Jiaqian Yu
email: jiaqian.yu@centralesupelec.fr

Matthew B Blaschko
email: matthew.blaschko@esat.kuleuven.be

A Convex Surrogate Operator for General Non-Modular Loss Functions

Empirical risk minimization frequently employs convex surrogates to underlying discrete loss functions in order to achieve computational tractability during optimization. However, classical convex surrogates can only tightly bound modular loss functions, submodular functions or supermodular functions separately while maintaining polynomial time computation. In this work, a novel generic convex surrogate for general nonmodular loss functions is introduced, which provides for the first time a tractable solution for loss functions that are neither supermodular nor submodular. This convex surrogate is based on a submodular-supermodular decomposition for which the existence and uniqueness is proven in this paper. It takes the sum of two convex surrogates that separately bound the supermodular component and the submodular component using slackrescaling and the Lovász hinge, respectively. It is further proven that this surrogate is convex, piecewise linear, an extension of the loss function, and for which subgradient computation is polynomial time. Empirical results are reported on a non-submodular loss based on the Sørensen-Dice difference function, and a real-world face track dataset with tens of thousands of frames, demonstrating the improved performance, efficiency, and scalability of the novel convex surrogate.

Introduction

Many learning problems involve the simultaneous prediction of multiple labels. A simple strategy is to empirically minimize the Hamming loss over the set of predictions [START_REF] Taskar | Maxmargin Markov networks[END_REF]. However, this does not always reflect the underlying risk of the prediction process, and may lead to suboptimal performance. Following the risk minimization principle [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], we may instead wish to minimize a loss function that more closely reflects the cost of a specific set of predictions. Alternatives to the Hamming loss are frequently employed in the discriminative learning literature: [START_REF] Cheng | Bayes optimal multilabel classification via probabilistic classifier chains[END_REF] uses a rank loss which is supermodular; [START_REF] Petterson | Submodular multi-label learning[END_REF] uses a non-submodular loss based on F-score; [START_REF] Doppa | HC-search for multi-label prediction: An empirical study[END_REF] uses modular losses e.g. Hamming loss and F1 loss which is nonsubmodular; and losses that are nonmodular are common in a wide range of problems, including Jaccard index based losses [START_REF] Blaschko | Learning to localize objects with structured output regression[END_REF][START_REF] Everingham | The Pascal visual object classes (VOC) challenge[END_REF][START_REF] Nowozin | Optimal decisions from probabilistic models: The intersection-over-union case[END_REF], or more general submodularsupermodular objectives [START_REF] Narasimhan | A submodularsupermodular procedure with applications to discriminative structure learning[END_REF].

This has motivated us to study the conditions for a loss function to be tractably upper bounded with a tight convex surrogate. For this, we make use of the discrete optimization literature, and in particular submodular analysis [START_REF] Fujishige | Submodular functions and optimization[END_REF][START_REF] Schrijver | Combinatorial Optimization: Polyhedra and Efficiency[END_REF]. Existing polynomial-time convex surrogates exist for supermodular [START_REF] Tsochantaridis | Large margin methods for structured and interdependent output variables[END_REF] or submodular losses [START_REF] Yu | Learning submodular losses with the Lovász hinge[END_REF], but not for more general non-modular losses. We may perform approximate inference in polynomial time via a greedy optimization procedure to compute a subgradient or cutting plane of a convex surrogate for a general increasing function, but this leads to poor performance of the training procedure in practice [START_REF] Finley | Training structural SVMs when exact inference is intractable[END_REF][START_REF] Joachims | Cuttingplane training of structural SVMs[END_REF]. A decomposition-based method for a general set function has been proposed in the literature [START_REF] Iyer | Algorithms for approximate minimization of the difference between submodular functions, with applications[END_REF], showing that under certain conditions a decomposition into a submodular plus a supermodular function can be efficiently found. Other relevant work includes the hardness results on submodular Hamming optimization and its approximation algorithms [START_REF] Gillenwater | Submodular Hamming metrics[END_REF].

