Bayesian Optimization for Hyperparameter Tuning of Deep Learning Models

to tune hyperparamters of deep studying fashions (Keras Sequential model), compared with a conventional strategy — 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 are costly to guage, lack an analytical type, or have unknown derivatives.
Within the context of hyperparameter optimization, the unknown operate will be:

• an goal operate,
• 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 check outcomes,
• computation value per epoch,
• mannequin measurement,
• reward quantity for reinforcement studying, and extra.

Not like conventional optimization strategies that depend on direct operate evaluations, Bayesian Optimization builds and refines a probabilistic mannequin of the target operate, utilizing this mannequin to intelligently choose the following analysis level.

The core concept revolves round two key elements:

1. Surrogate Mannequin (Probabilistic Mannequin)

The mannequin approximates the unknown goal operate (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 features. It present:

• a prediction of the operate worth at a given level μ(x) and
• a measure of uncertainty round that prediction σ(x), usually 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 operate determines a subsequent level (x_t+1)​ to guage by quantifying how “promising” a candidate level is for enhancing the target operate, 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 chance of enhancing upon the present finest noticed worth (f(x+)):

the place

• Φ: the cumulative distribution operate (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 operate (PDF) of the usual regular distribution.

EI is among the most generally used acquisition features. EI additionally considers the magnitude of the development not like PI.

Higher Confidence Certain (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 operate.

It guides the search in the direction of optimum areas whereas minimizing the variety of costly goal operate 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 essentially 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 operate.

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", path="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, usually a Gaussian Course of or a Tree-structured Parzen Estimator for Bayesian Optimization) utilizing all obtainable noticed datas factors (enter values and their corresponding output values) to approximate the true operate.

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 inner surrogate mannequin to mannequin the connection between hyperparameters and the target operate’s efficiency.

After every goal operate analysis by way of 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 (usually 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 operate.

This step is essential as a result of it identifies essentially the most promising subsequent candidate for analysis by balancing exploration (attempting 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 Certain to search out the following set of hyperparameters.

Step 4. Goal Operate Analysis

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

The Keras mannequin is educated 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. 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 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 strategy repeatedly performs these steps till the max_trials restrict is reached or different inner stopping standards similar to 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.

Study 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 total 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 section)

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, 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 constructive circumstances.
• Precision: Concurrently, precision skilled a lower. Ranging from round 0.83 (or 0.81), it settled at 0.6747 post-optimization. This implies that whereas extra constructive circumstances are being recognized, a better proportion of these identifications could be 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 usually 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',
    courses=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 robust precision and good total 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 total 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 circumstances.
• Precision: Grid Search (0.8304) achieved a lot greater precision than Bayesian Optimization (0.6747), indicating fewer false positives.
• Accuracy: Grid Search’s accuracy (0.7825) was notably greater than Bayesian Optimization’s (0.7175).

Common Comparability with Grid Search

1. Approaching the Search Area

Bayesian Optimization

• Clever/Adaptive: Bayesian Optimization builds a probabilistic mannequin (usually a Gaussian Course of) of the target operate (e.g., mannequin efficiency as a operate 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 permits it to make “clever” decisions about the place to pattern subsequent, balancing exploration (attempting 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 choosing the following.

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 number of the following set of hyperparameters to attempt. Every mixture is evaluated independently.
• Deterministic: Given the identical grid, it’s going to at all times discover the identical combos 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 operate (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 could actually 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 improve, the variety of combos grows exponentially. This results in very long term occasions and excessive computational value, making it impractical for giant search areas. That is sometimes called 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 the 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 may well 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 dear and you’ve got many hyperparameters to tune.
• When effectivity is paramount: To search out good hyperparameters rapidly, particularly in conditions with restricted computational assets or time.
• Black-box optimization issues: When the target operate is complicated, non-linear, and doesn’t have a recognized analytical type.

Grid Search

• Small, low-dimensional hyperparameter areas: When you’ve only some hyperparameters and a restricted variety of values for every, Grid Search could be a easy and efficient selection.
• When exhaustiveness is important: Should you 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 selected goal, on this case, maximizing recall.

It efficiently achieved a better recall charge (0.8400) in comparison with Grid Search, indicating its skill to search out extra constructive cases.
This functionality comes with an inherent trade-off, resulting in lowered precision and total accuracy.

Such an end result is very beneficial in functions the place minimizing false negatives is important (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 dear.

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

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

In abstract, whereas Grid Search presents an easy 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 selected, usually complicated, goal like maximizing recall in a high-dimensional area.

The optimum selection of tuning technique in the end is dependent upon the precise efficiency priorities and useful resource constraints of the appliance.

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

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