Note
Click here to download the full example code
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: 1.9774793331861478, (x - 2)^2: 0.0005071804337405452
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': 1.9774793331861478}
To get the best observed value of the objective function:
study.best_value
0.0005071804337405452
To get the best trial:
study.best_trial
FrozenTrial(number=99, values=[0.0005071804337405452], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 384118), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 385511), params={'x': 1.9774793331861478}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=99, state=TrialState.COMPLETE, value=None)
To get all trials:
study.trials
[FrozenTrial(number=0, values=[94.93658382968694], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 227471), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 227870), params={'x': -7.743540620826033}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=0, state=TrialState.COMPLETE, value=None), FrozenTrial(number=1, values=[81.30631706993397], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 228318), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 228655), params={'x': -7.017001556500584}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=1, state=TrialState.COMPLETE, value=None), FrozenTrial(number=2, values=[138.6409408651399], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 229089), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 229385), params={'x': -9.774588776901718}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=2, state=TrialState.COMPLETE, value=None), FrozenTrial(number=3, values=[37.468133026007834], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 229815), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 230098), params={'x': 8.121121876421661}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=3, state=TrialState.COMPLETE, value=None), FrozenTrial(number=4, values=[40.54309813584411], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 230516), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 230796), params={'x': 8.367346239670347}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=4, state=TrialState.COMPLETE, value=None), FrozenTrial(number=5, values=[60.65682097254059], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 231256), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 231532), params={'x': 9.788248902836926}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=5, state=TrialState.COMPLETE, value=None), FrozenTrial(number=6, values=[37.979090054501306], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 231980), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 232276), params={'x': -4.162717749053684}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=6, state=TrialState.COMPLETE, value=None), FrozenTrial(number=7, values=[94.17472755867425], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 232691), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 232956), params={'x': -7.704366417168833}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=7, state=TrialState.COMPLETE, value=None), FrozenTrial(number=8, values=[11.673934361586161], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 233507), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 233837), params={'x': -1.4167139712867627}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=8, state=TrialState.COMPLETE, value=None), FrozenTrial(number=9, values=[9.427758589025196], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 234374), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 234656), params={'x': 5.070465532948578}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=9, state=TrialState.COMPLETE, value=None), FrozenTrial(number=10, values=[2.99135823224877], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 235052), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 236338), params={'x': 3.7295543449827675}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=10, state=TrialState.COMPLETE, value=None), FrozenTrial(number=11, values=[4.532547930900754], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 236846), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 237896), params={'x': 4.1289781424196805}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=11, state=TrialState.COMPLETE, value=None), FrozenTrial(number=12, values=[1.3096726915944525], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 238399), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 239417), params={'x': 3.144409319952635}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=12, state=TrialState.COMPLETE, value=None), FrozenTrial(number=13, values=[0.04815632512554901], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 239928), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 241031), params={'x': 2.219445494657669}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=13, state=TrialState.COMPLETE, value=None), FrozenTrial(number=14, values=[0.5111782152563508], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 241598), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 242563), params={'x': 1.2850327173524996}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=14, state=TrialState.COMPLETE, value=None), FrozenTrial(number=15, values=[7.696689292324709], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 243016), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 243990), params={'x': -0.7742907728507318}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=15, state=TrialState.COMPLETE, value=None), FrozenTrial(number=16, values=[0.3049767900606813], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 244441), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 245479), params={'x': 1.4477529628321386}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=16, state=TrialState.COMPLETE, value=None), FrozenTrial(number=17, values=[25.908381916682693], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 245928), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 246953), params={'x': -3.0900276931154993}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=17, state=TrialState.COMPLETE, value=None), FrozenTrial(number=18, values=[1.2337521052704887], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 247403), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 248383), params={'x': 0.8892560577385584}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=18, state=TrialState.COMPLETE, value=None), FrozenTrial(number=19, values=[12.804721499672262], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 248833), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 249848), params={'x': 5.5783685528006}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=19, state=TrialState.COMPLETE, value=None), FrozenTrial(number=20, values=[0.020203514494314675], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 250324), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 251319), params={'x': 1.8578609325543654}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=20, state=TrialState.COMPLETE, value=None), FrozenTrial(number=21, values=[0.03711570528980411], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 251770), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 252785), params={'x': 2.192654367429872}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=21, state=TrialState.COMPLETE, value=None), FrozenTrial(number=22, values=[0.07966890603012913], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 253237), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 254327), params={'x': 2.2822568086514994}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=22, state=TrialState.COMPLETE, value=None), FrozenTrial(number=23, values=[14.979967491904295], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 254750), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 