OMP0083: Advanced Topics in Machine Learning 2022/2023
Introduction to Convex Optimization
Coursework
Due Date: 9 January 2022
Instructor: Massimiliano Pontil
TA: Isak Texas Falk
Coursework based on the notes by Saverio Salzo
Instructions There are 4 sections. Two assess the knowledge of the theory and two require im-
plementing algorithms. Provide sufficient explanations for all solutions. For the coding parts the
following libraries are required: numpy, sklearn, matplotlib.
Deadlines Submit your report by January 9 2022.
Policies Late submissions: past the online due date, late submissions will be penalized by 20% of
the original total score per late day (e.g., 40% for 2 days). Excused extensions will be given only for
significant issues and if requested well in advance the due date.
1 Questions with multiple answers [20%]
Question 1 [4%] Which one of these is a convex function?
a) max{ax+ b, x4 5, ex2}
b) min{ax+ b, x4 5, ex2}
c) ax+ b+ x4 5 ex2 .
Question 2 [4%] Consider the function
f(x) =
{x if x ∈]? 1, 0]
x2 if x ≥ 0.
Indicate which one of the two graphs below represents the subdifferential of f .
Question 3 [4%] The gradient of f(x) = Ax, x + x, b+ c, where A is a square matrix, not
necessarily symmetric is
a) Ax+Ax+ b
b) 2Ax+ b
c) A + b, where A denotes the transpose of A.
Question 4 [4%] Let g ∈ Γ0(R). The Fenchel conjugate of f(x) = g(2x) is
a) f(u) = g(u/2)
b) f(u) = g(2u)
c) f(u) = g(u).
Question 5 [4%] Referring to the ridge regression problem
and denoting by K the Gram matrix of the data, indicate the solution of the dual problem
a) uˉ = (K+ λnId)y
b) uˉ = (K2 + λnId)1y
c) uˉ = (K+ λnId)1y.
2
2 Theory on convex analysis and optimization [30%]
Problem 1. [4%] Compute (showing a complete derivation) the Fenchel conjugates of the following
functions.
1. f : R→ ]∞,+∞], with f(x) =
{
+∞ if x ≤ 0
log x if x > 0.
2. f(x) = x2.
3. ι[0,1] (the indicator function of the interval [0, 1]).
Problem 2. [7%] Let f : X → ]?∞,+∞] be a proper convex function.
1. Prove by induction the Jensen’s inequality, that is
for all x1, . . . , xn ∈ X and for all λ1, . . . , λn ∈ R+ with
∑n
i=1 λi = 1.
2. Using the characterization for differentiable functions prove that the function log is convex
3. Applying Jensen’s inequality to ? log, prove the following arithmetic-geometric inequality
for all x1, . . . , xn ∈ R+.
Problem 3. [4%] Given a polytope C = co(a1, . . . , am) in X. Prove that the maximum of a convex
function f on C is attained at one of the vertices a1, . . . , am.
Problem 4. [2%] Prove that the function f(x, y) = ∥x 2y∥2 is (jointly) convex
Problem 5. [4%] Provide minimal sufficient conditions for the existence and uniqueness of mini-
mizers for a convex function f : X → ]∞,+∞].
Problem 6. [9%] Consider the optimization problem.
min
∥Axb∥∞≤ε
1
2
∥x∥2 ,
where ε > 0, A ∈ Rn×d and b ∈ Rn. Solve the following points.
3
1. Compute the dual problem (using the Fenchel-Rockafellar duality theory). [Hint: put the
problem in the form f(x) + g(Ax)]
2. Does strong duality hold? Justify the answer. [Hint: use the qualification condition]
3. Write the KKT conditions.
4. Derive a rate of convergence on the primal iterates from the application of FISTA on the dual
problem. [Hint: recall the it is possible to bound the square of the distance to the primal
solution by the dual objective values]
3 Solving the Lasso problem [25%]
Consider the following problem
min
x∈Rd
1
2n
∥Ax? y∥2 + λ ∥x∥1 , (1)
where A ∈ Rn×d. Let us denote by ai ∈ Rd and aj ∈ Rn the i-th row and j-column of A respectively.
Then the objective function can be written in the form
1
2n
n∑
i=1
(?ai, x? ? yi)2 + λ ∥x∥1 , (2)
Then the proximal stochastic gradient algorithm is
xk+1 = proxγkλ∥·∥1
(
xk ? γk(?aik , x? ? yik)aik
)
, (PSGA)
where γk = n/(∥A∥2
√
k + 1) and (ik)k∈N is a sequence of i.i.d. random variables uniformly dis-
tributed on {1, . . . , n}.
Alternatively, problem (1) can be approached via a randomized coordinate proximal gradient
algorithm. Indeed, recalling that ?(1/2) ∥Ax? y∥2 = A?(Ax ? y) = (?aj , Ax ? y?)1≤j≤d one can
consider the following algorithm
xk+1j =
???softγjλ
(
xkj ? γjn ?aj , Axk ? y?
)
if j = jk
xkj otherwise,
(RCPGA)
where (jk)k∈N is a sequence of i.i.d. random variables uniformly distributed on {1, . . . , d} and γj =
n/ ∥aj∥2.
Generate the data according to the following python code with n = 1000, d = 500, s = 50,
σ = 0.06.
def generate_problem(n, d, s, std=0.06):
# Generate xs
4
# vectors with entries in [0.5, 1] and [-1, -0.5]
# repectively
assert s % 2 == 0, "s needs to be divisible by 2"
xsp = 0.5 * (np.random.rand(s // 2) + 1)
xsn = - 0.5 * (np.random.rand(s // 2) + 1)
xsparse = np.hstack([xsp, xsn, np.zeros(d - s)])
random.shuffle(xsparse)
# Generate A
A = np.random.randn(n, d)
# Generate eps
y = A @ xsparse + std * np.random.randn(n)
return xsparse, A, y
1. Implement algorithm (PSGA), recalling that proxγkλ∥·∥1 acts component-wise as a soft-
thresholding operator with threshold γkλ.
2. Implement algorithm (RCPGA).
3. Choose an appropriate regularization parameter λ and plot the objective function values vs
the number of iterations for both the algorithms described above. For algorithm (PSGA)
consider also the behavior of the objective values over the sequence of ergodic means, that is,
xˉk = (
∑k
i=0 γi)
?1∑k
i=0 γixi.
4 Support Vector Machines [25%]
The problem of SVM’s for classification is
where (xi, yi)1≤i≤n is the training set, xi ∈ Rd and yi ∈ {?1, 1} and Λ: Rd → H is the feature map
corresponding to the Gaussian kernel, that is,
where Ki,j = K(xi, xj) and ι[0, 1
λn
] is the indicator function of the interval [0,
1
λn ]. The problem can
be equivalently restated as
min
α∈Rn
1
2
?Kyα, α? ? ?1n, α?+
n∑
i=1
ι[0, 1
λn
](αi), (6)
5
where (Ky)i,j = yiKi,jyj . Note that the primal solution can be recovered via w =
∑n
i=1 yiαiΛ(xi) and
hence the decision function is
hw(x) = sign(?w,Λ(x)?) = sign
( n∑
i=1
yiαiK(xi, x)
)
. (7)
Figure 2: A realization of the two-moons dataset.
Generate the data (2D) according to the following python code:
from sklearn.datasets import make_moons
X, y = make_moons(n_samples=200, noise=0.05, random_state=0)
y = 2*y - 1
Choose appropriate values of λ and σ and address the following points.
1. Implement FISTA for solving the dual problem and plot the dual objective function.
2. Implement the randomized coordinate projected gradient algorithm on the dual problem (6)
and plot the dual objective function.
3. For each algorithm above, using a contour plot command, plot the decision boundary as well
as the two classes.
4.compare the performance of the two approaches in terms of speed and accuracy.