In optimization problems, determining whether a loss function is convex is crucial. Convexity simplifies the search for optimal solutions because it ensures the presence of a unique global minimum. (A “minimum” refers to the lowest value of a function within a specified domain or set of possible inputs. The “global minimum” is the lowest value of the function across its entire domain, meaning it’s the smallest value that the function can attain for any input.)
A function $f(x)$ is convex if, for any two points $x_{1}$ and $x_{2}$ in its domain, and for any $λ$ in the range [0, 1], the following inequality holds:
$f(λx_1 + (1 - λ)x_2) ≤ λf(x_1) + (1 - λ)f(x_2)$
In short, the line segment linking any two locations on a function’s graph must be above the graph to be convex.
We can determine the convexity using the following techniques:
We can determine convexity using the second derivative if our function is twice-differentiable:
Note: The function is neither convex nor concave if the second derivative has a variable sign, i.e., if it’s occasionally positive and occasionally negative.
Jensen’s inequality is a useful tool for determining convexity. It specifically aids in determining whether a function is convex by looking at how it interacts with expectations (averages) and the convexity of its subcomponents.
In relation to convexity and loss functions:
A loss function $L(θ)$ is convex if it satisfies Jensen’s inequality, meaning that, if it can be expressed as an expectation (average) of some value $g(θ)$, and $g(θ)$ is a convex function, then $L(θ)$ is convex.
In short, if $L(θ)=E[g(θ)]$ and $g(θ)$ is convex, then $L(θ)$ is convex.
To elaborate on Jensen’s inequality further:
If $f(x)$ is convex and $g$ is any arbitrary function, then the following is true for every random variable $X$:
$f(E[X])≤E[f(X)]$
This inequality ensures that the expected value of a convex function applied to a random variable is always greater than or equal to the convex function applied to the expected value of that random variable. It’s a powerful concept often used in the context of convex analysis and optimization.
Our loss function’s graph can be plotted to reveal some graphic insights. The function is probably convex if its curve is consistently above its tangent lines.
Remember: This isn’t a strict test, particularly for complicated functions.
Calculating $f(λx_1 +(1−λ)x_2)$ and $λf(x_1)+(1−λ)f(x_2)$ for various values of $λ$, $x_1$, and $x_2$ allows us to check the convexity of our loss function at various points. Convexity is suggested if the inequality holds for these values.
Here’s a simple Python code example that will show us how to use the second derivative test to determine whether a function is convex:
import numpy as npimport matplotlib.pyplot as pltdef loss_function(x):return x**2 # Replace this with your own loss functiondef is_convex(func, interval):x = np.linspace(interval[0], interval[1], 100)second_derivative = np.gradient(np.gradient(func(x), x), x)if np.all(second_derivative >= 0):return Trueelif np.all(second_derivative <= 0):return Falseelse:return Noneinterval = (-6, 6) # Define the interval where you want to check convexityresult = is_convex(loss_function, interval)if result is None:print("The function is neither convex nor concave.")elif result:print("The function is convex.")else:print("The function is not convex.")# Plot the function and its second derivativex = np.linspace(interval[0], interval[1], 100)plt.plot(x, loss_function(x), label="Loss Function")plt.plot(x, np.gradient(np.gradient(loss_function(x), x), x), label="Second Derivative")plt.legend()plt.xlabel("x")plt.ylabel("y")plt.title("Convexity Check")plt.grid(True)plt.savefig('output/output.png')plt.show()
Keep in mind that verifying convexity can be challenging, particularly for complex functions. Convexity can be taken for granted in many practical situations depending on the nature of the issue or specialized knowledge.