Neural Estimation of Submodular Functions with Applications to Differentiable Subset Selection
Anonymous Author(s
AfﬁliationAddress
Abstract
Submodular functions and variants, through their ability to characterize diversityand coverage, have emerged as a key tool for data selection and summarization.
Many recent approaches to learn submodular functions suffer from limited expressiveness.
In this work, we propose F LEX S UB N ET , a family of ﬂexible neuralmodels for both monotone and non-monotone submodular functions.
To ﬁt alatent submodular function from (set, value) observations, F LEX S UB N ET applies aconcave function on modular functions in a recursive manner.
We do not draw theconcave function from a restricted family, but rather learn from data using a highlyexpressive neural network that implements a differentiable quadrature procedure.
Such an expressive neural model for concave functions may be of independentinterest.
Next, we extend this setup to provide a novel characterization of monotoneα -submodular functions, a recently introduced notion of approximate submodularfunctions.
We then use this characterization to design a novel neural model for suchfunctions.
Finally, we consider learning submodular set functions under distantsupervision in the form of (perimeter-set, high-value-subset) pairs.
This yields anovel subset selection method based on an order-invariant, yet greedy sampler builtaround the above neural set functions.
Our experiments on synthetic and real datashow that F LEX S UB N ET outperforms several baselines.
Introduction
Owing to their strong characterization of diversity and coverage, submodular functions and theirextensions, viz. , weak and approximate submodular functions, have emerged as a powerful machineryin data selection tasks [39, 72, 29, 57, 77, 54, 5, 16, 64].
We propose trainable parameterized familiesof submodular functions under two supervision regimes.
In the ﬁrst setting the goal is to estimate thesubmodular function based on (set, value) pairs where the function outputs the value of an input set.
This problem is hard in the worst case for poly ( n ) value oracle queries [24].
This has applications inauction design where one may like to learn a player’s valuation function based on her bids [6].
In thesecond setting, the task is to learn the submodular function under the supervision of (perimeter-set,high-value-subset) pairs, where high-value-subset potentially maximizes the underlying functionagainst all other subsets of perimeter-set.
The trained function is expected to extract high-valuesubsets from freshly-speciﬁed perimeter sets.
This scenario has applications in itemset prediction inrecommendation [67, 68], data summarization [42, 2, 4, 11, 63], etc .
1.1 Our contributions
Driven by the above motivations, we propose (i) a novel family of highly expressive neural modelsfor submodular and α -submodular functions which can be estimated under the supervisions of both(set, value) and (perimeter-set, high-value-subset) pairs; (ii) a novel permutation adversarial trainingmethod for differentiable subset selection, which efﬁciently trains submodular and α -submodularfunctions based on (perimeter-set, high-value-subset) pairs.
We provide more details below
Neural models for
submodular functions
We
design
F
LEX S UB N ET
, a family
of ﬂexible neuralmodels
for monotone, non-monotone submodular functions and monotone α -submodular
functions
— Monotone submodular functions.
We model a monotone submodular function using a recursive
neural model which outputs a concave composed submodular function [20, 65, 44] at every step ofrecursion.
Speciﬁcally, it ﬁrst computes a linear combination of the submodular function computed inthe previous step and a modular function and, then applies a concave function to the result to outputthe next submodular function.
— Monotone α -submodular function.
Our proposed model for submodular function rests on theprinciple that a concave composed submodular function is always submodular.
However, to thebest of our knowledge, there is no known result for an α -submodular function.
We address thisgap by showing that an α -submodular function can be represented by applying a mapping ϕ on apositive modular set function, where ϕ satisﬁes a second-order differential inequality.
Subsequently,we provide a neural model representing the universal approximator of ϕ , which in turn is used formodeling an α -submodular function.
— Non-monotone submodular functions.
By applying a non-monotone concave function on modularfunction, we extend our model to non-monotone submodular functions.
Several recent models learn subclasses of submodular functions [45, 66, 68].
Bilmes and Bai [7]present a thorough theoretical characterization of the beneﬁts of network depth with ﬁxed concavefunctions, in a general framework called deep submodular functions (DSF).
DSF leaves open alldesign choices: the number of layers, their widths, the DAG topology, and the choice of concavefunctions (no implementation or experimental details are available).
