NIPS2010: optimization algorithms in machine learning

Optimization Algorithms in Machine Learning

Stephen Wright

University of Wisconsin-Madison

NIPS Tutorial, 6 Dec 2010

Stephen Wright (UW-Madison) Optimization in Machine Learning NIPS Tutorial, 6 Dec 2010 1 / 82

Optimization

Optimization is going through a period of growth and revitalization,
driven largely by new applications in many areas.
Standard paradigms (LP, QP, NLP, MIP) are still important, along
with general-purpose software, enabled by modeling languages that
make the software easier to use.
However, there is a growing emphasis on “picking and choosing”
algorithmic elements to ﬁt the characteristics of a given application
— building up a suitable algorithm from a “toolkit” of components.
It’s more important than ever to understand the fundamentals of
algorithms as well as the demands of the application, so that good
choices are made in matching algorithms to applications.

We present a selection of algorithmic fundamentals in this tutorial, with an
emphasis on those of current and potential interest in machine learning.


Topics

I. First-order Methods
II. Stochastic and Incremental Gradient Methods
III. Shrinking/Thresholding for Regularized Formulations
IV. Optimal Manifold Identiﬁcation and Higher-Order Methods.
V. Decomposition and Coordinate Relaxation


I. First-Order Methods

min f (x), with smooth convex f . Usually assume
2
µI f (x) LI for all x,

with 0 ≤ µ ≤ L. (L is thus a Lipschitz constant on the gradient f .)
µ > 0 ⇒ strongly convex. Have
1
f (y ) − f (x) − f (x)T (y − x) ≥ µ y − x 2
.
2
(Mostly assume · := · 2 .) Deﬁne conditioning κ := L/µ.
Sometimes discuss convex quadratic f :
1
f (x) = x T Ax, where µI A LI .
2


What’s the Setup?

Assume in this part of talk that we can evaluate f and f at each iterate
xi . But we are interested in extending to broader class of problems:
nonsmooth f ;
f not available;
only an estimate of the gradient (or subgradient) is available;
impose a constraint x ∈ Ω for some simple set Ω (e.g. ball, box,
simplex);
a nonsmooth regularization term may be added to the objective f .
Focus on algorithms that can be adapted to these circumstances.


Gradient

xk+1 = xk − αk f (xk ), for some αk > 0.

Different ways to identify an appropriate αk .
1 Hard: Interpolating scheme with safeguarding to identify an
approximate minimizing αk .
2 Easy: Backtracking. α, 1 α, 1 α, 1 α, ... until a sufficient decrease in
¯ 2¯ 4¯ 8¯
f is obtained.
3 Trivial: Don’t test for function decrease. Use rules based on L and µ.
Traditional analysis for 1 and 2: Usually yields global convergence at
unspecified rate. The “greedy” strategy of getting good decrease from the
current search direction is appealing, and may lead to better practical
results.
Analysis for 3: Focuses on convergence rate, and leads to accelerated
multistep methods.

Constant (Short) Steplength
By elementary use of Taylor’s theorem, obtain
αk
f (xk+1 ) ≤ f (xk ) − αk 1 − L f (xk ) 2 .
2
2
For αk ≡ 1/L, have
1
f (xk ) 2 .
f (xk+1 ) ≤ f (xk ) − 2
2L
It follows by elementary arguments (see e.g. Nesterov 2004) that
2L x0 − x 2
f (xk+1 ) − f (x ∗ ) ≤ .
k +1
The classic 1/k convergence rate!
By assuming µ > 0, can set αk ≡ 2/(µ + L) and get a linear (geometric)
rate: Much better than sublinear, in the long run
2k 2k
L−µ 2
xk − x ∗ 2
≤ x0 − x ∗ 2
= 1− x0 − x ∗ 2 .
L+µ κ+1

The 1/k 2 Speed Limit
Nesterov (2004) gives a simple example of a smooth function for which no
method that generates iterates of the form xk+1 = xk − αk f (xk ) can
converge at a rate faster than 1/k 2 , at least for its ﬁrst n/2 iterations.
Note that xk+1 ∈ x0 + span( f (x0 ), f (x1 ), . . . , f (xk )).
   
2 −1 0 0 ... ... 0 1
−1 2 −1 0 . . . . . . 0 0
   
A =  0 −1 2 −1 0 . . . 0 , e1 = 0
   
.. .. .. .
.
 
 . . .  .
0 ... 0 −1 2 0
and set f (x) = (1/2)x T Ax − e1 x. The solution has x ∗ (i) = 1 − i/(n + 1).
T

If we start at x0 = 0, each f (xk ) has nonzeros only in its ﬁrst k + 2
entries. Hence, xk+1 (i) = 0 for i = k + 3, k + 4, . . . , n. Can show
3L x0 − x ∗ 2
f (xk ) − f ∗ ≥ .
32(k + 1)2

Exact minimizing αk : Faster rate?

Take αk to be the exact minimizer of f along − f (xk ). Does this yield a
better rate of linear convergence?
Consider the convex quadratic f (x) = (1/2)x T Ax. (Thus x ∗ = 0 and
f (x ∗ ) = 0.) Here κ is the condition number of A.
We have f (xk ) = Axk . Exact minimizing αk :

xk A2 xk
T
1
αk = T A3 x
= arg min (xk − αAxk )T A(xk − αAxk ),
xk k α 2

1 1
which is in the interval L, µ . Can show that

2k
∗ 2
f (xk ) − f (x ) ≤ 1− [f (x0 ) − f (x ∗ )].
κ+1

No improvement in the linear rate over constant steplength.


Multistep Methods: Heavy-Ball

Enhance the search direction by including a contribution from the previous
step.
Consider ﬁrst constant step lengths:

xk+1 = xk − α f (xk ) + β(xk − xk−1 )

Analyze by deﬁning a composite iterate vector:

xk − x ∗
wk := .
xk−1 − x ∗

Thus
−α 2 f (x ∗ ) + (1 + β)I −βI
wk+1 = Bwk + o( wk ), B := .
I 0


B has same eigenvalues as

−αΛ + (1 + β)I −βI
, Λ = diag(λ1 , λ2 , . . . , λn ),
I 0

where λi are the eigenvalues of 2 f (x ∗ ). Choose α, β to explicitly
minimize the max eigenvalue of B, obtain
2
4 1 2
α= √ , β= 1− √ .
L (1 + 1/ κ)2 κ+1

Leads to linear convergence for xk − x ∗ with rate approximately

2
1− √ .
κ+1


Summary: Linear Convergence, Strictly Convex f

Best steepest descent: Linear rate approx (1 − 2/κ);
√
Heavy-ball: linear rate approx (1 − 2/ κ).
Big diﬀerence! To reduce xk − x ∗ by a factor , need k large enough that
k
2 κ
1− ≤ ⇐ k≥ | log | (steepest descent)
κ 2
√
2 k κ
1− √ ≤ ⇐ k≥ | log | (heavy-ball)
κ 2
√
A factor of κ diﬀerence. e.g. if κ = 100, need 10 times fewer steps.


Conjugate Gradient
Basic step is

xk+1 = xk + αk pk , pk = − f (xk ) + γk pk−1 .

We can identify it with heavy-ball by setting βk = αk γk /αk−1 . However,
CG can be implemented in a way that doesn’t require knowledge (or
estimation) of L and µ.
Choose αk to (approximately) miminize f along pk ;
Choose γk by a variety of formulae (Fletcher-Reeves, Polak-Ribiere,
etc), all of which are equivalent if f is convex quadratic. e.g.

γk = f (xk ) 2 / f (xk−1 ) 2 .

There is a rich convergence theory for f quadratic, including asymptotic
√
linear convergence with rate approx 1 − 2/ κ. (Like heavy-ball.)
See e.g. Chap. 5 of Nocedal & Wright (2006) and refs therein.

Accelerated First-Order Methods
Accelerate the rate to 1/k 2 for weakly convex, while retaining the linear
√
rate (related to κ) for strongly convex case.
Nesterov (1983, 2004) describes a method that requires κ.
0: Choose x0 , α0 ∈ (0, 1); set y0 ← x0 ./
1
k: xk+1 ← yk − L f (yk ); (*short-step gradient*)
2 2
solve for αk+1 ∈ (0, 1): αk+1 = (1 − αk+1 )αk + αk+1 /κ;
set βk = αk (1 − αk )/(αk 2 +α
k+1 );
set yk+1 ← xk+1 + βk (xk+1 − xk ).
Still works for weakly convex (κ = ∞).
FISTA (Beck & Teboulle 2007):
0: Choose x0 ; set y1 = x0 , t1 = 1;
1
k: xk ← yk − L f (yk );
1 2
tk+1 ← 2 1+ 1 + 4tk ;
tk −1
yk+1 ← xk + tk+1 (xk − xk−1 ).

yk+2

xk+2

yk+1
xk+1
xk

yk


Convergence Results: Nesterov

√
If α0 ≥ 1/ κ, have
k
1 4L
f (xk ) − f (x ∗ ) ≤ c1 min 1− √ , √ ,
κ ( L + c2 k)2

where constants c1 and c2 depend on x0 , α0 , L.
Linear convergence at “heavy-ball” rate in strongly convex case, otherwise
1/k 2 . FISTA also achieves 1/k 2 rate.
Analysis: Not intuitive. Based on bounding the diﬀerence between f and a
quadratic approximation to it, at x ∗ . FISTA analysis is 2-3 pages.


A Non-Monotone Gradient Method: Barzilai-Borwein
(Barzilai & Borwein 1988) BB is a gradient method, but with an unusual
choice of αk . Allows f to increase (sometimes dramatically) on some steps.
2
xk+1 = xk − αk f (xk ), αk := arg min sk − αzk ,
α

where
sk := xk − xk−1 , zk := f (xk ) − f (xk−1 ).
Explicitly, we have
T
sk zk
αk = T
.
zk zk
Note that for convex quadratic f = (1/2)x T Ax, we have
T
sk Ask
αk = T
∈ [L−1 , µ−1 ].
sk A2 sk
Hence, can view BB as a kind of quasi-Newton method, with the Hessian
−1
approximated by αk I .

Comparison: BB vs Greedy Steepest Descent


Many BB Variants
T T T T
can use αk = sk sk /sk zk in place of αk = sk zk /zk zk ;
alternate between these two formulae;
calculate αk as above and hold it constant for 2, 3, or 5 successive
steps;
take αk to be the exact steepest descent step from the previous
iteration.

Nonmonotonicity appears essential to performance. Some variants get
global convergence by requiring a suﬃcient decrease in f over the worst of
the last 10 iterates.
The original 1988 analysis in BB’s paper is nonstandard and illuminating
(just for a 2-variable quadratic).
In fact, most analyses of BB and related methods are nonstandard, and
consider only special cases. The precursor of such analyses is Akaike
(1959). More recently, see Ascher, Dai, Fletcher, Hager and others.

Primal-Dual Averaging

(see Nesterov 2009) Basic step:
k
1 γ
xk+1 = arg min [f (xi ) + f (xi )T (x − xi )] + √ x − x0 2
x k +1 k
i=0
γ
= arg min ¯T
gk x + √ x − x0 2 ,
x k
k
where gk :=
¯ i=0 f (xi )/(k + 1) — the averaged gradient.

The last term is always centered at the ﬁrst iterate x0 .
Gradient information is averaged over all steps, with equal weights.
γ is constant - results can be sensitive to this value.
The approach still works for convex nondiﬀerentiable f , where f (xi )
is replaced by a vector from the subgradient ∂f (xi ).


Convergence Properties
Nesterov proves convergence for averaged iterates:
k
1
xk+1 =
¯ xi .
k +1
i=0

Provided the iterates and the solution x ∗ lie within some ball of radius D
around x0 , we have
C
f (¯k+1 ) − f (x ∗ ) ≤ √ ,
x
k
where C depends on D, a uniform bound on f (x) , and γ (coeﬃcient
of stabilizing term).
Note: There’s averaging in both primal (xi ) and dual ( f (xi )) spaces.
Generalizes easily and robustly to the case in which only estimated
gradients or subgradients are available.
(Averaging smooths the errors in the individual gradient estimates.)

Extending to the Constrained Case: x ∈ Ω

How do these methods change when we require x ∈ Ω, with Ω closed and
convex?
Some algorithms and theory stay much the same, provided we can involve
Ω explicity in the subproblems.
Example: Primal-Dual Averaging for minx∈Ω f (x).
γ
xk+1 = arg min gk x + √ x − x0 2 ,
¯T
x∈Ω k
k
where gk :=
¯ i=0 f (xi )/(k + 1). When Ω is a box, this subproblem is
easy to solve.


