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Forward-Backward Greedy Algorithms for General Convex Smooth Functions
over A Cardinality Constraint
Abstract
We consider forward-backward greedy algorithms for solving sparse feature selection problems with general convex smooth functions.
A state-of-the-art greedy method, the ForwardBackward greedy algorithm (FoBa-obj) requires
to solve a large number of optimization problems, thus it is not scalable for large-size problems. The FoBa-gdt algorithm, which uses the
gradient information for feature selection at each
forward iteration, significantly improves the efficiency of FoBa-obj. In this paper, we systematically analyze the theoretical properties of both
forward-backward greedy algorithms. Our main
contributions are: 1) We derive better theoretical
bounds than existing analyses regarding FoBaobj for general smooth convex functions; 2) We
show that FoBa-gdt achieves the same theoretical
performance as FoBa-obj under the same condition: restricted strong convexity condition. Our
new bounds are consistent with the bounds of
a special case (least squares) and fills a previously existing theoretical gap for general convex
smooth functions; 3) We show that the restricted
strong convexity condition is satisfied if the number of independent samples is more than k̄ log d
where k̄ is the sparsity number and d is the dimension of the variable; 4) We apply FoBa-gdt
(with the conditional random field objective) to
the sensor selection problem for human indoor
activity recognition and our results show that
FoBa-gdt outperforms other methods (including
the ones based on forward greedy selection and
L1-regularization).
1. Introduction
Feature selection has been one of the most significant issues
in machine learning and data mining. Following the success of Lasso (Tibshirani, 1994), learning algorithms with
Preliminary work. Under review by the International Conference
on Machine Learning (ICML). Do not distribute.
sparse regularization (a.k.a. sparse learning) have recently
received significant attention. A classical problem is to estimate a signal β ∗ ∈ Rd from a feature matrix X ∈ Rn×d
and an observation y = Xβ ∗ + noise ∈ Rn , under the
assumption that β ∗ is sparse (i.e., β ∗ has k̄ d nonzero
elements). Previous studies have proposed many powerful
tools to estimate β ∗ . In addition, in certain applications, reducing the number of features has a significantly practical
value (e.g., sensor selection in our case).
The general sparse learning problems can be formulated as
follows (Jalali et al., 2011):
β̄ := arg min : Q(β; X, y)
β
s.t. :
kβk0 ≤ k̄. (1)
where Q(β; X, y) is a convex smooth function in terms of β
such as the least square loss (Tropp, 2004) (regression), the
Gaussian MLE (or log-determinant divergence) (Ravikumar et al., 2011) (covariance selection), and the logistic
loss (Kleinbaum & Klein, 2010) (classification). kβk0 denotes `0 -norm, that is, the number of nonzero entries of
β ∈ Rd . Hereinafter, we denote Q(β; X, y) simply as
Q(β).
From an algorithmic viewpoint, we are mainly interested
in three aspects for the estimator β̂: 1) estimation error
kβ̂ − β̄k; 2) objective error Q(β̂) − Q(β̄); and 3) feature
selection error, that is, the difference between supp(β̂) and
F̄ := supp(β̄), where supp(β) is a feature index set corresponding to nonzero elements in β. Since the constraint
defines a non-convex feasible region, the problem is nonconvex and generally NP-hard.
There are two types of approaches to solve this problem in the literature. Convex-relaxation approaches replace `0 -norm by `1 -norm as a sparsity penalty. Such
approaches include Lasso (Tibshirani, 1994), Danzig selector (Candès & Tao, 2007), and L1-regularized logistic regression (Kleinbaum & Klein, 2010). Alternative greedy-optimization approaches include orthogonal matching pursuit (OMP) (Tropp, 2004; Zhang, 2009),
backward elimination, and forward-backward greedy
method (FoBa) (Zhang, 2011a), which use greedy heuristic
procedure to estimate sparse signals. Both types of algorithms have been well studied from both theoretical and
empirical perspectives.
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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FoBa has been shown to give better theoretical properties
than LASSO and Dantzig selector for the least squared loss
function: Q(β) = 12 kXβ−yk2 (Zhang, 2011a). Jalali et al.
(2011) has recently extended it to general convex smooth
functions. Their method and analysis, however, pose computational and theoretical issues. First, since FoBa solves
a large number of single variable optimization problems in
every forward selection step, it is computationally expensive for general convex functions if the sub-problems have
no closed form solution. Second, though they have empirically shown that FoBa performs well for general smooth
convex functions, their theoretical results are weaker than
those for the least square case (Zhang, 2011a). More precisely, their upper bound for estimation error is looser and
their analysis requires more restricted conditions for feature selection and signal recovery consistency. The question of whether or not FoBa can achieve the same theoretical bound in the general case as in the least square case
motivates this work.
This paper addresses the theoretical and computational issues associated with the standard FoBa algorithm (hereinafter referred to as FoBa-obj because it solves single variable problems to minimize the objective in each forward
selection step). We study a new algorithm referred to as
“gradient” FoBa (FoBa-gdt) which significantly improves
the computational efficiency of FoBa-obj. The key difference is that FoBa-gdt only evaluates gradient information
in individual forward selection steps rather than solving a
large number of single variable optimization problems. Our
contributions are summarized as follows.
Theoretical Analysis of FoBa-obj and FoBa-gdt This paper presents three main theoretical contributions. First,
we derive better theoretical bounds for estimation error,
objective error, and feature selection error than existing
analyses for FoBa-obj for general smooth convex functions (Jalali et al., 2011) under the same condition: restricted strong convexity condition. Second, we show that
FoBa-gdt achieves the same theoretical performance as
FoBa-obj. Our new bounds are consistent with the bounds
of a special case, i.e., the least square case, and fills in the
theoretical gap between the general loss (Jalali et al., 2011)
and the least squares loss case (Zhang, 2011a). Our result
also implies an interesting result: when the signal noise ratio is big enough, the NP hard problem (1) can be solved
by using FoBa-obj or FoBa-gdt. Third, we show that the
restricted strong convexity condition is satisfied for a class
of commonly used machine learning objectives, e.g., logistic loss and least square loss, if the number of independent
samples is greater than k̄ log d where k̄ is the sparsity number and d is the dimension of the variable.
Application to Sensor Selection We have applied FoBagdt with the CRF loss function (referred to as FoBa-gdt-
CRF) to sensor selection from time-series binary location
signals (captured by pyroelectric sensors) for human activity recognition at homes, which is a fundamental problem in smart home systems and home energy management systems. In comparison with forward greedy and L1regularized CRFs (referred to as L1-CRF), FoBa-gdt-CRF
requires the smallest number of sensors for achieving comparable recognition accuracy.
1.1. Notation
Denote ej ∈ Rd as the j th natural basis in the space Rd .
The set difference A − B returns the elements that are in
A but outside of B. Given any integer s > 0, the restricted
strong convexity constants (RSCC) ρ− (s) and ρ+ (s) are
defined as follows: for any ktk0 ≤ s and t = β 0 − β, we
require
ρ+ (s) 2
ρ− (s) 2
ktk ≤ Q(β 0 ) − Q(β) − hOQ(β), ti ≤
ktk .
2
2
(2)
Similar definitions can be found in (Bahmani et al., 2011;
Jalali et al., 2011; Negahban et al., 2010; Zhang, 2009).
If the objective function takes the quadratic form Q(β) =
1
2
2 kXβ − yk , then the above definition is equivalent to the
restricted isometric property (RIP) (Candès & Tao, 2005):
ρ− (s)ktk2 ≤ kXtk2 ≤ ρ+ (s)ktk2 ,
where the well known RIP constant can be defined as δ =
max{1 − ρ− (s), ρ+ (s) − 1}. To give tighter values for
ρ+ (.) and ρ− (.), we only require (2) to hold for all β ∈
Ds := {kβk0 ≤ s | Q(β) ≤ Q(0)} throughout this paper.
Finally we define β̂(F ) as β̂(F ) := arg minsupp(β)⊂F :
Q(β). Note that the problem is convex as long as Q(β) is a
convex function. Denote F̄ := supp(β̄) and k̄ := |F̄ |.
We make use of order notation throughout this paper. If a
and b are both positive quantities that depend on n or p, we
write a = O(b) if a can be bounded by a fixed multiple of
b for all sufficiently large dimensions. We write a = o(b)
if for any positive constant φ > 0, we have a ≤ φb for all
sufficiently large dimensions. We write a = Ω(b) if both
a = O(b) and b = O(a) hold.
2. Related Work
Tropp (2004) investigated the behavior of the orthogonal
matching pursuit (OMP) algorithm for the least square
case, and proposed a sufficient condition (an `∞ type condition) for guaranteed feature selection consistency. Zhang
(2009) generalized this analysis to the case of measurement noise. In statistics, OMP is known as boosting
(Buhlmann, 2006) and similar ideas have been explored in
Bayesian network learning (Chickering & Boutilier, 2002).
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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Shalev-Shwartz et al. (2010) extended OMP to the general
convex smooth function and studied the relationship between objective value reduction and output sparsity. Other
greedy methods such as ROMP (Needell & Vershynin,
2009) and CoSaMp (Needell & Tropp, 2008) were studied
and shown to have theoretical properties similar to those
of OMP. Zhang (2011a) proposed a Forward-backward
(FoBa) greedy algorithm for the least square case, which is
an extension of OMP but has stronger theoretical guarantees as well as better empirical performance: feature selection consistency is guaranteed under the sparse eigenvalue
condition, which is an `2 type condition weaker than the
`∞ type condition. Note that if the data matrix is a Gaussian random matrix, the `2 type condition requires the measurements n to be of the order of O(s log d) where s is the
sparsity of the true solution and d is the number of features,
while the `∞ type condition requires n = O(s2 log d);
see (Zhang & Zhang, 2012; Liu et al., 2012). Jalali et al.
(2011) and Johnson et al. (2012) extended the FoBa algorithm to general convex functions and applied it to sparse
inverse covariance estimation problems.
