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title subtitle author institute date lang
Julia, my new computing friend?
Julia introduction for MATLAB users
Lilian Besson and Pierre Haessig
SCEE Team, IETR, CentraleSupélec, Rennes
Thursday 14th of June, 2018
english

Note: a PDF is available here online.


« Julia, my new friend for computing and optimization? »

  • Intro to the Julia programming language, for MATLAB users

  • Date: 14th of June 2018

  • Who: Lilian Besson & Pierre Haessig (SCEE & AUT team @ IETR / CentraleSupélec campus Rennes)


Agenda for today [25 min]

  1. What is Julia [3 min]
  2. Comparison with MATLAB [3 min]
  3. Examples of problems solved Julia [5 min]
  4. Longer example on optimization with JuMP [10min]
  5. Links for more information ? [2 min]

1. What is Julia ?

  • Developed and popular from the last 7 years
  • Open-source and free programming language (MIT license)
  • Interpreted and compiled, very efficient
  • But easy syntax, dynamic typing, inline documentation etc
  • Multi-platform (Windows, Mac OS X, GNU/Linux etc)
  • MATLAB-like imperative style
  • MATLAB-like syntax for linear algebra etc
  • Designed to be simple to learn and use
  • Easy to run your code in parallel (multi-core & cluster)
  • Used worldwide: research, data science, finance etc…

Ressources


Comparison with MATLAB

Julia 😃 MATLAB 😢
Cost Free ✌️ Hundreds of euros / year
License Open-source 1 year user license (no longer after your PhD!)
Comes from A non-profit foundation, and the community MathWorks company
Scope Mainly numeric Numeric only
Performances Very good performance Faster than Python, slower than Julia
Packaging Pkg manager included. Based on git + GitHub, very easy to use Toolboxes already included but 💰 have to pay if you wat more!
Editor/IDE Jupyter is recommended (Juno is also good) Good IDE already included
Parallel computations Very easy, low overhead cost Possible, high overhead
Usage Generic, worldwide 🌎 Research in academia and industry
Fame Young but starts to be known Old and known, in decline
Support? Community$^1$ (StackOverflow, mailing lists etc). By MathWorks
Documentation OK and growing, inline/online OK, inline/online

Note$^1$: JuliaPro offer paid licenses, if professional support is needed.


How to install Julia ⬇️

  • You can try online for free on JuliaBox.com

  • On Linux, Mac OS or Windows:

  • Takes about 4 minutes... and it's free !

You also need Python 3 to use Jupyter , I suggest to use Anaconda.com/download if you don't have Python yet.

download_julia.png 40%

  1. Select the binary of your platform 📦
  2. Run the binary 🏃 !
  3. Wait 🕜
  4. Done 👌 ! Test with julia in a terminal

Different tools to use Julia

  • Use julia for the command line for short experiments 50%
  • Use the Juno IDE to edit large projects

Demo time ⌚️ !

Demo time ⌚️ !


📦 How to install modules in Julia ?

  • Installing is easy !
julia> Pkd.add("IJulia")  # installs IJulia
  • Updating also!
julia> Pkg.update()

🔍 How to find the module you need ?


📦 Overview of famous Julia modules

  • Plotting:
  • The JuliaDiffEq collection for differential equations
  • The JuliaOpt collection for optimization
  • The JuliaStats collection for statistics
  • And many more!

Find more specific packages on GitHub.com/svaksha/Julia.jl/


Many packages, and a quickly growing community

bg original 50%

Julia is still in development, in version v0.6 but version 1.0 is planned soon!


