Gradient descent, Newton-like

Vanilla Newton methods

From school one might remember that classic so-called Newton method for iteratively improving a guessed location \(x_{k}\) for a root of a 1-dimensional function \(g:\mathbb{R}\to\mathbb{R}\):

\[ x_{k+1}=x_{k}-\frac{g\left(x_{k}\right)}{g'\left(x_{k}\right)} \]

Under the assumption that it is twice continuously differentiable, we can find stationary points—zeros of the derivative—of a function \(f:\mathbb{R}\to\mathbb{R}\). (i.e. extrema and saddle points, which might in fact be minima.)

\[ x_{k+1}=x_{k}-\frac{f'\left(x_{k}\right)}{f''\left(x_{k}\right)} \]

How and why does this work again? Loosely, the argument is that we can use the Taylor expansion of a function to produce a locally quadratic model for the function, which might be not at all globally quadratic.

Returning to the objective with a multidimensional argument \(f:\mathbb{R}^n\to\mathbb{R},\) Taylor’s theorem for twice continuously differentiable functions tells us that

\[\begin{aligned} f(x+p)&= f(x)+\nabla^T f (x+p)p +\frac{1}{2}p^T \nabla^2f(x+p)p + p^TR(x+p)p\\ \end{aligned}\] and \(\lim _{p \rightarrow 0} R(x+p)=0\).

Putting the remainder in Cauchy form, this tells us

\[ f(x+p)=f(x)+\nabla f^T(x)p+\frac{1}{2}p^T\nabla^2f(x+tp)p \] for some \(t\in(0,1)\).

Generally, we might take a local quadratic model for \(f\) at \(x+p\) as

\[ \nabla f(x+p)\simeq m_x(p)= f(x)+\nabla f^T(x)p+\frac{1}{2}p^T\nabla^2f(x)p \]

for some \(t\in(0,1)\).

If we ignore some subtleties and imagined this quadratic model was true, it would have an extremum where the directional derivative of this model vanished, i.e. \(\nabla m_x(p)\equiv 0\) where \(\nabla m_x(p):=\nabla f(x)+\nabla^2 f(x)p\). This would be satisfied at

\[\begin{aligned} \nabla f(x)+\nabla^2 f(x)p &= 0\\ \nabla f(x)&=-\nabla^2 f(x)p\\ p&=-[\nabla^2 f(x)]^{-1}\nabla f(x)\\ \end{aligned}\]

From this we get the multidimensional Newton update rule, by choosing for each \(x_{k}\) the locally optimal \(p\) quadratic approximation:

\[ x_{k+1}= x_{k}-\left[ \nabla^2 f\left( x_{k}\right)\right]^{-1} \nabla f\left( x_{k}\right), n \geq 0 \]

In practice the quadratic model is not true (or we would be done in one step) and we would choose a more robust method that by using a trust region or line-search algorithm to handle the divergence between the model and its quadratic approximation. Either way the quadratic approximation would be used, which would mean that the inverse Hessian matrix \(\left[ \nabla^2 f\left( x_{k}\right)\right]^{-1}\) would be important.

🏗 Unconstrained assumption, need for saddle points not just extrema with Lagrange methods, positive-definiteness etc.

Quasi Newton methods

Secant conditions and approximations to the Hessian.

Let’s say we are designing a second-order update method.

See e.g. Nocedal and Wright (2006).

In BFGS the approximate Hessian \(B_k\) is used in place of the true Hessian.

Given the displacement \(s_{k}\) and the change of gradients \(y_{k}\), the secant equation requires that the symmetric positive definite matrix \(B_{k+1}\) map \(s_{k}\) into \(y_{k}.\)

\[ H_k(x^{(k)} − x^{(k−1)}) = \nabla f(x^{(k)}) − \nabla f(x^{(k−1)}) \]

…

We hope to get from this the secant condition.

\[ B_{k+1} s_{k}=y_{k} \]

TODO: compare and contrast to the Wolfe conditions on step length.

🏗

Hessian free

Second order optimisation that does not require the Hessian matrix to be given explicitly. I should work out the boundaries of this definition. Does it include Quasi Newton? Intersect with? Overlap? 🏗

Andre Gibiansky has an example for coders.

Natural gradient descent.

I haven’t actually worked out what this is yet, nor whether it relates or is filed correctly there. See natural gradient descent.

Stochastic

See stochastic 2nd order gradient descent for some bonus complications.

References