All-to-all attention has replaceddomain-driven topology design in much of NLP [12].
Set transformers [43] would therefore be anatural alternative to compare against DSF, but need more memory and computation.
Here we explorethe following third, somewhat extreme trade-off: we restrict the topology to a single recursive chain,thus providing an effectively plug-and-play model with no topology and minimal hyperparameterchoices (mainly the length of the chain).
However, we compensate with a more expressive, trainableconcave function that is shared across all nodes of the chain.
Our experiments show that our strategyimproves ease of training and predictive accuracy beyond both set transformers and various DSFinstantiations with ﬁxed concave functions.
Related work is discussed in more detail in Appendix B.
Permutation insensitive differentiable subset selection.
It is common [66, 60, 54] to select asubset from a given dataset by sequentially sampling elements using a softmax distribution obtainedfrom the outputs of a set function on various sets of elements.
At the time of learning set functionsbased on (perimeter-set, high-value-subset) pairs, this protocol naturally results in order-sensitivityin the training process for learning set functions.
To mitigate this, we propose a novel max-minoptimization.
Such a formulation sets forth a game between an adversarial permutation generatorand the set function learner — where the former generates the worst-case permutations to induceminimum likelihood of training subsets and the latter keeps maximizing the likelihood function untilthe estimated parameters become permutation-agnostic.
To this end, we use a Gumbel-Sinkhornneural network [49] as a neural surrogate of hard permutations, that expedites the underlying trainingprocess and allows us to avoid combinatorial search on large permutation spaces.
Experiments.
We ﬁrst experiment with several submodular set functions and synthetically generated examples, which show that F LEX S UB N ET recovers the function more accurately than severalbaselines.
Later experiments with several real datasets on product recommendation reveal that
F LEX S UB N ET can predict the purchased items more effectively and efﬁciently than several baselines.
Design of F LEX S UB N ET
In this section, we ﬁrst present our notations and then propose a family of ﬂexible neural networkmodels for monotone and non-monotone submodular functions and α -submodular functions.
2.1 Notation and preliminary results
We denote V = { 1 , 2 , .., | V |} as the ground set or universal set of elements and S, T ⊆
V assubsets of V .
Each element s ∈ V may be associated with a feature vector z s .
Given a set function
F : 2 V → R , we deﬁne the marginal utility F ( s | S ) :
= F ( S ∪ { s } ) − F ( S ) .
The function F iscalled monotone if F ( s | S ) ≥ 0 whenever S ⊂ V and s ∈ V \ S ; F is called α -submodular withα >
if F ( s | S ) ≥ αF ( s | T ) whenever S ⊆ T and s ∈ V \ T [29, 75, 14].
As a special case, F issubmodular if α = 1 and F is modular if F ( s | S )
=
F ( s | T ) .
Here, F ( · ) is called normalized if
F ( ∅ ) = 0 .
Similarly, a real function f : R → R is normalized if f (0) = 0 .
Unless otherwise stated,we only consider normalized set functions in this paper.
We quote a key result often used for neuralmodeling for submodular functions [20, 65, 44].
Proposition 1.
Given the set function F : 2 V → R + and a real valued function φ : R → R , (i) theset function φ ( F ( · )) is monotone submodular if F is monotone submodular and φ is an increasingconcave function; and, (ii) φ ( F ( · )) is non-monotone submodular if F is positive modular and φ isnon-monotone.
2.2 Monotone submodular functions
Overview.
Our model for monotone submodular functions consists of a neural network whichcascades the underlying functions in a recursive manner for N steps.
Speciﬁcally, to compute thesubmodular function F ( n ) ( · ) at step n , it ﬁrst linearly combines the submodular function F ( n − 1) ( · )computed in the previous step and a trainable positive modular function m ( n ) ( · ) and then, applies amonotone concave activation function φ on it.
Recursive model.
We model the submodular function F θ ( · ) as follows:
[Equation]
where the iterations are indexed by 1 ≤ n ≤ N , { m ( n ) θ ( · ) } is a sequence of positive modularfunctions, driven by a neural network with parameter θ .
λ ∈ [0 , 1] is a tunable or trained parameter.