256009), params={'x': -1.8703962964926855}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=23, state=TrialState.COMPLETE, value=None), FrozenTrial(number=24, values=[17.12889642233407], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 256463), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 257744), params={'x': 6.138707095499036}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=24, state=TrialState.COMPLETE, value=None), FrozenTrial(number=25, values=[4.338685653896856], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 258216), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 259405), params={'x': -0.08295118855360029}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=25, state=TrialState.COMPLETE, value=None), FrozenTrial(number=26, values=[26.95460398401351], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 259869), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 260955), params={'x': 7.1917823513715895}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=26, state=TrialState.COMPLETE, value=None), FrozenTrial(number=27, values=[0.40556076796629736], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 261534), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 262586), params={'x': 2.636836531588992}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=27, state=TrialState.COMPLETE, value=None), FrozenTrial(number=28, values=[35.77817640692684], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 263177), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 264222), params={'x': -3.981486136983587}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=28, state=TrialState.COMPLETE, value=None), FrozenTrial(number=29, values=[2.954425061212697], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 264700), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 265941), params={'x': 0.28115589385985995}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=29, state=TrialState.COMPLETE, value=None), FrozenTrial(number=30, values=[4.411935692117412], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 266416), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 267504), params={'x': 4.100460828512975}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=30, state=TrialState.COMPLETE, value=None), FrozenTrial(number=31, values=[0.024738640198740496], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 267963), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 269206), params={'x': 2.157285219263415}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=31, state=TrialState.COMPLETE, value=None), FrozenTrial(number=32, values=[0.03727420584008413], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 269704), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 270813), params={'x': 1.8069347109393195}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=32, state=TrialState.COMPLETE, value=None), FrozenTrial(number=33, values=[4.641824802301216], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 271272), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 272371), params={'x': -0.15448945281735282}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=33, state=TrialState.COMPLETE, value=None), FrozenTrial(number=34, values=[17.04382116441494], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 272833), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 274006), params={'x': -2.128416302217467}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=34, state=TrialState.COMPLETE, value=None), FrozenTrial(number=35, values=[0.08479538009225489], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 274476), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 275588), params={'x': 1.7088035369509875}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=35, state=TrialState.COMPLETE, value=None), FrozenTrial(number=36, values=[1.7577342762286663], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 276045), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 277159), params={'x': 3.3257957143650247}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=36, state=TrialState.COMPLETE, value=None), FrozenTrial(number=37, values=[7.422459600124119], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 277645), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 278776), params={'x': 4.724419130773406}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=37, state=TrialState.COMPLETE, value=None), FrozenTrial(number=38, values=[21.086008086968622], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 279236), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 280351), params={'x': 6.591950357633303}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=38, state=TrialState.COMPLETE, value=None), FrozenTrial(number=39, values=[1.0626779465051337], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 280811), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 281970), params={'x': 0.9691372804756524}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=39, state=TrialState.COMPLETE, value=None), FrozenTrial(number=40, values=[64.89189351584767], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 282447), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 283580), params={'x': -6.055550478759827}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=40, state=TrialState.COMPLETE, value=None), FrozenTrial(number=41, values=[0.014468726321186998], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 284039), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 285174), params={'x': 2.1202860188101136}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=41, state=TrialState.COMPLETE, value=None), FrozenTrial(number=42, values=[0.4117085729727283], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 285655), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 286963), params={'x': 2.641645208018207}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=42, state=TrialState.COMPLETE, value=None), FrozenTrial(number=43, values=[7.999880113417535], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 287465), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 288585), params={'x': -0.8284059315129317}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=43, state=TrialState.COMPLETE, value=None), FrozenTrial(number=44, values=[2.396202657577876], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 289048), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 290225), params={'x': 0.45203273368656927}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=44, state=TrialState.COMPLETE, value=None), FrozenTrial(number=45, values=[7.488698486505567], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 290685), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 291817), params={'x': 4.736548645009909}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=45, state=TrialState.COMPLETE, value=None), FrozenTrial(number=46, values=[0.0098773670944578], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 292280), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 293451), params={'x': 1.900615055997109}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=46, state=TrialState.COMPLETE, value=None), FrozenTrial(number=47, values=[1.7794300532767888], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 293916), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 295083), params={'x': 3.3339527927467256}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=47, state=TrialState.COMPLETE, value=None), FrozenTrial(number=48, values=[8.729984166029597], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 295554), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 296704), params={'x': -0.9546546610440951}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=48, state=TrialState.COMPLETE, value=None), FrozenTrial(number=49, values=[4.093125122950313], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 297167), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 298355), params={'x': 4.023147331004421}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=49, state=TrialState.COMPLETE, value=None), FrozenTrial(number=50, values=[36.70108352298587], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 298821), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 299967), params={'x': 8.058141920010282}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=50, state=TrialState.COMPLETE, value=None), FrozenTrial(number=51, values=[0.025577540448447372], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 300432), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 301633), params={'x': 1.8400702014993848}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=51, state=TrialState.COMPLETE, value=None), FrozenTrial(number=52, values=[0.5881163506493359], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 302106), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 303288), params={'x': 2.766887443272698}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=52, state=TrialState.COMPLETE, value=None), FrozenTrial(number=53, values=[0.07554475140528312], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 303753), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 304959), params={'x': 1.7251459452631577}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=53, state=TrialState.COMPLETE, value=None), FrozenTrial(number=54, values=[1.9750342714743423], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 305453), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 306633), params={'x': 0.5946408745540013}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=54, state=TrialState.COMPLETE, value=None), FrozenTrial(number=55, values=[0.01017624905381714], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 307103), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 308263), params={'x': 2.1008773961490737}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=55, state=TrialState.COMPLETE, value=None), FrozenTrial(number=56, values=[10.45176025229884], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 308738), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 309982), params={'x': 5.23291822542712}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=56, state=TrialState.COMPLETE, value=None), FrozenTrial(number=57, values=[1.5772660619129106], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 310468), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 311635), params={'x': 3.2558925359730866}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=57, state=TrialState.COMPLETE, value=None), FrozenTrial(number=58, values=[0.4379320938110458], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 312106), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 313311), params={'x': 1.3382356206238917}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=58, state=TrialState.COMPLETE, value=None), FrozenTrial(number=59, values=[3.065182847271498], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 313783), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 314964), params={'x': 3.7507663599896754}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=59, state=TrialState.COMPLETE, value=None), FrozenTrial(number=60, values=[4.116377879132751], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 315438), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 316614), params={'x': -0.028885871391673756}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=60, state=TrialState.COMPLETE, value=None), FrozenTrial(number=61, values=[0.009971921752672941], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 317084), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 318309), params={'x': 1.9001404899237286}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=61, state=TrialState.COMPLETE, value=None), FrozenTrial(number=62, values=[0.012792403901923542], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 318781), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 319963), params={'x': 2.1131035096799544}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=62, state=TrialState.COMPLETE, value=None), FrozenTrial(number=63, values=[0.7070574521287807], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 320439), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 321666), params={'x': 1.1591329164910897}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=63, state=TrialState.COMPLETE, value=None), FrozenTrial(number=64, values=[139.25157967990492], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 322150), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 323350), params={'x': -9.800490654201838}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=64, state=TrialState.COMPLETE, value=None), FrozenTrial(number=65, values=[0.37188841994406496], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 323847), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 325040), params={'x': 2.6098265490646213}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=65, state=TrialState.COMPLETE, value=None), FrozenTrial(number=66, values=[0.017086808339836668], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 325568), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 326772), params={'x': 2.1307165190013744}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=66, state=TrialState.COMPLETE, value=None), FrozenTrial(number=67, values=[5.073215463578275], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 327274), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 328472), params={'x': 4.252379955420105}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=67, state=TrialState.COMPLETE, value=None), FrozenTrial(number=68, values=[5.8430025338092255], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 328946), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 330219), params={'x': -0.41723034355628297}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=68, state=TrialState.COMPLETE, value=None), FrozenTrial(number=69, values=[12.285030732594796], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 330690), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 331912), params={'x': -1.50500081777377}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=69, state=TrialState.COMPLETE, value=None), FrozenTrial(number=70, values=[1.6222118377929695], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 332411), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 333640), params={'x': 0.7263391982976906}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=70, state=TrialState.COMPLETE, value=None), FrozenTrial(number=71, values=[0.07279224027706353], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 334153), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 335370), params={'x': 2.269800371158128}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=71, state=TrialState.COMPLETE, value=None), FrozenTrial(number=72, values=[1.5271563281074492], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 335874), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 337066), params={'x': 3.2357816668438844}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=72, state=TrialState.COMPLETE, value=None), FrozenTrial(number=73, values=[0.006507730627244775], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 337677), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 338871), params={'x': 2.080670506551309}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=73, state=TrialState.COMPLETE, value=None), FrozenTrial(number=74, values=[0.9233265127074033], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 339347), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 340624), params={'x': 2.96089880461337}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=74, state=TrialState.COMPLETE, value=None), FrozenTrial(number=75, values=[0.5032972747922578], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 