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# Solvers, Optimizers, and Automatic Differentiation¶

## Overview¶

In this lecture we introduce a few of the Julia libraries that we’ve found particularly useful for quantitative work in economics

### Setup¶

```
using InstantiateFromURL
activate_github("QuantEcon/QuantEconLectureAllPackages", tag = "v0.9.8");
```

```
using LinearAlgebra, Statistics, Compat
using ForwardDiff, Flux, Optim, JuMP, Ipopt, BlackBoxOptim, Roots, NLsolve
using LeastSquaresOptim, Flux.Tracker
using Flux.Tracker: update!
using Optim: converged, maximum, maximizer, minimizer, iterations #some extra functions
```

## Introduction to Automatic Differentiation¶

Automatic differentiation (AD, sometimes called algorithmic differentiation) is a crucial way to increase the performance of both estimation and solution methods

There are essentially four ways to calculate the gradient or Jacobian on a computer

Calculation by hand

- Where possible, you can calculate the derivative on “pen and paper” and potentially simplify the expression
- Sometimes, though not always, the most accurate and fastest option if there are algebraic simplifications
- The algebra is error prone for non-trivial setups

Finite differences

- Evaluate the function at least $ N $ times to get the gradient – Jacobians are even worse
- Large $ \Delta $ is numerically stable but inaccurate, too small of $ \Delta $ is numerically unstable but more accurate
- Avoid if you can, and use packages (e.g. DiffEqDiffTools.jl ) to get a good choice of $ \Delta $

Symbolic differentiation

- If you put in an expression for a function, some packages will do symbolic differentiation
- In effect, repeated applications of the chain rule, product rule, etc.
- Sometimes a good solution, if the package can handle your functions

Automatic Differentiation

- Essentially the same as symbolic differentiation, just occurring at a different time in the compilation process
- Equivalent to analytical derivatives since it uses the chain rule, etc.

We will explore AD packages in Julia rather than the alternatives

### Automatic Differentiation¶

To summarize here, first recall the chain rule (adapted from Wikipedia)

$$ \frac{dy}{dx} = \frac{dy}{dw} \cdot \frac{dw}{dx} $$Consider functions composed of calculations with fundamental operations with known analytical derivatives, such as $ f(x_1, x_2) = x_1 x_2 + \sin(x_1) $

To compute $ \frac{d f(x_1,x_2)}{d x_1} $

$$ \begin{array}{l|l} \text{Operations to compute value} & \text{Operations to compute \$\frac{\partial f(x_1,x_2)}{\partial x_1}\$} \\ \hline w_1 = x_1 & \frac{d w_1}{d x_1} = 1 \text{ (seed)}\\ w_2 = x_2 & \frac{d w_2}{d x_1} = 0 \text{ (seed)} \\ w_3 = w_1 \cdot w_2 & \frac{\partial w_3}{\partial x_1} = w_2 \cdot \frac{d w_1}{d x_1} + w_1 \cdot \frac{d w_2}{d x_1} \\ w_4 = \sin w_1 & \frac{d w_4}{d x_1} = \cos w_1 \cdot \frac{d w_1}{d x_1} \\ w_5 = w_3 + w_4 & \frac{\partial w_5}{\partial x_1} = \frac{\partial w_3}{\partial x_1} + \frac{d w_4}{d x_1} \end{array} $$### Using Dual Numbers¶

One way to implement this (used in forward-mode AD) is to use dual numbers

Take a number $ x $ and augment it with an infinitesimal $ \epsilon $ such that $ \epsilon^2 = 0 $, i.e. $ x \to x + x' \epsilon $

All math is then done with this (mathematical, rather than Julia) tuple $ (x, x') $ where the $ x' $ may be hidden from the user

With this definition, we can write a general rule for differentiation of $ g(x,y) $ as

$$ g \big( \left(x,x'\right),\left(y,y'\right) \big) = \left(g(x,y),\partial_x g(x,y)x' + \partial_y g(x,y)y' \right) $$This calculation is simply the chain rule for the total derivative