2. Main differences in syntax between Julia and MATLAB

Ref: cheatsheets.quantecon.org

Julia MATLAB
File ext. .jl .m
Comment # blabla... % blabla...
Indexing a[1] to a[end] a(1) to a(end)
Slicing a[1:100] (view) a(1:100) (⚠️ copy)
Operations Linear algebra by default Linear algebra by default
Block Use end to close all blocks Use endif endfor etc
Help ?func help func
And a & b a && b
Or `a b`
Datatype Array of any type multi-dim doubles array
Array [1 2; 3 4] [1 2; 3 4]
Size size(a) size(a)
Nb Dim ndims(a) ndims(a)
Last a[end] a(end)
Tranpose a.' a.'
Conj. transpose a' a'
Matrix x a * b a * b
Element-wise x a .* b a .* b
Element-wise / a ./ b a ./ b
Element-wise ^ a ^ 3 a .^ 3
Zeros zeros(2, 3, 5) zeros(2, 3, 5)
Ones ones(2, 3, 5) ones(2, 3, 5)
Identity eye(10) eye(10)
Range range(0, 100, 2) or 1:2:100 1:2:100
Maximum max(a) max(max(a)) ?
Random matrix rand(3, 4) rand(3, 4)
L2 Norm norm(v) norm(v)
Inverse inv(a) inv(a)
Solve syst. a \ b a \ b
Eigen vals V, D = eig(a) [V,D]=eig(a)
FFT/IFFT fft(a), ifft(a) fft(a),ifft(a)

Very close to MATLAB for linear algebra!


3. Scientific problems solved with Julia

Just to give examples of syntax and modules

  1. 1D numerical integration and plot
  2. Solving a $2^{\text{nd}}$ order Ordinary Differential Equation

3.1. 1D numerical integration and plot

Exercise : evaluate and plot this function on [-1, 1] : $$\mathrm{Ei}(x) := \int_{-x}^{\infty} \frac{\mathrm{e}^u}{u} ;\mathrm{d}u$$

How to?

Use packages and everything is easy!


using QuadGK

function Ei(x, minfloat=1e-3, maxfloat=100)
    f = t -> exp(-t) / t  # inline function
    if x > 0
        return quadgk(f, -x, -minfloat)[1]
             + quadgk(f, minfloat, maxfloat)[1]
    else
        return quadgk(f, -x, maxfloat)[1]
    end
end

X = linspace(-1, 1, 1000)   # 1000 points
Y = [ Ei(x) for x in X ]

using Winston
plot(X, Y)
title("The function Ei(x)")
xlabel("x"); ylabel("y")
savefig("figures/Ei_integral.png")

bg original 65%


3.2. Solving a $2^{\text{nd}}$ order ODE

Goal : solve and plot the differential equation of a pendulum: $$\theta''(t) + b ,\theta'(t) + c ,\sin(\theta(t)) = 0$$ For $b = 1/4$, $c = 5$, $\theta(0) = \pi - 0.1$, $\theta'(0)=0$, $t\in[0,10]$

How to?

Use packages!


using DifferentialEquations

b, c = 0.25, 5.0

# macro magic!
pend2 = @ode_def Pendulum begin= ω  # ← yes, this is UTF8, θ and ω in text= (-b * ω) - (c * sin(θ))
end

prob = ODEProblem(pend, y0, (0.0, 10.0))
sol = solve(prob)         # ↑ solve on interval [0,10]
t, y = sol.t, hcat(sol.u...)'

using Winston
plot(t, y[:, 1], t, y[:, 2])
title("2D Differential Equation")
savefig("figures/Pendulum_solution.png")

bg original 70%


Examples

  1. Iterative computation: signal filtering
  2. Optimization: robust regression on RADAR data

Ex. 1: Iterative computation

Objective:

  • show the efficiency of Julia's Just-in-Time (JIT) compilation
  • but also its fragility...

Iterative computation: signal filtering

The classical saying:

« Vectorized code often runs much faster than the corresponding code containing loops. » (cf. MATLAB doc)

does not hold for Julia, because of its Just-in-Time compiler.

Example of a computation that cannot be vectorized

Smoothing of a signal ${u_k}_{k\in\mathbb{N}}$:

$$ y_k = ay_{k-1} + (1-a) u_k, k\in\mathbb{N}^+ $$

Parameter $a$ tunes the smoothing (none: $a=0$, strong $a\to1^-$).