341100), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 342373), params={'x': 1.2905655246661198}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=75, state=TrialState.COMPLETE, value=None), FrozenTrial(number=76, values=[0.007567307287312408], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 342852), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 344132), params={'x': 1.9130097287777972}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=76, state=TrialState.COMPLETE, value=None), FrozenTrial(number=77, values=[2.082870310817886], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 344609), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 345880), params={'x': 0.5567847316433054}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=77, state=TrialState.COMPLETE, value=None), FrozenTrial(number=78, values=[0.05874536597852126], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 346376), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 347623), params={'x': 2.24237443342589}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=78, state=TrialState.COMPLETE, value=None), FrozenTrial(number=79, values=[5.811689485572498], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 348132), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 349384), params={'x': 4.410744591526132}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=79, state=TrialState.COMPLETE, value=None), FrozenTrial(number=80, values=[2.7746834767866155], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 349885), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 351115), params={'x': 3.6657381177083677}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=80, state=TrialState.COMPLETE, value=None), FrozenTrial(number=81, values=[0.05814553588210845], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 351623), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 352842), params={'x': 1.758866145300772}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=81, state=TrialState.COMPLETE, value=None), FrozenTrial(number=82, values=[0.7838164641933648], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 353355), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 354629), params={'x': 1.114665902501567}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=82, state=TrialState.COMPLETE, value=None), FrozenTrial(number=83, values=[0.2687591687270852], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 355146), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 356381), params={'x': 2.5184198768634216}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=83, state=TrialState.COMPLETE, value=None), FrozenTrial(number=84, values=[3.376765840539572], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 356889), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 358184), params={'x': 0.16240215483921183}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=84, state=TrialState.COMPLETE, value=None), FrozenTrial(number=85, values=[0.013026087041595087], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 358694), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 359937), params={'x': 1.885868115578533}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=85, state=TrialState.COMPLETE, value=None), FrozenTrial(number=86, values=[0.8593104273205773], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 360450), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 361723), params={'x': 2.92698998231943}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=86, state=TrialState.COMPLETE, value=None), FrozenTrial(number=87, values=[0.15273947832613732], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 362246), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 363494), params={'x': 1.6091810159087236}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=87, state=TrialState.COMPLETE, value=None), FrozenTrial(number=88, values=[5.764268050824518], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 364027), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 365524), params={'x': -0.40088901260023235}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=88, state=TrialState.COMPLETE, value=None), FrozenTrial(number=89, values=[3.0031499341311916], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 366045), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 367283), params={'x': 3.732959876665121}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=89, state=TrialState.COMPLETE, value=None), FrozenTrial(number=90, values=[0.0008767342620403281], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 367797), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 369048), params={'x': 2.0296096987833434}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=90, state=TrialState.COMPLETE, value=None), FrozenTrial(number=91, values=[0.010668802880789037], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 369599), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 370868), params={'x': 2.1032898972832728}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=91, state=TrialState.COMPLETE, value=None), FrozenTrial(number=92, values=[1.2058193886541333], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 371393), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 372644), params={'x': 0.9019019221152724}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=92, state=TrialState.COMPLETE, value=None), FrozenTrial(number=93, values=[0.2365520859745248], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 373165), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 374476), params={'x': 1.513633794374522}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=93, state=TrialState.COMPLETE, value=None), FrozenTrial(number=94, values=[0.789378113444695], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 374996), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 376254), params={'x': 2.888469534336825}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=94, state=TrialState.COMPLETE, value=None), FrozenTrial(number=95, values=[0.02795154993119124], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 376773), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 378090), params={'x': 2.1671871703546395}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=95, state=TrialState.COMPLETE, value=None), FrozenTrial(number=96, values=[2.301847788587151], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 378726), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 379970), params={'x': 0.48281583563920916}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=96, state=TrialState.COMPLETE, value=None), FrozenTrial(number=97, values=[2.3137157809203983], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 380491), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 381783), params={'x': 3.521090326351594}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=97, state=TrialState.COMPLETE, value=None), FrozenTrial(number=98, values=[13.030537773518446], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 382314), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 383596), params={'x': 5.609783618656172}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=98, state=TrialState.COMPLETE, value=None), FrozenTrial(number=99, values=[0.0005071804337405452], datetime_start=datetime.datetime(2022, 11, 8, 12, 27, 45, 384118), datetime_complete=datetime.datetime(2022, 11, 8, 12, 27, 45, 385511), params={'x': 1.9774793331861478}, distributions={'x': UniformDistribution(high=10.0, low=-10.0)}, user_attrs={}, system_attrs={}, intermediate_values={}, trial_id=99, state=TrialState.COMPLETE, 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: 1.9960273417660763, (x - 2)^2: 1.578201344356158e-05
Total running time of the script: ( 0 minutes 0.384 seconds)