In this paper, we propose a novel convex surrogate for general non-modular loss functions, which is solvable for the first time for non-supermodular and nonsubmodular loss functions. In Section 2, we introduce the basic concepts used in this paper. In Section 3, we define a decomposition for a general non-modular loss function into supermodular and submodular components (Section 3.1), propose a novel convex surrogate operator based on this decomposition (Section 3.2), and demonstrate that it is convex, piecewise linear, an extension of the loss function, and for which subgradient computation is polynomial time (Section 3.3). In Section 4, we introduce the Sørensen-Dice loss, which is neither submodular nor supermodular. In Section 5 we demonstrate the feasibility, efficiency and scalability of our convex surrogate with the Sørensen-Dice loss on a synthetic problem, and a range of non-modular losses on a real-world face-track dataset comprising tens of thousands of video frames.

Non-modular loss functions

In empirical risk minimization for a set of binary predictions, we wish to minimize some functional of

R(h) = 1 n n i=1 ∆(y i , sign(h(x i ))). (1)
For an arbitrary loss function ∆ : {-1, +1} p × {-1, +1} p → R, we define a convex surrogate with an operator B,

B∆ : {-1, +1} p × R p → R. (2)
We may then minimize the empirical expectation of B∆(y, h(x)) with respect to functions h : X → R p . For well behaved function classes for h, minimization of the convex surrogate becomes tractable, provided that subgradient computation of B∆ is efficiently solvable.

Any loss function ∆ of this form may be interpreted as a set function where inclusion in a set is defined by a corresponding prediction being incorrect:

∆(y, ỹ) = l({i|y i = ỹi }) (3)
for some set function l.

In our analysis of convex surrogates for non-modular loss functions, we will employ several results for the Structured Output SVM [START_REF] Tsochantaridis | Large margin methods for structured and interdependent output variables[END_REF], which assumes that a structured prediction is made by taking an inner product of a feature representation of inputs and outputs: sign(h(x)) = arg max y w, φ(x, y) . The slack rescaling variant of the Structured Output SVM is as follows:

min w,ξ 1 2 w 2 + C n i=1 ξ i , ∀i, ∀ỹ ∈ Y : (4) w, φ(x i , y i ) -w, φ(x i , ỹ) ≥ 1 - ξ i ∆(y i , ỹ) (5)
In the sequel, we consider a feature function such that w, φ(x, y) = p j=1 w j , x j y j . Each w j is then a vector of length d, and w ∈ R d•p . Therefore p individual prediction functions parametrized by w j are simultaneously optimized, although we may also consider cases in which we constrain w j = w i ∀i, j. More generally, we may consider h : X → R p , which may have non-linearities, e.g. deep neural networks.

Mathematical preliminaries Definition 1.

A set function l maps from the powerset of some base set V to the reals l :

P(V ) → R. Definition 2. A set function l is non-negative if l(A) -l(∅) ≥ 0, ∀A ⊆ V .
We denote the set of all such loss functions satisfying Equation (3) F. Following standard conventions in submodular analysis, we assume that l(∅) = 0. In this paper we consider l is non-negative, which we will denote l ∈ F + .

Definition 3 (Submodular function). A set function l

: P(V ) → R is submodular iff for all B ⊆ A ⊂ V and x ∈ V \ A, l(B ∪ {x}) -l(B) ≥ l(A ∪ {x}) -l(A) (6)
A function is supermodular iff its negative is submodular, and a function is modular (e.g. Hamming loss) iff it is both submodular and supermodular. We denote the set of all submodular functions as S, and the set of all supermodular functions as G.

Definition 4. A set function l is symmetric if l(A) = c(|A|) for some function c : Z * → R. Proposition 1. A symmetric set function l is sub- modular iff c is concave [1, Proposition 6.1].
Definition 5 (Increasing function). A set function l : P(V ) → R is increasing if and only if for all subsets A ⊂ V and elements x ∈ V \ A, l(A) ≤ l(A ∪ {x}).

We note that the set of increasing supermodular functions is identical to G + . We will propose a convex surrogate operator for a general non-negative loss function, based on the fact that set functions can always be expressed as the sum of a submodular function and a supermodular function:

Proposition 2. For all set functions l, there always exists a decomposition into the sum of a submodular function f ∈ S and a supermodular function g ∈ G:

l = f + g (7)
A proof of this proposition is given in [19, Lemma 4]. Proposition 3. For an arbitrary decomposition l = f + g where g is not increasing, there exists a modular function m g s.t.

l = (f -m g ) + (g + m g ) (8)
with f := f -m g ∈ S, and g := g + m g ∈ G + is increasing.

