How to train continuous/soft classification model?

Question

The classic classification problem is like finding the function $F:\mathbb{R}^n\mapsto \{0,1\}$. The label set will be [Apple,Banana,Banana,...,Apple].

What if I want to train a function $F:\mathbb{R}\mapsto[0,1]$? My samples could be something like "this sample has 80% probability to be Apple and 20% to be Banana".

It seems a multi-output neural network works, as we can apply the softmax loss with cross entropy loss. What about random forest or other algorithms? I have tried some common algorithms in scikit-learn without any luck.

For example, this code:

import numpy as np
from sklearn.ensemble import RandomForestClassifier

N_FEATURES = 10
N_SAMPLES = 1000
N_CLASSES = 2

train_x = np.random.rand(N_SAMPLES, N_FEATURES)
train_y = np.random.rand(N_SAMPLES, N_CLASSES)
train_y = np.apply_along_axis(lambda x: x/x.sum(), 1, train_y)

model = RandomForestClassifier(n_estimators=10).fit(train_x, train_y)

Yields a ValueError: Unknown label type: 'continuous-multioutput'.