Example: Nesterov’s Constant Step Scheme for minx∈Ω f (x). Requires
just only calculation to be changed from the unconstrained version.
0: Choose x0 , α0 ∈ (0, 1); set y0 ← x0 , q ← 1/κ = µ/L.
1 1
k: xk+1 ← arg miny ∈Ω 2 y − [yk − L f (yk )] 2 ;
2
2 2
solve for αk+1 ∈ (0, 1): αk+1 = (1 − αk+1 )αk + qαk+1 ;
2
set βk = αk (1 − αk )/(αk + αk+1 );
set yk+1 ← xk+1 + βk (xk+1 − xk ).

Convergence theory is unchanged.


Regularized Optimization (More Later)

FISTA can be applied with minimal changes to the regularized problem

min f (x) + τ ψ(x),
x

where f is convex and smooth, ψ convex and “simple” but usually
nonsmooth, and τ is a positive parameter.
Simply replace the gradient step by
2
L 1
xk = arg min x − yk − f (yk ) + τ ψ(x).
x 2 L

(This is the “shrinkage” step; when ψ ≡ 0 or ψ = · 1, can be solved
cheaply.)
More on this later.


Further Reading

1 Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course,
Kluwer Academic Publishers, 2004.
2 A. Beck and M. Teboulle, “Gradient-based methods with application to signal
recovery problems,” in press, 2010. (See Teboulle’s web site).
3 B. T. Polyak, Introduction to Optimization, Optimization Software Inc, 1987.
4 J. Barzilai and J. M. Borwein, “Two-point step size gradient methods,” IMA
Journal of Numerical Analysis, 8, pp. 141-148, 1988.
5 Y. Nesterov, “Primal-dual subgradient methods for convex programs,”
Mathematical Programming, Series B, 120, pp. 221-259, 2009.


II. Stochastic and Incremental Gradient Methods

Still deal with (weakly or strongly) convex f . But change the rules:
Allow f nonsmooth.
Can’t get function values f (x).
At any feasible x, have access only to an unbiased estimate of an
element of the subgradient ∂f .
Common settings are:
f (x) = Eξ F (x, ξ),
where ξ is a random vector with distribution P over a set Ξ. Also the
special case:
m
f (x) = fi (x),
i=1

where each fi is convex and nonsmooth.


Applications

This setting is useful for machine learning formulations. Given data
xi ∈ Rn and labels yi = ±1, i = 1, 2, . . . , m, find w that minimizes
m
τ ψ(w ) + (w ; xi , yi ),
i=1

where ψ is a regularizer, τ > 0 is a parameter, and is a loss. For linear
classifiers/regressors, have the specific form (w T xi , yi ).
Example: SVM with hinge loss (w T xi , yi ) = max(1 − yi (w T xi ), 0) and
ψ = · 1 or ψ = · 2 .
2

Example: Logistic regression: (w T xi , yi ) = log(1 + exp(yi w T xi )). In
regularized version may have ψ(w ) = w 1 .


Subgradients

For each x in domain of f , g is a subgradient of f at x if

f (z) ≥ f (x) + g T (z − x), for all z ∈ domf .

Right-hand side is a supporting hyperplane.
The set of subgradients is called the subdiﬀerential, denoted by ∂f (x).
When f is diﬀerentiable at x, have ∂f (x) = { f (x)}.
We have strong convexity with modulus µ > 0 if
1
f (z) ≥ f (x)+g T (z−x)+ µ z−x 2
, for all x, z ∈ domf with g ∈ ∂f (x).
2
Generalizes the assumption 2 f (x) µI made earlier for smooth
functions.


f

supporting hyperplanes

x


“Classical” Stochastic Approximation

Denote by G (x, ξ) ths subgradient estimate generated at x. For
unbiasedness need Eξ G (x, ξ) ∈ ∂f (x).
Basic SA Scheme: At iteration k, choose ξk i.i.d. according to distribution
P, choose some αk > 0, and set

xk+1 = xk − αk G (xk , ξk ).

Note that xk+1 depends on all random variables up to iteration k, i.e.
ξ[k] := {ξ1 , ξ2 , . . . , ξk }.
When f is strongly convex, the analysis of convergence of E ( xk − x ∗ 2 ) is
fairly elementary - see Nemirovski et al (2009).


Rate: 1/k
Deﬁne ak = 1 E ( xk − x ∗ 2 ). Assume there is M > 0 such that
2
E ( G (x, ξ) 2 ) ≤ M 2 for all x of interest. Thus

1
xk+1 − x ∗ 2
2
2
1
= xk − αk G (xk , ξk ) − x ∗ 2
2
1 1 2
= xk − x ∗ 2 − αk (xk − x ∗ )T G (xk , ξk ) + αk G (xk , ξk ) 2 .
2
2 2
Taking expectations, get
1 2
ak+1 ≤ ak − αk E [(xk − x ∗ )T G (xk , ξk )] + αk M 2 .
2
For middle term, have
E [(xk − x ∗ )T G (xk , ξk )] = Eξ[k−1] Eξk [(xk − x ∗ )T G (xk , ξk )|ξ[k−1] ]
= Eξ[k−1] (xk − x ∗ )T gk ,

... where
gk := Eξk [G (xk , ξk )|ξ[k−1] ] ∈ ∂f (xk ).
By strong convexity, have
1
(xk − x ∗ )T gk ≥ f (xk ) − f (x ∗ ) + µ xk − x ∗ 2
≥ µ xk − x ∗ 2 .
2
Hence by taking expectations, we get E [(xk − x ∗ )T gk ] ≥ 2µak . Then,
substituting above, we obtain
1 2
ak+1 ≤ (1 − 2µαk )ak + αk M 2
2

When
1
αk ≡ ,
kµ
a neat inductive argument (exercise!) reveals the 1/k rate:

Q M2
ak ≤ , for Q := max x1 − x ∗ 2 , .
2k µ2


But... What if we don’t know µ? Or if µ = 0?