We apply a linear layer with positive weights on the each feature vector z s to compute the valueof m ( n ) θ ( · ) and then compute m ( n ) θ ( S ) =
(cid:80) s ∈ S m ( n ) θ ( s ) .
Moreover, φ θ is an increasing concavefunction which, as we shall see later, is modeled using neural networks.
Under these conditions, onecan use Proposition 1(i) to easily show that F θ ( S ) is a monotone submodular function (Appendix C).
2.3 Monotone- α -submodular functions
Our characterization for submodular functions in Eq.
(1) are based on Proposition 1(i), which impliesthat a concave composed submodular function is submodular.
However, to the best of our knowledge,a similar characterization of α -submodular functions is lacking in the literature.
To address this gap,we ﬁrst introduce a novel characterization of monotone α -submodular functions and then use it todesign a recursive model for such functions.
Novel characterization of α -submodular function.
In the following, we show how we can characterize an α -submodular function using a differential inequality (proven in Appendix C).
Theorem 2.
Given the function ϕ : R → R + and a modular function m : V → [0 , 1] , the setfunction F ( S ) = ϕ
( (cid:80) s ∈ S m ( s )) is monotone α -submodular for | S | ≤ k , if ϕ ( x ) is increasing in xand d 2 ϕ ( x ) d x
[Equation]
The above theorem also implies that given α = 1 , then F ( S ) =
ϕ ( (cid:80) s ∈ S m ( s )) is monotonesubmodular if ϕ ( x ) is concave in x , which reduces to a particular case of Proposition 1(i).
Once wedesign F ( S ) by applying ϕ on a modular function, our next goal is to design more expressive andﬂexible modeling in a recursive manner similar to Eq.
To this end, we extend Proposition 1 to thecase for α -submodular functions.
Proposition 3.
Given a monotone α -submodular function F ( · ) , φ ( F ( S )) is monotone α -submodular,if φ ( · ) is an increasing concave function.
Here, α remains the same for F and φ ( F ( · )) .
Note that, when F ( S ) is submodular, i.e. , α = 1 , the above result reduces to Proposition 1 in thecontext of concave composed modular functions.
Recursive model.
Similar to Eq.
(1) for submodular functions, our model for α -submodular functionsis also driven by an recursive model which maintains an α -submodular function and updates itsvalue recursively for N iterations.
However, in contrast to F LEX S UB N ET , where the underlyingsubmodular function is initialized with a positive modular function, we initialize the correspondingα -submodular function F (0) ( S ) with ϕ θ ( m (0) θ ( S )) , where ϕ θ is a trainable function satisfying theconditions of Theorem 2.
Then we recursively apply a trainable monotone concave function on
F ( n − 1) ( S ) for n ∈ [ N ] to build a ﬂexible model for α -submodular function.
Formally, we have:
[Equation]
with λ ∈ [0 , 1] .
Here, m θ , ϕ θ and φ θ are realized using neural networks parameterized by θ .
Then,using Proposition 3 and Theorem 2, we can show that F θ ( S ) is α -submodular in S (proven in
Proposition 6 in Appendix C).
2.4 Non-monotone submodular functions
In contrast to monotone set functions, non monotone submodular functions can be built by applyingconcave function on top of only one modular function rather than submodular function (Proposition 1(i) vs. (ii)).
To this end, we model a non-monotone submodular function F θ ( · ) as follows:
[Equation]
where m θ ( · ) is positive modular function but ψ θ ( · ) can be a non-monotone concave function.
Bothm θ and ψ θ are trainable functions realized using neural networks parameterized by θ .
One can use
Proposition 1(ii) to show that F θ ( S ) is a non-monotone submodular function.
2.5 Neural parameterization of m θ , φ θ , ψ θ , ϕ θ
We complete the neural parameterization introduced in Sections 2.2–2.4.
Each model has two typesof component functions: (i) the modular set function m θ ; and, (ii) the concave functions φ θ (Eq.
(1)and (2)) and
ψ θ (Eq.
and
the non-concave function ϕ θ (Eq.
While modeling m θ is simpleand straightforward, designing neural models for the other components, i.e. , φ θ , ψ θ and ϕ θ is nontrivial.
