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to tune hyperparamters of deep studying fashions (Keras Sequential mannequin), as compared with a standard method — Grid Search.
Bayesian Optimization
Bayesian Optimization is a sequential design technique for world optimization of black-box features.
It’s notably well-suited for features which might be costly to guage, lack an analytical kind, or have unknown derivatives.Within the context of hyperparameter optimization, the unknown operate could 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 measurement,
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 thought revolves round two key parts:
1. Surrogate Mannequin (Probabilistic Mannequin)
The mannequin approximates the unknown goal operate (f(x)) to a surrogate mannequin comparable 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 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 embody:
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 likely one of the most generally used acquisition features. 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 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 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 thought 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 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 accessible noticed datas factors (enter values and their corresponding output values) to approximate the true operate.
The surrogate mannequin supplies the imply prediction (μ(x)) (most probably 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 prepare 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 (usually an affordable, native optimizer like L-BFGS and even random search) to seek out the subsequent 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 comparable to Anticipated Enchancment or Higher Confidence Sure to seek 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 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() 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 comparable to early_stopping_callback are met.
Right here, we set recall as our key metrics to penalize the misclassification as False Optimistic prices us essentially 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 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 section)
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 optimistic 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 optimistic instances are being recognized, a better 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 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’,
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
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 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 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 (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 most probably 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 should all the time 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 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 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 improve, the variety of combos 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 nicely with high-dimensional hyperparameter areas.
3. Ensures and Exploration
Bayesian Optimization
Probabilistic assure: It goals to seek out the worldwide optimum effectively, but it surely doesn’t supply 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 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:
Massive, 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 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 identified analytical kind.
Grid Search
Small, low-dimensional hyperparameter areas: When you may have only some hyperparameters and a restricted variety of values for every, Grid Search could be a easy and efficient selection.
When exhaustiveness is important: For those who 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 price (0.8400) in comparison with Grid Search, indicating its skill to seek out extra optimistic situations.This functionality comes with an inherent trade-off, resulting in lowered precision and general 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 direction of promising areas, makes it a most well-liked 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 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 subtle and environment friendly methodology 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 methodology finally relies on the particular efficiency priorities and useful resource constraints of the applying.
Creator: Kuriko IWAIPortfolio / LinkedIn / GithubMay 26, 2025
All photographs, except in any other case famous, are by the writer.The article makes use of artificial knowledge, licensed below Apache 2.0 for business use.
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