An AD library using dual numbers concurrently calculates the function and its derivatives, repeating the chain rule until it hits a set of intrinsic rules such as

$$ \begin{align*} x + y \to \left(x,x'\right) + \left(y,y'\right) &= \left(x + y,\underbrace{x' + y'}_{\partial(x + y) = \partial x + \partial y}\right)\\ x y \to \left(x,x'\right) \times \left(y,y'\right) &= \left(x y,\underbrace{x'y + y'x}_{\partial(x y) = y \partial x + x \partial y}\right)\\ \exp(x) \to \exp(\left(x, x'\right)) &= \left(\exp(x),\underbrace{x'\exp(x)}_{\partial(\exp(x)) = \exp(x)\partial x} \right) \end{align*} $$### ForwardDiff.jl¶

We have already seen one of the AD packages in Julia

```
using ForwardDiff
h(x) = sin(x[1]) + x[1] * x[2] + sinh(x[1] * x[2]) # multivariate.
x = [1.4 2.2]
@show ForwardDiff.gradient(h,x) # use AD, seeds from x
#Or, can use complicated functions of many variables
f(x) = sum(sin, x) + prod(tan, x) * sum(sqrt, x)
g = (x) -> ForwardDiff.gradient(f, x); # g() is now the gradient
@show g(rand(20)); # gradient at a random point
# ForwardDiff.hessian(f,x') # or the hessian
```

We can even auto-differentiate complicated functions with embedded iterations

```
function squareroot(x) #pretending we don't know sqrt()
z = copy(x) # Initial starting point for Newton’s method
while abs(z*z - x) > 1e-13
z = z - (z*z-x)/(2z)
end
return z
end
squareroot(2.0)
```

```
using ForwardDiff
dsqrt(x) = ForwardDiff.derivative(squareroot, x)
dsqrt(2.0)
```

```
using Flux
using Flux.Tracker
using Flux.Tracker: update!
f(x) = 3x^2 + 2x + 1
# df/dx = 6x + 2
df(x) = Tracker.gradient(f, x)[1]
df(2) # 14.0 (tracked)
```

```
A = rand(2,2)
f(x) = A * x
x0 = [0.1, 2.0]
f(x0)
Flux.jacobian(f, x0)
```

As before, we can differentiate complicated functions

```
dsquareroot(x) = Tracker.gradient(squareroot, x)
```

From the documentation, we can use a machine learning approach to a linear regression

```
W = rand(2, 5)
b = rand(2)
predict(x) = W*x .+ b
function loss(x, y)
ŷ = predict(x)
sum((y .- ŷ).^2)
end
x, y = rand(5), rand(2) # Dummy data
loss(x, y) # ~ 3
```

```
W = param(W)
b = param(b)
gs = Tracker.gradient(() -> loss(x, y), Params([W, b]))
Δ = gs[W]
# Update the parameter and reset the gradient
update!(W, -0.1Δ)
loss(x, y) # ~ 2.5
```

## Optimization¶

There are a large number of packages intended to be used for optimization in Julia

Part of the reason for the diversity of options is that Julia makes it possible to efficiently implement a large number of variations on optimization routines

The other reason is that different types of optimization problems require different algorithms

#### Univariate Functions on Bounded Intervals¶

Univariate optimization defaults to a robust hybrid optimization routine called Brent’s method

```
using Optim
using Optim: converged, maximum, maximizer, minimizer, iterations #some extra functions
result = optimize(x-> x^2, -2.0, 1.0)
```

Always check if the results converged, and throw errors otherwise

```
converged(result) || error("Failed to converge in $(iterations(result)) iterations")
xmin = result.minimizer
result.minimum
```

The first line is a logical OR between `converged(result)`

and `error("...")`

If the convergence check passes, the logical sentence is true, and it will proceed to the next line; if not, it will throw the error

Or to maximize

```
f(x) = -x^2
result = maximize(f, -2.0, 1.0)
converged(result) || error("Failed to converge in $(iterations(result)) iterations")
xmin = maximizer(result)
fmax = maximum(result)
```

**Note:** Notice that we call `optimize`

results using `result.minimizer`

, and `maximize`

results using `maximizer(result)`

#### Unconstrained Multivariate Optimization¶

There are a variety of algorithms and options for multivariate optimization

From the documentation, the simplest version is

```
f(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
x_iv = [0.0, 0.0]
results = optimize(f, x_iv) # i.e. optimize(f, x_iv, NelderMead())
```

The default algorithm in `NelderMead`

, which is derivative-free and hence requires many function evaluations

To change the algorithm type to L-BFGS

```
results = optimize(f, x_iv, LBFGS())
println("minimum = $(results.minimum) with argmin = $(results.minimizer) in "*
"$(results.iterations) iterations")
```

Note that this has fewer iterations

As no derivative was given, it used finite differences to approximate the gradient of `f(x)`

However, since most of the algorithms require derivatives, you will often want to use auto differentiation or pass analytical gradients if possible

```
f(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
x_iv = [0.0, 0.0]
results = optimize(f, x_iv, LBFGS(), autodiff=:forward) # i.e. use ForwardDiff.jl
println("minimum = $(results.minimum) with argmin = $(results.minimizer) in "*
"$(results.iterations) iterations")
```

Note that we did not need to use `ForwardDiff.jl`

directly, as long as our `f(x)`

function was written to be generic (see the generic programming lecture )

Alternatively, with an analytical gradient

```
f(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
x_iv = [0.0, 0.0]
function g!(G, x)
G[1] = -2.0 * (1.0 - x[1]) - 400.0 * (x[2] - x[1]^2) * x[1]
G[2] = 200.0 * (x[2] - x[1]^2)
end
results = optimize(f, g!, x0, LBFGS()) # or ConjugateGradient()
println("minimum = $(results.minimum) with argmin = $(results.minimizer) in "*
"$(results.iterations) iterations")
```

For derivative-free methods, you can change the algorithm – and have no need to provide a gradient

```
f(x) = (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
x_iv = [0.0, 0.0]
results = optimize(f, x_iv, SimulatedAnnealing()) # or ParticleSwarm() or NelderMead()
```

However, you will note that this did not converge, as stochastic methods typically require many more iterations as a tradeoff for their global-convergence properties

See the maximum likelihood example and the accompanying Jupyter notebook

### JuMP.jl¶

The JuMP.jl package is an ambitious implementation of a modelling language for optimization problems in Julia

In that sense, it is more like an AMPL (or Pyomo) built on top of the Julia language with macros, and able to use a variety of different commerical and open source solvers

If you have a linear, quadratic, conic, mixed-integer linear, etc. problem then this will likely be the ideal “meta-package” for calling various solvers

For nonlinear problems, the modelling language may make things difficult for complicated functions (as it is not designed to be used as a general-purpose nonlinear optimizer)

See the quick start guide for more details on all of the options

The following is an example of calling a linear objective with a nonlinear constraint (provided by an external function)

Here `Ipopt`

stands for `Interior Point OPTimizer`

, a nonlinear solver in Julia

```
using JuMP, Ipopt
# solve
# max( x[1] + x[2] )
# st sqrt(x[1]^2 + x[2]^2) <= 1
function squareroot(x) # pretending we don't know sqrt()
z = x # Initial starting point for Newton’s method
while abs(z*z - x) > 1e-13
z = z - (z*z-x)/(2z)
end
return z
end
m = Model(with_optimizer(Ipopt.Optimizer))
# need to register user defined functions for AD
JuMP.register(m,:squareroot, 1, squareroot, autodiff=true)
@variable(m, x[1:2], start=0.5) # start is the initial condition
@objective(m, Max, sum(x))
@NLconstraint(m, squareroot(x[1]^2+x[2]^2) <= 1)
@show JuMP.optimize!(m)
```

And this is an example of a quadratic objective

```
# solve
# min (1-x)^2 + 100(y-x^2)^2)
# st x + y >= 10
using JuMP,Ipopt
m = Model(with_optimizer(Ipopt.Optimizer)) # settings for the solver
@variable(m, x, start = 0.0)
@variable(m, y, start = 0.0)
@NLobjective(m, Min, (1-x)^2 + 100(y-x^2)^2)
JuMP.optimize!(m)
println("x = ", value(x), " y = ", value(y))
# adding a (linear) constraint
@constraint(m, x + y == 10)
JuMP.optimize!(m)
println("x = ", value(x), " y = ", value(y))
```

### BlackBoxOptim.jl¶

Another package for doing global optimization without derivatives is BlackBoxOptim.jl

To see an example from the documentation

```
using BlackBoxOptim
function rosenbrock2d(x)
return (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
end
results = bboptimize(rosenbrock2d; SearchRange = (-5.0, 5.0), NumDimensions = 2);
```