Convex methods, such as LASSO (Zhao & Yu, 2006) and
Dantzig selector (Candès & Tao, 2007), were proposed for
sparse learning. The basic idea behind these methods is
to use the `1 -norm to approximate the `0 -norm in order
to transform problem (1) into a convex optimization problem. They usually require restricted conditions referred to
as irrepresentable conditions (stronger than the RIP condition) for guaranteed feature selection consistency (Zhang,
2011a). A multi-stage procedure on LASSO and Dantzig
selector (Liu et al., 2012) relaxes such condition, but it is
still stronger than RIP.
Algorithm 1 FoBa ( FoBa-obj FoBa-gdt )
Require: δ > 0 > 0
Ensure: β (k)
1: Let F (0) = ∅, β (0) = 0, k = 0,
2: while TRUE do
3:
%% stopping determination
4:
(k)
if Q(β (k) ) − minα,j ∈F
+ αej ) < δ
/ (k) Q(β
5:
6:
7:
kOQ(β (k) )k∞ < then
break
end if
%% forward step
8:
i(k) = arg mini∈F
/ (k) {minα Q(β
(k)
+ αei )}
(k)
i(k) = arg maxi∈F
)i |
/ (k) : |∇Q(β
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
F (k+1) = F (k) ∪ {i(k) }
β (k+1) = β̂(F (k+1) )
δ (k+1) = Q(β (k) ) − Q(β (k+1) )
k =k+1
%% backward step
while TRUE do
(k)
if mini∈F (k+1) Q(β (k) − βi ei ) − Q(β (k) ) ≥
δ (k) /2 then
break
end if
(k)
i(k) = arg mini Q(β (k) − βi ei )
k =k−1
F (k) = F (k+1) − {i(k+1) }
β (k) = β̂(F (k) )
end while
end while
3. The Gradient FoBa Algorithm
This section introduces the standard FoBa algorithm, that
is, FoBa-obj, and its variant FoBa-gdt. Both algorithms
start from an empty feature pool F and follow the same
procedure in every iteration consisting of two steps: a forward step and a backward step. The forward step evaluates
the “goodness” of all features outside of the current feature
set F , selects the best feature to add to the current feature
pool F , and then optimizes the corresponding coefficients
of all features in the current feature pool F to obtain a new
β. The elements of β in F are nonzero and the rest are zeros. The backward step iteratively evaluates the “badness”
of all features outside of the current feature set F , removes
“bad” features from the current feature pool F , and recomputes the optimal β over the current feature set F . Both algorithms use the same definition of “badness” for a feature:
the increment of the objective after removing this feature.
Specifically, for any features i in the current feature pool F ,
the “badness” is defined as Q(β − βi ei ) − Q(β), which is
a positive number. It is worth to note that the forward step
selects one and only one feature while the backward step
may remove zero, one, or more features. Finally, both algorithms terminate when no “good” feature can be identified
in the forward step, that is, the “goodness” of all features
outside of F is smaller than a threshold.
The main difference between FoBa-obj and FoBa-gdt lies
in the definition of “goodness” in the forward step and their
respective stopping criterion. FoBa-obj evaluates the goodness of a feature by its maximal reduction of the objective
function. Specifically, the “goodness” of feature i is defined as Q(β) − minα Q(β + αei ) (a larger value indicates
a better feature). This is a direct way to evaluate the “goodness” since our goal is to decrease the objective as much
as possible under the cardinality condition. However, it
may be computationally expensive since it requires solving
a large number of one-dimensional optimization problems,
which may or may not be solved in a closed form. To improve computational efficiency in such situations, FoBa-gdt
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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uses the partial derivative of Q with respect to individual
coordinates (features) as its “goodness’ measure: specifically, the “goodness” of feature i is defined as |∇Q(β)i |.
Note that the two measures of “goodness” are always nonnegative. If feature i is already in the current feature set
F , its “goodness” score is always zero, no matter which
measure to use. We summarize the details of FoBa-obj
and FoBa-gdt in Algorithm 1: the plain texts correspond
to the common part of both algorithms, and the ones with
solid boxes and dash boxes correspond to their individual
parts. The superscript (k) denotes the k th iteration incremented/decremented in the forward/backward steps.
Gradient-based feature selection has been used in a forward greedy method (Zhang, 2011b). FoBa-gdt extends it
to a Forward-backward procedure (we present a detailed
theoretical analysis of it in the next section). The main
workload in the forward step for FoBa-obj is on Step 4,
whose complexity is O(T D), where T represents the iterations needed to solve minα : Q(β (k) + αej ) and D is
the number of features outside of the current feature pool
set F (k) . In comparison, the complexity of Step 4 in FoBa
is just O(D). When T is large, we expect FoBa-gdt to be
much more computationally efficient. The backward steps
of both algorithms are identical. The computational costs
of the backward step and the forward step are comparable
in FoBa-gdt (but not FoBa-obj), because their main work
loads are on Step 10 and Step 21 (both are solving β̂(.))
respectively and the times of running Step 21 is always less
than that of Step 10.
4. Theoretical Analysis
This section first gives the termination condition of Algorithms 1 with FoBa-obj and FoBa-gdt because the number of iterations directly affect the values of RSCC (ρ+ (.),
ρ− (.), and their ratio), which are the key factors in our
main results. Then we discuss the values of RSCC in a
class of commonly used machine learning objectives. Next
we present the main results of this paper, including upper
bounds on objective, estimation, and feature selection errors for both FoBa-obj and FoBa-gdt. We compare our results to those of existing analyses of FoBa-obj and show
that our results fill the theoretical gap between the least
square loss case and the general case.
4.1. Upper Bounds on Objective, Estimation, and
Feature Selection Errors
We first study the termination conditions of FoBa-obj and
FoBa-gdt, as summarized in Theorems 1 and 2 respectively.
+ (1)
2
Theorem 1. Take δ > 4ρ
ρ− (s)2 kOQ(β̄)k∞ in Algorithm 1
with FoBa-obj where s can be any positive integer satisfy-
ing s ≤ n and
" s
(s − k̄) > (k̄ + 1)
ρ+ (s)
+1
ρ− (s)
!
2ρ+ (1)
ρ− (s)
#2
.
(3)
Then the algorithm terminates at some k ≤ s − k̄.
√
+ (1)
kOQ(β̄)k∞ in Algorithm 1
Theorem 2. Take > 2 ρ2ρ
− (s)
with FoBa-gdt, where s can be any positive integer satisfying s ≤ n and Eq.(3). Then the algorithm terminates at
some k ≤ s − k̄.
To simply the results, we first assume that the condition number κ(s) := ρ+ (s)/ρ− (s) is bounded (so is
ρ+ (1)/ρ− (s) because of ρ+ (s) ≥ ρ+ (1)). Then both
FoBa-obj and FoBa-gdt terminate at some k proportional to
the sparsity k̄, similar to OMP (Zhang, 2011b) and FoBaobj (Jalali et al., 2011; Zhang, 2011a). Note that the value
of k in Algorithm 1 is exactly the cardinality of F (k) and
the sparsity of β (k) . Therefore, Theorems 1 and 2 imply
that if κ(s) is bounded, FoBa-obj and FoBa-gdt will output a solution with sparsity proportional to that of the true
solution β̄.
Most existing works simply assume that κ(s) is bounded
or have similar assumptions. We make our analysis more
complete by discussing the values of ρ+ (s), ρ− (s), and
their ratio κ(s). Apparently, if Q(β) is strongly convex and
Lipschitzian, then ρ− (s) is bounded from below and ρ+ (s)
is bounded from above, thus restricting the ratio κ(s). To
see that ρ+ (s), ρ− (s), and κ(s) may still be bounded under milder conditions, we consider a common structure for
Q(β) used in many machine learning formulations:
n
Q(β) =
1X
li (Xi. β, yi ) + R(β)
n i=1
(4)
where (Xi. , yi ) is the ith training sample with Xi. ∈ Rd
and yi ∈ R, li (., .) is convex with respect to the first argument and could be different for different i, and both li (., .)
and R(.) are twice differentiable functions. li (., .) is typically the loss function, e.g., the quadratic loss li (u, v) =
(u − v)2 in regression problems and the logistic loss
li (u, v) = log(1 + exp{−uv}) in classification problems.
R(β) is typically the regularization, e.g., R(β) = µ2 kβk2 .
Theorem 3. Let s be a positive integer less than n, and
+
λ− , λ+ , λ−
R , and λR be positive numbers satisfying
λ− ≤ ∇21 li (Xi. β, yi ) ≤ λ+ ,
+
2
λ−
R I ∇ R(β) λR I
(∇21 li (., .) is the second derivative with respect to the
first argument) for any i and β ∈ Ds . Assume that
−
λ−
> 0 and the sample matrix X ∈ Rn×d
R + 0.5λ
has independent sub-Gaussian isotropic random rows or
columns (in the case of columns, all columns should also
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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√
satisfy kX.j k = n). If the number of samples satisfies
n ≥ Cs log d, then
+
ρ+ (s) ≤λ+
R + 1.5λ
(5a)
−
≥λ−
R + 0.5λ
+
λ+
R + 1.5λ
≤ −
λR + 0.5λ−
(5b)
ρ− (s)
κ(s)
=: κ
space limitations (the proofs are provided in Supplement).
The main results for FoBa-obj and FoBa-gdt are presented
in Theorems 4 and 5 respectively.
Theorem 4. Let s be any number that satisfies (3) and
choose δ as in Theorem 1 for Algorithm 1 with FoBa-obj.
Consider the output β (k) and its support set F (k) . We have
(5c)
hold with high probability1 , where C is a fixed constant.
Furthermore, define k̄, β̄, and δ (or ) in Algorithm 1 with
FoBa-obj (or FoBa-gdt) as in Theorem 1 (or Theorem 2).
Let
√
s = k̄ + 4κ2 ( κ + 1)2 (k̄ + 1)
(6)
and n ≥ Cs log d. We have that s satisfies (3) and Algorithm 1 with
√ FoBa-obj (or FoBa-gdt) terminates within at
most 4κ2 ( κ + 1)2 (k̄ + 1) iterations with high probability.
Roughly speaking, if the number of training samples is
large enough, i.e., n ≥ Ω(k̄ log d) (actually it could be
much smaller than the dimension d of the train data), we
have the following with high probability: Algorithm 1 with
FoBa-obj or FoBa-gdt outputs a solution with sparsity at
most Ω(k̄) (this result will be improved when the nonzero
elements of β̄ are strong enough, as shown in Theorems 4
and 5); s is bounded by Ω(k̄); and ρ+ (s), ρ− (s), and κ(s)
are bounded by constants. One important assumption is
that the sample matrix X has independent sub-Gaussian
isotropic random rows or columns. In fact, this assumption
is satisfied by many natural examples, including Gaussian
and Bernoulli matrices, general bounded random matrices
whose entries are independent bounded random variables
with zero mean and unit variances. Note that from the definition of “sub-Gaussian isotropic random vectors” (Vershynin, 2011, Definitions 19 and 22), it even allows the
dependence within rows or columns but not both. Another
−
important assumption is λ−
> 0, which means
R + 0.5λ
−
−
that either λR or λ is positive (both of them are nonnegative from the convexity assumption). WePcan simply verify
n
that 1) for the quadratic case Q(β) = n1 i=1 (Xi. β −yi )2 ,
−
−
we have λ = 1 and λR = 0; 2) for the logistic
Pn case with
bounded data matrix X, that is Q(β) = n1 i=1 log(1 +
exp{−Xi. βyi }) + µ2 kβk2 , we have λ−
R = µ > 0 and
λ− > 0 because Ds is bounded in this case.
Now we are ready to present the main results: the upper
bounds of estimation error, objective error, and feature selection error for both algorithms. ρ+ (s), ρ+ (1), and ρ− (s)
are involved in all bounds below. One can simply treat them
as constants in understanding the following results, since
we are mainly interested in the scenario when the number
of training samples is large enough. We omit proofs due to
1
“With high probability” means that the probability converges
to 1 with the problem size approaching to infinity.
16ρ2+ (1)δ ¯
∆,
ρ2− (s)
2ρ+ (1)δ ¯
Q(β (k) ) − Q(β̄) ≤
∆,
ρ− (s)
ρ− (s)2 (k)
¯
|F − F̄ | ≤|F̄ − F (k) | ≤ 2∆,
8ρ+ (1)2
kβ (k) − β̄k2 ≤
√
where γ =
γ}|.
4
ρ+ (1)δ
ρ− (s)
¯ := |{j ∈ F̄ − F (k) : |β̄j | <
and ∆
Theorem 5. Let s be any number that satisfies (3) and
choose as in Theorem 2 for Algorithm 1 with FoBa-gdt.
Consider the output β (k) and its support set F (k) . We have
kβ (k) − β̄k2 ≤
82 ¯
∆,
ρ2− (s)
Q(β (k) ) − Q(β̄) ≤
2 ¯
∆,
ρ− (s)
ρ− (s)2 (k)
¯
|F − F̄ | ≤|F̄ − F (k) | ≤ 2∆,
8ρ+ (1)2
where γ =
√
2 2
ρ− (s)
¯ := |{j ∈ F̄ − F (k) : |β̄j | < γ}|.
and ∆
Although FoBa-obj and FoBa-gdt use different criteria to
evaluate the “goodness” of each feature, they actually guarantee the same properties. Choose 2 and δ in the order of Ω(k∇Q(β̄)k2∞ ). For both algorithms, we have that
the estimation error kβ (k) − β̄k2 and the objectiev error
¯
Q(β (k) ) − Q(β̄) are bounded by Ω(∆k∇Q(
β̄)k2∞ ), and
(k)
the feature selection errors |F − F̄ | and |F̄ − F (k) | are
¯ k∇Q(β̄)k∞ and ∆
¯ are two key factors
bounded by Ω(∆).
in these bounds. k∇Q(β̄)k∞ roughly represents the noise
¯ defines the number of weak channels of the true
level2 . ∆
solution β̄ in F̄ . One can see that if all channels of β̄ on F̄
are strong enough, that is, |β̄j | > Ω(k∇Q(β̄)k∞ ) ∀j ∈ F̄ ,
¯ turns out to be 0. In other words, all errors (estimation
∆
error, objective error, and feature selection error) become 0,
when the signal noise ratio is big enough. Note that under
this condition, the original NP hard problem (1) is solved
exactly, which is summarized in the following corollary:
2
To see this, we can consider the least square case (with
standard noise assumption and each column of the measuren×d
ment
is normalized to 1): k∇Q(β̄)k∞ ≤
p matrix X ∈ R
−1
Ω( n log dσ) holds with high probability, where σ is the standard derivation.
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Corollary 1. Let s be any number that satisfies (3) and
choose δ (or ) as in Theorem 1 (or 2) for Algorithm 1 with
FoBa-gdt (or FoBa-obj). If
|β̄j |
8ρ+ (1)
≥ 2
ρ− (s)
k∇Q(β̄)k∞
∀j ∈ F̄ ,
then problem (1) can be solved exactly.
One may argue that since it is difficult to set δ or , it is still
hard to solve (1). In practice, one does not have to set δ or
and only needs to run Algorithm 1 without checking the
stopping condition until all features are selected. Then the
most recent β (k̄) gives the solution to (1).
4.2. Comparison for the General Convex Case
Jalali et al. (2011) analyzed FoBa-obj for general convex
smooth functions and here we compare our results to theirs.
They chose the true model β ∗ as the target rather than the
true solution β̄. In order to simplify the comparison, we assume that the distance between the true solution and the
true model is not too great3 , that is, we have β ∗ ≈ β̄,
supp(β ∗ ) = supp(β̄), and k∇Q(β ∗ )k∞ ≈ k∇Q(β̄)k∞ .
We compare our results from Section 4.1 and the results in
(Jalali et al., 2011). In the estimation error comparison, we
have from our results:
kβ (k) − β ∗ k ≈ kβ (k) − β̄k
¯ 1/2 k∇Q(β̄)k∞ ) ≈ Ω(∆
¯ 1/2 k∇Q(β ∗ )k∞ )
≤Ω(∆
and from the results in (Jalali et al., 2011): kβ (k) − β ∗ k ≤
Ω(k̄k∇Q(β ∗ )k∞ ). Note that ∆1/2 ≤ k̄ 1/2 k̄. Therefore, under our assumptions with respect to β ∗ and β̄, our
analysis gives a tighter bound. Notably, when there are
a large number of strong channels in β̄ (or approximately
¯ k̄.
β ∗ ), we will have ∆
Let us next consider the condition required for feature selection consistency, that is, supp(F(k) ) = supp(F̄) =
supp(β ∗ ). We have from our results:
kβ̄j k ≥ Ω(k∇Q(β̄)k∞ ) ∀j ∈ supp(β ∗ )
and from the results in (Jalali et al., 2011):
kβj∗ k ≥ Ω(k̄k∇Q(β ∗ )k∞ ) ∀j ∈ supp(β ∗ ).
When β ∗ ≈ β̄ and k∇Q(β ∗ )k∞ ≈ k∇Q(β̄)k∞ , our
results guarantee feature selection consistency under a
weaker condition.
3
This assumption is not absolutely fair, but holds in many
cases, such as in the least square case, which will be made clear
in Section 4.3.
4.3. A Special Case: Least Square Loss
1
2 kXβ
We next consider the least square case: Q(β) =
−
yk2 and shows that our analysis for the two algorithms in
Section 4.1 fills in a theoretical gap between this special
case and the general convex smooth case.
Following previous studies (Candès & Tao, 2007; Zhang,
2011b; Zhao & Yu, 2006), we assume that y = Xβ ∗ + ε
where the entries in ε are independent random sub-gaussian
variables, β ∗ is the true model with the support set F̄ and
the sparsity number k̄ := |F̄ |, and X ∈ Rn×d is normalized as kX.i k2 = 1 for all columns i = 1, · · · , d. We then
have following inequalities with high probability (Zhang,
2009):
p
k∇Q(β ∗ )k∞ = kX T εk∞ ≤ Ω( n−1 log d), (7)
p
k∇Q(β̄)k∞ ≤ Ω( n−1 log d),
(8)
p
∗
kβ̄ − β k2 ≤ Ω( n−1 k̄),
(9)
q
kβ̄ − β ∗ k∞ ≤ Ω( n−1 log k̄),
(10)
implying that β̄ and β ∗ are quite close when the true model
is really sparse, that is, when k̄ n.
An analysis for FoBa-obj in the least square case (Zhang,
2011a) has indicated that the following estimation error
bound holds with high probability:
kβ (k) − β ∗ k2 ≤ Ω(n−1 (k̄+
p
log d|{j ∈ F̄ : |βj∗ | ≤ Ω( n−1 log d)}|))
(11)
as well as the following condition for feature selection consistency:
p
supp(β (k) ) = supp(β ∗ ), if |βj∗ | ≥ Ω( n−1 log d) ∀j ∈ F̄ .
(12)
Applying the analysis for general convex smooth cases in
(Jalali et al., 2011) to the least square case, one obtains the
following estimation error bound from Eq. (7)
kβ (k) − β ∗ k2 ≤ Ω(k̄ 2 kOQ(β ∗ )k2∞ ) ≤ Ω(n−1 k̄ 2 log d)
and the following condition of feature selection consistency:
q
supp(β (k) ) = supp(β ∗ ), if |βj∗ | ≥ Ω( k̄n−1 log d) ∀ j ∈ F̄ .
One can observe that the general analysis gives a looser
bound for estimation error and requires a stronger condition
for feature selection consistency than the analysis for the
special case.
Our results in Theorems 4 and 5 bridge this gap when combined with Eqs. (9) and (10). The first inequalities in The-
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orems 4 and 5 indicate that
kβ (k) − β ∗ k2 ≤ (kβ (k) − β̄k + kβ̄ − β ∗ k)2
≤Ω n−1 (k̄ + log d | {j ∈ F̄ − F (k) :
p
[from Eq. (9)]
|β̄j | < Ω(n−1/2 log d)}|)
≤Ω n−1 (k̄ + log d | {j ∈ F̄ − F (k) :
p
|βj∗ | < Ω(n−1/2 log d)}|)
[from Eq. (10)]
which is consistent with the results in Eq. (11). The last
inequality in Theorem 5 also implies that feature selectionpconsistency is guaranteedp
as well, as long as |β̄j | >
Ω( n−1 log d) (or |βj∗ | > Ω( n−1 log d)) for all j ∈ F̄ .
This requirement agrees with the results in Eq. (12).
5. Application: Sensor Selection for Human
Activity Recognition
Machine learning technologies for smart home systems and
home energy management systems have recently attracted
much attention. Among the many promising applications
such as optimal energy control, emergency alerts for elderly persons living alone, and automatic life-logging, a
fundamental challenge for these applications is to recognize human activity at homes, with the smallest number of
sensors. The data mining task here is to minimize the number of sensors without significantly worsening recognition
accuracy. We used pyroelectric sensors, which return binary signals in reaction to human motion.
Fig. 1 shows our experimental room layout and sensor locations. The numbers represent sensors, and the ellipsoid
around each represents the area covered by it. We used
40 sensors, i.e., we observe a 40-dimensional binary time
series. A single person lives in the room for roughly one
month, and data is collected on the basis of manually tagging his activities into the pre-determined 14 categories
summarized in Table 5. For data preparation reasons, we
use the first 20% (roughly one week) samples in the data,
and divide it into 10% for training and 10% for testing. The
numbers of training and test samples are given in Table 1.
Pyroelectric sensors are preferable over cameras for two
practical reasons: cameras tend to create a psychological
barrier and pyroelectric sensors are much cheaper and easier to implement at homes. Such sensors only observe
noisy binary location information. This means that, for
high recognition accuracy, history (sequence) information
must be taken into account. The binary time series data
follows a linear-chain conditional random field (CRF) (Lafferty et al., 2001; Sutton & McCallum, 2006). Linear-chain
CRF gives a smooth and convex loss function; see Supple-
Figure 1. Room layout and sensor locations.
Table 1. Activities in the sensor data set
ID
Activity
train/test samples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Sleeping
Out of Home (OH)
Using Computer
Relaxing
Eating
Cooking
Showering (Bathing)
No Event
Using Toilet
Hygiene (brushing teeth, etc.)
Dishwashing
Beverage Preparation
Bath Cleaning/Preparation
Others
81442/87433
66096/42478
64640/46254
25290/65312
6403/6016
5249/4633
3930/4977
3443/3486
2464/2625
1614/1617
1495/1831
1388/1441
492/320
6549/2072
Total
-
270495/270495
ment D.2 for more details of CRF.
Our task then is sensor selection on the basis of noisy binary time series data, and to do this we apply our FoBagdt-CRF (FoBa-gdt with CRF objective function). Since it
is very expensive to evaluate the CRF objective value and
its gradient, FoBa-obj becomes impractical in this case (a
large number of optimization problems in the forward step
make it computationally very expensive). Here, we consider a sensor to have been “used” if at least one feature related to it is used in the CRF. Note that we have 14 activitysignal binary features (i.e., indicators of sensor/activity simultaneous activations) for each single sensor, and therefore we have 40 × 14 = 560 such features in total. In
addition, we have 14 × 14 = 196 activity-activity binary
features (i.e., indicators of the activities at times t − 1 and
t). As explained in Section D.2, we only enforced sparsity
on the first type of features.
First we compare FoBa-gdt-CRF with Forward-gdt-CRF
(Forward-gdt with CRF loss function) and L1-CRF4 in
4
L1-CRF solves the optimization problem with CRF loss + L1
regularization. Since it is difficult to search the whole space L1
regularization parameter value space, we investigated a number of
discrete values.
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Table 2. Sensor IDs selected by FoBa-gdt-CRF.
# of sensors=10
# of sensors=15
{1, 4, 5, 9, 10, 13, 19, 28, 34, 38}
{# of sensors=10} + {2, 7, 36, 37, 40}
w.r.t. activity 5 (Eating) significantly decreases from the
case # of sensors=10 to # of sensors=15 by including
sensor 2.
6. Additional Experiments
Figure 2. Top: comparisons of FoBa-gdt-CRF, Forward-gdt-CRF
and L1-CRF. Bottom: test error rates (FoBa-gdt-CRF) for individual activities.
terms of test recognition error over the number of sensors
selected (see the top of Fig. 2). We can observe that
• The performance for all methods improves when the umber of sensors increases.
• FoBa-gdt-CRF and Forward-gdt-CRF achieve comparable performance. However, FoBa-gdt-CRF reduces the
error rate slightly faster, in terms of the number of sensors.
• FoBa-gdt-CRF achieves its best performance with 14-15
sensors while Forward-gdt-CRF needs 17-18 sensors to
achieve the same error level. We obtain sufficient accuracy by using fewer than 40 sensors.
• FoBa-gdt-CRF consistently requires fewer features than
Forward-gdt-CRF to achieve the same error level when
using the same number of sensors.
We also analyze the test error rates of FoBa-gdt-CRF for
individual activities. We consider two cases with the number of sensors being 10 and 15, and entered their test error
rates for each individual activity in the bottom of Fig. 2.
We observe that:
• The high frequency activities (e.g., activities {1,2,3,4})
are well recognized in both cases. In other words, FoBagdt-CRF is likely to select sensors (features) which contribute to the discrimination of high frequency activities.
• The error rates for activities {5, 7, 9} significantly improve when the number of sensors increases from 10 to
15. Activities 7 and 9 are Showering and Using Toilet,
and the use of additional sensors {36, 37, 40} seems to
have contributed to this improvement. Also, a dinner table was located near sensor 2, which is why the error rate
Although this paper mainly focuses on the theoretical analysis for FoBa-obj and FoBa-gdt, we conduct additional experiments to study the behaviors of FoBa-obj and FoBa-gdt
in Supplement (part D). We mainly consider two models:
logistic regression and linear-chain CRFs, and the comparison among four algorithms (FoBa-gdt, FoBa-obj, and
their corresponding forward greedy algorithms) on both
synthetic data and real world data. We empirically show,
in application to logistic regression and conditional random fields (CRFs) (Lafferty et al., 2001), that FoBa-gdt
achieves similar empirical performance as FoBa-obj with a
much lower computational cost. Note that comparisons of
FoBa-obj and L1-regularization methods can be found in
previous studies (Jalali et al., 2011; Zhang, 2011a).
7. Conclusion
This paper considers two forward-backward greedy methods, a state-of-the-art greedy method FoBa-obj and its variant FoBa-gdt which is more efficient than FoBa-obj, for
solving sparse feature selection problems with general convex smooth functions. We systematically analyze the theoretical properties of both algorithms. Our main contributions include: 1) We derive better theoretical bounds
for FoBa-obj and FoBa-gdt than existing analyses regarding FoBa-obj in (Jalali et al., 2011) for general smooth
convex functions. Our result also suggests that the NP
hard problem (1) can be solved by FoBa-obj and FoBagdt if the signal noise ratio is big enough; 2) Our new
bounds are consistent with the bounds of a special case
(least squares) (Zhang, 2011a) and fills a previously existing theoretical gap for general convex smooth functions
(Jalali et al., 2011); 3) We show that the condition to satisfy
the restricted strong convexity condition in commonly used
machine learning problems; 4) We apply FoBa-gdt (with
the conditional random field objective) to the sensor selection problem for human indoor activity recognition and our
results show that FoBa-gdt can successfully remove unnecessary sensors and is able to select more valuable sensors
than other methods (including the ones based on forward
greedy selection and L1-regularization). As for the future
work, we plan to extend FoBa algorithms to minimize a
general convex smooth function over a low rank constraint.
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Supplemental Material: Proofs
This supplemental material provides the proofs for our main results including the properties of FoBa-obj (Theorems 1, and
4) and FoBa-gdt (Theorems 2 and 5), and analysis for the RSCC condition (Theorem 3).
Our proofs for Theorems 1, 2, 4, and 5 borrowed many tools from early literatures including Zhang (2011b), Jalali et al.
(2011), Johnson et al. (2012), Zhang (2009), Zhang (2011a). The key novelty in our proof is to develop a key property for
the backward step (see Lemmas 3 and 7), which gives an upper bound for the number of wrong features included in the
feature pool. By taking a full advantage of this upper bound, the presented proofs for our main results (i.e., Theorems 1,
2, 4, and 5) are significantly simplified. It avoids the complicated induction procedure in the proof for the forward greedy
method (Zhang, 2011b) and also improves the existing analysis for the same problem in Jalali et al. (2011).
A. Proofs of FoBa-obj
First we prove the properties of the FoBa-obj algorithm, particularly Theorems 4 and 1.
Lemma 1 and Lemma 2 build up the dependence between the objective function change δ and the gradient |∇Q(β)j |.
Lemma 3 studies the effect of the backward step. Basically, it gives an upper bound of |F (k) − F̄ |, which meets our
intuition that the backward step is used to optimize the size of the feature pool. Lemma 4 shows that if δ is big enough,
which means that the algorithm terminates early, then Q(β (k) ) cannot be smaller than Q(β̄). Lemma 5 studies the forward
step of FoBa-obj. It shows that the objective decreases sufficiently in every forward step, which together with Lemma 4
(that is, Q(β (k) ) > Q(β̄)), implies that the algorithm will terminate within limited number of iterations. Based on these
results, we provide the complete proofs of Theorems 4 and 1 at the end of this section.
Lemma 1. If Q(β) − minη Q(β + ηej ) ≥ δ, then we have
|∇Q(β)j | ≥
p
2ρ− (1)δ.
Proof. From the condition, we have
−δ ≥ min Q(β + ηej ) − Q(β)
η
ρ− (1) 2
≥ minhηej , ∇Q(β)i +
η (from the definition of ρ− (1))
η
2
2
ρ− (1)
∇Q(β)j
|∇Q(β)j |2
= min
η−
−
η
2
ρ− (1)
2ρ− (1)
|∇Q(β)j |2
=−
.
