A review of vector calculus and concepts in optimization


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  • The following topics will be covered in this lecture:
    • Concepts in analytical and numerical differentiation
    • Newton’s method in one variable
    • Tangent vectors, tangent spaces and vector fields
    • The Jacobian, the inverse function theorem and Newton’s method in multiple variables
    • Gradients, Hessians, and concepts in optimization

Concepts in analytical differentiation

Tangent line approximation by derivative.

Courtesy of Pbroks13, CC BY-SA 3.0, via Wikimedia Commons

  • The derivative represents the slope of a tangent line to a curve.
  • In the figure to the left, we see the function \( f \) represented by the blue curve.
  • The derivative \( f'(x) \) at a given point gives the infinitesimal rate of change at that point with respect to small changes in \( x \), denoted \( \delta_x \).
  • Suppose we have a point \( x_0 \), a nearby point that differs by only a small amount in \( x \) \[ x_1 = x_0+\delta_{x_1}, \]
  • The function \[ f(x_1) \approx f(x_0) + f'(x_0)\delta_{x_1} \] is what is known as the tangent line approximation to the function \( f \).
  • Such an approximation exists when \( f \) is sufficiently smooth and is accurate when \( \delta_{x_1} \) is small, so that the difference of \( x_1 \) from the fixed value \( x_0 \) is small.
  • We can see graphically how the approximation becomes worse as we take \( \delta_{x_1} \) too large.

Concepts in analytical differentiation

  • More generally, the tangent line approximation is one kind of general Taylor approximation.

  • Suppose we have a point \( x_0 \) fixed, and define \( x_1 \) as a small perturbation \[ x_1 = x_0+\delta_{x_1}, \]

  • If a function \( f \) has \( k \) continuous derivatives we can write \[ f(x_1) = f(x_0) + f'(x_0)\delta_{x_1} + \frac{f''(x_0)}{2!}\delta_{x_1}^2 + \cdots + \frac{f^{(k)}(x_0)}{k!} \delta_{x_1}^k + \mathcal{O}\left(\delta_{x_1}^{k+1}\right) \]

  • The \( \mathcal{O}\left(\delta_{x_1}^{k+1}\right) \) refers to terms in the remainder, that grows or shrinks like the size of the perturbation to the power \( k+1 \).

    • This is why this approximation works well when \( \delta_{x_1} \) is a small perturbation.
  • Another important practical example of using this Taylor approximation, when the function \( f \) has two continuous derivatives, is \[ f(x_0 + \delta_{x_1}) \approx f(x_0) + f'(x_0)\delta_{x_1} + f''(x_0) \frac{\delta_{x_1}^2}{2} \] which will be used shortly for obtaining solutions to several kinds of equations.

  • Particularly, this is strongly related to our second derivative test from univariate calculus.

An approach to numerical derivation

  • At the moment, we consider how Taylor's expansion can be used at first order again to approximate the derivative.

  • Recall, we write

    \[ \begin{align} f(x_1) &= f(x_0) + f'(x_0) \delta_{x_1} + \mathcal{O}\left( \delta_{x_1}^2\right) \\ \Leftrightarrow \frac{f(x_1) - f(x_0)}{ \delta_{x_1}} &= f'(x_0) + \mathcal{O}\left( \delta_{x_1}\right) \end{align} \]

  • This says that for a small value of \( \delta_{x_1} \), we can obtain the numerical approximation of \( f'(x_0) \) proportional to the accuracy of the largest decimal place of \( \delta_{x_1} \) by the difference on the left hand side.

  • This gives a forward finite difference equation approximation to the derivative.

  • We can similarly define a backward finite difference equation with \( \pmb{x}_1 := \pmb{x}_0 -\pmb{\delta}_{\pmb{x}_1} \).

  • In each case, we use the perturbation to parameterize the tangent-line approximation.

Newton's method in one variable

  • We have seen earlier the basic linear inverse problem,

    \[ \begin{align} \mathbf{A}\pmb{x} = \pmb{b} \end{align} \] where \( \pmb{b} \) is an observed quantity and \( \pmb{x} \) are the unknown variables related to \( \pmb{b} \) by the relationships in \( \mathbf{A} \).

    • We observed that a unique solution exists when all the relationships expressed by the columns are unique, corresponding to all non-zero eigenvalues.
  • A similar problem exists when the relationship between \( \pmb{x} \) and \( \pmb{b} \) is non-linear, but we still wish to find some such \( \pmb{x} \).

Nonlinear inverse problem (scalar case)
Suppose we know the nonlinear, scalar function \( f \) that gives a relationship \[ \begin{align} f(x^\ast) = b \end{align} \] for an observed \( b \) but an unknown \( x^\ast \). Finding a value of \( x^\ast \) that satisfies \( f(x^\ast)=b \) is known as a nonlinear inverse problem.
  • Define a function \[ \begin{align} \tilde{f}(x) = f(x)-b. \end{align} \]

  • Thus solving the nonlinear inverse problem in one variable is equivalent to finding the appropriate \( x^\ast \) for which \[ \begin{align} \tilde{f}(x^\ast)= 0 . \end{align} \]

  • Finding a zero of a function, or root finding, is thus equivalent to a nonlinear inverse problem.