The choice αk = 1/(kµ) requires strong convexity, with knowledge of the
modulus µ. An underestimate of µ can greatly degrade the performance of
the method (see example in Nemirovski et al. 2009).
Now describe a Robust Stochastic Approximation approach, which has a
√
rate 1/ k (in function value convergence), and works for weakly convex
nonsmooth functions and is not sensitive to choice of parameters in the
step length.
This is the approach that generalizes to mirror descent.


Robust SA

At iteration k:
set xk+1 = xk − αk G (xk , ξk ) as before;
set
k
i=1 αi xi
xk =
¯ k
.
i=1 αi

For any θ > 0 (not critical), choose step lengths to be

θ
αk = √ .
M k
Then f (¯k ) converges to f (x ∗ ) in expectation with rate approximately
x
(log k)/k 1/2 . The choice of θ is not critical.


Analysis of Robust SA
The analysis is again elementary. As above (using i instead of k), have:

1
αi E [(xi − x ∗ )T gi ] ≤ ai − ai+1 + αi2 M 2 .
2
By convexity of f , and gi ∈ ∂f (xi ):

f (x ∗ ) ≥ f (xi ) + giT (x ∗ − xi ),

thus
1
αi E [f (xi ) − f (x ∗ )] ≤ ai − ai+1 + αi2 M 2 ,
2
so by summing iterates i = 1, 2, . . . , k, telescoping, and using ak+1 > 0:
k k
1
αi E [f (xi ) − f (x ∗ )] ≤ a1 + M 2 αi2 .
2
i=1 i=1


Thus dividing by i=1 αi :

k k
i=1 αi f (xi ) a1 + 1 M 2 2
i=1 αi
E k
− f (x ∗ ) ≤ 2
k
.
i=1 αi i=1 αi

By convexity, we have
k
i=1 αi f (xi )
f (¯k ) ≤
x k
,
i=1 αi

so obtain the fundamental bound:
1 k
a1 + 2 M 2 2
i=1 αi
E [f (¯k ) − f (x ∗ )] ≤
x k
.
i=1 αi


θ
By substituting αi = √ ,
M i
we obtain

1 k 1
a1 + 2 θ2
E [f (¯k ) − f (x ∗ )] ≤
x θ k
i=1 i
1
√
M i=1 i

a1 + θ2 log(k + 1)
≤ θ
√
M k
a1
=M + θ log(k + 1) k −1/2 .
θ
That’s it!
Other variants: constant stepsizes αk for a ﬁxed “budget” of iterations;
periodic restarting; averaging just over the recent iterates. All can be
analyzed with the basic bound above.


Mirror Descent

The step from xk to xk+1 can be viewed as the solution of a subproblem:
1
xk+1 = arg min G (xk , ξk )T (z − xk ) + z − xk 2
2,
z 2αk
a linear estimate of f plus a prox-term. This provides a route to handling
constrained problems, regularized problems, alternative prox-functions.
For the constrained problem minx∈Ω f (x), simply add the restriction z ∈ Ω
to the subproblem above. In some cases (e.g. when Ω is a box), the
subproblem is still easy to solve.
We may use other prox-functions in place of (1/2) z − x 2 above. Such
2
alternatives may be particularly well suited to particular constraint sets Ω.
Mirror Descent is the term used for such generalizations of the SA
approaches above.


Mirror Descent cont’d
Given constraint set Ω, choose a norm · (not necessarily Euclidean).
Define the distance-generating function ω to be a strongly convex function
on Ω with modulus 1 with respect to · , that is,

(ω (x) − ω (z))T (x − z) ≥ x − z 2
, for all x, z ∈ Ω,

where ω (·) denotes an element of the subdifferential.
Now define the prox-function V (x, z) as follows:

V (x, z) = ω(z) − ω(x) − ω (x)T (z − x).

This is also known as the Bregman distance. We can use it in the
subproblem in place of 1 · 2 :
2

1
xk+1 = arg min G (xk , ξk )T (z − xk ) + V (z, xk ).
z∈Ω αk


Bregman distance is the deviation from linearity:

ω

V(x,z)

x z


Bregman Distances: Examples
For any Ω, we can use ω(x) := (1/2) x − x 2 , leading to prox-function
¯ 2
2.
V (x, z) = (1/2) x − z 2
For the simplex Ω = {x ∈ Rn : x ≥ 0, n xi = 1}, we can use instead
i=1
the 1-norm · 1 , choose ω to be the entropy function
n
ω(x) = xi log xi ,
i=1

leading to Bregman distance
n
V (x, z) = zi log(zi /xi ).
i=1

These are the two most useful cases.
Convergence results for SA can be generalized to mirror descent.

Incremental Gradient
(See e.g. Bertsekas (2011) and references therein.) Finite sums:
m
f (x) = fi (x).
i=1

Step k typically requires choice of one index ik ∈ {1, 2, . . . , m} and
evaluation of fik (xk ). Components ik are selected sometimes randomly or
cyclically. (Latter option does not exist in the setting f (x) := Eξ F (x; ξ).)

There are incremental versions of the heavy-ball method:
xk+1 = xk − αk fik (xk ) + β(xk − xk−1 ).
Approach like dual averaging: assume a cyclic choice of ik , and
approximate f (xk ) by the average of fi (x) over the last m iterates:
m
αk
xk+1 = xk − fik−l+1 (xk−l+1 ).
m
l=1


Achievable Accuracy

Consider the basic incremental method:

xk+1 = xk − αk fik (xk ).

How close can f (xk ) come to f (x ∗ ) — deterministically (not just in
expectation).
Bertsekas (2011) obtains results for constant steps αk ≡ α.

cyclic choice of ik : lim inf f (xk ) ≤ f (x ∗ ) + αβm2 c 2 .
k→∞

random choice of ik : lim inf f (xk ) ≤ f (x ∗ ) + αβmc 2 .
k→∞

where β is close to 1 and c is a bound on the Lipschitz constants for fi .
(Bertsekas actually proves these results in the more general context of
regularized optimization - see below.)


Applications to SVM

SA techniques have an obvious application to linear SVM classiﬁcation. In
fact, they were proposed in this context and analyzed independently by
researchers in the ML community for some years.
Codes: SGD (Bottou), PEGASOS (Shalev-Schwartz et al, 2007).
Tutorial: Stochastic Optimization for Machine Learning, Tutorial by N.
Srebro and A. Tewari, ICML 2010 for many more details on the
connections between stochastic optimization and machine learning.
Related Work: Zinkevich (ICML, 2003) on online convex programming.
Aiming to approximate the minimize the average of a sequence of convex
functions, presented sequentially. No i.i.d. assumption, regret-based
analysis. Take steplengths of size O(k −1/2 ) in gradient fk (xk ) of latest
convex function. Average regret is O(k −1/2 ).