Proof. Any modular function can be written as

m g (A) = j∈A w j (9)
for some coefficient vector w ∈ R |V | . For each j ∈ V , we may set

w j = -min A⊆V g(A ∪ {j}) -g(A). (10)
The resulting modular function will ensure that g +m g is increasing following Definition 5.

This proof indicates that a decomposition l = f + g is not-unique due to a modular factor. We subsequently demonstrate that decompositions can vary by more than a modular factor: Proposition 4 (Non-uniqueness of decomposition up to modular transformations.). For any set function, there exist multiple decompositions into submodular and supermodular components such that these components differ by more than a modular factor:

∃f 1 , f 2 ∈ S, g 1 , g 2 ∈ G (l = f 1 + g 1 = f 2 + g 2 ) ∧ (g 1 + m g1 = g 2 + m g2 ) (11)
where ∧ denotes "logical and," m g1 and m g2 are constructed as in Equations ( 9) and (10).

Proof. Let m be a submodular function that is not modular. For a given decomposition l = f 1 + g 1 , we may construct

f 2 := f 1 + m and g 2 := g 1 -m. As m is not modular, there is no modular m 1 such that g 1 -m 1 = g 1 -m = g 2 .

A convex surrogate for general non-modular losses

We will show in this section the unique decomposition for a general non-negative loss starting from any arbitrary submodular-supermodular decomposition, which allows us to define a convex surrogate operator based on such a canonical decomposition.

A canonical decomposition

In this section, we define an operator D such that g * := Dl ∈ G + is unique and f * := l -Dl ∈ S is then unique. We have demonstrated in the previous section that we may consider there to be two sources of nonuniqueness in the decomposition l = f + g: a modular component and a non-modular component related to the curvature of g (respectively f ). We define D such that these two sources of non-uniqueness are resolved using a canonical decomposition l = f * + g * . Definition 6. We define an operator D : F → G + as

Dl = arg min g∈G+ A⊆V g(A), s.t. l -g ∈ S. (12)
We note that minimizing the values of g will simultaneously remove the non-uniqueness due both to the modular non-uniqueness described in Proposition 3, as well as the non-modular non-uniqueness described in Proposition 4. We formally prove this in Proposition 5.

Proposition 5. Dl is unique for all l ∈ F that have a finite base set V .

Proof. We note that the arg min in Equation ( 12) is equivalent to a linear program: g is uniquely determined by a vector in R 2 |V | -1 the coefficients of which correspond to g(A) for all A ∈ P(V ) \ ∅, and we wish to minimize the sum of the entries subject to a set of linear constraints enforcing supermodularity of g, non-negativity of g, and submodularity of l -g.

From [18, Theorem 2], an LP of the form min

x∈R d r T x (13) s.t. Cx ≥ q (14)
has a unique solution if there is no y ∈ R d simultaneously satisfying

C J y ≥ 0, r T y ≤ 0, y = 0 ( 15
)
where J = {i|C i x * = q i } is the active set of constraints at an optimum x * . We note that as r is a vector of all ones (cf. Equation ( 12)), r T y ≤ 0 constrains y to lie in the non-positive orthant. However, as the linear program is minimizing the sum of x subject to lower bounds on each entry of x (e.g. positivity constraints), we know that C J y ≥ 0 will bound y to lie in the non-negative orthant. This means, at most, these constraints overlap at y = 0, but this is expressly forbidden by the last condition in Equation [START_REF] Krause | Submodular function maximization[END_REF].

Although Equation ( 12) is a linear programming problem, we do not consider this definition to be constructive in general as the size of the problem is exponential in |V | (see [START_REF] Iyer | Algorithms for approximate minimization of the difference between submodular functions, with applications[END_REF]). However, it may be possible to verify that a given decomposition satisfies this definition for some loss functions of interest. Furthermore, for some classes of set functions, the LP has lower complexity, e.g. for symmetric set functions the resulting LP is of linear size, and loss functions that depend only on the number of false positives and false negatives (such as the Sørensen-Dice loss discussed in Section 4) result in a LP of quadratic size.