As mentioned before, because we cannot rely on the structural complexity of our ‘network’(which is a simple linear recursion) or a judiciously pre-selected library of concave functions [7], weneed to invest more capacity in the concave function, effectively making it universal.
Monotone submodular function.
Our model for monotone submodular function described in Eq.
(1)consists of two neural components: the sequence of modular functions { m ( n ) θ } and φ θ .
— Parameterization of m θ .
We model the modular function m ( n ) θ : 2 V → R + in Eq.
(1) asm ( n ) θ ( S ) =
(cid:80) s ∈ S θ · z s , where z s is the feature vector for each element s ∈ S and both θ,
z s arenon-negative.
— Parameterization of φ θ .
Recall that φ θ is an increasing concave function.
We model it usingthe fact that a differentiable function is concave if its second derivative is negative.
We focus oncapturing the second derivative of the underlying function using a complex neural network of arbitrarycapacity, providing a negative output.
Hence, we use a positive neural network h θ to model thesecond derivative
[Equation]
Now, since φ θ is increasing, we have:
[Equation]
Here, φ θ ( · ) is normalized, i.e. , φ θ (0) = 0 , which ensures that { F ( n ) θ } in Eq.
(1) are also normalized.
An offset in Eq.
(5) allows a nonzero initial value of φ θ ( · ) , if required.
Note that monotonicity andconcavity of φ θ can be achieved by the restricting positivity of h θ .
Such a constraint can be ensuredby setting h θ ( · ) =
ReLU (Λ ( h ) θ ( · )) , where Λ ( h ) θ ( · ) is any complex neural network.
Hence, φ θ ( · )represents a class of universal approximators of normalized increasing concave functions, if Λ ( h ) θ ( · ) isan universal approximator of continuous functions [30].
For a monotone submodular function of theform of concave composed modular function, we have the following result (Proven in Appendix C).
Proposition 4.
Given an universal set V , a constant (cid:15) > 0 and a submodular function F ( S ) =
φ ( (cid:80) s ∈ S m
( z s ))
where z s ∈ R d , S ⊂ V , 0 ≤ m ( z ) < ∞ for all z ∈ R d .
Then there exists twofully connected neural networks m θ
and h θ 2 of width d + 4 and 5 respectively, each with ReLUactivation function, such that the following conditions hold:
[Equation]
Monotone α -submodular model.
An α -submodular model described in Eq.
(2) has three trainablecomponents: (i) the sequence of modular functions { m ( n ) θ ( · ) } , (ii) the concave function φ θ ( · )and (iii)
ϕ θ ( · ) .
For the ﬁrst two components, we reuse the parameterizations used for monotonesubmodular functions.
In the following, we describe our proposed neural parameterization of ϕ θ .
— Parameterization of ϕ θ ( · ) .
From Theorem 2, we note that ϕ θ ( · ) is increasing and satisﬁes d 2 ϕ θ ( x ) d x≤κ ( α ) d ϕ θ ( x ) d x where, κ ( α ) = 1 k log (1 /α ) .
It implies that
[Equation]
Driven by the last inequality, we have
[Equation]
Parameterizing non-monotone submodular model.
As suggested by Eq.
(3), our model for nonmonotone submodular function contains a non-monotone concave function ψ θ and a modular functionm θ .
We model m θ using the same parameterization used for the monotone set functions.
Weparameterize the ψ θ as follows.
— Parameterization of ψ θ .
Modeling a
generic possibly non monotone) submodular function
requires
a general form of concave function ψ θ which is not necessarily increasing.
The trick is to designψ θ ( · ) in such a way that its second derivative is negative everywhere, whereas its ﬁrst derivative canhave any sign.
For x ∈ [0 , x max ] , we have:
[Equation]
where h θ , g θ ≥ 0 .
Moreover, we assume that (cid:82) ∞
h θ ( b ) d b and (cid:82) ∞
g θ ( b ) d b are convergent.
We usex max in the upper limit of the second integral to ensure that ψ θ is normalized
, i.e. , ψ θ (0) = 0 .
Next,we compute the ﬁrst derivative of ψ θ ( x ) as:
[Equation]
which can have any sign, since both integrals are positive.
Here, the second derivative of ψ θ becomes
[Equation]
which implies that ψ θ is concave.