  • The Newton-Raphson method is one classical approach which has inspired many modern techniques.

Newton's method in one variable

  • We are searching for the point \( x^\ast\in \mathbb{R} \) for which the modified equation \( \tilde{f}\left(x^\ast\right) = 0 \), and we suppose we have a good initial guess \( x_0 \).
  • We define the tangent approximation as, \[ t(\delta_x) = \tilde{f}(x_0) + \tilde{f}'(x_0) \delta_x \] for some small perturbation value of \( \delta_x \).
  • Recall, \( \tilde{f}'(x_0) \) refers to the value of the derivative of \( \tilde{f} \) at the point \( x_0 \) – suppose this value is nonzero.
  • In this case, we will examine where the tangent line intersects zero to find a better approximation of \( x^\ast \).
  • Suppose that for \( \delta_{x_0} \) we have \[ \begin{matrix} t(\delta_{x_0}) = 0 & \Leftrightarrow & 0= \tilde{f}(x_0) + \tilde{f}'(x_0) \delta_{x_0} & \Leftrightarrow &\delta_{x_0} = \frac{-\tilde{f}(x_0)}{\tilde{f}'(x_0)} \end{matrix} \]
  • The above solution makes sense as long as \( f'(x_0) \) is not equal to zero;
    • if not, this says that the tangent line intersects zero at \( x_1 = x_0 + \delta_{x_0} \), giving a new approximation of \( x^\ast \).
Animation of Newton iterations.

Courtesy of Ralf Pfeifer, CC BY-SA 3.0, via Wikimedia Commons

  • The process of recursively solving for a better approximation of \( x^\ast \) terminates when we reach a certain tolerated level of error in the solution or the process times out, failing to converge.
  • This method has a direct analog in multiple variables, for which we will need to extend our notion of the derivative and Taylor’s theorem to multiple dimensions.

Newton's method – example

  • As a quick example, let's consider the Newton algorithm built-in to Scipy.

    • Scipy is another standard library like Numpy, but which contains various scientific methods and solvers rather than general linear algebra.
  • Specifically, we will import the built-in newton function from the optimize sub-module of scipy.

from scipy.optimize import newton
  • In the following, we define the cubic function \( f(x):=x^3 \), but we are interested in the value \( x^\ast \) for which \( f\left(x^\ast\right)=1 \)

    • The augmented function \( \tilde{f}(x):= x^3 - 1 \) defines the root-finding problem from the nonlinear inverse problem:
def f(x): return (x**3 - 1)
  • The newton function can be supplied an analytical derivative, if this can be computed, to improve the accuracy versus, e.g., a finite-differences approximation.

    • In the below, we supply this as a simple lambda function in the arguments of newton:
root = newton(f, 1.5, fprime=lambda x: 3 * x**2)

Tangent vectors

  • To expand our discussion to multiple variables, we will review some fundamental concepts of vector calculus.

  • Suppose we have a vector valued function, with a single argument:

    \[ \begin{align} \pmb{x}:&\mathbb{R} \rightarrow \mathbb{R}^{N};\\ \pmb{x}(t) :=& \begin{pmatrix} x_1(t) & \cdots & x_{N}(t)\end{pmatrix}^\top; \end{align} \]

    • prototypically, we will think of \( \pmb{x}(t) \) as a curve in state-space, with its position at each time \( t\in\mathbb{R} \) defined by the equation above.
Tangent vector
Suppose \( \pmb{x}(t) \) is defined as above and that each of the component functions \( x_i(t) \) are differentiable. The tangent vector to the state trajectory \( \pmb{x} \) is defined as \[ \vec{x}:= \frac{\mathrm{d}}{\mathrm{d}t} \pmb{x}:= \begin{pmatrix}\frac{\mathrm{d}}{\mathrm{d}t} x_1(t) & \cdots & \frac{\mathrm{d}}{\mathrm{d}t} x_{N}(t)\end{pmatrix}^\top \]
  • In the above, the interpretation of the derivative defining a tangent line is extended into multiple variables;

    • in this case, the tangent line is embedded in a higher-dimensional space of multiple variables.

Tangent spaces

  • An important extension of the tangent vector is the notion of the tangent space;
    • this can be defined in terms of all differential perturbations generated at a point:
    Tangent spaces
    Let \( \pmb{x}\in\mathbb{R}^{N} \) and \( \gamma(t) \) be an arbitrary differentiable curve \( \pmb{\gamma}:\mathbb{R}\rightarrow \mathbb{R}^{N} \) such that \( \pmb{\gamma}(0)= \pmb{x} \) with a tangent vector defined as \( \vec{\gamma}(0):= \frac{\mathrm{d}}{\mathrm{d}t}|_0 \pmb{\gamma} \). The tangent space at \( T_{\pmb{x}} \) is defined by the linear span of all tangent vectors as such through \( \pmb{x} \).
  • In the above, we consider only the simplest definition of the tangent space;
    • in this case the tangent space, \( T_{\pmb{x}} \equiv \mathbb{R}^{N} \), is simply the space of all perturbations to the point \( \pmb{x} \).
    • However, this idea is extended into far greater generality: