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Lecture Topic: Optimisation beyond 1D

Optimisation beyond 1D

Beyond optimisation in 1D, we will study two directions.

First, the equivalent in nth dimension, x∗ ∈ Rn such that f (x∗) ≤ f (x) for allx ∈ Rn.

Second, constrained optimisation, i.e. x∗ ∈ Rn such that f (x∗) ≤ f (x) for allx ∈ Rn where gi (x) ≤ 0, i = 1 . . .m.

For arbitrary f , gi : Rn → R, this is undecidable.

We hence focus on (in some sense) smooth f , gi , where it is still NP-Hard todecide, whether a point is a local optimum.

Only for smooth and convex f , gi and under additional assumptions, can onereason about global optima.

The methods presented are used throughout all of modern machine learning andmuch of operations research.

Jakub Marecek and Sean McGarraghy (UCD) Numerical Analysis and Software October 23, 2015 2 / 1

Optimisation: Key ConceptsConstrained minimisation: x∗ ∈ Rn such that f (x∗) ≤ f (x) for all x ∈ Rn wheregi (x) ≤ 0, i = 1 . . .m.

Jacobian ∇f : the m × n matrix of all first-order partial derivatives of avector-valued function g : Rn → Rm.

Hessian H: a square matrix of second-order partial derivatives of a scalar-valuedfunction f , H(f )(x) = J(∇f )(x).

Gradient methods: consider f (x + ∆x) ≈ f (x) +∇f (x)∆x and go in the“antigradient direction”

Newton-type methods: consider the quadratic approximationf (x + ∆x) ≈ f (x) +∇f (x)∆x + 1

2 ∆xTH(x)∆x and multiply the “antigradientdirection” with the inverse Hessian

A witness: Checking whether a point x∗ ∈ R satisfies f ′(x∗) = 0 is beyond 1Dmuch easier than checking x∗ is a local (!) minimum.

Function Classes

Function f : Rn → R is Lipschitz-continuous with constant L, L finite, if andonly if: ||f (x)− f (y)|| ≤ L||x − y || for any x , y ∈ Rn.

Any Lipschitz-continuous function can be approximated by an infinitelydifferentiable function within arbitrarily small accuracy.

We denote by C k,pL (Q) the class of functions defined on Q ⊆ Rn, which are k

times continuously differentiable on Q and whose pth defivative isLipschitz-continuous on Q with constant L.

Function f belongs to C 2,1L (Rn) if and only if ||f ′′(x)|| ≤ L for all x ∈ Rn.

Gradient

Definition

If a scalar-valued function f : Rn → R has first-order partial derivatives withrespect to each xi , then the n-dimensional equivalent of the first derivative f ′(x)is the gradient vector

∇f (x) = ∇f =

(∂f (x)

∂f∂x1∂f∂x2...∂f∂xn

Directional Derivative

When the partial derivatives are not well-defined, we may consider:

Definition

Let f : Rn −→ R. The directional derivative of f at x ∈ Rn in the direction v is

dv f (x) =∂f

∂v:= v · ∇f (x) (dot product)

vi∂f

∂xi.

where v = (v1, . . . , vn)t ∈ Rn.

Jacobian

Definition

If the function g : Rn → Rm has first-order partial derivatives with respect to eachxi , then the m × n matrix:

J =∂f

∂x1· · · ∂f

∂f1∂x1

· · · ∂f1∂xn

.... . .

...∂fm∂x1

· · · ∂fm∂xn

is the Jacobian.

Hessian

Definition

The n-dimensional equivalent of the second derivative f ′′(x) is the Hessian matrix:

Hf (x) =

(∂2f (x)

∂xi∂xj

∂2f (x)∂x1∂x1

∂2f (x)∂x1∂x2

· · · ∂2f (x)∂x1∂xn

∂2f (x)∂x2∂x1

∂2f (x)∂x2∂x2

· · · ∂2f (x)∂x2∂xn

......

. . ....

∂2f (x)∂xn∂x1

∂2f (x)∂xn∂x2

· · · ∂2f (x)∂xn∂xn

Note that Hf (x∗) is a symmetric matrix since ∂2f (x)∂xi∂xj

= ∂2f (x)∂xj∂xi

for all i , j . We omit

the subscript where not needed.

Taylor Series

Definition

The Taylor series expansion of f (x) about some xk ∈ Rn is:

f (x) ≈ f (xk) + (∇f (xk))t(x − xk) +1

2(x − xk)tHf (xk)(x − xk) + · · · ,

where f (x) ∈ R, x and xk ∈ Rn, and Hf (xk) ∈ MnR.

Also we define

Definition

∂v2:=

n∑i=1

vi∂f

n∑j=1

vj∂2f

∂xi∂xj

n∑i ,j=1

vivj∂2f

∂xi∂xj

= v tHf (x)v .

Minima

Theorem

Let U be an open subset of Rn, f : U −→ R be a twice continuously differentiablefunction on U, and let x∗ be a critical point of f , i.e., ∇f (x∗) = 0. Then

(a) x∗ is a local maximum of f if ∂2f∂v2 < 0 for all nonzero v ∈ Rn;

(b) x∗ is a local minimum of f if ∂2f∂v2 > 0 for all nonzero v ∈ Rn;

(c) x∗ is a saddle point of f if there exist v ,w ∈ Rn such that

∂v2< 0 <

∂w2.

It is clear that this involves examining the sign of v tHf (x)v for various v . It canbe shown that this theorem leads to a practical test as follows.

The Approaches

Derivative-free methods

Gradient methods

Quasi-Newton methods

Newton-type methods

Interior-point methods

Derivative-Free Methods

For functions for x∗ ∈ Rn, the convergence of derivative-free methods is provablyslow.

Theorem (Nesterov)

For L-Lipschitz function f , ε ≤ 12L, and f provided as an oracle that allows f (x)

to be evaluated for x , derivative-free methods require(⌊L

calls to the oracle to reach ε accuracy.

To put the lower bound into perspective, consider a single computer, which cansustain the performance of 1011 operations per second (“100 gigaFLOPS”) and afunction, which can be evaluated in n operations:

For L = 2, n = 10, 10% accuracy, you need 1011 operations, or 1 second.

For L = 2, n = 10, 1% accuracy, you need 1021 operations, or 325 years.

For L = 2, n = 10, 0.1% accuracy, you need 1031 operations, or 1012 years.

Gradient Methods

Let us consider a local, unconstrained minimum x∗ of a multi-variate function,i.e., f (x∗) ≤ f (x)∀x with ||x − x∗|| ≤ ε.From the definition, at a local minimum x∗, we expect the variation in f due to asmall variation ∆x in x to be non-negative: ∇f (x∗)∆x =

∑ni=1

∂f (x∗)∂xi

∆x ≥ 0.

By considering ∆x coordinate wise, we get: ∇f (x∗) = 0.

Gradient Methods

∑ni=1

∂f (x∗)∂xi

∆x ≥ 0.

Gradient Methods

∑ni=1

∂f (x∗)∂xi

∆x ≥ 0.

Gradient Methods

In gradient methods, you consider xk+1 = xk − hk∇f (xk), where hk is one of:

Constant step hk = h or hk = h/√k + 1

Full relaxation hk = arg minh≥0 f (xk − h∇f (xk))

Armijo line search: find xk+1 such that the ratio ∇f (xk )(xk−xk+1)f (xk )−f (xk+1)

is within

some interval

Gradient Methods

For all of the choices above with f ∈ C 1,1L (Rn), one has

f (xk)− f (xk+1) ≥ ωL ||∇f (xk)||2. We hence want to bound the norm of the

gradient. It turns out:

kmini=0||∇f (xi )|| ≤

1√k + 1

ωL (f (x0)− f ∗)

This means that the norm of the gradient is less than ε, if the number ofiterations is greater than L

ωε2 (f (x0)− f ∗)− 1.

Gradient Methods

Theorem (Nesterov)

For f ∈ C 2,2M (Rn), lIn � H(f ∗) � LIn, a certain gradient method starting from

x0, r0 = ||x0 − x∗|| ≤ 2lM := r converges as follows:

||xk − x∗|| ≤ r r0

r − ρ0

(1− 2l

L + 3l

This is called the (local) linear (rate of) convergence.

Newton-Type Methods

In finding a solution to a system of non-linear equations F (x) = 0,x ∈ Rn,F : Rn → Rn,

displacement ∆x as a solution to F (x) +∇F (x)∆x = 0, which is known as theNewton system.

Assuming [∇F ]−1 exists, we can use:

xk+1 = xk −[∇F (xk)

]−1F (xk).

When we move from finding zeros of F (x) to minimisingf (x), x ∈ Rn, f : Rn → R by finding zeros of ∇f (x) = 0, we obtain:

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇F (xk)

]−1F (xk).

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇F (xk)

]−1F (xk).

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇F (xk)

]−1F (xk).

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇F (xk)

]−1F (xk).

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇F (xk)

]−1F (xk).

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

Alternatively, let us consider a quadratic approximation of f at point xk , i.e.,

f (xk) + 〈∇f (xk), x − xk〉+ 12〈H(xk)(x − xk), x − xk〉.