We finally note that from Equation (3), for every ∆(y, •) we may consider its equivalence to a set function l = g * + f * , and denote the resulting decomposition of ∆(y,

•) = ∆ G (y, •) + ∆ S (y, •) (16)
into its supermodular and submodular components, respectively.1

Definition of the convex surrogate

Now that we have defined a unique decomposition l = g * + f * , we will use this decomposition to construct a surrogate B∆ that is convex, piecewise linear, an extension of ∆, and for which subgradient computation is polynomial time. We construct a surrogate B by taking the sum of two convex surrogates applied to ∆ G and ∆ S independently. These surrogates are slackrescaling [START_REF] Tsochantaridis | Large margin methods for structured and interdependent output variables[END_REF] applied to ∆ G and the Lovász hinge [START_REF] Yu | Learning submodular losses with the Lovász hinge[END_REF] applied to ∆ S .

Definition 7 (Slack-rescaling operator [START_REF] Yu | Learning submodular losses with the Lovász hinge[END_REF]). The slack-rescaling operator S is defined as:

S∆(y, h(x)) := max ỹ∈Y ∆(y, ỹ) (1 + h(x), ỹ -h(x), y ) . (17)
The Lovász hinge of a submodular function builds on the Lovász extension [START_REF] Lovász | Submodular functions and convexity[END_REF]:

Definition 8 (Lovász hinge [START_REF] Yu | Learning submodular losses with the Lovász hinge[END_REF]). The Lovász hinge, L, is defined as the unique operator such that, for a submodular set function l related to ∆ as in Eq. (3):

L∆(y, h(x)) :=   max π p j=1 s πj (l ({π 1 , • • • , π j }) -l ({π 1 , • • • , π j-1 }))   + (18)
where (•) + = max(•, 0), π is a permutation,

s πj = 1 -h πj (x)y πj , (19)
and h πj (x) is the π j th dimension of h(x).

Algorithm 1 Cutting plane algorithm

1: Input: (x 1 , y 1 ), • • • , (x n , y n ), C, 2: S i = ∅, ∀i = 1, • • • , n 3: repeat 4: for i = 1, • • • , n do 5: ŷL = arg max ỹ H L (y i ) = arg max ỹ L∆ S 6: ŷS = arg max ỹ H S (y i ) = arg max ỹ S∆ G 7: H(ŷ) = H L (ŷ L ) + H S (ŷ S )
8:

ξ i = max{0, H(y i )} 9:
if H(ŷ) > ξ i + then 10:

S i := S i ∪ {y i } 11:
w ← optimize Equation ( 4) with constraints defined by ∪ i S i 12:

end if 13: until no S i has changed during an iteration 14: return (w, ξ) Definition 9 (General non-modular convex surrogate). For an arbitrary non-negative loss function ∆, we define

B D ∆ := L∆ S + S∆ G (20)
where ∆ S and ∆ G are as in Equation ( 16), and D is the decomposition of l defined by Definition 6.

We use a cutting plane algorithm to solve the maxmargin problem as shown in Algorithm 1.

Properties of B D

In the remainder of this section, we show that B D has many desirable properties. Specifically, we show that B D is closer to the convex closure of the loss function than slack rescaling and that it generalizes the Lovász hinge (Theorems 1 and 2). Furthermore, we formally show that B D ∆ is convex (Theorem 3), an extenstion of ∆ for a general class of loss functions (Theorem 4), and polynomial time computable (Theorem 5).

The sum in Equation ( 21) is precisely the sum that should be minimized in Equation [START_REF] Gillenwater | Submodular Hamming metrics[END_REF]. We now show that this sum cannot be minimized further while allowing f m to be non-modular. If there exists any g s.t.

f := l -g is submodular but not modular, by definition there exists at least one subset S s ⊆ V and one j ∈ S s such that

f (S s \ {j}) + f ({j}) > f (S s ) + f (∅). ( 22
)
Then by subtracting each time one element from the subset S s , we have

f (S s ) < f (S s \ {j}) + f ({j}) ≤ f (S s \ ({j} ∪ {k})) + f ({k}) + f ({j}) ≤ • • • ≤ j∈Ss f ({j})), ∀k ∈ S s \ ({j} ( 23
)
which implies

g(S s ) > l(S s ) - j∈Ss l({j}) (24)
By taking the sum of the inequalities as in Equation 24for all subsets S, we have that

S⊆V g(S) > S⊆V   l(S) - j∈S l({j})   = S⊆V g m (S)
which means S⊆V g(S) > S⊆V g m (S) for any g. By Definition 6, g * = g m = Dl, thus f * := l-Dl = f m is modular.