💥 Iteration (for loop) cannot be avoided.

NB : Matlab also has JIT but it may not work well in all cases.


Signal filtering in Julia 👌

function smooth(u, a)
    y = zeros(u)

    y[1] = (1-a)*u[1]
    for k=2:length(u)  # this loop is NOT slow!
        y[k] = a*y[k-1] + (1-a)*u[k]
    end

    return y
end

figures/signal_filtering.png 50%


Performance of the signal filter

Implementation Time for $10 ,\mathrm{Mpts}$ notes
Julia $50-70,\mathrm{ms}$ Fast! Easy! 👌
Octave native $88000,\mathrm{ms}$ slow!! 🐌
Python native $4400,\mathrm{ms}$ slow! 🐌
SciPy's lfilter $70,\mathrm{ms}$ many lines of C
Python + @numba.jit $50,\mathrm{ms}$ since $2012$
@numba.jit # <- factor ×100 speed-up!
def smooth_jit(u, a):
    y = np.zeros_like(u)
    y[0] = (1-a)*u[0]
    for k in range(1, len(u)):
        y[k] = a*y[k-1] + (1-a)*u[k]
    return y

Conclusion on the performance

For this simple iterative computation:

  • Julia performs very well, much better than native Python
  • but it's possible to get the same with fresh Python tools (Numba)
  • more realistic examples are needed

Fragility of Julia's JIT Compilation 💥

The efficiency of the compiled code relies on type inference.

function smooth1(u, a)
    y = 0
    for k=1:length(u)
        y = a*y + (1-a)*u[k]
    end
    return y
end
function smooth2(u, a)
    y = 0.0   # <- difference is here!
    for k=1:length(u)
        y = a*y + (1-a)*u[k]
    end
    return y
end

An order of magnitude difference 🐌vs🏃

julia> @time smooth1(u, 0.9);
  0.212018 seconds (30.00 M allocations: 457.764 MiB ...)
julia> @time smooth2(u, 0.9);
  0.024883 seconds (5 allocations: 176 bytes)

Fortunately, Julia gives a good diagnosis tool 🛠

julia> @code_warntype smooth1(u, 0.9);
...  # ↓ we spot a detail
y::Union{Float64, Int64}
...

y is either Float64 or Int64 when it should be just Float64.

Cause: initialization y=0 vs. y=0.0!


Ex. 2: Optimization in Julia

Objective: demonstrate JuMP, a Modeling Language for Optimization in Julia.

Some research groups migrate to Julia just for this package!

Cf. JuMP.ReadTheDocs.io for documentation!


Optimization problem

Example problem: identifying the sea clutter in Weather Radar data.

  • is a robust regression problem
    • $\hookrightarrow$ is an optimization problem!
References An « IETR-colored » example, inspired by:
  • Radar data+photo: P.-J. Trombe et al., « Weather radars – the new eyes for offshore wind farms?,» Wind Energy, 2014.
  • Regression methods: S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. (Example 6.2).

Weather radar: the problem of sea clutter

Given $n$ data points $(x_i, y_i)$, fit a linear trend:

$$\hat{y} = a.x + b$$

An optimization problem with two parameters: $a$ (slope), $b$ (intercept)


Regression as an optimization problem

The parameters for the trend $(a,b)$ should minimize a criterion $J$ which penalizes the residuals $r_i = y_i - \hat{y} = y_i - a.x + b$:

$$J(a,b) = \sum_i \phi(r_i)$$

where $\phi$ is the penaly function, to be chosen:

  • $\phi(r) = r^2$: quadratic deviation $\rightarrow$ least squares regression
  • $\phi(r) = \lvert r \rvert$: absolute value deviation
  • $\phi(r) = h(r)$: Huber loss
  • ...

