Fit a sine curve with a tanh network

A network with one hidden layer of tanh units can bend a straight line into a smooth curve. In this task you train a 1-16-1 tanh network to fit \sin(x) from noisy samples. The checks then compare the trained network with the true curve at new points that your code never sees.

Task

Write two functions.

1. train(x, y, rng) trains a network and returns the trained model.
2. predict(model, x) returns the network output for each value in x.

The data comes from make_sine(n, rng) in support.py. Each x is uniform from -\pi to \pi, and y = \sin(x) + \varepsilon with Gaussian noise of standard deviation 0.1.

Use 1 input, 16 tanh hidden units and 1 linear output: \hat y = \tanh(x W_1 + b_1) W_2 + b_2. Train it on the mean squared error \frac{1}{N} \sum_i (\hat y_i - y_i)^2 with full-batch gradient descent. Adam reaches the target with the least tuning.

During training, call report(step, loss) from practice_check once for each update step. The first step is 0. The page draws these values as a live loss curve, and the checks count them.

Input and output

• train returns the model. The model can be any Python value, for example a list of weight arrays.
• predict returns a float array of shape (M,) for an input of shape (M,).

Constraints

• Budget: at most 1500 update steps, so at most 1500 calls to report.
• Target: a mean squared error of at most 0.01 against the true curve \sin(x), at 2000 new values of x from -\pi to \pi. This page does not give their seed.
• Take every random number from rng. The same seed must give the same model.
• Do not change x or y, and do not change their shape.
• predict must work for one value, shape (1,).
• Use numpy only. One full submit has a limit of 20 seconds.

Examples

x, y = make_sine(200, np.random.default_rng(0))
model = train(x, y, np.random.default_rng(1))
predict(model, np.array([0.0, np.pi / 2, -np.pi / 2]))

[-0.0165  1.0418 -1.0157]

The output is rounded to 4 decimals. The true values are 0, 1 and minus 1. The reference uses Adam with the learning rate 0.01. Its first reported loss is 4.2061 and its last is 0.0101, after 1500 steps. The noise alone gives a mean squared error of 0.0106 on these 200 points, so a training loss far below 0.01 means that the network fits the noise.