to tune hyperparamters of deep studying fashions (Keras Sequential model), 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 are costly to guage, lack an analytical kind, 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 take a look at outcomes,
- computation price per epoch,
- mannequin dimension,
- reward quantity for reinforcement studying, and extra.
In contrast to 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 subsequent analysis level.
The core concept revolves round two key parts:
1. Surrogate Mannequin (Probabilistic Mannequin)
The mannequin approximates the unknown goal operate (f(x)) to a surrogate mannequin resembling 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), typically represented as a confidence interval.
Mathematically, for a Gaussian Course of, the predictions at an unobserved level (x∗), given noticed information (X, y), are usually distributed:
the place
- μ(x∗): the imply prediction and
- σ²(x∗): the predictive variance.
2. Acquisition Perform
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.
Frequent acquisition features embrace:
Likelihood of Enchancment (PI)
PI selects the purpose that has the very best chance of enhancing upon the present greatest 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 greatest 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 advance 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 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 operate.
It guides the search in 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 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 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", 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 ought 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 practice the probabilistic mannequin (surrogate mannequin, typically a Gaussian Course of or a Tree-structured Parzen Estimator for Bayesian Optimization) utilizing all out there noticed datas factors (enter values and their corresponding output values) to approximate the true operate.
The surrogate mannequin supplies the imply prediction (μ(x)) (almost certainly 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 through practice run, the noticed information 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 subsequent level (x_t+1) that maximizes the chosen acquisition operate.
This step is essential as a result of it identifies probably 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 resembling Anticipated Enchancment or Higher Confidence Certain to search out the subsequent set of hyperparameters.
Step 4. Goal Perform 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 information. 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.Selection('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 information 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 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 inner stopping standards resembling early_stopping_callback are met.
Right here, we set recall as our key metrics to penalize the misclassification as False Constructive prices us probably the most within the fraud detection case.
Study Extra: KerasTuner Supply 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 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 improvement (coaching / validation mixed) to check part)
Greatest 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 constructive instances.
- 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 instances are being recognized, the next proportion of these identifications could be false positives.
- Accuracy: The general accuracy of the mannequin additionally noticed a decline, transferring from an preliminary 0.7640 (or 0.7475) all the way down to 0.7175. That is in keeping 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',
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 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
Greatest 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 strong 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 instances.
- 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).
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 operate (e.g., mannequin efficiency as a operate of hyperparameters). It makes use of this mannequin to foretell which hyperparameter mixtures are almost certainly 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 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 sometimes operates sequentially, evaluating one level at a time and updating its mannequin earlier than choosing the subsequent.
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 choice of the subsequent set of hyperparameters to strive. Every mixture is evaluated independently.
- Deterministic: Given the identical grid, it would all the time 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 operate (e.g., coaching a Machine Studying mannequin) is computationally costly or time-consuming.
- Scalability: Usually scales higher to higher-dimensional hyperparameter areas than Grid Search, although it may 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 improve, the variety of mixtures grows exponentially. This results in very long term instances and excessive computational price, making it impractical for big search areas. That is sometimes called the “curse of dimensionality.”
- Scalability: Doesn’t scale properly 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 an alternative, it converges probabilistically in 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 search out greatest 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 will 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 dear and you’ve got many hyperparameters to tune.
- When effectivity is paramount: To seek out good hyperparameters rapidly, particularly in conditions with restricted computational sources or time.
- Black-box optimization issues: When the target operate 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 generally is a easy and efficient alternative.
- When exhaustiveness is important: For those who 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 area 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 skill to search out extra constructive situations.
This functionality comes with an inherent trade-off, resulting in lowered precision and total accuracy.
Such an final result is very priceless in purposes the place minimizing false negatives is important (e.g., medical analysis, fraud detection).
Its effectivity, stemming from probabilistic modeling that guides the search in direction of promising areas, makes it a most well-liked methodology for optimizing pricey 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 affords an easy and exhaustive method, Bayesian Optimization stands out as a extra refined and environment friendly methodology able to find superior outcomes with fewer evaluations, notably when optimizing for a selected, typically complicated, goal like maximizing recall in a high-dimensional area.
The optimum alternative of tuning methodology finally is determined by the particular efficiency priorities and useful resource constraints of the applying.
Writer: Kuriko IWAI
Portfolio / LinkedIn / Github
Could 26, 2025
All photos, except in any other case famous, are by the creator.
The article makes use of artificial information, licensed beneath Apache 2.0 for business use.