Similar to h θ ( · ) , we can model g θ ( · ) =
ReLU (Λ gθ ( · )) .
Such arepresentation makes ψ θ ( · ) a universal approximator of normalized concave functions.
As suggestedby Eq.
(3), ψ θ takes m θ ( S ) as input.
Therefore, in practice, we set x max = max S m θ ( S ) .
2.6 Parameter estimation from (set, value) pairs
In this section, our goal is to learn θ from a set of pairs { ( S i , y i ) | i ∈
[ I ] } , such that y i ≈ F θ ( S i ) .
Hence, our task is to solve the following optimization problem:
[Equation]
In the following, we discuss methods to solve this problem.
Backpropagation through double integral.
Each of our proposed set function models consists ofone or more double integrals.
Thus computation of the gradients of the loss function in Eq.
(12) requires gradients of these integrals.
To compute them, we leverage the methods proposed by Wehenkeland Louppe [71], which speciﬁcally provide forward and backward procedures for neural networksinvolving integration.
Speciﬁcally, we use the Clenshaw-Curtis (CC) quadrature or Trapezoidal Rulefor numerical computation of φ θ ( · ) and ψ θ ( · ) .
On the other hand, we compute the gradients ∇ θ φ θ ( · )and ∇ θ ψ θ ( · ) by leveraging Leibniz integral rule [17].
In practice, we replace the upper limit ( b = ∞ )of the inner integral in Eqs.
(5), (8) and (9) with b max = x max during training.
Decoupling into independent integrals.
The aforementioned end-to-end training can be challengingin practice, since the gradients are also integrals which can make loss convergence elusive.
Moreover,it is also inefﬁcient, since for each step of CC quadrature of the outer integral, we need to performnumerical integration of the entire inner integral.
To tackle this limitation, we decouple the underlyingdouble integral into two single integrals, parameterized using two neural networks.
— Monotone submodular functions.
During training our monotone submodular function model in
Eq.
(1), we model φ θ in Eq.
(5) as:
[Equation]
In contrast to end-to-end training of θ where φ θ was realized only via neural network of h θ , here weuse two decoupled networks, i.e. , φ (cid:48)
θ and h β .
Then, we learn θ and β by minimizing the regularizedsum of squared error, i.e. ,
[Equation]
[Equation]
Here, G ( n ) ( S ) = λF ( n − 1) ( S ) + (1 − λ ) m ( n ) θ ( S ) is the input to φ θ in the recursion (1).
Recall that
N is the number of steps in the recursion.
Since φ θ is monotone, we need φ (cid:48) θ ≥ 0 which is ensured byφ (cid:48) θ ( x ) =
ReLU (Λ φ (cid:48)
θ ( x )) .
In principle, we would like to have φ (cid:48) θ ( x ) =
(cid:82) ∞ x
h β ( a ) d a for all x ∈ R .
In practice, we approximate this by penalizing the regularizer values in the domain of interest—thevalues which are fed as input to φ θ .
While there are potential hazards to such an approximation, in our experiments, the beneﬁts outweighed the risks.
The above parameterization involves only single integrals.
The use of an auxiliarynetwork h β allows more ﬂexibility, leading to improved robustness during training optimization.
Theabove minimization task achieves approximate concavity of φ via training, whereas backpropagating through double integrals enforces concavity by design.
Appendix D extends this method forα -submodular and non-monotone submodular functions.
Differentiable subset selection
In Section 2.6, we tackled the problem of learning F θ from (set, value) pairs.
In this section, weconsider learning F θ from a given set of (perimeter-set, high-value-subset) pair instances.
3.1 Learning F θ from (perimeter-set, high-value-subset)
Let us assume that the training set U consists of { ( V, S ) } pairs, where V is some arbitrary subsetof the universe of element, and
S ⊆
V is a high-value subset of V .
Given a set function model F θ ,our goal is to estimate θ so that, across all possible cardinality constrained subsets in S (cid:48) ⊆
V with| S (cid:48)
| = | S | , F θ ( · ) attains its maximum value at S .
