Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components

1 Introduction . Given a ground set V , a function f : 2V → R is submodular if for every A , B ⊆ V it satisfies f ( A ) + f ( B ) ≥ f ( A ∩B ) + f ( A ∪B ) . Submodular functions are ubiquitous in combinatorial optimization and machine learning , arising , e.g. , in image segmentation [ 22 ] , hypergraph clustering [ 30 ] , data subset selection [ 44 ] , document summarization [ 31 ] , and dense subgraph discovery [ 43 ] . Algorithms for minimizing submodular functions are well studied . Strongly-polynomial time algorithms for general submodular minimization exist [ 36 , 17 , 18 ] , but their runtimes are impractical in most cases . The past decade has witnessed several advances in faster algorithms for decomposable submodular function minimization ( DSFM ) , i.e. , minimizing a sum of simpler submodular functions [ 9 , 24 , 39 , 29 , 35 , 10 , 21 , 19 ] . This problem is defined by identifying a set E ⊆ 2V of subsets of the ground set . The goal is then to solve minimizeS⊆V f ( S ) = ∑ e∈E fe ( S ∩ e ) , ( 1 ) where for each e ∈ E , fe is a submodular function supported on a subset e ⊆ V . Functions of this form often arise as energy functions in computer vision [ 22 , 23 , 13 ] , and generalized cut functions for hypergraph clustering and learning [ 12 , 28 , 30 , 32 , 41 , 42 ] , among other applications . This paper focuses on cardinality-based DSFM ( Card-DSFM ) , where every component fe in the sum is a concave cardinality function , i.e. , fe ( A ) = ge ( |A| ) for some concave function ge . A single concave cardinality function is effectively a function of one variable and is trivial to minimize . However , set functions obtained via sums of these functions are much more complex and have broad modeling power , making this one of the most widely studied and applied variants since the earliest ∗This research was performed while the author was a postdoctoral associate at Cornell University . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . work on DSFM [ 24 , 39 ] . In terms of theory , previous research has addressed specialized runtimes and solution techniques [ 24 , 39 , 22 , 41 ] . In practice , cardinality-based decomposable submodular functions frequently arise as higher-order energy functions in computer vision [ 22 ] and set cover functions [ 39 ] . The most widely used generalized hypergraph cut functions are also cardinality based [ 30 , 32 , 41 , 42 , 1 ] . Even previous research on algorithms for the more general DSFM problem tends to focus on cardinality-based examples in experimental results [ 9 , 39 , 19 , 29 ] . We present the first approximation algorithms for Card-DSFM , using a purely combinatorial approach that relies on approximately reducing ( 1 ) to a sparse graph cut problem . The fact that concave cardinality functions are representable by graph cuts has previously been noted in different contexts [ 22 , 39 , 41 ] —this can be accomplished by combining simple graph gadgets , whose cut properties together model the function . However , previous techniques are limited in that they ( i ) focus only on exact reduction techniques , ( ii ) do not consider the density of the resulting graph , and ( iii ) do not address any questions regarding the optimality of such a reduction . Here we develop new approximate reduction methods leading to a sparse graph , which we show is optimally sparse in terms of the standard gadget reduction strategy . We show that representing a concave cardinality function with a sparse graph is equivalent to approximating a concave function with a piecewise linear curve , where the number of linear pieces determines the sparsity of the reduced graph . We develop a new algorithm for the resulting piecewise linear approximation problem , and prove new bounds on the number of edges needed to model different concave functions that arise in practice . Combining our reduction strategy with state of the art algorithms for maximum flow [ 15 , 26 , 40 , 14 ] leads to fast runtime guarantees for finding approximate solutions to Card-DSFM . Our algorithm is also easy to implement and achieves substantial practical improvements on benchmark image segmentation experiments [ 19 , 9 , 29 ] and algorithms for hypergraph clustering [ 42 ] . 