Lemma 2. For a loss function ∆ such that ∆ S is increasing, we have

S∆ G = S(∆ -∆ S ) = S∆ -S∆ S . (25)
Proof. By Definition 7, for every single cutting plane determined by some ỹ, we have

S (∆(y, ỹ) -∆ S (y, ỹ)) = (∆(y, ỹ) -∆ S (y, ỹ)) (1 + h(x), ỹ -h(x), y ) = ∆(y, ỹ) (1 + h(x), ỹ -h(x), y ) -∆ S (y, ỹ) (1 + h(x), ỹ -h(x), y ) = S∆(y, ỹ) -S∆ S (y, ỹ). ( 26
)
As this property holds for all cutting planes, it also holds for the supporting hyperplanes that define the convex surrogate and S(∆ -∆ S ) = S∆ -S∆ S .

Definition 10. A convex surrogate function B∆(y, •) is an extension when

B∆(y, •) = ∆(y, •) (27)
on the vertices of the 0-1 unit cube under the mapping to R p : i = {1, . . . , p}, [u] i = 1 -h πi (x)y πi Proof. By Definition 6, B D ∆ is the sum of the two convex surrogates, which is a convex surrogate.

Theorem 4. B D ∆ is an extension of ∆ iff ∆ S is non-negative.
Proof. From [30, Proposition 1], S∆ is an extension for any supermodular increasing ∆; L∆ is an extension iff ∆ is submodular and non-negative as in this case, L coincides with the Lovász extenstion [START_REF] Lovász | Submodular functions and convexity[END_REF]. By construction from Definition 6 we have ∆ G and ∆ S for g ∈ G + and f ∈ S, respectively. Thus Equation 27 holds for both S∆ G and L∆ S if ∆ S is non-negative. Then B D ∆ taking the sum of the two extensions, Equation 27 also holds for every vertex of the unit cube as ∆ = ∆ G +∆ S , which means B D is also an extension of ∆ .

Theorem 5. The subgradient computation of B D ∆ is polynomial time given polynomial time oracle access to f * and g * .

Proof. Given f * and g * we know that the subgradient computation of L∆ S and S∆ G are each polynomial time. Thus taking the sum of the two is also polynomial time.

Sørensen-Dice loss

The Sørensen-Dice criterion [START_REF] Dice | Measures of the amount of ecologic association between species[END_REF][START_REF] Sørensen | A method of establishing groups of equal amplitude in plant sociology based on similarity of species and its application to analyses of the vegetation on Danish commons[END_REF] is a popular criterion for evaluating diverse prediction problems such as image segmentation [START_REF] Sabuncu | A generative model for image segmentation based on label fusion[END_REF] and language processing [START_REF] Rychlỳ | A lexicographer-friendly association score[END_REF].

In this section, we introduce the Sørensen-Dice loss based on the Sørensen-Dice coefficient. We prove that the Sørensen-Dice loss is neither supermodular nor submodular, and we will show in the experimental results section that our novel convex surrogate can yield improved performance on this measure.

Definition 11 (Sørensen-Dice Loss). Denote y ⊆ V is a set of positive labels, e.g. foreground pixels, the Sørensen-Dice loss on given a groundtruth y and a predicted output ỹ is defined as

∆ D (y, ỹ) = 1 - 2|y ∩ ỹ| |y| + |ỹ| . ( 28
)
Proposition 6. ∆ D (y, ỹ) is neither submodular nor supermoduler under the isomorphism (y * , ỹ) → A := {i|y * i = ỹi }, ∆ J (y * , ỹ) ∼ = l(A).

