncnorm
from typing import List, Dict, Tuple, Any
 
class TPESampler:
    """
    Tree-structured Parzen Estimator for configuration sampling.
    """
    
    def __init__(
        self,
        search_space: Dict[str, Dict],
        gamma: float = 0.25,  # Top percentile for "good"
        n_candidates: int = 24,  # Candidates to evaluate
        bandwidth_factor: float = 1.0,
    ):
        self.search_space = search_space
        self.gamma = gamma
        self.n_candidates = n_candidates
        self.bandwidth_factor = bandwidth_factor
        
        self.observations: List[Tuple[Dict, float]] = []
    
    def _split_observations(self) -> Tuple[List[Dict], List[Dict]]:
        """Split observations into good (l) and bad (g) groups."""
        if len(self.observations) < 2:
            return [], []
        
        sorted_obs = sorted(self.observations, key=lambda x: x[1])
        n_good = max(1, int(len(sorted_obs) * self.gamma))
        
        good = [config for config, _ in sorted_obs[:n_good]]
        bad = [config for config, _ in sorted_obs[n_good:]]
        
        return good, bad
    
    def _estimate_density_continuous(
        self, 
        values: List[float], 
        param_config: Dict
    ) -> callable:
        """Fit truncated Gaussian KDE to continuous values."""
        low, high = param_config["low"], param_config["high"]
        
        if len(values) == 0:
            # Uniform prior
            return lambda x: 1.0 / (high - low)
        
        values = np.array(values)
        bandwidth = self.bandwidth_factor * np.std(values) + 1e-8
        
        def density(x):
            if x < low or x > high:
                return 0.0
            # Sum of truncated Gaussians centered at each observation
            pdf_sum = 0.0
            for v in values:
                a, b = (low - v) / bandwidth, (high - v) / bandwidth
                pdf_sum += truncnorm.pdf(x, a, b, loc=v, scale=bandwidth)
            return pdf_sum / len(values)
        
        return density
    
    def _sample_from_density(
        self, 
        density: callable, 
        param_config: Dict, 
        n_samples: int
    ) -> np.ndarray:
        """Sample from density using rejection sampling."""
        low, high = param_config["low"], param_config["high"]
        samples = []
        
        # Simple rejection sampling
        max_attempts = n_samples * 100
        attempts = 0
        
        while len(samples) < n_samples and attempts < max_attempts:
            x = np.random.uniform(low, high)
            acceptance = density(x) / (2.0 / (high - low))  # Approximate
            if np.random.random() < min(1.0, acceptance):
                samples.append(x)
            attempts += 1
        
        # Fall back to uniform if rejection sampling fails
        while len(samples) < n_samples:
            samples.append(np.random.uniform(low, high))
        
        return np.array(samples)
    
    def sample(self) -> Dict[str, Any]:
        """Sample a new configuration using TPE."""
        if len(self.observations) < 10:
            # Not enough data, sample uniformly
            return self._sample_uniform()
        
        good, bad = self._split_observations()
        
        if len(good) < 2 or len(bad) < 2:
            return self._sample_uniform()
        
        config = {}
        
        for param_name, param_config in self.search_space.items():
            if param_config["type"] == "continuous":
                good_vals = [c[param_name] for c in good]
                bad_vals = [c[param_name] for c in bad]
                
                l_density = self._estimate_density_continuous(good_vals, param_config)
                g_density = self._estimate_density_continuous(bad_vals, param_config)
                
                # Sample candidates from l, select best l/g ratio
                candidates = self._sample_from_density(
                    l_density, param_config, self.n_candidates
                )
                
                best_candidate = candidates[0]
                best_ratio = -float('inf')
                
                for c in candidates:
                    l_val = l_density(c) + 1e-10
                    g_val = g_density(c) + 1e-10
                    ratio = l_val / g_val
                    if ratio > best_ratio:
                        best_ratio = ratio
                        best_candidate = c
                
                config[param_name] = best_candidate
                
            elif param_config["type"] == "categorical":
                choices = param_config["choices"]
                good_counts = {c: 1 for c in choices}  # Laplace smoothing
                bad_counts = {c: 1 for c in choices}
                
                for c in good:
                    good_counts[c[param_name]] += 1
                for c in bad:
                    bad_counts[c[param_name]] += 1
                
