# 2022-04-04 Differentiation¶

## Last time¶

• Nonlinear models

• Computing derivatives

• numeric

• analytic by hand

## Today¶

• Project feedback

• Computing derivatives in maintainable code

• Forward and reverse

• Algorithmic (automatic) differentiation

using LinearAlgebra
using Plots
default(linewidth=4, legendfontsize=12)

function vander(x, k=nothing)
if isnothing(k)
k = length(x)
end
m = length(x)
V = ones(m, k)
for j in 2:k
V[:, j] = V[:, j-1] .* x
end
V
end

function vander_chebyshev(x, n=nothing)
if isnothing(n)
n = length(x) # Square by default
end
m = length(x)
T = ones(m, n)
if n > 1
T[:, 2] = x
end
for k in 3:n
#T[:, k] = x .* T[:, k-1]
T[:, k] = 2 * x .* T[:,k-1] - T[:, k-2]
end
T
end

runge(x) = 1 / (1 + 10*x^2)
runge_noisy(x, sigma) = runge.(x) + randn(size(x)) * sigma

"""Minimize loss(c) via gradient descent with initial guess c0
using learning rate gamma.  Declares convergence when gradient
is less than tol or after 500 steps.
"""
c = copy(c0)
chist = [copy(c)]
lhist = [loss(c)]
for it in 1:500
c -= gamma * g
push!(chist, copy(c))
push!(lhist, loss(c))
if norm(g) < tol
break
end
end
(c, hcat(chist...), lhist)
end

grad_descent (generic function with 1 method)


# Constructive project feedback/questions¶

• You can start now: look at a few projects in #sharing on Zulip

## Examples¶

• Explain (in a sentence or two) something you learned:

It was neat to see how interpolation showed up in this graphics application, and how B-splines were used to have both sharp corners and smooth curves in the same representation.

• Ask a question about the software, the underlying math, or about how people interact with or are affected by the software.

• Give a suggestion for improving or extending the project, or a new perspective by which to understand its context.

# Nonlinear models¶

Instead of the linear model

$f(x,c) = V(x) c = c_0 + c_1 \underbrace{x}_{T_1(x)} + c_2 T_2(x) + \dotsb$
let’s consider a rational model with only three parameters
$f(x,c) = \frac{1}{c_1 + c_2 x + c_3 x^2} = (c_1 + c_2 x + c_3 x^2)^{-1} .$
We’ll use the same loss function
$L(c; x,y) = \frac 1 2 \lVert f(x,c) - y \rVert^2 .$

We will also need the gradient

$\nabla_c L(c; x,y) = \big( f(x,c) - y \big)^T \nabla_c f(x,c)$
where

(26)\begin{align} \frac{\partial f(x,c)}{\partial c_1} &= -(c_1 + c_2 x + c_3 x^2)^{-2} = - f(x,c)^2 \\ \frac{\partial f(x,c)}{\partial c_2} &= -(c_1 + c_2 x + c_3 x^2)^{-2} x = - f(x,c)^2 x \\ \frac{\partial f(x,c)}{\partial c_3} &= -(c_1 + c_2 x + c_3 x^2)^{-2} x^2 = - f(x,c)^2 x^2 . \end{align}

# Fitting a rational function¶

f(x, c) = 1 ./ (c[1] .+ c[2].*x + c[3].*x.^2)
f2 = f(x, c).^2
[-f2 -f2.*x -f2.*x.^2]
end
function loss(c)
r = f(x, c) - y
0.5 * r' * r
end
r = f(x, c) - y
vec(r' * gradf(x, c))
end

gradient (generic function with 1 method)

x = LinRange(-1, 1, 200)
y = runge_noisy(x, .5)
c, _, lhist = grad_descent(loss, gradient, [1., 0, 1.],
gamma=1e-2)
@show c
plot(lhist, yscale=:log10)

c = [1.189773116107557, 0.37072648768485034, 6.165752708060916]


# Compare fits on noisy data¶

scatter(x, y)
V = vander_chebyshev(x, 7)
plot!(x -> runge(x), color=:black, label="Runge")
plot!(x, V * (V \ y), label="Chebyshev fit")
plot!(x -> f(x, c), label="Rational fit")


# How do we compute those derivatives as the model gets complicated?¶

$\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}$
• How should we choose $$h$$?