2 Background on Graph Reductions . A nonnegative set function f : 2V → R+ on a ground set V is graph reducible if there exists a directed graph G = ( Vf , Ef ) on a larger node set Vf = V ∪ A ∪ { s , t } , which includes an auxiliary node setA as well as source and sink nodes s and t , whose s-t cut structure models f . More precisely , f is graph reducible if for every S ⊆ V we have f ( S ) = minimize T⊆A cutG ( { s } ∪ S ∪ T ) , ( 2 ) where cutG is the cut function on the graph G. In words , this says that evaluating f ( S ) is equivalent to finding the minimum s-t cut penalty that could be obtained by placing S on the s-side of the cut and V \S on the t-side . Given such a graph , f can be minimized by finding the minimum s-t cut set U∗ ⊆ V ∪ A in G and taking S∗ = V ∩ U∗ . When ( 2 ) holds , we say that G models f . Graph reduction strategy for Card-DSFM . In general , the function f ( S ) = ∑ e∈E fe ( S ∩ e ) is not a concave cardinality function , even if individual components fe in the sum are . However , in order to show f is graph reducible , it suffices to find a small reduced graph Ge for each e ∈ E that models fe in the sense of ( 2 ) , and combine these together into a larger graph . A number of related approaches for this task have been developed [ 39 , 22 , 41 ] . We review our own previous reduction strategy [ 41 ] , which we build on in this paper . Modeling fe can be accomplished by combining a set of cardinality-based ( CB ) gadgets [ 41 ] . For a set e ⊆ V of size k = |e| , a CB-gadget is a small graph parameterized by positive weights ( a , b ) . Each v ∈ e defines a node , and the gadget also involves a single auxiliary node ve . For each v ∈ e , there is a directed edge ( v , ve ) with weight a · ( k − b ) , and a directed edge ( ve , v ) with weight a · b . The resulting graph gadget models the function fa , b ( S ) = a ·min { |S| · ( k − b ) , ( k − |S| ) · b } . ( 3 ) To see why , consider where we must place the auxiliary node ve when solving a minimum s-t cut problem involving this small graph gadget . If we place i = |S| nodes on the s-side , then placing ve on the s-side has a cut penalty of ab ( k − i ) , whereas placing ve on the t-side gives a penalty of ai ( k − b ) . To minimize the cut as in ( 2 ) for a fixed S ⊆ e , we choose the smaller of the two cut penalties . We previously showed that any cardinality-based submodular function fe on a ground set e can be modeled by combining |e| − 1 CB-gadgets [ 41 ] . Analogously , Stobbe and Krause [ 39 ] showed that a concave cardinality function on a ground set e can be decomposed into a sum of |e| − 1 threshold potentials [ 39 ] and can be represented by graph cuts , though they provided no explicit reduction strategy . Priori to this , Kohli et al . [ 22 ] highlighted a similar type of gadget for modeling a class of robust potential functions for image segmentation , noting that multiple gadgets could be combined to model arbitrary concave functions . Using any of these techniques , the function fe can be modeled in the sense of ( 2 ) using a graph G with O ( |e| ) nodes and O ( |e|2 ) edges . 3 Sparse Reductions via Piecewise Linear Approximation . We now turn to a new and refined problem of finding sparse and approximate reduction strategies for Card-DSFM , which we prove can be cast as approximating a concave function with a piecewise linear curve . All proofs are given in the supplement , where we also derive a similar graph reduction scheme that is slightly more efficient for modeling symmetric functions , i.e. , fe ( A ) = fe ( e\A ) . 3.1 The sparse approximate reduction problem . We first introduce a special parameterized function with broad modeling power . Definition 1 A k-node combined gadget function ( CGF ) of order J is a function of the form : ` ( x ) = z0 · ( k − x ) + zk · x+ ∑J j=1 aj min { x · ( k − bj ) , ( k − x ) · bj } , ( 4 ) parameterized by scalars z0 ≥ 0 , zk ≥ 0 , and vectors a = ( aj ) ∈ RJ > 0 and b = ( bj ) ∈ RJ > 0 , where bj < bj+1 for j ∈ { 1 , 2 , . . . , J − 1 } and bJ < k. If we define a set function on a ground set e of size k by fe ( S ) = ` ( |S| ) , then fe is exactly the function that is modeled by combining J CB-gadgets with parameters ( aj , bj ) Jj=1 , and additionally placing a directed edge from a source node s to each v ∈ e of weight z0 , and an edge from v ∈ e to a sink node t with weight zk . Previous reduction techniques amount to proving that any concave function g on an interval [ 0 , k ] ( representing a submodular function fe ( S ) = g ( |S| ) ) matches some k-node CGF of order k − 1 at integer inputs [ 41 , 39 ] . We focus on a new sparse reduction problem . Definition 2 Let g : [ 0 , k ] → R+ be a nonnegative concave function and fix ε ≥ 0 . The sparsest approximate reduction ( SpAR ) problem seeks a k-node CGF ` with minimum order J so that g ( i ) ≤ ` ( i ) ≤ ( 1 + ε ) g ( i ) for all i ∈ { 0 , 1 , 2 , . . . k } . ( 5 ) This is equivalent to finding the minimum number of CB-gadgets needed to approximately model a concave cardinality function . Thus , solving this problem provides the sparsest