                # Compute l/g ratio for each choice
                ratios = {}
                for choice in choices:
                    l_prob = good_counts[choice] / sum(good_counts.values())
                    g_prob = bad_counts[choice] / sum(bad_counts.values())
                    ratios[choice] = l_prob / (g_prob + 1e-10)
                
                # Sample proportionally to ratio
                total_ratio = sum(ratios.values())
                probs = [ratios[c] / total_ratio for c in choices]
                config[param_name] = np.random.choice(choices, p=probs)
        
        return config
    
    def _sample_uniform(self) -> Dict[str, Any]:
        """Sample uniformly from search space."""
        config = {}
        for param_name, param_config in self.search_space.items():
            if param_config["type"] == "continuous":
                if param_config.get("log", False):
                    log_val = np.random.uniform(
                        np.log(param_config["low"]),
                        np.log(param_config["high"])
                    )
                    config[param_name] = np.exp(log_val)
                else:
                    config[param_name] = np.random.uniform(
                        param_config["low"], param_config["high"]
                    )
            elif param_config["type"] == "categorical":
                config[param_name] = np.random.choice(param_config["choices"])
        return config
    
    def update(self, config: Dict[str, Any], performance: float):
        """Record observation for future sampling."""
        self.observations.append((config, performance))

Multi-Fidelity Surrogate Modeling

BOHB evaluates configurations at multiple fidelity levels. A key design question is: which observations should inform the TPE model?

Strategies:

Observation Aggregation Strategies

•Highest fidelity only: Use only observations from maximum budget. Most accurate but fewest observations.
•All fidelities pooled: Treat all observations equally. Many observations but mixing fidelity levels may confuse the model.
•Fidelity-weighted: Weight observations by fidelity level when building densities. Balances quantity and quality.
•Per-fidelity models: Build separate TPE models for each fidelity. More complex but handles fidelity-dependent structure.

BOHB's Approach:

The original BOHB uses a pragmatic strategy:

Per-bracket models: Each bracket maintains its own observation history
Highest completed fidelity: For each configuration, use only its highest-fidelity observation
Cross-bracket sharing: Optionally share information across brackets at the same fidelity

This balances the need for relevant observations (same fidelity) with sample efficiency (cross-bracket sharing).

Fidelity Correlation Matters

When low-fidelity and high-fidelity performance are highly correlated, using all fidelities improves sample efficiency. When correlation is weak, mixing fidelities can mislead the model. BOHB's per-bracket approach provides a reasonable middle ground.

Complete BOHB Implementation

bohb.py
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"""
BOHB: Bayesian Optimization and Hyperband
"""
import numpy as np
from typing import Dict, List, Callable, Any, Tuple
from dataclasses import dataclass, field
import logging
 
logger = logging.getLogger(__name__)
 
@dataclass
class BOHBConfig:
    config_id: str
    bracket: int
    hyperparameters: Dict[str, Any]
    performances: Dict[int, float] = field(default_factory=dict)
    status: str = "active"
 
class BOHB:
    """
    BOHB: Combines Hyperband scheduling with TPE sampling.
    """
    
    def __init__(
        self,
        search_space: Dict[str, Dict],
        max_budget: float = 81,
        eta: int = 3,
        random_fraction: float = 0.33,
        evaluator: Callable[[Dict, float], float] = None,
    ):
        self.search_space = search_space
        self.max_budget = max_budget
        self.eta = eta
        self.random_fraction = random_fraction
        self.evaluator = evaluator
        
        self.s_max = int(np.log(max_budget) / np.log(eta))
        
        # Per-bracket TPE samplers
        from tpe_sampler import TPESampler
        self.samplers = {
            s: TPESampler(search_space) for s in range(self.s_max + 1)
        }
        
        self.all_configs: List[BOHBConfig] = []
        self.config_counter = 0
    
    def _sample_config(self, bracket: int) -> Dict[str, Any]:
        """Sample configuration using TPE or random."""
        if np.random.random() < self.random_fraction:
            return self._sample_random()
        return self.samplers[bracket].sample()
    
    def _sample_random(self) -> Dict[str, Any]:
        """Uniform random sampling."""
        config = {}
        for name, spec in self.search_space.items():
            if spec["type"] == "continuous":
                if spec.get("log", False):
                    val = np.exp(np.random.uniform(
                        np.log(spec["low"]), np.log(spec["high"])
                    ))
                else:
                    val = np.random.uniform(spec["low"], spec["high"])
                config[name] = val
            elif spec["type"] == "categorical":
                config[name] = np.random.choice(spec["choices"])
        return config
    
    def run_bracket(self, s: int) -> List[BOHBConfig]:
        """Run one Successive Halving bracket with TPE sampling."""
        n = int(np.ceil((self.s_max + 1) / (s + 1) * (self.eta ** s)))
        r = self.max_budget * (self.eta ** (-s))
        num_rounds = s + 1
        
        logger.info(f"Bracket s={s}: n={n}, r={r:.1f}")
        