Further Reading
1 A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, “Robust stochastic
approximation approach to stochastic programming,” SIAM Journal on
Optimization, 19, pp. 1574-1609, 2009.
2 D. P. Bertsekas, “Incremental gradient, subgradient, and proximal methods for
convex optimization: A Survey,” Chapter 4 in Optimization and Machine Learning,
upcoming volume edited by S. Nowozin, S. Sra, and S. J. Wright (2011).
3 A. Juditsky and A. Nemirovski, “ First-order methods for nonsmooth convex
large-scale optimization. I: General-purpose methods,” Chapter 5 in Optimization
and Machine Learning (2011).
4 A. Juditsky and A. Nemirovski, “ First-order methods for nonsmooth convex
large-scale optimization. I: Utilizing problem structure,” Chapter 6 in Optimization
and Machine Learning (2011).
5 O. L. Mangasarian and M. Solodov, “Serial and parallel backpropagation
convergencevia nonmonotone perturbed minimization,” Optimization Methods and
Software 4 (1994), pp. 103–116.
6 D. Blatt, A. O. Hero, and H. Gauchman, “A convergent incremental gradient
method with constant step size,” SIAM Journal on Optimization 18 (2008), pp.
29–51.

III. Shrinking/Thresholding for Regularized Optimization
In many applications, we seek not an exact minimizer of the underlying
objective, but rather an approximate minimizer that satisﬁes certain
desirable properties:
sparsity (few nonzeros);
low-rank (if a matrix);
low “total-variation”;
generalizability. (Vapnik: “...tradeoﬀ between the quality of the
approximation of the given data and the complexity of the
approximating function.”)
“Desirable” properties depend on context and application .
A common way to obtain structured solutions is to modify the objective f
by adding a regularizer τ ψ(x), for some parameter τ > 0.

min f (x) + τ ψ(x).

Often want to solve for a range of τ values, not just one value in isolation.

Basics of Shrinking
Regularizer ψ is often nonsmooth but “simple.” Shrinking / thresholding
approach (a.k.a. forward-backward splitting) is useful if the problem is
easy to solve when f is replaced by a quadratic with diagonal Hessian:
1
min g T (z − x) + z −x 2
2 + τ ψ(z).
z 2α
Equivalently,
1
z − (x − αg ) 2 + τ ψ(z).
min 2
2α z

Deﬁne the shrinking operator as the arg min:
1 2
Sτ (y , α) := arg min z −y 2 + τ ψ(z).
z 2α
Typical algorithm:
xk+1 = Sτ (xk − αk gk , αk ),
with for example gk = f (xk ).

“Practical” Instances of ψ

Cases for which the subproblem is simple:
ψ(z) = z 1 . Thus Sτ (y , α) = sign(y ) max(|y | − ατ, 0). When y
complex, have

max(|y | − τ α, 0)
Sτ (y , α) = y.
max(|y | − τ α, 0) + τ α

ψ(z) = g ∈G z[g ] 2 or ψ(z) = g ∈G z[g ] ∞, where z[g ] , g ∈ G
are non-overlapping subvectors of z. Here

max(|y[g ] | − τ α, 0)
Sτ (y , α)[g ] = y .
max(|y[g ] | − τ α, 0) + τ α [g ]


Z is a matrix and ψ(Z ) = Z ∗ is the nuclear norm of Z : the sum of
singular values. Threshold operator is
1 2
Sτ (Y , α) := arg min Z −Y F +τ Z ∗
Z 2α

with solution obtained from the SVD Y = UΣV T with U, V
orthonormal and Σ = diag(σi )i=1,2,...,m . Setting
˜
Σ = diag(max(σi − τ α, 0)i=1,2,...,m ), the solution is
˜
Sτ (Y , α) = U ΣV T .

(Actually not cheap to compute, but in some cases (e.g. σi − τ α < 0
for most i) approximate solutions can be found in reasoable time.)


Connections

The thresholding operator generalizes:
Gradient methods for unconstrained minimization. Here ψ ≡ 0 and
Sτ (y , α) = y .
Projected gradient for minx∈Ω f (x) with Ω closed and convex. Here ψ
is the indicator function for Ω (zero on Ω, ∞ elsewhere), and

Sτ (y , α) = PΩ (y ),

where PΩ is projection onto Ω.


Applications
LASSO for variable selection. Originally stated as
1
Ax − b 2 such that x 1 ≤ T ,
min 2
2 x
for parameter T > 0. Equivalent to an “ 2 - 1 ” formulation:
1 2
min Ax − b 2 +τ x 1, for some τ > 0.
x 2
Group LASSO for selection of variable “groups.”
1 2
min Ax − b 2 + x[g ] 2 ,
x 2
g ∈G

with each [g ] a subset of indices {1, 2, . . . , n}.
When groups [g ] are disjoint, easy to solve the subproblem.
Still true if · 2 is replaced by · ∞ .
When groups overlap, can replicate variables, to have one copy of
each variable in each group — thus reformulate as non-overlapping.

Compressed Sensing. Sparse signal recovery from noisy measurements.
Given matrix A (with more columns than rows) and observation vector y ,
seek a sparse x (i.e. few nonzeros) such that Ax ≈ y . Solve

1 2
min Ax − b 2 +τ x 1.
x 2

Under “restricted isometry” properties on A (“tall, thin” column
submatrices are nearly orthonormal), x 1 is a good surrogate for
card(x).
Assume that A is not stored explicitly, but matrix-vector
multiplications are available. Hence can compute f and f .
Often need solution for a range of τ values.


1 -Regularized Logistic Regression. Feature vectors xi , i = 1, 2, . . . , m
with labels ±1. Seek odds function parametrized by w ∈ Rn :
Tx
p+ (x; w ) := (1 + e w )−1 , p− (x; w ) := 1 − p+ (x; w ).

