University of Michigan - Ann Arbor

The derivative of a function $$f: \mathbb{R} ightarrow \mathbb{R}$$ at a point $$x$$ represents the rate of change in the function with respect to changes in its arguments. Formally, it is defined as the limit of the ratio between a perturbation $$h$$ and the change in the function value as $$h$$ approaches zero: $$f'(x) = \lim_{h ightarrow 0} \frac{f(x+h) - f(x)}{h}$$. The derivative essentially measures how rapidly a function's value would increase or decrease given an infinitesimally small adjustment to its parameter.

Derivative of a Scalar Function

On a straight line, the function's derivative...

Gradient descent is a fundamental optimization algorithm that leverages gradients to minimize a model's loss function. Because the gradient of a function points in the direction of steepest ascent, moving the model's parameters in the opposite direction iteratively lowers the loss. Each step of such gradient-based optimization algorithms requires calculating the exact gradient of the loss with respect to the parameters.

Gradient Descent

https://www.wired.com/2015/04/crash-course-derivatives/

A crash course of derivatives

Learning rate is a positive scalar determining the size of the step in method of steepest descent. 

Learning Rate

A derivative of a derivative is known as second derivative. Second derivative is often used to measure curvature.

Second Derivative

Hessian matrix is defined as a matrix containing second derivatives of function having multiple input dimensions. It's defined such that
$H(f)(x)_{i,j} = \frac{\partial^2}{\partial x_i \partial x_j}f(x)$

Hessian Matrix 

Denote the function as $f(x)$, $g$ is the gradient and $H$ is is the Hessian at $x^{(0)}$. We calculate the new point $x = x^{(0)} - \epsilon g$. We can obtian that
$f(x^{(0)} - \epsilon g) \approx f(x^{(0)}) - \epsilon g^Tg + \frac{1}{2} \epsilon^2 g^THg$ 
According to the above equation, the optimal step size when $g^THg$ is positive is
$\epsilon^* = \frac{g^Tg}{g^THg}$

Optimal Step Size according to Taylor Series Approximation

A Lipschitz continuous function is a functionfwhose rateof change is bounded by a Lipschitz constant $L$:
$L = \forall x \forall y. |f(x)-f(y)| \leq L ||x-y||_2$

Lipschitz Continuous

Functions that are composed of other differentiable functions can be differentiated using systematic rules. For differentiable functions $$f(x)$$ and $$g(x)$$ and a constant $$C$$, the primary rules are:
- Constant multiple rule: $$\frac{d}{dx} [C f(x)] = C \frac{d}{dx} f(x)$$
- Sum rule: $$\frac{d}{dx} [f(x) + g(x)] = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)$$
- Product rule: $$\frac{d}{dx} [f(x) g(x)] = f(x) \frac{d}{dx} g(x) + g(x) \frac{d}{dx} f(x)$$
- Quotient rule: $$\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{g(x) \frac{d}{dx} f(x) - f(x) \frac{d}{dx} g(x)}{g^2(x)}$$

Differentiation Rules

Several fundamental functions have well-known derivatives that serve as building blocks for more complex calculus operations. For any constant $$C$$ and any power $$n 
eq 0$$, the standard derivatives are:
- Constant: $$\frac{d}{dx} C = 0$$
- Power: $$\frac{d}{dx} x^n = n x^{n-1}$$
- Exponential: $$\frac{d}{dx} e^x = e^x$$
- Natural Logarithm: $$\frac{d}{dx} \ln x = x^{-1}$$

Derivatives of Common Functions

For functions of a single variable, the chain rule is used to compute the derivative of deeply nested functions. Suppose that $$y = f(g(x))$$ is a composite function, and that the underlying functions $$y = f(u)$$ and $$u = g(x)$$ are both differentiable. The chain rule states that the derivative of $$y$$ with respect to $$x$$ is the product of the derivative of $$y$$ with respect to $$u$$ and the derivative of $$u$$ with respect to $$x$$:

$$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$$

Chain Rule for Single-Variable Functions

When a multivariate function outputs a vector rather than a scalar, the most natural representation of its derivative is the Jacobian matrix. Specifically, the derivative of a vector $$\mathbf{y}$$ with respect to an input vector $$\mathbf{x}$$ is a matrix that compiles the partial derivatives of each individual component of $$\mathbf{y}$$ with respect to each component of $$\mathbf{x}$$.

Jacobian Matrix

The partial derivative of a multivariate function $$y = f(x_1, x_2, \ldots, x_n)$$ with respect to its $$i^	extrm{th}$$ parameter $$x_i$$ measures how the function changes as $$x_i$$ increases while treating all other variables as constants. Formally, it is defined as the limit: $$\frac{\partial y}{\partial x_i} = \lim_{h ightarrow 0} \frac{f(x_1, \ldots, x_{i-1}, x_i+h, x_{i+1}, \ldots, x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{h}$$.

Partial Derivative

The gradient of a scalar-valued function with respect to a vector $$\mathbf{x}$$ is itself a vector-valued output that has the identical shape as the input vector $$\mathbf{x}$$. This property ensures that each element of the gradient vector directly corresponds to the partial derivative of the scalar output with respect to the matching element in the input vector.

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