2ρ− (1)
It indicates that |∇Q(β)j | ≥
p
2ρ− (1)δ.
Lemma 2. If Q(β) − minj,η Q(β + ηej ) ≤ δ, then we have
k∇Q(β)k∞ ≤
p
2ρ+ (1)δ.
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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Proof. From the condition, we have
δ ≥Q(β) − min Q(β + ηej )
η,j
= max Q(β) − Q(β + ηej )
η,j
ρ+ (1) 2
η
η,j
2
2
ρ+ (1)
∇Q(β)j
|∇Q(β)j |2
η−
= max −
−
η,j
2
ρ+ (1)
2ρ+ (1)
2
|∇Q(β)j |
= max
j
2ρ+ (1)
k∇Q(β)k2∞
=
2ρ+ (1)
≥ max −hηej , ∇Q(β)i −
It indicates that k∇Q(β)k∞ ≤
p
2ρ+ (1)δ.
Lemma 3. (General backward criteria). Consider β (k) with the support F (k) in the beginning of each iteration in Algorithm 1 with FoBa-obj (Here, β (k) is not necessarily the output of this algorithm). We have for any β̄ ∈ Rd with the support
F̄
(k)
kβF (k) −F̄ k2 = k(β (k) − β̄)F (k) −F̄ k2 ≥
δ (k)
δ
|F (k) − F̄ | ≥
|F (k) − F̄ |.
ρ+ (1)
ρ+ (1)
(13)
Proof. We have
(k)
|F (k) − F̄ | min Q(β (k) − βj ej ) ≤
j∈F (k)
X
(k)
Q(β (k) − βj ej )
j∈F (k) −F̄
≤
X
(k)
Q(β (k) ) − OQ(β (k) )j βj
j∈F (k) −F̄
≤|F (k) − F̄ |Q(β (k) ) +
+
ρ+ (1) (k) 2
(βj )
2
ρ+ (1)
k(β (k) − β̄)F (k) −F̄ k2 .
2
The second inequality is due to OQ(β (k) )j = 0 for any j ∈ F (k) − F̄ and the third inequality is from β̄F (k) −F̄ = 0. It
follows that
δ (k)
ρ+ (1)
(k)
k(β (k) − β̄)F (k) −F̄ k2 ≥ |F (k) − F̄ |( min Q(β (k) − βj ej ) − Q(β (k) )) ≥ |F (k) − F̄ |
,
(k)
2
2
j∈F
which implies that the claim using δ (k) ≥ δ.
Lemma 4. Let β̄ = arg minsupp(β)∈F̄ Q(β). Consider β (k) in the beginning of each iteration in Algorithm with FoBa-obj.
Denote its support as F (k) . Let s be any integer larger than |F (k) − F̄ |. If takes δ >
we have Q(β
(k)
) ≥ Q(β̄).
4ρ+ (1)
2
ρ− (s)2 kOQ(β̄)k∞
in FoBa-obj, then
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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Proof. We have
Q(β (k) ) − Q(β̄)
ρ− (s) (k)
kβ − β̄k2
2
ρ− (s) (k)
≥hOQ(β̄)F (k) −F̄ , (β (k) − β̄)F (k) −F̄ i +
kβ − β̄k2
2
ρ− (s) (k)
≥ − kOQ(β̄)k∞ k(β (k) − β̄)F (k) −F̄ k1 +
kβ − β̄k2
2
q
ρ− (s) (k)
≥ − kOQ(β̄)k∞ |F (k) − F̄ |kβ (k) − β̄k +
kβ − β̄k2
2
r
ρ+ (1) ρ− (s) (k)
(k)
2
≥ − kOQ(β̄)k∞ kβ − β̄k
+
kβ − β̄k2 (from Lemma 3)
δ!
2
p
ρ− (s) kOQ(β̄)k∞ ρ+ (1)
√
kβ (k) − β̄k2
=
−
2
δ
≥hOQ(β̄), β (k) − β̄i +
>0. (from δ >
4ρ+ (1)
kOQ(β̄)k2∞ )
ρ− (s)2
It proves the claim.
Proof of Theorem 4
Proof. From Lemma 4, we only need to consider the case Q(β (k) ) ≥ Q(β̄). We have
0 ≥ Q(β̄) − Q(β (k) ) ≥ hOQ(β (k) ), β̄ − β (k) i +
ρ− (s)
kβ̄ − β (k) k2 ,
2
which implies that
ρ− (s)
kβ̄ − β (k) k2 ≤ − hOQ(β (k) ), β̄ − β (k) i
2
= − hOQ(β (k) )F̄ −F (k) , (β̄ − β (k) )F̄ −F (k) i
≤kOQ(β (k) )F̄ −F (k) k∞ k(β̄ − β (k) )F̄ −F (k) k1
q
p
≤ 2ρ+ (1)δ |F̄ − F (k) ||β̄ − β (k) k. (from Lemma 2)
We obtain
kβ̄ − β
(k)
p
q
2 2ρ+ (1)δ
k≤
|F̄ − F (k) |.
ρ− (s)
It follows that
8ρ+ (1)δ
|F̄ − F (k) | ≥kβ̄ − β (k) k2
ρ− (s)2
≥k(β̄ − β (k) )F̄ −F (k) k2
=kβ̄F̄ −F (k) k2
≥γ 2 |{j ∈ F̄ − F (k) : |β̄j | ≥ γ}|
=
16ρ+ (1)δ
|{j ∈ F̄ − F (k) : |β̄j | ≥ γ}|,
ρ− (s)2
which implies that
|F̄ − F (k) | ≥ 2|{j ∈ F̄ − F (k) : |β̄j | ≥ γ}| = 2(|F̄ − F (k) | − |{j ∈ F̄ − F (k) : |β̄j | < γ}|)
⇒|F̄ − F (k) | ≤ 2|{j ∈ F̄ − F (k) : |β̄j | < γ}|.
(14)
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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The first inequality is obtained from Eq. (14)
kβ̄ − β
(k)
p
p
q
q
2ρ+ (1)δ
4 ρ+ (1)δ
(k)
k≤
|F̄ − F | ≤
|j ∈ F̄ − F (k) : |β̄j | < γ|.
ρ− (s)
ρ− (s)
2
Next, we consider
Q(β̄) − Q(β (k) )
ρ− (s)
kβ̄ − β (k) k2
2
ρ− (s)
=hOQ(β (k) )F̄ −F (k) , (β̄ − β (k) )F̄ −F (k) i +
kβ̄ − β (k) k2 (from OQ(β (k) )F (k) = 0)
2
ρ− (s)
≥ − kOQ(β (k) )k∞ kβ̄ − β (k) k1 +
kβ̄ − β (k) k2
2
q
p
ρ− (s)
≥ − 2ρ+ (1)δ |F̄ − F (k) |kβ̄ − β (k) k +
kβ̄ − β (k) k2
2
2
q
p
ρ− (s)
(k)
(k)
≥
kβ̄ − β k − 2ρ+ (1)δ |F̄ − F |/ρ− (s) − 2ρ+ (1)δ|F̄ − F (k) |/(2ρ− (s))
2
≥hOQ(β (k) ), β̄ − β (k) i +
≥ − 2ρ+ (1)δ|F̄ − F (k) |/(2ρ− (s))
≥−
2ρ+ (1)δ
|{j ∈ F̄ − F (k) : |β̄j | < γ}|.
ρ− (s)
It proves the second inequality. The last inequality in this theorem is obtained from the following:
2ρ+ (1)δ (k)
|F − F̄ |
2ρ+ (1)2
≤k(β (k) − β̄)F (k) −F̄ k2
≤kβ
(k)
(from Lemma 3)
2
− β̄k
8ρ+ (1)δ
|F̄ − F (k) |
ρ− (s)2
16ρ+ (1)δ
|{j ∈ F̄ − F (k) : |β̄j | < γ}|.
≤
ρ− (s)2
≤
It completes the proof.
Lemma 5. (General forward step of FoBa-obj) Let β = arg minsupp(β)⊂F Q(β). For any β 0 with support F 0 and i ∈
{i : Q(β) − minη Q(β + ηei ) ≥ (Q(β) − minj,η Q(β + ηej ))}, we have
|F 0 − F |(Q(β) − min Q(β + ηei )) ≥
η
ρ− (s)
(Q(β) − Q(β 0 )).
ρ+ (1)
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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Proof. We have that the following holds with any η
|F 0 − F | min
Q(β + η(βj0 − βj )ej )
j∈F 0 −F,η
X
≤
Q(β + η(βj0 − βj )ej )
j∈F 0 −F
X
≤
Q(β) + η(βj0 − βj )∇Q(β)j +
j∈F 0 −F
ρ+ (1)
(βj − βj0 )2 η 2
2
≤|F 0 − F |Q(β) + ηh(β 0 − β)F 0 −F , ∇Q(β)F 0 −F i +
ρ+ (1) 0
kβ − βk2 η 2
2
ρ+ (1) 0
kβ − βk2 η 2
2
ρ− (s)
ρ+ (1) 0
≤|F 0 − F |Q(β) + η(Q(β 0 ) − Q(β) −
kβ − β 0 k2 ) +
kβ − βk2 η 2 .
2
2
=|F 0 − F |Q(β) + ηhβ 0 − β, ∇Q(β)i +
By minimizing η, we obtain that
|F 0 − F |( min
0
j∈F −F,η
Q(β + η(βj0 − βj )ej ) − Q(β))
≤ min η(Q(β 0 ) − Q(β) −
η
≤−
ρ+ (1) 0
ρ− (s)
kβ − β 0 k2 ) +
kβ − βk2 η 2
2
2
(Q(β 0 ) − Q(β) − ρ−2(s) kβ − β 0 k2 )2
2ρ+ (1)kβ 0 − βk2
4(Q(β) − Q(β 0 )) ρ−2(s) kβ − β 0 k2
2ρ+ (1)kβ 0 − βk2
ρ− (s)
=
(Q(β 0 ) − Q(β)).