We will use the diminishing returns definition of submodularity in Definition 3 to first prove the following lemma:

∆ D (y, ỹ) = 1 - 2m -2n 2m -n + p = n + p 2m -n + p ( 29
)
For a given groundtruth y i.e. m, we have if

B ⊆ A, then n B ≤ n A , and p B ≤ p A .
Considering i is an extra false negative, we calculate the marginal gain on A and B respectively:

∆ D (A ∪ {i}) -∆ D (A) = n A + 1 + p A 2m -n A -1 + p A - n A + p A 2m -n A + p A (30) = 2m + 2p A (2m -n A + p A -1)(2m -n A + p A ) (31) ∆ D (B ∪ {i}) -∆ D (B) = 2m + 2p B (2m -n B + p B -1)(2m -n B + p B ) . (32)
Numerically, we have following counter examples which prove that ∆ D restricted to false negatives is neither submodular nor supermodular. We set m = 10,

n A = [1 : 8], n B = n A -1 ≤ n A , p A = 8, p B = 5 ≤ p A ,

and we plot the values of Equation (31) and

Equation 32 as a function of n A . We can see from Figure 1 that there exists a cross point between these two plots, which indicates that submodularity (Definition 3) does not hold for ∆ D or its negative.

Lemma 3 implies Proposition 6 as the restriction of a submodular function is itself submodular.

Experimental Results

We demonstrate the correctness and feasibility of the proposed convex surrogate on experiments using Dice loss, as well as on a face classification problem from video sequences with a family of non-modular losses.

Dice loss

We test the proposed surrogate on a binary set prediction problem. Two classes of 2-dimensional data are generated by different Gaussian mixtures as shown in Fig 2 . We use the B D during training time with the non-modular loss ∆ D to construct a convex surrogate. We compare it to slack rescaling S with an approximate optimization procedure based on greedy maximization. We additionally train an SVM (denoted 0-1 in the results table) for comparison. During test time, we evaluate with ∆ D and with Hamming loss to calculate the empirical error values as shown in Table 1.

We can see from the result that training ∆ D with B D yields the best result while using ∆ D during test time. B D performs better than S in both cases due to the failure of the approximate maximization procedure necessary to maintain computational feasibility [START_REF] Krause | Submodular function maximization[END_REF].

Face classification in video sequences

We also evaluate the proposed convex surrogate operator on a real-world face track dataset [START_REF] Everingham | Hello! My name is... Buffy" -automatic naming of characters in TV video[END_REF][START_REF] Everingham | Taking the bite out of automatic naming of characters in TV video[END_REF][START_REF] Sivic | Who are you?" -learning person specific classifiers from video[END_REF]. The frames of the dataset are from the TV series "Buffy the Vampire Slayer". This dataset contains 1437 tracks and 27504 frames in total.

We focus on a binary classification task to recognize the leading role: "Buffy" is positive-labelled, "not Buffy" is negative-labelled. Example images are shown in Fig. 4. Each track is represented as a bag of frames, for which the size of the tracks varies from 1 frame to more than 100 frames, and each image is represented as a Fisher Vector Face descriptor of dimension 1937.

We have used different non-supermodular and nonsubmodular loss functions in our experiments as shown For the cases that the submodular component is non-negative, i.e. using ∆ 1 and ∆ 2 , the lowest empirical error is achieved when using B D .

I = {i|y i = ỹi } gives the set of incorrect prediction elements; α is a parameter that allows us to define the value of l(V ). Due to the fact that the size of the tracks varies widely, we further normalize the loss function with respect to the track size. We use α = 2 for ∆ 2 and α = 0.5 for ∆ 4 in the experiments.

As we can see explicitly in Fig. 3, no ∆ is supermodular or submodular. ∆ 1 , ∆ 2 and ∆ 3 are increasing loss functions, while ∆ 4 is non-increasing. For ∆ 1 and ∆ 2 , we notice that the values of the set functions on a single element are non-zero i.e. l 1 ({j}) > 0, l 2 ({j}) > 0, ∀j ∈ V ; while for the loss ∆ 3 and ∆ 1 these values are zero i.e. l 3 ({j}) = l 4 ({j}) = 0, ∀j ∈ V .

Fig. 3 shows the corresponding decomposition of each loss into the supermodular and submodular components as specified in Definition 6. We denote each loss function as

l k = f k + g k , for k = {1, 2, 3, 4}.
By construction, all supermodular g k , for k = {1, 2, 3, 4}, are non-negative increasing. For the submodular component, f 1 is non-negative increasing, f 2 is non-negative and non-increasing, while f 3 and f 4 are both non-positive decreasing.

