Bayesian Optimization for Hyperparameter Tuning of Deep Learning Models

to tune hyperparamters of deep studying fashions (Keras Sequential model), as compared with a standard method — Grid Search.

Bayesian Optimization

Bayesian Optimization is a sequential design technique for international optimization of black-box features.

It’s notably well-suited for features which can be costly to guage, lack an analytical kind, or have unknown derivatives.
Within the context of hyperparameter optimization, the unknown perform might be:

• an goal perform,
• accuracy worth for a coaching or validation set,
• loss worth for a coaching or validation set,
• entropy gained or misplaced,
• AUC for ROC curves,
• A/B take a look at outcomes,
• computation value per epoch,
• mannequin measurement,
• reward quantity for reinforcement studying, and extra.

In contrast to conventional optimization strategies that depend on direct perform evaluations, Bayesian Optimization builds and refines a probabilistic mannequin of the target perform, utilizing this mannequin to intelligently choose the following analysis level.

The core thought revolves round two key parts:

1. Surrogate Mannequin (Probabilistic Mannequin)

The mannequin approximates the unknown goal perform (f(x)) to a surrogate mannequin reminiscent of Gaussian Course of (GP).

A GP is a non-parametric Bayesian mannequin that defines a distribution over features. It present:

• a prediction of the perform worth at a given level μ(x) and
• a measure of uncertainty round that prediction σ(x), typically represented as a confidence interval.

Mathematically, for a Gaussian Course of, the predictions at an unobserved level (x∗), given noticed knowledge (X, y), are usually distributed:

the place

• μ(x∗): the imply prediction and
• σ²(x∗): the predictive variance.

2. Acquisition Perform

The acquisition perform determines a subsequent level (x_t+1)​ to guage by quantifying how “promising” a candidate level is for enhancing the target perform, by balancing:

• Exploration (Excessive Variance): Sampling in areas with excessive uncertainty to find new promising areas and
• Exploitation (Excessive Imply): Sampling in areas the place the surrogate mannequin predicts excessive goal values.

Widespread acquisition features embrace:

Likelihood of Enchancment (PI)
PI selects the purpose that has the very best likelihood of enhancing upon the present finest noticed worth (f(x+)):

the place

• Φ: the cumulative distribution perform (CDF) of the usual regular distribution, and
• ξ≥0 is a trade-off parameter (exploration vs. exploitation).

ξ controls a trade-off between exploration and exploitation, and a bigger ξ encourages extra exploration.

Anticipated Enchancment (EI)
Quantifies the anticipated quantity of enchancment over the present finest noticed worth:

Assuming a Gaussian Course of surrogate, the analytical type of EI is outlined:

the place ϕ is the likelihood density perform (PDF) of the usual regular distribution.

EI is likely one of the most generally used acquisition features. EI additionally considers the magnitude of the advance not like PI.

Higher Confidence Sure (UCB)
UCB balances exploitation (excessive imply) and exploration (excessive variance), specializing in factors which have each a excessive predicted imply and excessive uncertainty:

κ≥0 is a tuning parameter that controls the stability between exploration and exploitation.

A bigger κ places extra emphasis on exploring unsure areas.

Bayesian Optimization Technique (Iterative Course of)

Bayesian Optimization iteratively updates the surrogate mannequin and optimizes the acquisition perform.

It guides the search in direction of optimum areas whereas minimizing the variety of costly goal perform evaluations.

Now, allow us to see the method with code snippets utilizing KerasTuner for a fraud detection job (binary classification the place y=1 (fraud) prices us essentially the most.)

Step 1. Initialization

Initializes the method by sampling the hyperparameter house randomly or low-discrepancy sequencing (ususally selecting up 5 to 10 factors) to get an thought of the target perform.

These preliminary observations are used to construct the primary model of the surrogate mannequin.

As we construct Keras Sequential mannequin, we first outline and compile the mannequin, then outline theBayesianOptimization tuner with the variety of preliminary factors to evaluate.

import keras_tuner as kt
import tensorflow as tf
from tensorflow import keras
from keras.fashions import Sequential
from keras.layers import Dense, Dropout, Enter

# initialize a Keras Sequential mannequin
mannequin = Sequential([
    Input(shape=(self.input_shape,)),
    Dense(
        units=hp.Int(
            'neurons1', min_value=20, max_value=60, step=10),
             activation='relu'
    ),
    Dropout(
        hp.Float(
             'dropout_rate1', min_value=0.0, max_value=0.5, step=0.1
    )),
    Dense(
        units=hp.Int(
            'neurons2', min_value=20, max_value=60, step=10),
            activation='relu'
    ),
    Dropout(
         hp.Float(
              'dropout_rate2', min_value=0.0, max_value=0.5, step=0.1
    )),
    Dense(
         1, activation='sigmoid', 
         bias_initializer=keras.initializers.Constant(
             self.initial_bias_value
        )
    )
])