ρ+ (1)
≤−
(15)
(from (a + b)2 ≥ 4ab)
It follows that
|F 0 − F |(Q(β) − min Q(β + ηei ))
η,j
0
≥|F − F |(Q(β) −
≥|F 0 − F |(Q(β) −
≥
min
Q(β + ηej ))
min
Q(β + η(βj0 − βj )ej ))
j∈F 0 −F,η
j∈F 0 −F,η
ρ− (s)
(Q(β) − Q(β 0 )),
ρ+ (1)
(from Eq. (15))
which proves the claim.
Proof of Theorem 1
Proof. Assume that the algorithm terminates at some number larger than s − k̄. Then we consider the first time k = s − k̄.
Denote the support of β (k) as F (k) . Let F 0 = F̄ ∪ F (k) and β 0 = arg minsupp(β)⊂F 0 Q(β). One can easily verify that
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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|F̄ ∪ F (k) | ≤ s. We consider the very recent (k − 1) step and have
δ (k) =Q(β (k−1) ) − min Q(β (k−1) + ηei )
η,i
ρ− (s)
(Q(β (k−1) ) − Q(β 0 ))
ρ+ (1)|F 0 − F (k−1) |
ρ− (s)
≥
(Q(β (k) ) − Q(β 0 ))
ρ+ (1)|F 0 − F (k−1) |
ρ− (s)
ρ− (s) (k)
0 2
0
(k)
0
kβ
−
β
k
≥
hOQ(β
),
β
−
β
i
+
2
ρ+ (1)|F 0 − F (k−1) |
2
ρ− (s)
≥
kβ (k) − β 0 k2 (from OQ(β 0 )F 0 = 0)
2ρ+ (1)|F 0 − F (k−1) |
≥
(16)
where the first inequality is due to Lemma 5 and the second inequality is due to Q(β (k) ) ≤ Q(β (k−1) ). From Lemma 3,
we have
2ρ+ (1)
2ρ+ (1)
kβ (k) − β̄k2 = 0
kβ (k) − β̄k2
(17)
δ (k) ≤ (k)
|F − F̄ |
|F − F̄ |
Combining Eq. (16) and (17), we obtain that
k(β (k) − β̄)k2 ≥
ρ− (s)
2ρ+ (1)
2
|F 0 − F̄ |
kβ (k) − β 0 k2 ,
− F (k−1) |
|F 0
which implies that
kβ (k) − β̄k ≥ tkβ (k) − β 0 k,
where
ρ− (s)
t :=
2ρ+ (1)
s
ρ− (s)
=
2ρ+ (1)
s
|F (k) − F (k) ∩ F̄ |
|F 0 − F (k) | + 1
ρ− (s)
=
2ρ+ (1)
s
|F (k) − F (k) ∩ F̄ |
|F̄ − F (k) ∩ F̄ | + 1
ρ− (s)
=
2ρ+ (1)
s
|F (k) | − |F (k) ∩ F̄ |
|F̄ | − |F (k) ∩ F̄ | + 1
ρ− (s)
=
2ρ+ (1)
s
k − |F (k) ∩ F̄ |
k̄ − |F (k) ∩ F̄ | + 1
|F 0 − F̄ |
− F (k−1) |
|F 0
s
ρ− (s)
(s − k̄)
≥
(from k = s − k̄ and s ≥ 2k̄ + 1)
2ρ+ (1) (k̄ + 1)
s
ρ+ (s)
≥
+ 1 (from the assumption on s).
ρ− (s)
It follows
tkβ (k) − β 0 k ≤ kβ (k) − β̄k ≤ kβ (k) − β 0 k + kβ 0 − β̄k
(from Eq. (3))
which implies that
(t − 1)kβ (k) − β 0 k ≤ kβ 0 − β̄k.
(18)
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Next we have
Q(β (k) ) − Q(β̄) =Q(β (k) ) − Q(β 0 ) + Q(β 0 ) − Q(β̄)
ρ+ (s) (k)
ρ− (s) 0
kβ − β 0 k2 −
kβ − β̄k2
2
2
≤(ρ+ (s) − ρ− (s)(t − 1)2 )/2kβ (k) − β 0 k2
≤
≤0.
Since the sequence Q(β (k) ) is strictly decreasing, we know that Q(β (k+1) ) < Q(β (k) ) ≤ Q(β̄). However, from Lemma 4
we also have Q(β̄) ≤ Q(β (k+1) ). Thus it leads to a contradiction. This indicates that the algorithm terminates at some
integer not greater than s − k̄.
B. Proofs of FoBa-gdt
Next we consider Algorithm 1 with FoBa-gdt; specifically we will prove Theorem 5 and Theorem 2. Our proof strategy is
similar to FoBa-obj.
Lemma 6. If |OQ(β)j | ≥ , then we have
Q(β) − min Q(β + ηej ) ≥
η
2
.
2ρ+ (1)
Proof. Consider the LHS:
Q(β) − min Q(β + ηej )
η
= max Q(β) − Q(β + ηej )
η
ρ+ (1) 2
η
η
2
ρ+ (1) 2
= max −ηOQ(β)j −
η
η
2
|OQ(β)j |2
≥
2ρ+ (1)
2
≥
.
2ρ+ (1)
≥ max −hOQ(β), ηej i −
It completes the proof.
This lemma implies that δ (k0 ) ≥
2
2ρ+ (1)
for all k0 = 1, · · · , k, and
Q(β (k0 −1) ) − min Q(β (k0 −1) + ηej ) ≥ δ (k0 ) ≥
η
2
,
2ρ+ (1)
if |OQ(β (k0 −1) )j | ≥ .
Lemma 7. (General backward criteria). Consider β (k) with the support F (k) in the beginning of each iteration in Algorithm 1 with FoBa-obj. We have for any β̄ ∈ Rd with the support F̄
(k)
kβF (k) −F̄ k2 = k(β (k) − β̄)F (k) −F̄ k2 ≥
δ (k)
2
|F (k) − F̄ | ≥
|F (k) − F̄ |.
ρ+ (1)
2ρ+ (1)2
(19)
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Proof. We have
(k)
|F (k) − F̄ | min Q(β (k) − βj ej ) ≤
j∈F (k)
X
(k)
Q(β (k) − βj ej )
j∈F (k) −F̄
≤
X
(k)
Q(β (k) ) − OQ(β (k) )j βj
j∈F (k) −F̄
≤|F (k) − F̄ |Q(β (k) ) +
+
ρ+ (1) (k) 2
(βj )
2
ρ+ (1)
k(β (k) − β̄)F (k) −F̄ k2 .
2
The second inequality is due to OQ(β (k) )j = 0 for any j ∈ F (k) − F̄ and the third inequality is from β̄F (k) −F̄ = 0. It
follows that
δ (k)
ρ+ (1)
(k)
k(β (k) − β̄)F (k) −F̄ k2 ≥ |F (k) − F̄ |( min Q(β (k) − βj ej ) − Q(β (k) )) ≥ |F (k) − F̄ |
,
2
2
j∈F (k)
which implies the claim using δ (k) ≥
2
2ρ+ (1) .
Lemma 8. Let β̄ = arg minsupp(β)∈F̄ Q(β). Consider β (k) in the beginning of each iteration in Algorithm 1 with FoBaobj. Denote its support as F (k) . Let s be any integer larger than |F (k) − F̄ |. If take >
then we have Q(β
(k)
√
2 2ρ+ (1)
ρ− (s) kOQ(β̄)k∞
) ≥ Q(β̄).
Proof. We have
0 >Q(β (k) ) − Q(β̄)
ρ− (s) (k)
kβ − β̄k2
2
ρ− (s) (k)
≥hOQ(β̄)F (k) −F̄ , (β (k) − β̄)F (k) −F̄ i +
kβ − β̄k2
2
ρ− (s) (k)
≥ − kOQ(β̄)k∞ k(β (k) − β̄)F (k) −F̄ k1 +
kβ − β̄k2
2
q
ρ− (s) (k)
kβ − β̄k2
≥ − kOQ(β̄)k∞ |F (k) − F̄ |kβ (k) − β̄k +
2
√
2ρ+ (1) ρ− (s) (k)
≥ − kOQ(β̄)k∞ kβ (k) − β̄k2
+
kβ − β̄k2 (from Lemma 7)
!
2
√
ρ− (s) kOQ(β̄)k∞ 2ρ+ (1)
=
−
kβ (k) − β̄k2
2
√
2 2ρ+ (1)
>0. (from >
kOQ(β̄)k∞ )
ρ− (s)
≥hOQ(β̄), β (k) − β̄i +
It proves our claim.
Proof of Theorem 5
Proof. From Lemma 8, we only need to consider the case Q(β (k) ) ≥ Q(β̄). We have
0 ≥ Q(β̄) − Q(β (k) ) ≥ hOQ(β (k) ), β̄ − β (k) i +
ρ− (s)
kβ̄ − β (k) k2 ,
2
in FoBa-obj,
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which implies that
ρ− (s)
kβ̄ − β (k) k2 ≤ − hOQ(β (k) ), β̄ − β (k) i
2
= − hOQ(β (k) )F̄ −F (k) , (β̄ − β (k) )F̄ −F (k) i
≤kOQ(β (k) )F̄ −F (k) k∞ k(β̄ − β (k) )F̄ −F (k) k1
q
≤ |F̄ − F (k) ||β̄ − β (k) k.
We obtain
kβ̄ − β
(k)
2
k≤
ρ− (s)
q
|F̄ − F (k) |.
It follows that
42
|F̄ − F (k) | ≥kβ̄ − β (k) k2
ρ− (s)2
≥k(β̄ − β (k) )F̄ −F (k) k2
=kβ̄F̄ −F (k) k2
≥γ 2 |{j ∈ F̄ − F (k) : |β̄j | ≥ γ}|
=
82
|{j ∈ F̄ − F (k) : |β̄j | ≥ γ}|,
ρ− (s)2
which implies that
|F̄ − F (k) | ≥ 2|{j ∈ F̄ − F (k) : |β̄j | ≥ γ}| = 2(|F̄ − F (k) | − |{j ∈ F̄ − F (k) : |β̄j | < γ}|)
⇒|F̄ − F (k) | ≤ 2|{j ∈ F̄ − F (k) : |β̄j | < γ}|.