Formally, we wish to estimate θ so
that, for allpossible ( V, S ) ∈ U , we have:
[Equation]
3.2 Probabilistic greedy model
The problems of maximizing both monotone submodular and monotone α -submodular functionsrely on a greedy heuristic [51].
It sequentially chooses elements maximizing the marginal gain andhence, cannot directly support backpropagation.
Tschiatschek et al.
[68] tackle this challenge with aprobabilistic model which greedily samples elements from a softmax distribution with the marginalgains as input.
Having chosen the ﬁrst j elements of high-value-subset from perimeter-set V , itdraws the ( j + 1) th element from the remaining candidates in V with probability proportional to themarginal utility of the candidate.
If the elements of S under permutation π are written as π ( S ) j forj ∈ [ | S | ] , then the probability of selecting the elements of the set S in the sequence π ( S ) is given by
[Equation]
[Equation]
Here, F θ is the underlying submodular function, τ is a temperature parameter and S 0 = ∅ .
Bottleneck in estimating θ .
Note that, the above model (16) generates the elements in a sequentialmanner and is sensitive to π .
Tschiatschek et al.
[68] attempt to remove this sensitivity by maximizingthe sum of probability (16) over all possible π :
[Equation]
However, enumerating all such permutations for even a medium-sized subset is expensive bothin terms of time and space.
They attempt to tackle this problem by approximating the mode andthe mean of Pr( · ) over the permutation space Π | S | .
But, it still require searching over the entirepermutation space, which is extremely time consuming.
3.3 Proposed approach
Here, we describe our proposed method of permutation adversarial parameter estimation that avoidsenumerating all possible permutations of the subset elements, while ensuring that the learned parameter θ ∗ remains permutation invariant.
Max-min optimization problem.
We ﬁrst set up a max-min game between a permutation generatorand the maximum likelihood estimator (MLE) of θ .
Here, for each subset S , the permutation generatorproduces an adversarial permutation π ∈ Π | S | which induces a low value of the underlying likelihood.
On the other hand, MLE learns θ in the face of these adversarial permutations.
Hence, we have thefollowing optimization problem:
[Equation]
[Equation]
Differentiable surrogate for permutations.
Solving the inner minimization in (18) requires searching for π over Π | S | , which appears to confer no beneﬁt beyond (17).
To sidestep this limitation, werelax the inner optimization problem by using a doubly stochastic matrix P ∈ P | S | as an approximation for the corresponding permutation π .
Suppose Z S =
[ z s ] s ∈ S is the feature matrix where eachrow corresponds to an element of S .
Then Z π ( S )
≈ P Z S .
Thus, Pr θ ( π ( S ) | V ) can be evaluatedby iterating down the rows of P Z S and, therefore, written in the form Pr θ ( P , S ) .
Thus, we get acontinuous approximation to (18):
[Equation]
(19) so that we can learn θ by continuous optimization.
We generate the soft permutation matrices Pusing a Gumbel-Sinkhorn network [49].
Given a subset S , it takes a seed matrix B S as input andgenerates P S in a recursive manner:
[Equation]
Here, t is a temperature parameter; and, D c and D r provide column-wise and row-wise normalization.
Thanks to these two operations, P k tends to a doubly stochastic matrix for sufﬁciently large k .
Denoting P ∞ =
lim k →∞ P k , one can show [49] that
[Equation]
where the temperature t controls the ‘hardness’ of the “soft permutation” P .
As t → 0 , P tends tobe a hard permutation matrix [9].
The above procedure demands different seeds B S across differentpairs of ( V, S ) ∈ U , which makes the training computationally expensive in terms of both timeand space.
Therefore, we model B S by feeding the feature matrix Z S into an additional neuralnetwork G ω which is shared across different ( V, S ) pairs.
Then the optimization (19) reduces tomax θ min ω (cid:80) ( V,S ) ∈U
log
Pr θ ( ω, S ) , with ω taking the place of P .
Experiments
In this section, we ﬁrst evaluate F LEX S UB N ET using a variety of synthesized set functions andshow that it is able to ﬁt them under the supervision of ( set, value ) pairs more accurately than thestate-of-the-art methods.
These experiments are consistent with recent work on learning submodularfunctions, except that our planted functions are more challenging.