reduction in terms of a standard reduction strategy . Without loss of generality , Definition 2 is restricted to considering functions that upper bound g. Any other approximating function could be scaled to produce an upper bound leading to the same guarantees for Card-DSFM . Importantly , we care about approximating the concave function g only at integer inputs , since we are ultimately concerned with modeling the set function fe ( S ) = g ( |S| ) . Satisfying g ( x ) ≤ ` ( x ) ≤ ( 1 + ε ) g ( x ) for all x ∈ [ 0 , k ] is a much stronger requirement and may require more CB-gadgets than is actually necessary to approximate fe . Connection to piecewise linear approximation Our first result establishes a precise one-to-one correspondence between a certain class of piecewise linear functions and combined gadget functions . Lemma 1 The k-node CGF in ( 4 ) is nonnegative , piecewise linear , concave , and has exactly J + 1 linear pieces . Conversely , let ` ′ : [ 0 , k ] → R+ be concave and piecewise linear with J + 1 linear pieces , and let mi be the slope of the ith linear piece and Bi be the ith breakpoint . Then ` ′ is uniquely characterized as the k-node CGF parameterized by ai = 1k ( mi −mi+1 ) and bi = Bi for i ∈ { 1 , 2 , . . . J } , z0 = ` ′ ( 0 ) /k , and zk = ` ′ ( k ) /k . This result tells us that solving SpAR ( Def 2 ) for g is equivalent to finding a concave piecewise linear function with the smallest number of linear pieces approximating g at integer points . Although some techniques for approximating a concave function with a piecewise linear curve already exist [ 34 ] and could be used as heuristics , these are ultimately unable to find optimal solutions for SpAR . First of all , these methods focus on approximating concave functions over continuous intervals rather than just at integer inputs . More importantly , they do not provide instance optimal approximations , but only give upper bounds on the number of linear pieces needed . We therefore turn to new methods that will allow us to exactly solve our sparse approximation problem . Algorithm 1 GREEDYPLCOVER ( g , ε ) Input : ε ≥ 0 , concave function g Output : piecewise linear ` with fewest linear pieces such that g ( i ) ≤ ` ( i ) ≤ ( 1 + ε ) g ( i ) L ← ∅ , u← 0 //u = smallest integer not covered by approximating line while u ≤ k do u′ , L← NEXTLINE ( g , u , ε ) //line covering widest range [ u , u ’ -1 ] L ← L ∪ { L } , u← u′ end while Return ` ( x ) = minL∈L L ( x )

Summary: This work studies minimizing decomposable submodular functions, which are of the form f(S) = \sum_{e \in E}f_e(S \cap e), where each f_e is a ``simple'' submodular function. In many practical applications, f_e(A) is cardinality based, that is, f_e(A) = g_e(|A|) for some concave function g_e. Several discrete/continuous algorithms have been proposed to minimize such decomposable submodular functions. One combinatorial approach to tackle this problem is to introduce a gadget graph G_e so that, for any A, g_e(|A|) is equal to the minimum cut of G_e subject to A (in addition to some special vertex) being contained in one side of the cut. It is known that any cardinality-based submodular function can be represented this way as a gadget graph of O(|e|) vertices and O(|e|^2) edges.  The main contribution of this work is providing a method for constructing a gadget graph that ``approximates'' cardinality-based submodular function instead of exactly simulating it. Specifically, it is shown that, to obtain a multiplicative error of 1 \pm epsilon, we only need O(epsilon^{-1} |e| log |e|) edges. A lower bound of O(epsilon^{-1/2} |e| log |e|) is also shown. When g(x) = x(k-x), which often appears in practice, it is shown that the number of edges can be further improved to O(\epsilon^{-1/2}log log epsilon^{-1} \cdot |e|). These results are obtained by approximating the concave function g_e with a piecewise linear function l and then representing l as a gadget graph.  Empirical results on image segmentation tasks show that the proposed approximate reduction outputs a much smaller gadget graph compared to the exact reduction without much sacrificing the quality of the solution. For example, for some choice of the parameters, the size of the output graph becomes 0.013 (1/0.013?) times smaller while the multiplicative error is merely 1 \pm 4*10^-3. It is then confirmed that the trade-off of the approximation ratio and runtime of the proposed method is much better than those of other methods. It is also empirically investigated that the quality of hypergraph clustering does not deteriorate much by replacing the objective submodular functions with gadget graphs obtained by the proposed approximate reduction.