Scaled, negative log likelihood function L(w ) is
 
1
L(w ) = −  log p− (xi ; w ) + log p+ (xi ; w )
m
yi =−1 yi =1
 
m
1 T
=−  w T xi − log(1 + e w xi ) .
m
yi =−1 i=1

To get a sparse w (i.e. classify on the basis of a few features) solve:

min L(w ) + λ w 1.
w


Matrix Completion. Seek a matrix X ∈ Rm×n with low rank that
matches certain observations, possibly noisy.
1 2
min A(X ) − b 2 + τ ψ(X ),
X 2
where A(X ) is a linear mapping of the components of X (e.g.
element-wise observations).
Can have ψ as the nuclear norm — see discussion above for solution of
subproblems via SVD.
At NIPS 2010: “Practical Large-Scale Optimization for Max-Norm
Regularization” by Lee et al. discuss ψ(X ) = X max :

X max := inf{ U 2,∞ , V 2,∞ | X = UV T },

where U 2,∞ is the maximum 2 norm of a row of U. The shrinking
operation can be solved eﬃciently using the “squash” operator.


Basic Algorithm

(Fukushima and Mine, 1981) for solving minx f (x) + τ ψ(x).
0: Choose x0
k: Choose αk > 0 and set

xk+1 = Sτ (xk − αk f (xk ); αk )
1
= arg min f (xk )T (z − xk ) + z − xk 2
2 + τ ψ(z).
z 2αk

Straightforward, but can be fast when the regularization is strong (i.e.
solution is “highly constrained”).
Can show convergence for steps αk ∈ (0, 2/L), where L is the bound on
2 f . (Like a short-step gradient method.)


Enhancements

Alternatively, since αk plays the role of a steplength, can adjust it to get
better performance and guaranteed convergence.

“Backtracking:” decrease αk until sufficient decrease condition holds.
Use Barzilai-Borwein strategies to get nonmonotonic methods. By
enforcing sufficient decrease every 10 iterations (say), still get global
convergence.

The approach can be accelerated using optimal gradient techniques. See
earlier discussion of FISTA, where we solve the shrinking problem with
αk = 1/L in place of a step along − f with this steplength.
Note that these methods reduce ultimately to gradient methods on a
reduced space: the optimal manifold defined by the regularizer ψ.
Acceleration or higher-order information can help improve performance.


Continuation in τ

Performance of basic shrinking methods is quite sensitive to τ .
Typically higher τ ⇒ stronger regularization ⇒ optimal manifold has lower
dimension. Hence, it’s easier to identify the optimal manifold, and basic
shrinking methods can sometimes do so quickly.
For smaller τ , a simple “continuation” strategy can help:
0: Given target value τf , choose initial τ0 > τf , starting point x and
¯
factor σ ∈ (0, 1).
k: Find approx solution x(τk ) of minx f (x) + τ ψ(x), starting from x ;
¯
if τk = τf then STOP;
Set τk+1 ← max(τf , στk ) and x ← x(τk );
¯


Solution x(τ ) is often desired on a range of τ values anyway, so
eﬀorts for larger τ are not wasted.
Accelerated methods such as FISTA are less sensitive to the “small τ ”
issue.
Not much analysis of this approach has been done. Better heuristics
and theoretical support are needed.


Stochastic Gradient + Regularization
Solve the regularized problem, but have only estimates of f (xk ).
We can combine dual averaging, stochastic gradient, and shrinking: see
Xiao (2010).
min φτ (x) := Eξ f (x; ξ) + τ ψ(x)
x
At iteration k choose ξk randomly and i.i.d from the ξ distribution, and
choose gk ∈ ∂f (xk ; ξk ). Use these to deﬁne the averaged subgradient
gk = k gi /(k + 1), and solve the subproblem
¯ i=1
γ
xk+1 = arg min gk x + τ ψ(x) + √ x − x0 2 .
¯T
x k
Same as earlier, but with regularizer ψ included explicitly.
Can prove convergence results for averaged iterates xk : roughly
¯
C
E φτ (¯k ) − φ∗ ≤ √ ,
x τ
k
where the expectation of φ is taken over the random number stream
ξ0 , ξ1 , . . . , ξk−1 .

Further Reading

1 F. Bach. “Sparse methods for machine learning: Theory and algorithms” Tutorial
at NIPS 2009. (Slides on web.)
2 M. Fukushima and H. Mine. “A generalized proximal point algorithm for certain
non-convex minimization problems.” International Journal of Systems Science, 12,
pp. 989–1000, 1981.
3 P. L. Combettes and V. R. Wajs. “Signal recovery by proximal forward-backward
splitting.” Multiscale Modeling and Simulation, 4, pp. 1168–1200, 2005.
4 S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo. “Sparse reconstruction by
separable approximation.” IEEE Transactions on Signal Processing, 57, pp.
2479–2493, 2009.
5 E. Cand`s, J. Romberg, and T. Tao. “Stable signal recovery for incomplete and
e
inaccurate measurements.” Communications in Pure and Applied Mathematics,
59, pp. 1207–1223, 2006.
6 L. Xiao. “Dual averaging methods for regularized stochastic learning and online
optimization.” TechReport MSR-TR-2010-23, Microsoft Research, March 2010.


IV. Optimal Manifold Identiﬁcation

When constraints x ∈ Ψ or a nonsmooth regularizer ψ(x) are present,
identiﬁcation of the manifold on which x ∗ lies can improve algorithm
performance, by focusing attention on a reduced space. We can thus
evaluate partial gradients and Hessians, restricted to just this space.
For nonsmooth regularizer ψ, the active manifold is a smooth surface
passing through x ∗ along which ψ is smooth.
Example: for ψ(x) = x 1 , have manifold consisting of z with

≥ 0 if xi∗ > 0

zi ≤ 0 if xi∗ < 0

= 0 if xi∗ = 0.



For a polyhedral Ω, the active manifold is the face on which x ∗ lies.
Example: For Ω = [0, 1]n , active manifold consists of z with

= 1
 if xi∗ = 1
zi = 0 if xi∗ = 0

∈ [0, 1] if xi∗ ∈ (0, 1).


M

x*

Can parametrize M with a single variable.