We compare different convex surrogates during training for these non-modular functions. And we additionally train on the Hamming loss (labelled 0-1) as a comparison. As training non-supermodular loss with slack rescaling is NP-hard, we have employed the simple application of the greedy approach as in [START_REF] Krause | Submodular function maximization[END_REF].

10-fold-cross-validation has been carried out and we obtain an average performance and standard error as shown in Table 2.

From Table 2 we can see that when the submodular The primal-dual gap as a function of the number of cutting-plane iterations using different convex surrogates for the four non-modular functions in Equations (33) to 36. The primal-dual gap from B D is drawn in red; the gap from S is drawn in green, and gap from Hamming loss (labelled 0-1, and equivalent to a SVM) in blue. Our convex surrogate operator B D can achieve a comparable convergence rate to an SVM, demonstrating that optimization is very fast in practice and the method scales well to large datasets. p = 10 p = 50 p = 100 B D 0.002 ± 0.000 0.018 ± 0.003 0.060 ± 0.008 S 0.002 ± 0.000 0.016 ± 0.002 0.057 ± 0.002 Table 3: The comparison of the computation time (s) for one loss augmented inference.

component of the decomposition is non-negative, i.e. in the case of using ∆ 1 and ∆ 2 , the lowest empirical error is achieved by using our convex surrogate operator B D .

Fig. 5 shows the primal-dual gap as a function of the cutting plane iterations for each experiment using different loss functions and different convex surrogate operators. We can see that in all cases, the convergence of B D is at a rate comparable to an SVM, supporting the wide applicability and scalability of the convex surrogate. We have also compared the expected time of one loss augmented inference. Table 3 shows the comparison using ∆ 1 with B D and S. As the cost per iteration is comparable to slack-rescaling, and the number of iterations to convergence is also comparable, there is consequently no computational disadvantage to using the proposed framework, while the statistical gains are significant.

Discussion and Conclusions

The experiments have demonstrated that the proposed convex surrogate is efficient, scalable, and reduces test time error for a range of loss functions, including the Sørensen-Dice loss, which is a popular evaluation metric in many problem domains. We see that slack rescaling with greedy inference can lead to poor performance for non-supermodular losses. This is especially apparent for the results of training with ∆ 1 , in which the test-time loss was approximately double that of the proposed method. Similarly, ignoring the loss function and simply training with 0-1 loss can lead to compara-tively poor performance, e.g. ∆ 1 and ∆ 2 . This clearly demonstrates the strengths of the proposed method for non-modular loss functions for which a decomposition with a non-negative submodular component is possible (∆ 1 and ∆ 2 , but not ∆ 3 or ∆ 4 ). The characterization and study of this family of loss functions is a promising avenue for future research, with implications likely to extend beyond empirical risk minimization with non-modular losses as considered in this paper. The primal-dual convergence results empirically demonstrate that the loss function is feasible to apply in practice, even on a dataset consisting of tens of thousands of video frames. The convex surrogate is directly amenable to other optimization techniques, such as stochastic gradient descent [START_REF] Bottou | The tradeoffs of large scale learning[END_REF], or Frank-Wolfe approaches [START_REF] Lacoste-Julien | Block-coordinate Frank-Wolfe optimization for structural SVMs[END_REF], as well as alternate function classes including neural networks.

In this work, we have introduced a novel convex surrogate for general non-modular loss functions. We have defined a decomposition for an arbitrary loss function into a supermodular non-negative function and a submodular function. We have proved both the existence and the uniqueness of this decomposition. Based on this decomposition, we have proposed a novel convex surrogate operator taking the sum of two convex surrogates that separately bound the supermodular component and the submodular component using slack-rescaling and the Lovász hinge, respectively. We have demonstrated that our new operator is a tighter approximation to the convex closure of the loss function than slack rescaling, that it generalizes the Lovász hinge, and is convex, piecewise linear, an extension of the loss function, and for which subgradient computation is polynomial time. Opensource code of 2 regularized risk minimization with this operator is available for download from https: //github.com/yjq8812/aistats2016.

Lemma 1 .

 1 If l ∈ G, then f * := l -Dl ∈ S ∩ G i.e. modular. Proof. First we set l = g m + f m where f m is modular and f m ({j}) = l({j}). Then any subset S ⊆ V f m (S) = j∈S l({j}), g m (S) = l(S) -

Theorem 1 .Theorem 2 .