Next, we use datasets gathered from
Amazon baby registry [23] to show that F LEX S UB N ET can learn to select data from (perimeter-set,high-value-subset) pairs more effectively than several baselines.
4.1 Training by (set, value) pairs
Dataset generation.
We generate | V | =10 4 samples, where we draw the feature vector z s for eachsample s ∈ V uniformly at random, i.e. , z s ∈ Unif [0 , 1] d with d =10 .
Then, we generate subsets S of different sizes by randomly gathering elements from the set V .
Finally, we compute the values of F ( S )for different submodular functions F .
Speciﬁcally, we consider seven planted set functions: (i) Log :
F ( S )
= log( (cid:80) s ∈ S 1 (cid:62) z s ) , (ii) LogDet : F ( S ) =
log det ( I + (cid:80) s
∈ S z s z (cid:62) s ) [62, 8], (iii) Facility location (FL) :
F ( S ) =
(cid:80) s (cid:48)
∈ V max s ∈ S z (cid:62) s z s (cid:48) /
( || z s |||| z s (cid:48)
|| )
[50, 18, 13, 52], (iv) Monotone graphcut (Gcut σ =0 . 1 ) :
In general, Gcut σ is computed using F ( S ) :
= (cid:80) u ∈ V,v ∈ S z (cid:62) u
z v − σ (cid:80) u,v ∈ S z (cid:62) u
z v .
It measures the weighted cut across ( S, V \ S ) when the weight of the edge ( u, v ) is computed as z (cid:62)
u
z v [31, 61, 33, 34, 35].
Here, σ trades off between diversity and representation.
We set σ = 0 .
Note that Gcut σ is monotone (non-monotone) submodular when
[Table]
[Equation] [Equation]
[Equation]
Evaluation protocol.
We sample | V | =10000 (set,value) instances as described above and split theminto train, dev and test folds of equal size.
We present the train and dev folds to the set functionmodel and measure the performance in terms of RMSE on test fold instances.
In the ﬁrst fourdatasets, we used our monotone submodular model described in Eq.
(1); in case of Log × Sqrt and
Log ×
LogDet, we used our α -submodular model described in Eq.
(2); and, for Gcut
σ =0 .
8 , we usedour non-monotone submodular model in Eq.
For α -submodular model, we tuned α using crossvalidation.
Appendix E provides hyperparameter tuning details for all methods.
Baselines.
We compare F LEX S UB N ET against four state-of-the-art models, viz.
, Set-transformer [43],
Deep-set [74], deep submodular function (DSF)
[7] and mixture submodular function (SubMix) [66].
Appendix E contains more details about the baselines.
Results.
We compare the performance of F LEX S UB N ET against the above four baselines in termsof RMSE on the test fold.
Table 1 summarizes the results.
We make the following observations.(1)
F LEX S UB N ET outperforms the baselines by a substantial margin across all datasets.
(2) Set–transformer outperforms all the other baselines.
While both Deep-set and Set-transformer ensurepermutation invariant aggregation of the feature vectors { z s | s ∈ S } , Set-transformer allows explicitinteractions between set elements via all-to-all attention layers.
As a result, it outperforms all otherbaselines even without explicitly using the knowledge of submodularity in its network architecture.
Note that, like Deep-set, F LEX S UB N ET does not directly model any interaction between the elements.
However, it explicitly models submodularity or α -submodularity in the network architecture, whichhelps it outperform the baselines.
(3) While, in principle, DSF function family contains that of
F LEX S UB N ET , standard multilayer instantations of DSF with ﬁxed concave functions cannot learnwell from (set,value) training.
[Table]
Next, we compare different variants of our approach: (i) F LEX S UB N ET , trained by decoupling the double integral into independent integrals (Eq.
(14)); (ii) F LEX S UB N ET , trainedusing end-to-end training via backpropagation through double integrals (Section 2.6); and (iii) F LEXS UB N ET with N = 1 , where N is the number of steps of the recursions (1).
Table 2 summarizesthe results which reveal the following observations.
(1) We achieve substantial performance gainvia decoupling into independent integrals.
Although decoupling integrals is an approximation ofend-to-end training, it also provides a more smooth loss surface than the loss on the double integral.(2) Running F LEX S UB N ET for only a single step signiﬁcantly deteriorates performance.
