Note
Click here to download the full example code
1. Lightweight, versatile, and platform agnostic architecture
Optuna is entirely written in Python and has few dependencies. This means that we can quickly move to the real example once you get interested in Optuna.
Quadratic Function Example
Usually, Optuna is used to optimize hyperparameters, but as an example, let’s optimize a simple quadratic function: \((x - 2)^2\).
First of all, import optuna.
import optuna
In optuna, conventionally functions to be optimized are named objective.
def objective(trial):
x = trial.suggest_float("x", -10, 10)
return (x - 2) ** 2
This function returns the value of \((x - 2)^2\). Our goal is to find the value of x
that minimizes the output of the objective function. This is the “optimization.”
During the optimization, Optuna repeatedly calls and evaluates the objective function with
different values of x.
A Trial object corresponds to a single execution of the objective
function and is internally instantiated upon each invocation of the function.
The suggest APIs (for example, suggest_float()) are called
inside the objective function to obtain parameters for a trial.
suggest_float() selects parameters uniformly within the range
provided. In our example, from \(-10\) to \(10\).
To start the optimization, we create a study object and pass the objective function to method
optimize() as follows.
study = optuna.create_study()
study.optimize(objective, n_trials=100)
You can get the best parameter as follows.
best_params = study.best_params
found_x = best_params["x"]
print("Found x: {}, (x - 2)^2: {}".format(found_x, (found_x - 2) ** 2))
Found x: 1.987658373658125, (x - 2)^2: 0.0001523157407624621
We can see that the x value found by Optuna is close to the optimal value of 2.
Note
When used to search for hyperparameters in machine learning, usually the objective function would return the loss or accuracy of the model.
Study Object
Let us clarify the terminology in Optuna as follows:
Trial: A single call of the objective function
Study: An optimization session, which is a set of trials
Parameter: A variable whose value is to be optimized, such as
xin the above example
In Optuna, we use the study object to manage optimization.
Method create_study() returns a study object.
A study object has useful properties for analyzing the optimization outcome.
To get the dictionary of parameter name and parameter values:
study.best_params
{'x': 1.987658373658125}
To get the best observed value of the objective function:
study.best_value
0.0001523157407624621
To get the best trial:
study.best_trial
FrozenTrial(number=43, state=TrialState.COMPLETE, values=[0.0001523157407624621], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 54425), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 58124), params={'x': 1.987658373658125}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=43, value=None)
To get all trials:
study.trials
[FrozenTrial(number=0, state=TrialState.COMPLETE, values=[50.47746406229292], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 917116), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 917541), params={'x': -5.1047494017940505}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=0, value=None), FrozenTrial(number=1, state=TrialState.COMPLETE, values=[1.6847137338367277], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 917998), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 918266), params={'x': 0.7020347717150788}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=1, value=None), FrozenTrial(number=2, state=TrialState.COMPLETE, values=[0.5826198110547585], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 918634), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 918882), params={'x': 1.2367046370802726}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=2, value=None), FrozenTrial(number=3, state=TrialState.COMPLETE, values=[86.54993829428828], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 919240), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 919467), params={'x': -7.303221930830645}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=3, value=None), FrozenTrial(number=4, state=TrialState.COMPLETE, values=[53.183517450639705], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 919824), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 920050), params={'x': 9.29270302772845}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=4, value=None), FrozenTrial(number=5, state=TrialState.COMPLETE, values=[20.61282750168229], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 920405), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 920684), params={'x': 6.5401351853972685}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=5, value=None), FrozenTrial(number=6, state=TrialState.COMPLETE, values=[138.42079318741204], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 921096), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 921364), params={'x': -9.76523663966909}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=6, value=None), FrozenTrial(number=7, state=TrialState.COMPLETE, values=[6.8182045459436775], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 921778), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 922015), params={'x': -0.6111691913669013}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=7, value=None), FrozenTrial(number=8, state=TrialState.COMPLETE, values=[53.17317352326416], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 922368), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 922605), params={'x': -5.291993796161936}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=8, value=None), FrozenTrial(number=9, state=TrialState.COMPLETE, values=[21.972584000805796], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 922982), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 923231), params={'x': -2.6874922934129497}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=9, value=None), FrozenTrial(number=10, state=TrialState.COMPLETE, values=[1.7562008146382877], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 923583), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 927465), params={'x': 3.3252172707289502}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=10, value=None), FrozenTrial(number=11, state=TrialState.COMPLETE, values=[0.03074506360887859], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 927849), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 931325), params={'x': 1.8246572966762558}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=11, value=None), FrozenTrial(number=12, state=TrialState.COMPLETE, values=[1.5482317228188045], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 931705), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 935085), params={'x': 3.2442795999367684}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=12, value=None), FrozenTrial(number=13, state=TrialState.COMPLETE, values=[0.06973480751904872], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 935452), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 938765), params={'x': 2.2640734888606744}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=13, value=None), FrozenTrial(number=14, state=TrialState.COMPLETE, values=[7.292991215140509], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 939127), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 942384), params={'x': 4.700553871919705}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=14, value=None), FrozenTrial(number=15, state=TrialState.COMPLETE, values=[10.525181237686928], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 942761), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 946109), params={'x': -1.2442535717306265}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=15, value=None), FrozenTrial(number=16, state=TrialState.COMPLETE, values=[0.06507172640207383], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 946472), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 949937), params={'x': 2.2550916039427285}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=16, value=None), FrozenTrial(number=17, state=TrialState.COMPLETE, values=[13.217660524641824], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 950303), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 953769), params={'x': 