The first inequality is obtained from Eq. (20)
kβ̄ − β
(k)
2
k≤
ρ− (s)
q
|F̄ −
F (k) |
√ q
2 2
≤
|j ∈ F̄ − F (k) : |β̄j | < γ|.
ρ− (s)
Next, we consider
Q(β̄) − Q(β (k) )
ρ− (s)
kβ̄ − β (k) k2
2
ρ− (s)
=hOQ(β (k) )F̄ −F (k) , (β̄ − β (k) )F̄ −F (k) i +
kβ̄ − β (k) k2 (from OQ(β (k) )F (k) = 0)
2
ρ− (s)
≥ − kOQ(β (k) )k∞ kβ̄ − β (k) k1 +
kβ̄ − β (k) k2
2
q
ρ− (s)
≥ − |F̄ − F (k) |kβ̄ − β (k) k +
kβ̄ − β (k) k2
2
2
q
ρ− (s)
(k)
(k)
≥
kβ̄ − β k − |F̄ − F |/ρ− (s) − 2 |F̄ − F (k) |/(2ρ− (s))
2
≥hOQ(β (k) ), β̄ − β (k) i +
≥ − 2 |F̄ − F (k) |/(2ρ− (s))
≥−
2
|{j ∈ F̄ − F (k) : |β̄j | < γ}|.
ρ− (s)
(20)
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It proves the second inequality. The last inequality in this theorem is obtained from
2
|F (k) − F̄ |
2ρ+ (1)2
≤k(β (k) − β̄)F (k) −F̄ k2
≤kβ
(k)
(from Lemma 7)
2
− β̄k
42
|F̄ − F (k) |
≤
ρ− (s)2
82
≤
|{j ∈ F̄ − F (k) : |β̄j | < γ}|.
ρ− (s)2
It completes the proof.
Next we will study the upper bound of “k” when Algorithm 1 terminates.
Lemma 9. (General forward step) Let β = arg minsupp(β)⊂F Q(β). For any β 0 with support F 0 and i ∈ {i : |OQ(β)i | ≥
maxj |OQ(β)j |}, we have
|F 0 − F |(Q(β) − min Q(β + ηei )) ≥
η
ρ− (s)
(Q(β) − Q(β 0 )).
ρ+ (1)
Proof. The following proof follows the idea of Lemma A.3 in (Zhang, 2011b). Denote i∗ as arg maxi |OQ(β)i |. For all
j ∈ {1, · · · , d}, we define
ρ+ (1) 2
Pj (η) = ηsgn(βj0 )OQ(β)j +
η .
2
It is easy to verify that minη Pj (η) = −
|OQ(β)j |2
2ρ+ (1)
and
min Pi (η) ≤ min Pi∗ (η) = min Pj (η) ≤ min Pj (η).
η
η
j,η
j
Denoting u = kβF0 0 −F k1 , we obtain that
X
u min Pj (η) ≤
j
|βj0 |Pj (η)
j∈F 0 −F
=η
X
βj0 OQ(β)j +
j∈F 0 −F
uρ+ (1) 2
η
2
uρ+ (1) 2
η
2
uρ+ (1) 2
=ηhβ 0 − β, OQ(β)i +
η (from OQ(β)F = 0)
2
ρ− (s) 0
uρ+ (1) 2
≤η(Q(β 0 ) − Q(β) −
kβ − βk2 ) +
η .
2
2
=ηhβF0 0 −F , OQ(β)F 0 −F i +
Taking η = (Q(β) − Q(β 0 ) +
ρ− (s)
0
2 kβ
− βk2 )/uρ+ (1) on both sides, we get
2
Q(β) − Q(β 0 ) + ρ−2(s) kβ − β 0 k2
(Q(β) − Q(β 0 ))ρ− (s)kβ − β 0 k2
u min Pj (η) ≤ −
≤−
,
j
2ρ+ (1)u
ρ+ (1)u
where it uses the inequality (a + b)2 ≥ 4ab. From Eq. (21), we have
min Pi (η) ≤ min Pj (η)
η
j
(Q(β) − Q(β 0 ))ρ− (s)kβ − β 0 k2
ρ+ (1)u2
ρ− (s)
≤−
(Q(β) − Q(β 0 )),
ρ+ (1)|F 0 − F |
≤−
(21)
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where we use
kβ−β 0 k2
u2
kβ−β 0 k2
k(β−β 0 )F 0 −F k2
=
≥
1
|F 0 −F | .
Finally, we use
min Q(β + ηei ) − Q(β) ≤ min Q(β + ηsgn(βi0 )ei ) − Q(β) ≤ min Pi (η)
η
η
η
to prove the claim.
Proof of Theorem 2
Proof. Please refer to the proof of Theorem 1.
C. Proofs of RSCC
Proof of Theorem 3
Proof. First from the random matrix theory (Vershynin, 2011), we have that for any random matrix X ∈ Rn×d with
independent sub-gaussian isotropic random rows or columns, there exists a constant C0 such that
√
n − C0
p
p
kXhk √
kXhk
≤ max
≤ n + C0 s log d
khk0 ≤s khk
khk0 ≤s khk
s log d ≤ min
(22)
holds with high probability.
From the mean value theorem, for any β 0 , β ∈ Ds , there exists a point θ in the segment of β 0 and β satisfying
Q(β 0 ) − Q(β) − h∇Q(β), α − βi =
=
1 0
(β − β)T
2
n
1 0
(β − β)T ∇2 Q(θ)(β 0 − β)
2
!
1X T 2
X ∇ li (Xi. θ, yi )Xi. + ∇2 R(θ) (β 0 − β)
n i=1 i. 1
(from θ ∈ Ds )
!
n
1X − T
−
λ Xi. Xi. + λR I (β 0 − β)
n i=1
−
λ
1
= (β 0 − β)T
X T X + λ−
I
(β 0 − β)
R
2
n
1
≥ (β 0 − β)T
2
=
λ−
λ−
kX(β 0 − β)k2 + R kβ 0 − βk2
2n
2
(23)
Similarly, one can easily obtain
Q(β 0 ) − Q(β) − h∇Q(β), α − βi ≤
λ+
λ+
kX(β 0 − β)k2 + R kβ 0 − βk2
2n
2
Using (22), we have
p
p
√
√
( n − C0 s log d)2 kβ 0 − βk2 ≤ kX(β 0 − β)k2 ≤ ( n + C0 s log d)2 kβ 0 − βk2
Together with (23) and (24), we obtain

!2 
r
1 −
s log d  0
λR + λ− 1 − C0
kβ − βk2 ≤ Q(β 0 ) − Q(β) − h∇Q(β), α − βi
2
n

!2 
r
1
s log d  0
+
≤ λ+
1 + C0
kβ − βk2 ,
R +λ
2
n
(24)
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which implies the claims (5a), (5a), and (5c) by taking n ≥ Cs log d where C = C02 /(1 −
√
2/2)2 .
Next we apply this result to prove the rest part. In Algorithm 1, since choose n ≥ Cs log d, we have that ρ− (s), ρ+ (s),
and κ(s) in Algorithm 1 with FoBa-obj satisfy the bounds in (5). To see why s chosen in (6) satisfies the condition in (3),
we verify the right hand side of (3):
" s
RHS = (k̄ + 1)
ρ+ (s)
+1
ρ− (s)
!
2ρ+ (1)
ρ− (s)
#2
√
≤ 4(k̄ + 1)( κ + 1)2 κ2 ≤ s − k̄.
√
Therefore, Algorithm 1 terminates at most 4κ2 ( κ + 1)2 (k̄ + 1) iterations with high probability from Theorem 1. The
claims for Algorithm 1 with FoBa-gdt can be proven similarly.
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Supplemental Material: Additional Experiments
D. Additional Experiments
We conduct additional experiments using two models: logistic regression and linear-chain CRFs. Four algorithms (FoBagdt, FoBa-obj, and their corresponding forward greedy algorithms, i.e., Forward-obj and Forward-gdt) are compared on
both synthetic data and real world data. Note that comparisons of FoBa-obj and L1-regularization methods can be found
in previous studies (Jalali et al., 2011; Zhang, 2011a).
It is hard to evaluate ρ− (s) and ρ+ (s) in practice, and we typically employ the following strategies to set δ in FoBa-obj
and FoBa-gdt:
• If we have the knowledge of F̄ (e.g., in synthetic simulations), we can compute the true solution β̄. δ and are given
as δ = Q(β̄) − minα,j ∈/ F̄ Q(β̄ + αej ) and = kOQ(β̄)k∞ .
• We can change the stopping criteria in both FoBa-obj and FoBa-gdt based on the sparsity of β (k) , that is, when the
number of nonzero entries in β (k) exceeds a given sparsity level, the algorithm terminates.
We employed the first strategy only in synthetic simulations for logistic regression and the second strategy in all other
cases.
D.1. Experiments on Logistic Regression
L2 -regularized logistic regression is a typical example of models with smooth convex loss functions:
Q(β) =
1
1X
log(1 + exp(−yi Xi. β)) + λ||β||2 .
n i
2
where Xi. ∈ Rd , that is, the ith row of X, is the ith sample and yi ∈ {1, −1} is the corresponding label. This is a binary
classification problem for finding a hyperplane defined by β in order to separate two classes of data in X. log(1 + exp(.))
defines the penalty for misclassification. The `2 -norm is used for regularization in order to mitigate overfitting. The
gradient of Q(β) is computed by
∂Q(β)
1 X −yi exp(−yi Xi. β) T
=
X + λβ.
∂β
n i 1 + exp(−yi Xi. β) i.
D.1.1. S YNTHETIC DATA
We first compare FoBa-obj and FoBa-gdt with Forward-obj and Forward-gdt5 using synthetic data generated as follows.