# compile the mannequin
mannequin.compile(
    optimizer=optimizer,
    loss='binary_crossentropy',
    metrics=[
        'accuracy',
        keras.metrics.Precision(name='precision'),
        keras.metrics.Recall(name='recall'),
        keras.metrics.AUC(name='auc')
    ]
)

# outline a tuner with the intial factors
tuner = kt.BayesianOptimization(
    hypermodel=custom_hypermodel,
    goal=kt.Goal("val_recall", route="max"), 
    max_trials=max_trials,
    executions_per_trial=executions_per_trial,
    listing=listing,
    project_name=project_name,
    num_initial_points=num_initial_points,
    overwrite=True,
)

num_initial_points defines what number of preliminary, randomly chosen hyperparameter configurations must be evaluated earlier than the algorithm begins to information the search.

If not given, KerasTuner takes a default worth: 3 * dimensions of the hyperparameter house.

Step 2. Surrogate Mannequin Coaching

Construct and practice the probabilistic mannequin (surrogate mannequin, typically a Gaussian Course of or a Tree-structured Parzen Estimator for Bayesian Optimization) utilizing all accessible noticed datas factors (enter values and their corresponding output values) to approximate the true perform.

The surrogate mannequin offers the imply prediction (μ(x)) (most definitely from the Gaussian course of) and uncertainty (σ(x)) for any unobserved level.

KerasTuner makes use of an inner surrogate mannequin to mannequin the connection between hyperparameters and the target perform’s efficiency.

After every goal perform analysis by way of practice run, the noticed knowledge factors (hyperparameters and validation metrics) are used to replace the inner surrogate mannequin.

Step 3. Acquisition Perform Optimization

Use an optimization algorithm (typically an inexpensive, native optimizer like L-BFGS and even random search) to search out the following level (x_t+1)​ that maximizes the chosen acquisition perform.

This step is essential as a result of it identifies essentially the most promising subsequent candidate for analysis by balancing exploration (making an attempt new, unsure areas of the hyperparameter house) and exploitation (refining promising areas).

KerasTuner makes use of an optimization technique reminiscent of Anticipated Enchancment or Higher Confidence Sure to search out the following set of hyperparameters.

Step 4. Goal Perform Analysis

Consider the true, costly goal perform (f(x)) on the new candidate level (x_t+1)​.

The Keras mannequin is skilled utilizing the offered coaching datasets and evaluated on the validation knowledge. We set val_recall as the results of this analysis.

def match(self, hp, mannequin=None, *args, **kwargs):
    mannequin = self.construct(hp=hp) if not mannequin else mannequin
    batch_size = hp.Alternative('batch_size', values=[16, 32, 64])
    epochs = hp.Int('epochs', min_value=50, max_value=200, step=50)
  
    return mannequin.match(
        batch_size=batch_size,
        epochs=epochs,
        class_weight=self.class_weights_dict,
        *args,
        **kwargs
    )

Step 5. Information Replace

Add the newly noticed knowledge level (x_(t+1​), f(x_(t+1)​)) to the set of observations.

Step 6. Iteration

Repeat Step 2 — 5 till a stopping criterion is met.

Technically, the tuner.search() methodology orchestrates your entire Bayesian optimization course of from Step 2 to five:

tuner.search(
    X_train, y_train,
    validation_data=(X_val, y_val),
    callbacks=[early_stopping_callback]
)

best_hp = tuner.get_best_hyperparameters(num_trials=1)[0]
best_keras_model_from_tuner = tuner.get_best_models(num_models=1)[0]

The tactic repeatedly performs these steps till the max_trials restrict is reached or different inner stopping standards reminiscent of early_stopping_callback are met.

Right here, we set recall as our key metrics to penalize the misclassification as False Constructive prices us essentially the most within the fraud detection case.

Be taught Extra: KerasTuner Source Code

Outcomes

The Bayesian Optimization course of aimed to boost the mannequin’s efficiency, primarily by maximizing recall.