 12 If l ∈ G + , then B D ∆ ≥ S∆ over the unit cube given in Definition 10, and therefore B D is closer to the convex closure of ∆ than S. Proof. By the definition of the Lovász hinge L [30], we know that for any modular function ∆ S we have L∆ S ≥ S∆ S over the unit cube. As a result of Lemma 1 and Lemma 2,we have B D ∆ = S∆ G + L∆ S = S(∆ -∆ S ) + L∆ S ≥ S∆ -S∆ S + S∆ S = S∆. If l ∈ S, then B D ∆ = L∆ Proof. For l ∈ S, we construct g * = 0, and f * = l is submodular. By Definition 6, g * (V ) is minimum, so g * = Dl. Then B D ∆ = L∆ + S0 = L∆ Theorem 3. B D ∆ is convex for arbitrary ∆.

Lemma 3 .

 3 ∆ D restricted to false negatives is neither submodular nor supermoduler. Proof. With the notation m := |y * | > 0, p := |ỹ \ y * |, and n := |y * \ ỹ|, we have that

  [1:8], nB=nA-1, pA=8, pB=5

Figure 1 :

 1 Figure 1: Plots of Eq. (31) (red) and Eq. 32 (blue) as a function of n A . As these two plots cross, neither function bounds the other.

Figure 2 :

 2 Figure 2: The data for the synthetic problem. The negative samples are drawn from a mixture of Gaussians.

Figure 3 :

 3 Figure 3: The plot of the four loss functions used in our experiments as in Equations (33) to (36). The x axis is the number of mispredictions for each track (we show here the loss functions corresponding to track length equal to 10 as an example), and the y axis is the value of loss function. The original losses are drawn in red; the supermodular components are drawn in green, and the submodular components in blue.

Figure 4 :

 4 Figure 4: Examples of the face track images. Fig. 4(a)shows the "Buffy" role thus a positive-labelled image and Fig.4(b) shows a negative-labelled image. An automated pipeline described in[START_REF] Everingham | Hello! My name is... Buffy" -automatic naming of characters in TV video[END_REF][START_REF] Everingham | Taking the bite out of automatic naming of characters in TV video[END_REF][START_REF] Sivic | Who are you?" -learning person specific classifiers from video[END_REF] was used for feature extraction (Fig.4(c)).

Figure 5 :

 5 Figure5: The primal-dual gap as a function of the number of cutting-plane iterations using different convex surrogates for the four non-modular functions in Equations (33) to 36. The primal-dual gap from B D is drawn in red; the gap from S is drawn in green, and gap from Hamming loss (labelled 0-1, and equivalent to a SVM) in blue. Our convex surrogate operator B D can achieve a comparable convergence rate to an SVM, demonstrating that optimization is very fast in practice and the method scales well to large datasets.

Table 2 :

 2 For the face classification task, the cross comparison of average loss values (with standard error) using different surrogate operator and losses as in Equation (33) to Equation (36) during training, respectively.

	Test
	p = 6 B D 0.1121 ± 0.0040 0.6027 ± 0.0125 ∆ D 0-1 0-1 0.1497 ± 0.0046 0.5370 ± 0.0114 S 0.3183 ± 0.0148 0.7313 ± 0.0209	loss functions ∆ 3 (∆ S negative) ∆ 4 (∆ S negative) B D 0.194 ± 0.006 0.238 ± 0.008 ∆ 1 ∆ 2 0.148 ± 0.005 0.108 ± 0.004 0-1 0.228 ± 0.007 0.284 ± 0.004 0.144 ± 0.004 0.107 ± 0.003
		S	0.398 ± 0.015	0.243 ± 0.005	0.143 ± 0.006	0.106 ± 0.003
	Table 1: For the synthetic data ex-
	periment, the cross comparison of aver-
	age loss values (with standard error) us-
	ing different surrogate operations during
	training, and different evaluation func-
	tions during test time. ∆ D is the Dice
	loss as in Eq. (28).

Note that ∆S and ∆G are due to Eq. (3) for f * and g * which explicitly depend on D. For simplicity of notation, we will use ∆G instead of ∆ DG

Acknowledgements

This work is partially funded by Internal Funds KU Leuven, ERC Grant 259112, and FP7-MC-CIG 334380. The first author is supported by a fellowship from the China Scholarship Council.