        # Sample configurations using TPE
        configs = []
        for i in range(n):
            hp = self._sample_config(s)
            config = BOHBConfig(
                config_id=f"b{s}_c{self.config_counter}",
                bracket=s,
                hyperparameters=hp
            )
            configs.append(config)
            self.all_configs.append(config)
            self.config_counter += 1
        
        survivors = list(range(n))
        
        for round_idx in range(num_rounds):
            budget = r * (self.eta ** round_idx)
            budget = min(budget, self.max_budget)
            n_keep = max(1, len(survivors) // self.eta)
            
            # Evaluate survivors
            results = []
            for idx in survivors:
                config = configs[idx]
                if self.evaluator:
                    perf = self.evaluator(config.hyperparameters, budget)
                else:
                    perf = self._simulate(config.hyperparameters, budget)
                
                config.performances[round_idx] = perf
                results.append((idx, perf))
                
                # Update TPE sampler
                self.samplers[s].update(config.hyperparameters, perf)
            
            # Keep top performers
            results.sort(key=lambda x: x[1])
            survivors = [idx for idx, _ in results[:n_keep]]
            
            for idx, _ in results[n_keep:]:
                configs[idx].status = "eliminated"
        
        for idx in survivors:
            configs[idx].status = "completed"
        
        return configs
    
    def run(self, n_iterations: int = 1) -> BOHBConfig:
        """Run BOHB for specified iterations."""
        for iteration in range(n_iterations):
            logger.info(f"
=== BOHB Iteration {iteration + 1} ===")
            
            for s in range(self.s_max, -1, -1):
                self.run_bracket(s)
        
        # Find best
        completed = [c for c in self.all_configs if c.status == "completed"]
        best = min(completed, key=lambda c: list(c.performances.values())[-1])
        
        logger.info(f"Best: {best.config_id}, perf={list(best.performances.values())[-1]:.4f}")
        return best
    
    def _simulate(self, hp: Dict, budget: float) -> float:
        """Simulated evaluator."""
        lr = hp.get("learning_rate", 0.001)
        quality = -((np.log10(lr) + 3) ** 2) / 4
        progress = 1 - np.exp(-budget / 30)
        return 0.5 + 0.2 * quality * (1 - 0.8 * progress) + np.random.normal(0, 0.01)

When to Use BOHB

BOHB Excels When

•Search space has exploitable structure
•Good configurations cluster together
•Many hyperparameters interact
•Sufficient budget for model building
•Continuous hyperparameters dominate

Prefer Plain Hyperband When

•Search space is highly discrete
•Good configs are scattered randomly
•Very few evaluations possible
•Parameters are independent
•Computational overhead is critical

Practical Recommendation

Start with BOHB as default for neural network hyperparameter optimization where learning rate, weight decay, and architecture parameters form a structured, continuous space. Fall back to vanilla Hyperband if BOHB shows no improvement after ~50 configurations or if the overhead becomes problematic.

Summary: BOHB

Key Takeaways

•BOHB = Hyperband + TPE: Intelligent sampling within Hyperband's efficient evaluation framework
•TPE models l(λ)/g(λ): Samples configurations likely to be in the 'good' region based on past observations
•Per-bracket modeling: Each bracket maintains separate TPE model for appropriate fidelity
•Random fraction: Balances exploration (random) with exploitation (TPE-guided)
•Best for structured spaces: When good hyperparameters cluster, BOHB dramatically improves over random sampling

Looking Ahead:

The final page covers Budget Allocation—strategies for distributing computational resources across hyperparameter optimization, including how to balance parallel workers, handle heterogeneous hardware, and manage optimization campaigns over time.

Page Complete

You now understand how BOHB combines Bayesian optimization with Hyperband, including TPE sampling, multi-fidelity modeling, and when to use BOHB over vanilla Hyperband.