Identification
Algorithms of shrinking / gradient projection type can identify the optimal
manifold M, or a good approximation to it, without knowing x ∗ .
HOW? Optimality conditions for the problem minx∈Ω f (x) are
x∗ − f (x ∗ ) ∈ x ∗ + NΩ (x ∗ ),
where NΩ (x) is the normal cone to Ω at x:
NΩ (x) = {s | s T (z − x) ≤ 0 for all z ∈ Ω}.
When M is defined appropriately, we find that
R(Ω, M) := {x + s | x ∈ M and s ∈ NΩ (x)}
has an interior in Rn . If x ∗ satisfies a nondegeneracy condition — that is,
− f (x ∗ ) ∈ ri NΩ (x ∗ ) (relative interior), we have
x− f (x) ∈ R(Ω, M)
for all x sufficiently close to x ∗ . From such points, projection recovers M:
PΩ (x − f (x)) ∈ M.

N
x*−Df(x*)

x*


The non-averaged iterates from gradient projection methods eventually lie
on the correct manifold M. The same is true for dual-averaging methods
(where the gradient term is averaged over all steps).
When we have a nonsmooth regularizer ψ, instead of Ω, the analogous
property is that the solution of the shrink subproblem, for some ﬁxed
positive α, lies on the optimal manifold M.
Under reasonable conditions on αk , the “basic” shrink method eventually
has all its iterates on M. Also true for dual-averaged methods.
In practice, often use heuristics for deciding when M (or a small superset)
has been reached. If a distance-to-solution bound is available, and if
Lipschitz constant for f is known, can make this decision more rigorous.


How Might This Help?
Consider again logistic regression with regularizer ψ(w ) = w 1 .
 
m
1 T
L(w ) = −  w T xi − log(1 + e w xi ) .
m
yi =−1 i=1

Requires calculation of Xw where X = [xiT ]m . (Can be cheap if w has
i=1
few nonzeros.) For gradient, have
T
1 −(1 + e w xi )−1 , yi = −1,
L(w ) = X T u, where ui = T
m (1 + e −w xi )−1 , yi = +1.

requires m exponentials, and a matrix-vector multiply by X (with a full
vector u).
If just a subset G of components needed, multiply by a column submatrix
T
X·G — much cheaper than full gradient if |G| n.

Reduced Hessian

T
2 1 e w xi
L(w ) = X T diag(v )X , where vi = .
n (1 + e w T xi )2
Often much cheaper to calculate |G| × |G| reduced Hessian than the full
Hessian.
Can use sampling (Byrd et al., 2010) to approximate the projected
Hessian: take a subset S ⊂ {1, 2, . . . , m} and use XSG in place of X·G .
Reduces evaluation cost by a factor |S|/m.


Higher-Order Shrinking Method
for problem minx f (x) + τ x 1. Step k:
Choose a subset Gk ⊃ {i | xk (i) = 0}
Evaluate Gk f (xk ) and solve (in closed form):
1 T
min f (xk )T d + d d + τ xk + d 1, s.t. d(i) = 0 for i ∈ Gk .
/
d 2αk
Repeat with a decreasing sequence of αk , until suﬃcient decrease;
+
Set xk = xk + d;
+
Deﬁne Ck ⊂ Gk by Ck := {i | xk (i) = 0}.
Calculate reduced Newton step in Ck components with (approximate)
reduced Hessian matrix HCk Ck and reduced gradient Ck f , evaluated
+
at xk . Take this reduced step if is gives an improvement in objective,
+
giving xk+1 . Otherwise, settle for xk+1 ← xk .
Can be generalized to other separable regularizers ψ(x). Choice of subset
must conform to separation in regularizer, and all components must be
checked periodically.

Newton-Like Methods

Newton-like methods are ubiquitous in smooth optimization — motivated
by second-order Taylor series.
(Basic) Newton’s Method steps obtained from

1
xk+1 = arg min f (xk ) + f (xk )T (z − xk ) + (z − xk )T Hk (z − xk ),
z 2
where Hk = 2 f (xk ). Near a local minimizer with second-order suﬃcient
conditions, converges superlinearly: xk+1 − x ∗ = o( xk − x ∗ ).
Can modify by
adding a prox-term e.g. multiple of z − xk 2;

Adding a trust-region constraint z − xk ≤ ∆k (equivalent);
doing a line search along d.


Choices of Hk
Hessian 2 f often expensive to evaluate, so can use approximations, e.g.
Re-use 2 f from a previous iterate.
Use a sampled approximation to 2 f (xk ) (see above).
Use a diagonal approximation — at least gets the scaling right. See
e.g. Barzilai-Borwein above.
quasi-Newton methods (BFGS, L-BFGS), which deﬁne Hk to be a
matrix that mimics the bevavior of the true Hessian over previous
steps. Requires only gradients f .
Other approximations that exploit the structure of the problem. e.g.
for nonlinear least squares f (x) = (1/2) r (x) 2 for r : Rn → Rm ,
2
Hessian is
m
2
f (x) = J(x)T J(x) + (1/2) ri (x) 2
ri (x),
i=1

where J is the m × n Jacobian of r . In Gauss-Newton method, use
H(x) = J(x)T J(x).

Higher-Order Information and Constraints

Higher-order methods can be extended to presence of constraints x ∈ Ω or
regularizers ψ(x) provided these elements can be incorporated explicitly
into the subproblems. e.g. for constraints
1
xk+1 = arg min f (xk ) + f (xk )T (z − xk ) + (z − xk )T Hk (z − xk ),
z∈Ω 2
and for regularizers
1
xk+1 = arg min f (xk ) + f (xk )T (z − xk ) + (z − xk )T Hk (z − xk ) + τ ψ(z).
z 2
These subproblems are typically harder to solve than the “shrink”
subproblem, unless Hk is simple (e.g. diagonal).
In practice, can do manifold identiﬁcation and reduction (see above), or
form simpler approximations to Ω (but then may need to incorporate
curvature information about Ω into Hk to ensure fast convergence).

Solving for xk+1

When Hk positive deﬁnite, can solve Newton equations explicitly by solving

Hk (z − xk ) = − f (xk ).

Alternatively, apply conjugate gradient to this system to get an inexact
solution. Each iterate requires a multiplication by Hk .
Can precondition CG, e.g. by using sample approximations, structured
approximations, or Krylov subspace information gathered at previous
evaluations of 2 f .
L-BFGS stores Hk in implicit form, by means of 2m vectors in Rn , for a
small parameter m (e.g. 5). Recovers solution of the equation above via
2m inner products.


