app.py

import streamlit as st
import numpy as np
import jax.numpy as jnp
import jax
import matplotlib.pyplot as plt

# Set random key
seed=321
key = jax.random.PRNGKey(seed)

st.title('Fitting simple models with JAX')
st.header('A quadratric regression example')

st.markdown('*\"Parametrised models are simply functions that depend on inputs and trainable parameters. There is no fundamental difference between the two, except that trainable parameters are shared across training samples whereas the input varies from sample to sample.\"* [(Yann LeCun, Deep learning course)](https://atcold.github.io/pytorch-Deep-Learning/en/week02/02-1/#Parametrised-models)')

st.latex(r'''h(\boldsymbol x, \boldsymbol w)= \sum_{k=1}^{K}\boldsymbol w_{k} \phi_{k}(\boldsymbol x)''')


# Sidebar inputs
number_of_observations = st.sidebar.slider('Number of observations', min_value=50, max_value=150, value=100)
noise_standard_deviation = st.sidebar.slider('Standard deviation of the noise', min_value = 0.0, max_value=2.0, value=1.0)
cost_function = st.sidebar.radio('What cost function you want to use for the fitting?', options=('RMSE-Loss', 'Huber-Loss'))

# Generate random data
np.random.seed(2)

w = jnp.array([3.0, -20.0, 32.0])  # coefficients                                    
X = np.column_stack((np.ones(number_of_observations), 
                     np.random.random(number_of_observations)))
X = jnp.column_stack((X, X[:,1] ** 2))   # add x**2 column

additional_noise = 8 * np.random.binomial(1, 0.03, size = number_of_observations)
y = jnp.array(np.dot(X, w) + noise_standard_deviation * np.random.randn(number_of_observations) \
        + additional_noise)
        
# Plot the data
fig, ax = plt.subplots(dpi=320)
ax.set_xlim((0,1))
ax.set_ylim((-5,26))
ax.scatter(X[:,1], y, c='#e76254' ,edgecolors='firebrick')

st.pyplot(fig)

st.subheader('Train a model')

st.markdown('*\"A Gradient Based Method is a method/algorithm that finds the minima of a function, assuming that one can easily compute the gradient of that function. It assumes that the function is continuous and differentiable almost everywhere (it need not be differentiable everywhere).\"* [(Yann LeCun, Deep learning course)](https://atcold.github.io/pytorch-Deep-Learning/en/week02/02-1/#Parametrised-models)')

st.markdown('Using gradient descent we find the minima of the loss adjusting the weights in each step given the following formula:')

st.latex(r'''\bf{w}\leftarrow \bf{w}-\eta \frac{\partial\ell(\bf{X},\bf{y}, \bf{w})}{\partial \bf{w}}''')


# Fitting by the respective cost_function
if cost_function == 'RMSE-Loss':

     def loss(w):
         return 1/X.shape[0] * jax.numpy.linalg.norm(jnp.dot(X, w) - y)**2
         
     st.write('You selected the RMSE loss function.')
     st.latex(r'''\ell(X, y, w)=\frac{1}{m}||Xw - y||_{2}^2''')
     st.latex(r'''\ell(X, y, w)=\frac{1}{m}\big(\sqrt{(Xw - y)\cdot(Xw - y)}\big)^2''')
     st.latex(r'''\ell(X, y, w)= \frac{1}{m}\sum_1^m (\hat{y}_i - y_i)^2''')
else:
     st.write("You selected the Huber loss function.")
     st.latex(r'''
\ell_{H} = 
\begin{cases} 
      (y^{(i)}-\hat{y}^{(i)})^2 & \text{for }\quad |y^{(i)}-\hat{y}^{(i)}|\leq \delta \\
      2\delta|y^{(i)}-\hat{y}^{(i)}| - \delta^2 & \text{otherwise}
\end{cases}''')

st.markdown('The training loop:')

code = '''NUM_ITER = 1000
# initialize parameters
w = np.array([3., -2., -8.])
for i in range(NUM_ITER):
     # update parameters
     w -= learning_rte * grad_loss(w)'''
     
st.code(code, language='python')