The tuning efforts yielded a trade-off throughout key metrics, leading to a mannequin with considerably improved recall on the expense of some precision and general accuracy in comparison with the preliminary state:

• Recall: 0.9055 (0.6595 -> 0.6450) — 0.8400
• Precision: 0.6831 (0.8338 -> 0.8113) — 0.6747
• Accuracy: 0.7427 (0.7640 -> 0.7475) — 0.7175
  (From growth (coaching / validation mixed) to check part)

Finest performing hyperparameter set:

• neurons1: 40
• dropout_rate1: 0.0
• neurons2: 20,
• dropout_rate2: 0.4
• optimizer_name: lion,
• learning_rate: 0.004019639999963362
• batch_size: 64
• epochs: 200
• beta_1_lion: 0.9
• beta_2_lion: 0.99

Optimum Neural Community Abstract:

Key Efficiency Metrics:

• Recall: The mannequin demonstrated a big enchancment in recall, rising from an preliminary worth of roughly 0.66 (or 0.645) to 0.8400. This means the optimized mannequin is notably higher at figuring out optimistic circumstances.
• Precision: Concurrently, precision skilled a lower. Ranging from round 0.83 (or 0.81), it settled at 0.6747 post-optimization. This means that whereas extra optimistic circumstances are being recognized, the next proportion of these identifications is perhaps false positives.
• Accuracy: The general accuracy of the mannequin additionally noticed a decline, shifting from an preliminary 0.7640 (or 0.7475) right down to 0.7175. That is according to the noticed trade-off between recall and precision, the place optimizing for one typically impacts the others.

Evaluating with Grid Search

We tuned a Keras Sequential mannequin with Grid Search on Adam optimizer for comparability:

import tensorflow as tf
from tensorflow import keras
from keras.fashions import Sequential
from keras.layers import Dense, Dropout, Enter
from sklearn.model_selection import GridSearchCV
from scikeras.wrappers import KerasClassifier

param_grid = {
    'model__learning_rate': [0.001, 0.0005, 0.0001],
    'model__neurons1': [20, 30, 40],
    'model__neurons2': [20, 30, 40],
    'model__dropout_rate1': [0.1, 0.15, 0.2],
    'model__dropout_rate2': [0.1, 0.15, 0.2],
    'batch_size': [16, 32, 64],
    'epochs': [50, 100],
}

input_shape = X_train.form[1]
initial_bias = np.log([np.sum(y_train == 1) / np.sum(y_train == 0)])
class_weights = class_weight.compute_class_weight(
    class_weight='balanced',
    lessons=np.distinctive(y_train),
    y=y_train
)
class_weights_dict = dict(zip(np.distinctive(y_train), class_weights))

keras_classifier = KerasClassifier(
    mannequin=create_model,
    model__input_shape=input_shape,
    model__initial_bias_value=initial_bias,
    loss='binary_crossentropy',
    metrics=[
        'accuracy',
        keras.metrics.Precision(name='precision'),
        keras.metrics.Recall(name='recall'),
        keras.metrics.AUC(name='auc')
    ]
)

grid_search = GridSearchCV(
    estimator=keras_classifier,
    param_grid=param_grid,
    scoring='recall',
    cv=3,
    n_jobs=-1,
    error_score='elevate'
)

grid_result = grid_search.match(
    X_train, y_train,
    validation_data=(X_val, y_val),
    callbacks=[early_stopping_callback],
    class_weight=class_weights_dict
)

optimal_params = grid_result.best_params_
best_keras_classifier = grid_result.best_estimator_

Outcomes

Grid Search tuning resulted in a mannequin with sturdy precision and good general accuracy, although with a decrease recall in comparison with the Bayesian Optimization method:

• Recall: 0.8214(0.7735 -> 0.7150)— 0.7100
• Precision: 0.7884 (0.8331 -> 0.8034) — 0.8304
• Accuracy:0.8005 (0.8092 -> 0.7700) — 0.7825

Finest performing hyperparameter set:

• neurons1: 40
• dropout_rate1: 0.15
• neurons2: 40
• dropout_rate2: 0.1
• learning_rate: 0.001
• batch_size: 16
• epochs: 100

Optimum Neural Community Abstract:

Grid Search Efficiency:

• Recall: Achieved a recall of 0.7100, a slight lower from its preliminary vary (0.7735–0.7150).
• Precision: Confirmed sturdy efficiency at 0.8304, an enchancment over its preliminary vary (0.8331–0.8034).
• Accuracy: Settled at 0.7825, sustaining a stable general predictive functionality, barely decrease than its preliminary vary (0.8092–0.7700).

Comparability with Bayesian Optimization:

• Recall: Bayesian Optimization (0.8400) considerably outperformed Grid Search (0.7100) in figuring out optimistic circumstances.
• Precision: Grid Search (0.8304) achieved a lot increased precision than Bayesian Optimization (0.6747), indicating fewer false positives.
• Accuracy: Grid Search’s accuracy (0.7825) was notably increased than Bayesian Optimization’s (0.7175).