4.2 Training by (perimeter-set, high-value-subset)
Datasets.
We use the Amazon baby registry dataset [23] which contains 17 product categories.
Among them, we only consider those categories where | V | > 50 , where V is the total number ofitems in the universal set.
These categories are: (i) Gear, (ii) Bath, (iii) Health, (iv) Diaper, (v) Toys,(vi)
Bedding, (vii) Feeding, (viii) Apparel and (ix) Media.
They are also summarized in Appendix F.
[Table]
Evaluation protocol.
Each dataset contains a universal set V and a set of subsets S = { S } .
Eachitem s is characterized by a short textual description such as “bath: Skip Hop Moby Bathtub Elbow
Rest, Blue : Bathtub Side Bumpers : Baby”.
From this text, we compute z s using BERT [12].
Wesplit S into equal-sized training ( S train ), dev ( S dev ) and test ( S test ) folds.
We disclose the trainingand dev sets to the submodular function models, which are trained using our proposed permutation adversarial approach (Section 3).
Then the trained model is used to sample item sequence S (cid:48) with preﬁx S (cid:48)≤ K = { s 1 , . . .
, s K } .
Here, s i is the item selected at the i th step of the greedy algorithm.
Weassess the quality of the sampled sequence using two metrics.
Mean Jaccard coefﬁcient (MJC).
Given a set T in the test set, we ﬁrst measure the overlap between Tand the ﬁrst | T | elements of S (cid:48) using Jaccard coefﬁcient, i.e. , JC ( T ) =
| S
(cid:48)≤| T | ∩ T | / | S (cid:48)≤| T | ∪ T | andthen average over all subsets in the test set, i.e. , T ∈ S test to compute mean Jaccard coefﬁcient [32].
Mean NDCG@10.
The greedy sampler outputs item sequence S (cid:48) .
For each T ∈ S test , we compute the
NDCG given by the order of ﬁrst 10 elements of S (cid:48) , where i ∈ S (cid:48) is assigned a gold relevance label1 , if i ∈ T and 0 , otherwise.
Finally, we average over all test subsets to compute Mean NDCG@10.
Our method vs. baselines.
We compare the subset selection ability of F LEX S UB N ET against severalbaselines which include two trainable submodular models: (i) Deep submodular function (DSF) and(ii)
Mixture of submodular functions (SubMix) described in Section 4.1; two popular non-trainablesubmodular functions which include (iii) Facility location (FL) [50, 18], (iv) Determinantal pointprocess (DPP) [8]; and, a non-submodular function (v) Disparity Min (DisMin) which is oftenused in data summarization [11].
Here, we use the monotone submodular model of F LEX S UB N ET .
Appendix F contains more details about the baselines.
We did not consider general purpose setfunctions, e.g. , Set-transformer, Deep-set, etc., because they cannot be maximized using greedy-likealgorithms and therefore, we cannot apply our proposed method in Section 3.
Results.
Table 3 provides a comparative analysis across all candidate set functions, which shows that:
[Figure]
(i) F LEX S UB N ET outperforms all the baselines across all the datasets;
(ii) DSF is the second best performer across all datasets; and, (iii) the
performance of non-trainable set functions is
poor, as
they are not trained
to mimic the set selection process.
Recall our key motivation was totrain parameters invariant to the order of the elements of the input subset.
Tschiatschek et al.
[68] achieve this goal using a Monte Carlo average ofthe likelihood over many samples.
In Figure 4, we compare our methodwith theirs for a representative dataset (Gear), which shows that our method is ≥ 4 × faster.
Conclusion
We introduced F LEX S UB N ET : a family of submodular functions, represented by neural networksthat implement quadrature-based numerical integration, and supports end-to-end backpropagationthrough these integration operators.
We designed a permutation adversarial subset selection method,which ensures that the estimated parameters are independent of greedy item selection order.
Onboth synthetic and real datasets, F LEX S UB N ET improves upon recent competitive formulations.
Ourwork opens up several avenues of future work.
One can extend our work for γ -weakly submodularfunctions [15].
Another extension of our work is to leverage other connections to convexity, e.g. , the
Lovasz extension [46, 1] similar to Karalias et al.
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