• Too big: discretization error dominates (think of truncating the Taylor series)

• Too small: rounding error dominates

# Automatic step size selection¶

• Walker and Pernice

• Dennis and Schnabel

diff(f, x; h=1e-8) = (f(x+h) - f(x)) / h

function diff_wp(f, x; eps=1e-8)
"""Diff using Walker and Pernice (1998) choice of step"""
h = eps * (1 + abs(x))
(f(x+h) - f(x)) / h
end

x = 1
diff_wp(sin, x) - cos(x)

-2.9698852266335507e-9


# Symbolic differentiation¶

using Symbolics

@variables x
Dx = Differential(x)

y = sin(x)
Dx(y)

$$$\frac{dsin(x)}{dx}$$$
expand_derivatives(Dx(y))

$$$\cos\left( x \right)$$$

# Awesome, what about products?¶

y = x
for _ in 1:2
y = cos(y^pi) * log(y)
end
expand_derivatives(Dx(y))

$$$\frac{\left( \frac{\cos\left( x^{\pi} \right)}{x} - 3.141592653589793 x^{2.141592653589793} \log\left( x \right) \sin\left( x^{\pi} \right) \right) \cos\left( \cos^{3.141592653589793}\left( x^{\pi} \right) \left( \log\left( x \right) \right)^{3.141592653589793} \right)}{\cos\left( x^{\pi} \right) \log\left( x \right)} - \left( \frac{3.141592653589793 \cos^{3.141592653589793}\left( x^{\pi} \right) \left( \log\left( x \right) \right)^{2.141592653589793}}{x} - 9.869604401089358 \cos^{2.141592653589793}\left( x^{\pi} \right) \left( \log\left( x \right) \right)^{3.141592653589793} x^{2.141592653589793} \sin\left( x^{\pi} \right) \right) \sin\left( \cos^{3.141592653589793}\left( x^{\pi} \right) \left( \log\left( x \right) \right)^{3.141592653589793} \right) \log\left( \log\left( x \right) \cos\left( x^{\pi} \right) \right)$$$
• The size of these expressions can grow exponentially

# Hand-coding derivatives¶

$df = f'(x) dx$
function f(x)
y = x
for _ in 1:2
a = y^pi
b = cos(a)
c = log(y)
y = b * c
end
y
end

f(1.9), diff_wp(f, 1.9)

(-1.5346823414986814, -34.032439961925064)

function df(x, dx)
y = x
dy = dx
for _ in 1:2
a = y ^ pi
da = pi * y^(pi - 1) * dy
b = cos(a)
db = -sin(a) * da
c = log(y)
dc = 1/y * dy
y = b * c
dy = db * c + b * dc
end
y, dy
end

df(1.9, 1)

(-1.5346823414986814, -34.03241959914048)


# We can go the other way¶

We can differentiate a composition $$h(g(f(x)))$$ as

(27)\begin{align} \operatorname{d} h &= h' \operatorname{d} g \\ \operatorname{d} g &= g' \operatorname{d} f \\ \operatorname{d} f &= f' \operatorname{d} x. \end{align}

What we’ve done above is called “forward mode”, and amounts to placing the parentheses in the chain rule like

$\operatorname d h = \frac{dh}{dg} \left(\frac{dg}{df} \left(\frac{df}{dx} \operatorname d x \right) \right) .$

The expression means the same thing if we rearrange the parentheses,

$\operatorname d h = \left( \left( \left( \frac{dh}{dg} \right) \frac{dg}{df} \right) \frac{df}{dx} \right) \operatorname d x$

which we can compute with in reverse order via

$\underbrace{\bar x}_{\frac{dh}{dx}} = \underbrace{\bar g \frac{dg}{df}}_{\bar f} \frac{df}{dx} .$

# A reverse mode example¶

$\underbrace{\bar x}_{\frac{dh}{dx}} = \underbrace{\bar g \frac{dg}{df}}_{\bar f} \frac{df}{dx} .$
function g(x)
a = x^pi
b = cos(a)
c = log(x)
y = b * c
y
end
(g(1.4), diff_wp(g, 1.4))

(-0.32484122107701546, -1.2559760384500684)

function gback(x, y_)
a = x^pi
b = cos(a)
c = log(x)
y = b * c
# backward pass
c_ = y_ * b
b_ = c * y_
a_ = -sin(a) * b_
x_ = 1/x * c_ + pi * x^(pi-1) * a_
x_
end
gback(1.4, 1)

-1.2559761698835525


# Automatic differentiation¶

using Zygote

WARNING: using Zygote.gradient in module Main conflicts with an existing identifier.