Assuming that H(xk) � 0, one should like to choose xk+1 by minimising theapproximation, i.e. solving

∇f (xk) + H(xk)(xk+1 − xk) = 0 for xk+1:

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

xk+1 = xk −[∇2f (xk)

]−1∇f (xk).

Newton-Type Methods

Theorem (Nesterov)

For f ∈ C 2,2M (Rn), where there exists a local minimum with positive definite

Hessian H(f ∗) � lIn, x0 close enough to x∗, i.e. ||x0 − x∗|| < 2l3M , Newton method

starting from x0 converges as follows:

||xk+1 − x∗|| ≤ M||xk − x∗||2

2l − 2M||xk − x∗||

This is called the (local) quadratic (rate of) convergence.

Newton-Type Methods

Newton method is only locally convergent, but the region of convergence is similarfor gradient and Newton methods.

One can try to address the possible divergence by considering damping:

xk+1 = xk − hk[∇2f (xk)

]−1∇f (xk), where hk ≥ 0 is a step-size, which usuallygoes to 1 as k goes to infinity, or other “regularisations”.

One can try to make a single iteration cheaper by either exploiting sparsity of theHessian or by considering some approximation of its inverse.

Newton-Type Methods

xk+1 = xk − hk[∇2f (xk)

Newton-Type Methods

xk+1 = xk − hk[∇2f (xk)

Quasi-Newton Methods

Quasi-Newton methods build up a sequence of approximations Hk of the inverseof the Hessian and use it in computing the step. Starting with H0 = In and somex0, in each iteration: xk+1 = xk + hkHk∇f (xk) for some step-length hk andHk+1 = Hk + ∆Hk where:

In rank-one methods, ∆Hk = (δk−Hkγk )(δk−Hkγk )T

〈δk−Hkγk ,γk 〉

In Broyden-Fletcher-Goldfarb-Shanno (BFGS):

∆Hk = Hkγk (δk )T +δk (γk )THk

〈Hkγk ,γk 〉 − βk Hkγk (γk )THk

〈Hkγk ,γk 〉

where δk = xk+1 − xk , γk = ∇f (xk+1)−∇f (xk), andβk = 1 + 〈γk , δk〉/〈Hkγk , γk〉. These methods are very successful in practice,although their rates of convergence are very hard to bound.

Lagrangian

Consider a constrained minimisation problemmin f (x) subject to g(x) ≤ 0, x ∈ Rn where f (x) : Rn → R, g(x) : Rn → Rm andthe global optimum at x∗ is f ∗.

Let us consider g as m inequalities gi (x) ≤ 0, i = 1 . . .m and let us introduceLagrange multipliers (also known as dual variables) y = (y1, y2, . . . , ym)T , yi ≥ 0,one scalar for each inequality gi .

The Lagrangian of the constrained minimisation problem is:L(x , y) = f (x) + yTg(x).

One can extend this to an additional constraint x ∈ X ⊆ Rn.

Lagrangian

The “Lagrangian primal” is LP(x) = maxy≥0 L(x , y), with LP(x) :=∞ if anyinequality is violated.

The “Lagrangian dual” is LD(y) = minx∈X L(x , y).

Its value clearly depends on the choice of y . For any y ≥ 0, however,

f ∗ ≥ LD(y), i.e.

f ∗ ≥ maxy≥0 minx∈X L(x , y), and

f ∗ = minx∈X maxy≥0 L(x , y).

minx∈X maxy≥0 L(x , y) ≥ maxy≥0 minx∈X L(x , y) (“weak duality”).

Any primal feasible solution provides an upper bound for the dual problem, andany dual feasible solution provides a lower bound for the primal problem.

Lagrangian

f ∗ ≥ LD(y), i.e.

Lagrangian

f ∗ ≥ LD(y), i.e.

Lagrangian

f ∗ ≥ LD(y), i.e.

Lagrangian

f ∗ ≥ LD(y), i.e.

Lagrangian

f ∗ ≥ LD(y), i.e.

Lagrangian

f ∗ ≥ LD(y), i.e.

Lagrangian

f ∗ ≥ LD(y), i.e.

Strong Duality and KKT Conditions

Assuming differentiability of L, Karush-Kuhn-Tucker (KKT) conditions arecomposed of stationarity (∇xL(x , y) = 0), primal feasibility (g(x) ≤ 0, x ∈ X ),dual feasibility (y ≥ 0), and“complementarity slackness” (yigi (x) = 0).

Under some “regularity” assumptions (also known as constraint qualifications), weare guaranteed that a point x satisfying the KKT condition exists.

If X ⊆ Rn is convex, f and g are convex, optimum f ∗ is finite, and the regularityassumptions hold, then we have

minx∈X maxy≥0 L(x , y) = maxy≥0 minx∈X L(x , y) (“strong duality”) and KKTconditions guarantee global optimality.