The data matrix X ∈ Rn×d consistes of two classes of data (each row is a sample) from i.i.d. Gaussian distribution
N (β ∗ , Id×d ) and N (−β ∗ , Id×d ), respectively, where the true parameter β ∗ is sparse and its nonzero entries follow i.i.d.
uniform distribution U[0, 1]. The two classes are of the same size. The norm of β ∗ is normalized to 5 and its sparseness
ranges from 5 to 14. The vector y is the class label vector with values being 1 or -1. We set λ = 0.01, n = 100, and
d = 500. The results are averaged over 50 random trials.
Four evaluation measures are considered in our comparison: F-measure (2|F (k) ∩ F̄ |/(|F (k) | + |F̄ |)), estimation error
(kβ (k) − β̄k/kβ̄k), objective value, and the number of nonzero elements in β (k) (the value of k when the algorithm
terminates). The results in Fig. 3 suggest that:
• The FoBa algorithms outperform the forward algorithms overall, especially in F-measure (feature selection) performance;
• The FoBa algorithms are likely to yield sparser solutions than the forward algorithms, since they allow for the removal
of bad features;
5
Forward-gdt is equivalent to the orthogonal matching pursuit in (Zhang, 2009).
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
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FoBa−obj
forward−obj
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objective
estimation error
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# of nonzero entries
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F−measure
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forward−gdt
FoBa−obj
forward−obj
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5
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forward−gdt
FoBa−obj
forward−obj
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forward−gdt
FoBa−obj
forward−obj
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sparsity
5
6
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8
9
10
11
12
13
14
sparsity
Figure 3. Comparison for F-measure, estimation error, objective value, and the number of nonzero elements. The horizontal axis is the
sparsity number of the true model ranging from 5 to 14.
Table 3. Computation time (seconds) on synthetic data (top: logistic regression; bottom: CRF).
S
FoBa-gdt
Forward-gdt
FoBa-obj
Forward-obj
5
8
11
14
1.96e-2
1.16e-2
1.53e-2
1.51e-2
0.65e-2
0.76e-2
1.05e-2
1.00e-2
1.25e+1
1.52e+1
1.83e+1
2.29e+1
1.24e+1
1.52e+1
1.80e+1
2.25e+1
10
15
20
25
30
1.02e+0
1.85e+0
2.51e+0
3.86e+0
5.13e+0
0.52e+0
1.05e+0
1.53e+0
2.00e+0
3.00e+0
3.95e+1
5.53e+1
6.89e+1
8.06e+1
9.04e+1
1.88e+1
2.65e+1
3.33e+1
3.91e+1
4.39e+1
• There are minor differences in estimation errors and objective values between the FoBa algorithms and the forward
algorithms;
• FoBa-obj and FoBa-gdt perform similarly. The former gives better objective values and the latter is better in estimation
error;
• FoBa-obj and Forward-obj decrease objective values more quickly than FoBa-gdt and Forward-gdt since they employ
the direct “goodness” measure (reduction of the objective value).
As shown in Table 3, FoBa-gdt and Forward-gdt are much more efficient than FoBa-obj and Forward-obj, because FoBa-obj
and Forward-obj solve a large number of optimization problems in each iteration.
D.1.2. UCI DATA
We next compare FoBa-obj and FoBa-gdt on UCI data sets. We use λ = 10−4 in all experiments. Note that the sparsity
numbers here are different from those in the past experiment. Here “S” denotes the number of nonzero entries in the output.
We use three evaluation measures: 1) training classification error, 2) test classification error, and 3) training objective value
(i.e., the value of Q(β (k) )). The datasets are available online6 . From Table 4, we observe that
• FoBa-obj and FoBa-gdt obtain similar training errors in all cases;
• FoBa-obj usually obtain a lower objective value than FoBa-gdt, but this does not imply that FoBa-obj selects better
features. Indeed, FoBa-gdt achieves lower test errors in several cases.
We also compare the computation time FoBa-obj and FoBa-gdt and found that FoBa-gdt was empirically at least 10 times
faster than FoBa-obj (detailed results omitted due to space limitations). In summary, FoBa-gdt has the same theoretical
6
http://www.csie.ntu.edu.tw/˜cjlin/libsvmtools/datasets
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Table 4. Comparison of different algorithms in terms of training error, testing error, and objective function value (from left to right) on
a1a, a2a, and w1a data sets (from top to bottom). The values of n denotes the number of training/testing samples.
S
objective
training error
testing error
a1a (FoBa-obj/FoBa-gdt), n = 1605/30956, d = 123
10
20
30
40
50
60
70
556/558
517/529
499/515
490/509
484/494
481/484
480/480
0.169/0.17
0.163/0.16
0.151/0.145
0.150/0.146
0.146/0.141
0.14/0.141
0.138/0.139
0.165/0.164
0.163/0.160
0.162/0.158
0.162/0.159
0.162/0.161
0.166/0.161
0.166/0.163
a2a (FoBa-obj/FoBa-gdt), n = 2205/30296, d = 123
10
20
30
40
50
60
70
870/836
801/810
775/790
758/776
749/764
743/750
740/742
0.191/0.185
0.18/0.177
0.172/0.168
0.162/0.167
0.163/0.166
0.162/0.161
0.162/0.158
0.174/0.160
0.163/0.157
0.165/0.156
0.163/0.155
0.162/0.157
0.163/0.160
0.163/0.161
w1a (FoBa-obj/FoBa-gdt), n = 2477/47272, d = 300
10
20
30
40
50
60
70
574/595
424/487
341/395
288/337
238/282
215/226
198/206
0.135/0.125
0.0969/0.0959
0.0813/0.0805
0.0704/0.0715
0.06/0.0658
0.0547/0.0553
0.0511/0.0511
0.142/0.127
0.11/0.104
0.0993/0.0923
0.089/0.0853
0.0832/0.0825
0.0814/0.0818
0.0776/0.0775
guarantee as FoBa-obj, and empirically FoBa-dgt achieves competitive in terms of empirical performance; in terms of
computational time, FoBa-gdt is much more efficient.
D.2. Simulations on Linear-Chain CRFs
Another typical class of model family with smooth convex loss functions is conditional random fields (CRFs) (Lafferty
et al., 2001; Sutton & McCallum, 2006), which is most commonly used in segmentation and tagging problems with respect
to sequential data. Let Xt ∈ RD , t = 1, 2, · · · , T be a sequence of random vectors. The corresponding labels of Xt ’s are
stored in a vector y ∈ RT . Let β ∈ RM be a parameter vector and fm (y, y 0 , xt ), m = 1, · · · , M , be a set of real-valued
feature functions. A linear-chain CRFs will then be a distribution Pβ (y|X) that takes the form
T
1 Y
Pβ (y|X) =
exp
Z(X) t=1
(
M
X
)
βm fm (yt , yt−1 , Xt )
m=1
where Z(X) is an instance-specific normalization function
Z(X) =
T
XY
y t=1
(
exp
M
X
)
βm fm (yt , yt−1 , Xt ) .
(25)
m=1
In order to simplify the introduction below, we assume that all Xt ’s are discrete random vectors with limited states. If we
have training data X i and y i (i = 1, · · · , I) we can estimate the value of the parameter β by maximizing the likelihood
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Forward-Backward Greedy Algorithms for General Convex Smooth Functions over A Cardinality Constraint
Q
i
Pβ (y i |X i ), which is equivalent to minimizing the negative log-likelihood:
min : Q(β) := − log
β
I
Y
!
i
i
Pβ (y |X )
i=1
=−
I X
T X
M
X
i
βm fm (yti , yt−1
, xit )
+
I
X
i=1 t=1 m=1
log Z(X i ).
i=1
The gradient is computed by
XX
X 1 ∂Z(X i )
∂Q(β)
i
=−
fm (yti , yt−1
, Xti ) +
∂βm
Z(X i ) ∂βm
t
i
i
Note that it would be extremely expensive (the complexity would be O(2T )) to directly calculate Z(X i ) from (25), since
i
)
the sample space of y is too large. So is ∂Z(X
∂βm . The sum-product algorithm can reduce complexity to polynomial time in
terms of T (see (Sutton & McCallum, 2006) for details).
In this experiment, we use a data generator provided in Matlab CRF7 . We omit a detailed description of the data generator
here because of space limitations. We generate a data matrix X ∈ {1, 2, 3, 4, 5}T ×D with T = 800 and D = 4 and the
corresponding label vector y ∈ {1, 2, 3, 4}T (four classes)8 . Because of our target application, i.e., sensor selection, we
only enforced sparseness to the features related to observation (56 in all) and the features w.r.t. transitions remained dense.
Fig. 4 shows the performance, averaged over 20 runs, with respect to training error, test error, and objective value. We
observe from Fig 4:
• FoBa-obj and Forward-obj gives lower objective values, training error, and test error than FoBa-gdt and Forward-gdt
when the sparsity level is low.
• The performance gap between different algorithms becomes smaller when the sparsity level increases.
• The performance of FoBa-obj is quite close to that of Forward-obj, while FoBa-gdt outperforms Forward-gdt particularly when the sparsity level was low.
The bottom of Table 3 shows computation time for all four algorithms. Similar to the logistic regression case, the gradient
based algorithms (FoBa-gdt and Forward-gdt) are much more efficient than the objective based algorithms FoBa-obj and
Forward-obj. From Table 3, FoBa-obj is computationally more expensive than Forward-obj, while they are comparable in
the logistic regression case. The main reason for this is that the evaluation of objective values in the CRF case is more
expensive than that in the logistic regression case. These results demonstrate the potential of FoBa-gdt.
0.32
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FoBa−gdt
forward−gdt
FoBa−obj
forward−obj
0.35
0.3
testing error
0.28
0.26
0.24
0.22
0.25
0.2
0.2
0.18
FoBa−gdt
forward−gdt
FoBa−obj
forward−obj
650
objective value on training data
FoBa−gdt
forward−gdt
FoBa−obj
forward−obj
0.34
training error
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30
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Figure 4. Comparison of different algorithms in terms of training error and test error on synthetic data for CRF. The horizontal axis is
the sparsity number of the true model ranging from 10 to 30.
7
8
http://www.di.ens.fr/˜mschmidt/Software/crfChain.html
We here choose a small sample size because of the high computational costs of FoBa-obj and Forward-obj.
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