Zygote.gradient(f, 1.9)

(-34.03241959914049,)


# How does Zygote work?¶

square(x) = x^2
@code_llvm square(1.5)

;  @ In[41]:1 within square
define double @julia_square_10931(double %0) #0 {
top:
; ┌ @ intfuncs.jl:312 within literal_pow
; │┌ @ float.jl:405 within *
%1 = fmul double %0, %0
; └└
ret double %1
}

@code_llvm Zygote.gradient(square, 1.5)

;  @ /home/jed/.julia/packages/Zygote/H6vD3/src/compiler/interface.jl:74 within gradient
define [1 x double] @julia_gradient_11254(double %0) #0 {
top:
;  @ /home/jed/.julia/packages/Zygote/H6vD3/src/compiler/interface.jl:76 within gradient
; ┌ @ /home/jed/.julia/packages/Zygote/H6vD3/src/compiler/interface.jl:41 within #56
; │┌ @ In[26]:1 within Pullback
; ││┌ @ /home/jed/.julia/packages/ZygoteRules/AIbCs/src/adjoint.jl:67 within #1822#back
; │││┌ @ /home/jed/.julia/packages/Zygote/H6vD3/src/lib/number.jl:6 within #254
; ││││┌ @ promotion.jl:380 within * @ float.jl:405
%1 = fmul double %0, 2.000000e+00
; └└└└└
;  @ /home/jed/.julia/packages/Zygote/H6vD3/src/compiler/interface.jl:77 within gradient
%.fca.0.insert = insertvalue [1 x double] zeroinitializer, double %1, 0
ret [1 x double] %.fca.0.insert
}


# Kinds of algorithmic differentation¶

• Source transformation: Fortran code in, Fortran code out

• Duplicates compiler features, usually incomplete language coverage

• Produces efficient code

• Hard to vectorize

• Loops are effectively unrolled/inefficient

• Just-in-time compilation: tightly coupled with compiler

• JIT lag

• Needs dynamic language features (JAX) or tight integration with compiler (Zygote, Enzyme)

• Some sharp bits

# Forward or reverse?¶

It all depends on the shape.

• One input, many outputs: use forward mode

• “One input” can be looking in one direction

• Many inputs, one output: use reverse mode

• Will need to traverse execution backwards (“tape”)

• Hierarchical checkpointing

• About square? Forward is usually a bit more efficient.

# Ill-conditioned optimization¶

$L(c; x, y) = \frac 1 2 \lVert \underbrace{f(x, c) - y}_{r(c)} \rVert_{C^{-1}}^2$

Gradient of $$L$$ requires the Jacobian $$J$$ of the model $$f$$.

$g(c) = \nabla_c L = r^T \underbrace{\nabla_c f}_{J}$

We can solve $$g(c) = 0$$ using a Newton method

$g(c + \delta c) = g(c) + \underbrace{\nabla_c g}_H\delta c + O((\delta c)^2)$

The Hessian requires the second derivative of $$f$$, which can cause problems.

$H = J^T J + r^T (\nabla_c J)$

# Newton-like methods for optimization¶

Solve

$H \delta c = -g(c)$

Update $$c \gets c + \gamma \delta c$$\$ using a line search or trust region.

# Outlook¶

• The optimization problem can be solved using a Newton method. It can be onerous to implement the needed derivatives.

• The Gauss-Newton method (see activity) is often more practical than Newton while being faster than gradient descent, though it lacks robustness.

• The Levenberg-Marquardt method provides a sort of middle-ground between Gauss-Newton and gradient descent.

• Many globalization techniques are used for models that possess many local minima.

• One pervasive approach is stochastic gradient descent, where small batches (e.g., 1 or 10 or 20) are selected randomly from the corpus of observations (500 in our current example), and a step of gradient descent is applied to that reduced set of observations. This helps to escape saddle points and weak local minima.

• Among expressive models $$f(x,c)$$, some may converge much more easily than others. Having a good optimization algorithm is essential for nonlinear regression with complicated models, especially those with many parameters $$c$$.

• Classification is a very similar problem to regression, but the observations $$y$$ are discrete, thus

• models $$f(x,c)$$ must have discrete output

• the least squares loss function is not appropriate.

• Why momentum really works