5.635610062237399}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=17, value=None), FrozenTrial(number=18, state=TrialState.COMPLETE, values=[14.650322437269471], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 954137), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 957753), params={'x': -1.8275739623512792}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=18, value=None), FrozenTrial(number=19, state=TrialState.COMPLETE, values=[1.4635228034772978], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 958125), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 961643), params={'x': 0.7902385344716514}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=19, value=None), FrozenTrial(number=20, state=TrialState.COMPLETE, values=[31.265489837537686], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 962099), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 965611), params={'x': 7.59155522529624}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=20, value=None), FrozenTrial(number=21, state=TrialState.COMPLETE, values=[0.26282178578241805], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 966017), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 969539), params={'x': 2.512661472886756}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=21, value=None), FrozenTrial(number=22, state=TrialState.COMPLETE, values=[4.387926806515784], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 969962), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 973497), params={'x': 4.094737884919205}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=22, value=None), FrozenTrial(number=23, state=TrialState.COMPLETE, values=[0.04767194822274313], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 973925), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 977460), params={'x': 2.2183390671014767}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=23, value=None), FrozenTrial(number=24, state=TrialState.COMPLETE, values=[0.01142677967388981], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 977901), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 981385), params={'x': 1.8931038837286882}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=24, value=None), FrozenTrial(number=25, state=TrialState.COMPLETE, values=[6.804218025694791], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 981844), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 985348), params={'x': 4.608489606207928}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=25, value=None), FrozenTrial(number=26, state=TrialState.COMPLETE, values=[3.3050086334143365], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 985823), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 989400), params={'x': 0.18203172926083871}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=26, value=None), FrozenTrial(number=27, state=TrialState.COMPLETE, values=[0.05817631247806992], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 989858), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 993351), params={'x': 1.7588023373287587}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=27, value=None), FrozenTrial(number=28, state=TrialState.COMPLETE, values=[5.744471383594953], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 993828), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 46, 997384), params={'x': -0.3967626882098597}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=28, value=None), FrozenTrial(number=29, state=TrialState.COMPLETE, values=[26.42009029833538], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 46, 997857), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 1503), params={'x': -3.1400476941693234}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=29, value=None), FrozenTrial(number=30, state=TrialState.COMPLETE, values=[0.4824889312713451], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 1930), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 5558), params={'x': 1.3053857680184309}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=30, value=None), FrozenTrial(number=31, state=TrialState.COMPLETE, 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value=None), FrozenTrial(number=36, state=TrialState.COMPLETE, values=[5.0336336583197285], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 26227), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 29908), params={'x': -0.24357608703599093}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=36, value=None), FrozenTrial(number=37, state=TrialState.COMPLETE, values=[15.388772631481123], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 30279), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 33888), params={'x': 5.922852614040084}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=37, value=None), FrozenTrial(number=38, state=TrialState.COMPLETE, values=[0.0463827199411849], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 34257), 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distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=40, value=None), FrozenTrial(number=41, state=TrialState.COMPLETE, values=[0.1839379007946642], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 46401), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 50040), params={'x': 2.428879820922673}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=41, value=None), FrozenTrial(number=42, state=TrialState.COMPLETE, values=[2.243976007677352], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 50412), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 54048), params={'x': 3.497990656739004}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=42, value=None), FrozenTrial(number=43, state=TrialState.COMPLETE, 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18, 5, 56, 47, 266869), params={'x': 1.8605655343933187}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=93, value=None), FrozenTrial(number=94, state=TrialState.COMPLETE, values=[1.6550638485796423], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 267244), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 271133), params={'x': 3.2864928482427107}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=94, value=None), FrozenTrial(number=95, state=TrialState.COMPLETE, values=[0.17334879801123101], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 271510), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 275394), params={'x': 1.5836482280436037}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=95, value=None), FrozenTrial(number=96, state=TrialState.COMPLETE, values=[4.38383713760443], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 275774), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 279742), params={'x': -0.09376148059047784}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=96, value=None), FrozenTrial(number=97, state=TrialState.COMPLETE, values=[8.010038637865053], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 280118), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 284069), params={'x': -0.8302011656179238}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=97, value=None), FrozenTrial(number=98, state=TrialState.COMPLETE, values=[0.0072892123145798515], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 284446), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 288390), params={'x': 1.9146231160408167}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=98, value=None), FrozenTrial(number=99, state=TrialState.COMPLETE, values=[0.5040382752534611], datetime_start=datetime.datetime(2023, 1, 18, 5, 56, 47, 288893), datetime_complete=datetime.datetime(2023, 1, 18, 5, 56, 47, 292841), params={'x': 1.2900434694620655}, user_attrs={}, system_attrs={}, intermediate_values={}, distributions={'x': FloatDistribution(high=10.0, log=False, low=-10.0, step=None)}, trial_id=99, value=None)]
To get the number of trials:
len(study.trials)
100
By executing optimize() again, we can continue the optimization.
study.optimize(objective, n_trials=100)
To get the updated number of trials:
len(study.trials)
200
As the objective function is so easy that the last 100 trials don’t improve the result. However, we can check the result again:
best_params = study.best_params
found_x = best_params["x"]
print("Found x: {}, (x - 2)^2: {}".format(found_x, (found_x - 2) ** 2))
Found x: 1.9972790141744907, (x - 2)^2: 7.403763862622424e-06
Total running time of the script: ( 0 minutes 0.872 seconds)