Bayesian Optimization for Hyperparameter Tuning of Deep Studying Fashions

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

Bayesian Optimization

Bayesian Optimization is a sequential design technique for world optimization of black-box capabilities.

It’s notably well-suited for capabilities which are costly to guage, lack an analytical kind, or have unknown derivatives.
Within the context of hyperparameter optimization, the unknown perform could 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 subsequent analysis level.

The core concept revolves round two key parts:

1. Surrogate Mannequin (Probabilistic Mannequin)

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

A GP is a non-parametric Bayesian mannequin that defines a distribution over capabilities. 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 Operate

The acquisition perform determines a subsequent level (x_t+1)​ to guage by quantifying how “promising” a candidate level is for bettering 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 capabilities embody:

Likelihood of Enchancment (PI)
PI selects the purpose that has the best chance of bettering 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 chance density perform (PDF) of the usual regular distribution.

EI is among the most generally used acquisition capabilities. EI additionally considers the magnitude of the development in contrast to 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 steadiness 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 the 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 activity (binary classification the place y=1 (fraud) prices us probably the most.)

Step 1. Initialization

Initializes the method by sampling the hyperparameter area randomly or low-discrepancy sequencing (ususally choosing up 5 to 10 factors) to get an concept 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 needs to be evaluated earlier than the algorithm begins to information the search.

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

Step 2. Surrogate Mannequin Coaching

Construct and prepare 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 gives the imply prediction (μ(x)) (almost definitely from the Gaussian course of) and uncertainty (σ(x)) for any unobserved level.

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

After every goal perform analysis through prepare run, the noticed knowledge factors (hyperparameters and validation metrics) are used to replace the interior surrogate mannequin.

Step 3. Acquisition Operate Optimization

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

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

KerasTuner makes use of an optimization technique similar to Anticipated Enchancment or Higher Confidence Sure to seek out the subsequent set of hyperparameters.

Step 4. Goal Operate Analysis

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

The Keras mannequin is skilled utilizing the supplied 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. Knowledge 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() technique orchestrates all the 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 inside stopping standards similar to early_stopping_callback are met.

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

Be taught Extra: KerasTuner Source Code

Outcomes

The Bayesian Optimization course of aimed to reinforce 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 major enchancment in recall, growing 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 constructive instances.
• 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 constructive instances are being recognized, the next proportion of these identifications is perhaps false positives.
• Accuracy: The general accuracy of the mannequin additionally noticed a decline, transferring from an preliminary 0.7640 (or 0.7475) right down to 0.7175. That is in step with 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='increase'
)

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 strategy:

• 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 strong 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 constructive instances.
• Precision: Grid Search (0.8304) achieved a lot larger precision than Bayesian Optimization (0.6747), indicating fewer false positives.
• Accuracy: Grid Search’s accuracy (0.7825) was notably larger than Bayesian Optimization’s (0.7175).

Normal Comparability with Grid Search

1. Approaching the Search Area

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 combos are almost 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 the direction of extra promising areas of the hyperparameter area. This enables it to make “clever” selections 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 sometimes operates sequentially, evaluating one level at a time and updating its mannequin earlier than deciding on the subsequent.

Grid Search:

• Exhaustive/Brute-force: Grid Search systematically tries each doable 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 choice of the subsequent set of hyperparameters to strive. Every mixture is evaluated independently.
• Deterministic: Given the identical grid, it can at all times discover the identical combos in the identical order.

2. Computational Value

Bayesian Optimization

• Extra Environment friendly: Designed to seek 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 Studying mannequin) is computationally costly or time-consuming.
• Scalability: Typically scales higher to higher-dimensional hyperparameter areas than Grid Search, although it could possibly nonetheless be computationally intensive for very excessive dimensions as a result of 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 combos grows exponentially. This results in very long term instances and excessive computational value, making it impractical for giant search areas. That is also known as the “curse of dimensionality.”
• Scalability: Doesn’t scale nicely with high-dimensional hyperparameter areas.

3. Ensures and Exploration

Bayesian Optimization

• Probabilistic assure: It goals to seek out the worldwide optimum effectively, nevertheless it does not supply a tough assure like Grid Seek for discovering the best possible inside a discrete set. As an alternative, it converges probabilistically in the direction of the optimum.
• Smarter exploration: Its steadiness of exploration and exploitation helps it keep away from getting caught in native optima and uncover optimum values extra successfully.

Grid Search

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

4. When to Use Which

Bayesian Optimization:

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

Grid Search

• Small, low-dimensional hyperparameter areas: When you could have only some hyperparameters and a restricted variety of values for every, Grid Search could be a easy and efficient alternative.
• When exhaustiveness is vital: If you happen to completely must 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 area and prioritizing a particular goal, on this case, maximizing recall.

It efficiently achieved the next recall fee (0.8400) in comparison with Grid Search, indicating its skill to seek out extra constructive situations.
This functionality comes with an inherent trade-off, resulting in lowered precision and general accuracy.

Such an final result is very helpful in functions the place minimizing false negatives is vital (e.g., medical analysis, fraud detection).
Its effectivity, stemming from probabilistic modeling that guides the search in the direction of promising areas, makes it a most popular technique for optimizing expensive 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 provides a simple and exhaustive strategy, Bayesian Optimization stands out as a extra refined and environment friendly technique able to find superior outcomes with fewer evaluations, notably when optimizing for a particular, typically complicated, goal like maximizing recall in a high-dimensional area.

The optimum alternative of tuning technique finally is determined by the particular efficiency priorities and useful resource constraints of the applying.

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

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