app.py

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


def parabola_fn(x):
    return x**0.5


def circle_fn(x):
    return (1 - x**2) ** 0.5


d_parabola_fn = jax.grad(parabola_fn)
d_circle_fn = jax.grad(circle_fn)


def loss_fn(params):
    x1 = params["x1"]
    x2 = params["x2"]
    # parpendicular line to the tangent of the parabola: y = m1 * x + c1
    m1 = -1 / d_parabola_fn(x1)
    c1 = parabola_fn(x1) - m1 * x1

    def perpendicular_parabola_fn(x):
        return m1 * x + c1

    # parpendicular line to the tangent of the circle: y = m2 * x + c2
    m2 = -1 / d_circle_fn(x2)
    c2 = circle_fn(x2) - m2 * x2

    def perpendicular_circle_fn(x):
        return m2 * x + c2

    # x_star and y_star are the intersection of the two lines
    x_star = (c2 - c1) / (m1 - m2)
    y_star = m1 * x_star + c1

    # three quantities should be equal to each other
    # 1. distance between intersection and parabola
    # 2. distance between intersection and circle
    # 3. distance between intersection and x=0 line

    d1 = (x_star - x1) ** 2 + (y_star - parabola_fn(x1)) ** 2
    d2 = (x_star - x2) ** 2 + (y_star - circle_fn(x2)) ** 2
    d3 = x_star**2

    aux = {
        "x_star": x_star,
        "y_star": y_star,
        "perpendicular_parabola_fn": perpendicular_parabola_fn,
        "perpendicular_circle_fn": perpendicular_circle_fn,
        "r": d1**0.5,
    }
    # final loss
    loss = (d1 - d2) ** 2 + (d1 - d3) ** 2 + (d2 - d3) ** 2

    return loss, aux


x = jnp.linspace(0, 1, 100)

st.title("Radius of the Circle: Optimization Playground")
st.markdown(
    r"""
            Inspired from: https://twitter.com/iwontoffendyou/status/1704935240907518367
            
            Optimize the radius of the circle such that it is tangent to the parabola, unit circle and the x=0 line
            
            Method:
            - The inner circle is tangent to the parabola at $x=x_1$ and tangent to the unit circle at $x=x_2$.
            - Let's call the center of the inner circle as $(x^*, y^*)$.
            - We know that ditances between $(x^*, y^*)$ and the parabola, unit circle and the x=0 line should be equal to each other.
            - First, we can find analytical forms of perpendicular lines shown in the figure. They have the form of $y = m * x + c$ where $m = -\frac{1}{f'(x)}$ and $c = f(x) - m * x$.
            - Perpendicular line to the parabola: $y = m_1 * x + c_1$
            - Perpendicular line to the unit circle: $y = m_2 * x + c_2$
            - The intersection of the two lines is $(x^*, y^*)$. $x^* = \frac{c_2 - c_1}{m_1 -m_2}$ and $y^* = m_1 * x^* + c_1$ or $y^* = m_2 * x^* + c_2$.
            - We define three distances: $d_1 = (x^* - x_1)^2 + (y^* - f(x_1))^2$, $d_2 = (x^* - x_2)^2 + (y^* - f(x_2))^2$ and $d_3 = {x^*}^2$.
            - Our loss function is $L = (d_1 - d_2)^2 + (d_1 - d_3)^2 + (d_2 - d_3)^2$.
            
            """
)

col1, col2 = st.columns(2)

x1 = col1.slider("initial x1 (x intersection with parabola)", 0.0, 1.0, 0.5)
x2 = col1.slider("initial x2 (x intersection with the circle)", 0.0, 1.0, 0.5)
n_epochs = col2.slider("n_epochs", 0, 1000, 50)
lr = col2.slider("lr", 0.0, 1.0, value=0.1, step=0.01)

# submit button
submit = st.button("submit")

# when submit button is clicked run the following code
params = {"x1": x1, "x2": x2}
losses = []
value_and_grad_fn = jax.value_and_grad(loss_fn, has_aux=True)

# initialize plot
fig, axes = plt.subplots(1, 2, figsize=(12, 6))
axes[0].set_xlim(0, 1)
axes[0].set_ylim(0, 1)
axes[0].set_aspect("equal")

value, aux = loss_fn(params)
(pbola_plot,) = axes[0].plot(x, parabola_fn(x), color="red")
(pbola_perpendicular_plot,) = axes[0].plot(x, aux["perpendicular_parabola_fn"](x), color="red", linestyle="--")
(cicle_plot,) = axes[0].plot(x, circle_fn(x), color="blue")
(circle_perpendicular_plot,) = axes[0].plot(x, aux["perpendicular_circle_fn"](x), color="blue", linestyle="--")
x_star, y_star = aux["x_star"], aux["y_star"]
(x0_perpendicular_plot,) = axes[0].plot([0, 1], [y_star, y_star], color="black", linestyle="--")
radius = aux["r"]
axes[0].add_patch(plt.Circle((x_star, y_star), radius, fill=False))

axes[1].set_xlim(0, n_epochs)
axes[1].set_ylim(0, value)
(loss_plot,) = axes[1].plot(losses, color="black")

pbar = st.progress(0)

with st.empty():
    st.pyplot(fig)
    if submit:
        for i in range(n_epochs):
            (value, _), grad = value_and_grad_fn(params)

            params["x1"] -= lr * grad["x1"]
            params["x2"] -= lr * grad["x2"]
            losses.append(value)

            _, aux = loss_fn(params)
            print(params, grad, lr)
            pbola_plot.set_data(x, parabola_fn(x))
            pbola_perpendicular_plot.set_data(x, aux["perpendicular_parabola_fn"](x))
            cicle_plot.set_data(x, circle_fn(x))
            circle_perpendicular_plot.set_data(x, aux["perpendicular_circle_fn"](x))
            x_star, y_star = aux["x_star"], aux["y_star"]
            x0_perpendicular_plot.set_data([0, 1], [y_star, y_star])
            radius = aux["r"]
            axes[0].add_patch(plt.Circle((x_star, y_star), radius, fill=False))

            loss_plot.set_data(range(len(losses)), losses)

            pbar.progress(i / n_epochs)

            axes[0].set_title(f"x1: {params['x1']:.3f}, x2: {params['x2']:.3f} \n r: {radius:.4f}")
            axes[1].set_title(f"epoch: {i}, loss: {value:.5f}")

            st.pyplot(fig)