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| 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}}''') | |
| 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') | |
| # Fitting by the respective cost_function | |
| w = jnp.array(np.random.random(3)) | |
| learning_rate = 0.05 | |
| NUM_ITER = 1000 | |
| 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''') | |
| progress_bar = st.progress(0) | |
| status_text = st.empty() | |
| grad_loss = jax.grad(loss) | |
| # Perform gradient descent | |
| progress_counter = 0 | |
| for i in range(1,NUM_ITER+1): | |
| if i %10==0: | |
| # Update progress bar. | |
| progress_counter += 1 | |
| progress_bar.progress(progress_counter) | |
| # Update parameters. | |
| w -= learning_rate * grad_loss(w) | |
| # Update status text. | |
| if (i)%100==0: | |
| # report the loss at the current epoch | |
| status_text.text( | |
| 'Trained loss at epoch %s is %s' % (i, loss(w))) | |
| # Plot the final line | |
| fig, ax = plt.subplots(dpi=120) | |
| ax.set_xlim((0,1)) | |
| ax.set_ylim((-5,26)) | |
| ax.scatter(X[:,1], y, c='#e76254' ,edgecolors='firebrick') | |
| ax.plot(X[jnp.dot(X, w).argsort(), 1], jnp.dot(X, w).sort(), 'k-', label='Final line') | |
| st.pyplot(fig) | |
| status_text.text('Done!') | |
| 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}''') | |