Common Comparability with Grid Search

1. Approaching the Search House

Bayesian Optimization

• Clever/Adaptive: Bayesian Optimization builds a probabilistic mannequin (typically a Gaussian Course of) of the target perform (e.g., mannequin efficiency as a perform of hyperparameters). It makes use of this mannequin to foretell which hyperparameter mixtures are most definitely to yield higher outcomes.
• Knowledgeable: It learns from earlier evaluations. After every trial, the probabilistic mannequin is up to date, guiding the search in direction of extra promising areas of the hyperparameter house. This permits it to make “clever” decisions about the place to pattern subsequent, balancing exploration (making an attempt new, unknown areas) and exploitation (specializing in areas which have proven good outcomes).
• Sequential: It usually operates sequentially, evaluating one level at a time and updating its mannequin earlier than deciding on the following.

Grid Search:

• Exhaustive/Brute-force: Grid Search systematically tries each potential mixture of hyperparameter values from a pre-defined set of values for every hyperparameter. You specify a “grid” of values, and it evaluates each level on that grid.
• Uninformed: It doesn’t use the outcomes of earlier evaluations to tell the collection of the following set of hyperparameters to attempt. Every mixture is evaluated independently.
• Deterministic: Given the identical grid, it’ll at all times discover the identical mixtures in the identical order.

2. Computational Value

Bayesian Optimization

• Extra Environment friendly: Designed to search out optimum hyperparameters with considerably fewer evaluations in comparison with Grid Search. This makes it notably efficient when evaluating the target perform (e.g., coaching a Machine Learning mannequin) is computationally costly or time-consuming.
• Scalability: Typically scales higher to higher-dimensional hyperparameter areas than Grid Search, although it might probably nonetheless be computationally intensive for very excessive dimensions because of the overhead of sustaining and updating the probabilistic mannequin.

Grid Search

• Computationally Costly: Because the variety of hyperparameters and the vary of values for every hyperparameter enhance, the variety of mixtures grows exponentially. This results in very future occasions and excessive computational value, making it impractical for big search areas. That is also known as the “curse of dimensionality.”
• Scalability: Doesn’t scale effectively with high-dimensional hyperparameter areas.

3. Ensures and Exploration

Bayesian Optimization

• Probabilistic assure: It goals to search out the worldwide optimum effectively, however it does not provide a tough assure like Grid Seek for discovering the very best inside a discrete set. As a substitute, it converges probabilistically in direction of the optimum.
• Smarter exploration: Its stability of exploration and exploitation helps it keep away from getting caught in native optima and uncover optimum values extra successfully.

Grid Search

• Assured to search out finest in grid: If the optimum hyperparameters are inside the outlined grid, Grid Search is assured to search out them as a result of it tries each mixture.
• Restricted exploration: It might probably miss optimum values in the event that they fall between the discrete factors outlined within the grid.

4. When to Use Which

Bayesian Optimization:

• Massive, high-dimensional hyperparameter areas: When evaluating fashions is pricey and you’ve got many hyperparameters to tune.
• When effectivity is paramount: To search out good hyperparameters rapidly, particularly in conditions with restricted computational sources or time.
• Black-box optimization issues: When the target perform is advanced, non-linear, and doesn’t have a identified analytical kind.

Grid Search

• Small, low-dimensional hyperparameter areas: When you might have only some hyperparameters and a restricted variety of values for every, Grid Search could be a easy and efficient alternative.
• When exhaustiveness is essential: In case you completely have to discover each single outlined mixture.

Conclusion

The experiment successfully demonstrated the distinct strengths of Bayesian Optimization and Grid Search in hyperparameter tuning.
Bayesian Optimization, by design, proved extremely efficient at intelligently navigating the search house and prioritizing a selected goal, on this case, maximizing recall.

It efficiently achieved the next recall fee (0.8400) in comparison with Grid Search, indicating its capability to search out extra optimistic cases.
This functionality comes with an inherent trade-off, resulting in diminished precision and general accuracy.

Such an end result is very priceless in functions the place minimizing false negatives is essential (e.g., medical prognosis, fraud detection).
Its effectivity, stemming from probabilistic modeling that guides the search in direction of promising areas, makes it a most popular methodology for optimizing pricey experiments or simulations the place every analysis is pricey.

In distinction, Grid Search, whereas exhaustive, yielded a extra balanced mannequin with superior precision (0.8304) and general accuracy (0.7825).

This means Grid Search was extra conservative in its predictions, leading to fewer false positives.

In abstract, whereas Grid Search gives an easy and exhaustive method, Bayesian Optimization stands out as a extra subtle and environment friendly methodology able to find superior outcomes with fewer evaluations, notably when optimizing for a selected, typically advanced, goal like maximizing recall in a high-dimensional house.

The optimum alternative of tuning methodology finally relies on the particular efficiency priorities and useful resource constraints of the appliance.

Writer: Kuriko IWAI
Portfolio / LinkedIn / Github
Might 26, 2025

All pictures, until in any other case famous, are by the creator.
The article makes use of artificial knowledge, licensed under Apache 2.0 for commercial use.