🔧 Choice of penalty function

The choice of the loss function influences:

  • the optimization result (fit quality)
    • e.g., in the presence of outliers
  • the properties of optimization problem: convexity, smoothness

Properties of each function

  • quadratic: convex, smooth, heavy weight for strong deviations
  • absolute value: convex, not smooth
  • Huber: a mix of the two

🛠 How to solve the regression problem?

Option 1: a big bag of tools

A specific package for each type of regression:

  • « least square toolbox » ($\rightarrow$ MultivariateStats.jl)
  • « least absolute value toolbox » ($\rightarrow$ quantile regression)
  • « Huber toolbox » (i.e., robust regression $\rightarrow$ ???)
  • ...

Option 2: the « One Tool »

$\Longrightarrow$ a Modeling Language for Optimization

  • more freedom to explore variants of the problem

Modeling Languages for Optimization

Purpose: make it easy to specify and solve optimization problems without expert knowledge.


JuMP: optimization modeling in Julia

  • The JuMP package offers a domain-specific modeling language for mathematical optimization.

JuMP interfaces with many optimization solvers: open-source (Ipopt, GLPK, Clp, ECOS...) and commercial (CPLEX, Gurobi, MOSEK...).

  • Other Modeling Languages for Optimization:

    • Standalone software: AMPL, GAMS
    • Matlab: YALMIP (previous seminar), CVX
    • Python: Pyomo, PuLP, CVXPy

Claim: JuMP is fast, thanks to Julia's metaprogramming capabilities (generation of Julia code within Julia code).


📈 Regression with JuMP — common part

  • Given x and y the $300$ data points:
m = Model(solver = ECOSSolver())

@variable(m, a)
@variable(m, b)

res = a*x .- y + b

res (« residuals ») is an Array of $300$ elements of type JuMP.GenericAffExpr{Float64,JuMP.Variable}, i.e., a semi-symbolic affine expression.

  • Now, we need to specify the penalty on those residuals.

Regression choice: least squares regression

$$\min \sum_i r_i^2$$

Reformulated as a Second-Order Cone Program (SOCP):

$$\min j, \quad \text{such that} ; \lVert r \rVert_2 \leq j$$

@variable(m, j)
@constraint(m, norm(res) <= j)
@objective(m, Min, j)

(SOCP problem $\Longrightarrow$ ECOS solver)


Regression choice: least absolute deviation

$$\min \sum_i \lvert r_i \rvert $$

Reformulated as a Linear Program (LP)

$$\min \sum_i t_i, \quad \text{such that} ; -t_i \leq r_i \leq t_i$$

@variable(m, t[1:n])
@constraint(m, res .<= t)
@constraint(m, res .>= -t)
@objective(m, Min, sum(t))

Solve! ⚙️

julia> solve(m)
[solver blabla... ⏳ ]
:Optimal  # hopefully
julia> getvalue(a), getvalue(b)
(-1.094, 127.52)  # for least squares

Observations:

  • least abs. val., Huber
  • least squares

JuMP: summary 📜

A modeling language for optimization, within Julia:

  • gives access to all classical optimization solvers
  • very fast (claim)
  • gives freedom to explore many variations of an optimization problem (fast prototyping)

🗒 More on optimization with Julia:

  • JuliaOpt: host organization of JuMP
  • Optim.jl: implementation of classics in Julia (e.g., Nelder-Mead)
  • JuliaDiff: Automatic Differentiation to compute gradients, thanks to Julia's strong capability for code introspection

Conclusion

Sum-up

Thanks for joining 👏 !

Your mission, if you accept it... 💥

  1. 👶 Padawan level: Train yourself a little bit on Julia $\hookrightarrow$ JuliaBox.com ? Or install it on your laptop! And ead introduction in the Julia manual!
  2. 👩‍🎓 Jedi level: Try to solve a numerical system, from your research or teaching, in Julia instead of MATLAB
  3. ⚔️ Master level: From now on, try to use open-source & free tools for your research (Julia, Python and others)… 🤑
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