An example for parallel execution of the objective is provided

## Systems of Equations and Least Squares¶

### Roots.jl¶

A root of a real function $ f $ on $ [a,b] $ is an $ x \in [a, b] $ such that $ f(x)=0 $

For example, if we plot the function

$$ f(x) = \sin(4 (x - 1/4)) + x + x^{20} - 1 \tag{1} $$

with $ x \in [0,1] $ we get

The unique root is approximately 0.408

The Roots.jl package offers `fzero()`

to find roots

```
using Roots
f(x) = sin(4 * (x - 1/4)) + x + x^20 - 1
fzero(f, 0, 1)
```

### NLsolve.jl¶

The NLsolve.jl package provides functions to solve for multivariate systems of equations and fixed points

From the documentation, to solve for a system of equations without providing a Jacobian

```
using NLsolve
f(x) = [(x[1]+3)*(x[2]^3-7)+18
sin(x[2]*exp(x[1])-1)] # returns an array
results = nlsolve(f, [ 0.1; 1.2])
```

In the above case, the algorithm used finite differences to calculate the Jacobian

Alternatively, if `f(x)`

is written generically, you can use auto-differentiation with a single setting

```
results = nlsolve(f, [ 0.1; 1.2], autodiff=:forward)
println("converged=$(NLsolve.converged(results)) at root=$(results.zero) in "*
"$(results.iterations) iterations and $(results.f_calls) function calls")
```

Providing a function which operates inplace (i.e. modifies an argument) may help performance for large systems of equations (and hurt it for small ones)

```
function f!(F, x) # modifies the first argument
F[1] = (x[1]+3)*(x[2]^3-7)+18
F[2] = sin(x[2]*exp(x[1])-1)
end
results = nlsolve(f!, [ 0.1; 1.2], autodiff=:forward)
println("converged=$(NLsolve.converged(results)) at root=$(results.zero) in "*
"$(results.iterations) iterations and $(results.f_calls) function calls")
```

## LeastSquaresOptim.jl¶

Many optimization problems can be solved using linear or nonlinear least squares

Let $ x \in R^N $ and $ F(x) : R^N \to R^M $ with $ M \geq N $, then the nonlinear least squares problem is

$$ \min_x F(x)^T F(x) $$While $ F(x)^T F(x) \to R $, and hence this problem could technically use any nonlinear optimizer, it is useful to exploit the structure of the problem

In particular, the Jacobian of $ F(x) $, can be used to approximate the Hessian of the objective

As with most nonlinear optimization problems, the benefits will typically become evident only when analytical or automatic differentiation is possible

If $ M = N $ and we know a root $ F(x^*) = 0 $ to the system of equations exists, then NLS is the defacto method for solving large **systems of equations**

An implementation of NLS is given in LeastSquaresOptim.jl

From the documentation

```
using LeastSquaresOptim
function rosenbrock(x)
[1 - x[1], 100 * (x[2]-x[1]^2)]
end
LeastSquaresOptim.optimize(rosenbrock, zeros(2), Dogleg())
```

**Note:** Because there is a name clash between `Optim.jl`

and this package, to use both we need to qualify the use of the `optimize`

function (i.e. `LeastSquaresOptim.optimize`

)

Here, by default it will use AD with `ForwardDiff.jl`

to calculate the Jacobian,
but you could also provide your own calculation of the Jacobian (analytical or using finite differences) and/or calculate the function inplace

```
function rosenbrock_f!(out, x)
out[1] = 1 - x[1]
out[2] = 100 * (x[2]-x[1]^2)
end
LeastSquaresOptim.optimize!(LeastSquaresProblem(x = zeros(2),
f! = rosenbrock_f!, output_length = 2))
# if you want to use gradient
function rosenbrock_g!(J, x)
J[1, 1] = -1
J[1, 2] = 0
J[2, 1] = -200 * x[1]
J[2, 2] = 100
end
LeastSquaresOptim.optimize!(LeastSquaresProblem(x = zeros(2),
f! = rosenbrock_f!, g! = rosenbrock_g!, output_length = 2))
```

## Additional Notes¶

Watch this video from one of Julia’s creators on automatic differentiation