For example, Slater’s constraint qualifications is: ∃x ∈ int(X ) such that g(x) < 0.

If f and g are linear, no further constraint qualifications is needed and KKTconditions suffice.

Penalties and Barriers

Notice that the Lagrangian, as defined above, is not Lipschitz-continuous and isnot differentiable. Let us consider a closed set G defined by g , and let us assumeit has non-empty interior.

A penalty φ for G is a continuous function, such that φ(x) = 0 for any x ∈ G andφ(x) ≥ 0 for any x 6∈ G . E.g.

∑mi=1 max{gi (x), 0} (non-smooth),∑m

i=1(max{gi (x), 0})2 (smooth).

A barrier φ for G is a continuous function, such that φ(x)→∞ as x approachesthe boundary of G and is bounded from below elsewhere. E.g.∑m

(−gi (x))p , p ≥ 1 (power), −∑m

i=1 ln(−g(x)) (logarithmic).

One can consider (variants of) the Lagrangian of a constrained problem, whichinvolve a barrier for the inequalities.

Using such Lagrangians, one can develop interior-point methods.

(−gi (x))p , p ≥ 1 (power), −∑m

Interior-Point Methods

Interior-point methods solve progressively less relaxed first-order optimalityconditions of a problem, which is equivalent to a constrained optimisation problemand uses barriers.

Consider a constrained minimisation min f (x) subject to g(x) ≤ 0 wheref (x) : Rn → R, g(x) : Rn → Rm are convex and twice differentiable.

A nonnegative slack variable z ∈ Rm can be used to replace the inequality byequality g(x) + z = 0.

Negative z can be avoided by using a barrier µ∑m

i=1 ln zi .

The (variant of) Lagrangian is: L(x , y , z ;µ) = f (x) + yT (g(x) + z)µ∑m

i=1 ln zi .

Now we can differentiate:

∇xL(x , y , z ;µ) =∇f (x) +∇g(x)T y (8.1)

∇yL(x , y , z ;µ) =g(x) + z (8.2)

∇zL(x , y , z ;µ) =yµZ−1e, (8.3)

where Z = diag(z1, z2, . . . , zm) and e = [1, 1, . . . 1]T .

The first-order optimality conditions obtained by setting the partial derivatives tozero are:

∇f (x) +∇g(x)T y = 0 (8.4)

g(x) + z = 0 (8.5)

YZe = µe (8.6)

y , z ≥ 0 (8.7)

where Y = diag(y1, y2, . . . , ym) and the parameter µ is reduced to 0 in the largenumber of iterations.

This can be solved using the Newton method, where at each step, one solves alinear system:[

−H(x , y) B(x)T

B(x) XY−1

] [∆x−∆y

[∇f (x) + B(x)T y−g(x)− µY−1e

]where H(x , y) = ∇2f (x) +

∑mi=1 yi∇2gi (x) ∈ Rn×n and B(x) = ∇g(x) ∈ Rm×n.

This is a a saddle point system, which has often positive semidefinite A. Forconvex f , g , H(x, y) is positive semidefinite and diagonal matrix ZY 1 is alsopositive definite. A variety of methods works very well.

Condition Numbers

Assume the instance d := (A; b; c) is given. One can formalise the followingnotion, due to Renegar:

C (d) :=||d ||

inf{||∆d || : instance d + ∆d is infeasible or unbounded }.

The system (8.4–8.7) will have the condition number C (d)/µ.

Condition Numbers

Assume the instance d := (A; b; c) is given. One can formalise the followingnotion, due to Renegar:

C (d) :=||d ||

inf{||∆d || : instance d + ∆d is infeasible or unbounded }.

The system (8.4–8.7) will have the condition number C (d)/µ.

Condition Numbers in Analyses

For a variety of methods, including:

Ellipsoid method

Perceptron method

Von Neumann method

assuming A is invertible, one can show a bound on the number of iterations islogarithmic in C (d).

This highlights the need for preconditioners.

Condition Numbers in Analyses

For a variety of methods, including:

Ellipsoid method

Perceptron method

Von Neumann method

assuming A is invertible, one can show a bound on the number of iterations islogarithmic in C (d).

This highlights the need for preconditioners.

Conclusions

Constrained optimisation is the work-horse of operations research.

Interior-point methods have been used on problems in dimensions 109.

Still, there are many open problems, including Smale’s 9th problem:

Is feasibility of a linear system of inequalities Ax ≥ b in P over reals,

i.e., solvable in polynomial time on the BSS machine?

Conclusions

lecture topic: optimisation beyond 1d - ibm · pdf fileoptimisation beyond 1d beyond...

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