“Higher-Order” Methods in ML
Several approaches tried (in addition to sampling and reduced-space
techniques discussed above).
Bordes et al. (2009, corrected 2010) for SVM
m
τwT w + (w ; xi , yi ),
i=1

scales the stochastic gradient step with a diagonal Hk , obtained from
finite differences of the last estimated gradient over the last step.
Schraudolph et al. (AISTATS 2007) “online BFGS” uses conventional
quasi-Newton update formulae (e.g. L-BFGS) based on estimated
gradient differences over previous steps.

Since the gradients are so inexact (based on just one data point), both in
update and right-hand side of the step equations, these methods are really
stochastic gradient with interesting scaling, rather than quasi-Newton in
the conventional sense.

Further Reading

1 L. Bottou and A. Moore, “Learning with large datasets,” Tutorial at NIPS 2007
(slides on web).
2 J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006.
3 R. H. Byrd, G. M. Chin, W. Neveitt, and J. Nocedal. “On the use of stochastic
hessian information in unconstrained optimization.” Technical Report,
Northwestern University, June 2010.
4 M. Fisher, J. Nocedal, Y. Tremolet, and S. J. Wright. “Data assimilation in
weather forecasting: A case study in PDE-constrained optimization.” Optimization
and Engineering, 10, pp. 409–426, 2009.
5 A. S. Lewis and S. J. Wright. “Identifying activity.” Technical report, ORIE,
Cornell University. Revised April 2010.
6 S. J. Wright, “Accelerated block-coordinate relaxation for regularized
optimization,” Technical Report, August 2010.
7 W. Hare and A. Lewis. “Identifying active constraints via partial smoothness and
prox-regularity.” Journal of Convex Analysis, 11, pp. 251–266, 2004.


V. Decomposition / Coordinate Relaxation

For min f (x), at iteration k, choose a subset Gk ⊂ {1, 2, . . . , n} and take
a step dk only in these components. i.e. ﬁx dk (i) = 0 for i ∈ Gk .
/
Gives more manageable subproblem sizes, in practice.
Can
take a reduced gradient step in the Gk components;
take multiple “inner iterations”
actually solve the reduced subproblem in the space deﬁned by Gk .


Constraints and Regularizers Complicate Things

For minx∈Ω f (x), need to put enough components into Gk to stay feasible,
as well as make progress.
Example: min f (x1 , x2 ) with x1 + x2 = 1. Relaxation with Gk = {1} or
Gk = {2} won’t work.
For separable regularizer (e.g. Group LASSO) with

ψ(x) = ψg (x[g ] ),
g ∈G

need to ensure that Gk is a union of the some index subsets [g ]. i.e. the
relaxation components must be consonant with the partitioning.


Decomposition and Dual SVM

Decomposition has long been popular for solving the dual (QP)
formulation of SVM, since the number of variables (= number of training
examples) is sometimes very large.

SMO: Each Gk has two components.
LIBSVM: SMO approach (still |Gk | = 2), with diﬀerent heuristic for
choosing Gk .
LASVM: Again |Gk | = 2, with focus on online setting.
SVM-light: Small |Gk | (default 10).
GPDT: Larger |Gk | (default 400) with gradient projection solver as
inner loop.


Choice of Gk and Convergence Results
Some methods (e.g. Tseng and Yun, 2010) require Gk to be chosen so
that the improvement in subproblem objective obtained over the subset Gk
is at least a ﬁxed fraction of the improvement available over the whole
space. Undesirable, since to check it, usually need to evaluate the full
gradient f (xk ).
Alternative is a generalized Gauss-Seidel requirement, where each
coordinate is “touched” at least once every T iterations:
Gk ∪ Gk+1 ∪ . . . ∪ Gk+T −1 = {1, 2, . . . , n}.
Can show global convergence (e.g. Tseng and Yun, 2009; Wright, 2010).
There are also results on
global linear convergence rates
optimal manifold identiﬁcation
fast local convergence for an algorithm that takes reduced steps on
the estimated optimal manifold.
All deterministic analyses.

Stochastic Coordinate Descent
Analysis tools of stochastic gradient may be useful. If steps have the form
xk+1 = xk − αk gk , where

[ f (xk )]i if i ∈ Gk
gk (i) =
0 otherwise,

With suitable random selection of Gk can ensure that gk (appropriately
scaled) is an unbiased estimate of f (xk ). Hence can apply SGD
techniqes discussed earlier, to choose αk and obtain convergence.
Nesterov (2010) proposes another randomized approach for the
unconstrained problem with known separate Lipschitz constants Li :

∂fi ∂fi
(x + hei ) − (x) ≤ Li |h|, i = 1, 2, . . . , n.
∂xi ∂xi

(Works with blocks too, instead of individual components.)

At step k:
n
Choose index ik ∈ {1, 2, . . . , n} with probability pi := Li /( j=1 Lj );
Take gradient step in ik component:
1 ∂f
xk+1 = xk − ei .
Lik ∂xik k

Basic convergence result:
C
E [f (xk )] − f ∗ ≤ .
k
As for SA (earlier) but without any strong convexity assumption.
Can also get linear convergence results (in expectation) by assuming
strong convexity in f , according to diﬀerent norms.
Can also accelerate in the usual fashion (see above), to improve expected
convergence rate to O(1/k 2 ).


Further Reading

1 P. Tseng and S. Yun, “A coordinate gradient descent method for linearly
constrained smooth optimization and support vector machines training.”
Computational Optimization and Applications, 47, pp. 179–206, 2010.
2 P. Tseng and S. Yun, “A coordinate gradient descent method for nonsmooth
separable minimization.” Mathematical Programming, Series B, 117. pp.
387–423, 2009.
3 S. J. Wright, “Accelerated block-coordinate relaxation for regularized
optimization.” Technical Report, UW-Madison, August 2010.
4 Y. Nesterov, “Eﬃciency of coordinate descent methods on huge-scale optimization
problems.” CORE Discussion Paper 2010/2, CORE, UCL, January 2010.


Conclusions

We’ve surveyed a number of topics in algorithmic fundamentals, with an
eye on recent developments, and on topics of relevance (current or future)
to machine learning.
The talk was far from exhaustive. Literature on Optimization in ML is
huge and growing.
There is much more to be gained from the interaction between the two
areas.

FIN


NIPS2010: optimization algorithms in machine learning

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