Unsupervised Learning for Combinatorial Optimization with Principled Objective Relaxation
Authors: Haoyu Peter Wang
Nan Wu
Hang Yang
Cong Hao
Pan Li
Venue: NeurIPS 2022 Accept (2022)
Keywords: learning for combinatorial optimization
model-based optimization
graph neural networks

Abstract: Using machine learning to solve combinatorial optimization (CO) problems is challenging, especially when the data is unlabeled. This work proposes an unsupervised learning framework for CO problems. Our framework follows the standard relaxation-plus-rounding approach and adopts neural networks to parameterize the relaxed solutions so that simple back-propagation can train them end-to-end. Our key contribution is the observation that if the relaxed objective satisfies entry-wise concavity, a low optimization loss guarantees the quality of the obtained integral solutions. This observation significantly generalizes the applicability of the previous framework inspired by Erdos' probabilistic method (Karalias & Loukas, 2020). Our framework is particularly suitable to guide the design of objective models in the applications where the objectives are not given explicitly while requiring being modeled and learned first. We evaluate our framework by solving a synthetic graph optimization problem, and two real-world applications including resource allocation in circuit design and approximate computing. Our framework largely outperforms the baselines based on reinforcement learning and Gumbel-softmax tricks. 

Unsupervised Learning for Combinatorial Optimization with Principled Objective Relaxation
Haoyu Wang Purdue University wang5272@purdue.co edu
Nan Wu
Hang Yang
UCSB
Georgia Tech.
nanwuQucsb.edd edu
hyang628ggatech. edu
Cong Hao Georgia Tech. callie. hww.gatech. edu
Pan Li
Purduc University
panli@purdue- edu
Abstract
Using machine learning to solve combinatorial optimization (CO) problems is challenging, especially when the data is unlabeled. This work proposes an unsu- pervised learning framework for CO problems. Our framework follows standard relaxation-plee plusooooddd-----iggggg approach and adopts neural networks to parameterize the relaxed solutions that simple back preck--rppagttion end. Our key contribution the observation that if the relaxed objective satisfies entry-wise concavity, low optimization loss guarantees the quality of the final integral solutions. This observation significantly broadens the applicability of the previous framework inspired by Erdōs' probabilistic method [T]. In particular, this observation can guide the design of objective models in applications where the objectives
train the model end-to-
given explicitly while requiring being modeled in prior. We evaluate our framework by solving synthetic graph optimization problem, and two real-world applications including resource allocation circuit design and approximate computing. Our framework largely outperforms the baselines based on naïve relaxation, reinforcement learning, and Gumbel-softmax tricks.
1 Introduction
Combinatorial optimization (CO) with the goal of finding the optimal solution from discrete space is fundamental problem in many scientific and engineering applications 4 Most CO problems NP-complete. Traditional methods efficient in practice often either depend on heuristics or produce approximation solutions. Designing these approaches requires considerable insights into the problem. Recently, machine learning has paved new way to develop CO algorithms, which asks to use neural networks (NNs) to extract heuristies from the data (5 Several learning for CO (LCO) approaches have therefore been developed, providing solvers for the problems including SAT 88 TO mixed
integer linear programming 13), vertex covering [14]] and routing problems [16 23
Another promising way to use machine learning techniques is to learn proxies of the CO objectives whose evaluation could be expensive and time-consuming 22 For example, optimize hardware/system design, evaluating the objectives such as computation latency, power efficiency (28). and resource consumption [1 requires running complex simulators. Also, in molecule design, evaluating the objective properties such as protein fluorescence or DNA binding affinity needs either costly simulations living experiments (3). In these cases, proxies of the objectives can be learned first to reduce the evaluation cost [3 and then optimize these proxies to solve the design problems. Later, we name the CO problem with proxy objectives as Proxy-based CO (PCO) problems.
OOr code and the datasets are available at:
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

Our Principle: m
((X:0)
Example:
Tuelankooees related
min X1:C3
Unde confguratio::
Ne
gi (C)x, so(C),gg
Reasstionn
arC 91(
C- (61
min (+(R;C) Sem
Chere (c
anrryaaten Concave and wace Opimum rcoveryyeeluliiiii under difkere C
WD Coverant
(iatyonac Cancare
Dndemminietic
(i(C)) = (380- 100 305)
Rounding
[x(C)= (580 10C, 903)
9a(C)) (30) + 302)
MAR:cH
sx(C) (50) 50s)/33 /33//333/ 60
Gumbch-aufirax
Relasation
coustoit!
dOty Chump
(For urXoowa,(XX.C). build cnzy-wisc concare NNS proxies)
principle (for the case without constraints as an illustration) and an example. Consider an optimization objective ((X; C), X E {0, 1J". Here, is the problem configuration such as an attributed graph in a graph optimization problem. We relax the objective to an entry-wise concave fx C), X E [0, 1]" The soft solution by minimizing discrete solution (in Def. D with performance guarantee. When C(; is not explicitly given, we will learn its relaxed proxies f, by NN. This corresponds to PCO problem. In the toy example on the right, we first learn different relaxed proxies f, 's with or without the entry-wise concave constraint. We compare their landscapes in the top-right figure. We further optimize both proxies and round the obtained soft solutions to integral solutions. The bottom-right figure shows the optimum-recovery probabilities of different methods under different C's.
Figure 1:
The entry-wise concave
(x(x;) C) will be rounded
Moreover, we claim that learning for PCO is often of greater significance than learning for traditional CO problems because commercial CO solvers such as Gurobi cannot applied in PCO due to the missing closed-form objectives and the lack of heuristics for such proxy objectives. Generic solvers such simulated annealing (35) may be applied while they are extremely slow.
In this work, we propose an unsupervised LCO framework. Our findings are applied to general CO problems while exhibiting extraordinary promise for PCO problems. Unsupervised LCO has recently attracted great attentions 91] 20,36 due to its several great advantages. Traditional supervised learning often gets criticized for the dependence on huge amounts of labeled data (38). Reinforcement learning (RL) on the other hand suffers from notoriously unstable training [39]. In contrast, unsupervised LCO is superior in its faster training, good generalization, and strong capability of dealing with large-scale problems [T]. Moreover, unsupervised learning has never been systematically investigated for PCO problems. Previous works for PCO problems, e.g., hardware design 30].1], were all based on RL. A systematic unsupervised learning framework for PCO problems is still under development.
Our framework follows relaxation eelaxation-plu--eeee plus-rounding approach. We optimize a carefully-designe continuous relaxation of the cost model (penalized with constraints if any) and obtain soft solution. Then, we decode the soft assignment to have the final discrete solution. This follows a common approach to design an approximation algorithm 41) However, the soft assignment here is given by a NN model optimized based on the historical (unlabeled) data via gradient descent. Note that learning from historical data is expected to facilitate the model understanding the data distribution, which helps with extracting heuristics, avoiding local minima and achieving fast inference. An illustration of our principle with a txy-example is shown in Fig. I We also provide the pipeline to empirically evaluate our elaxation-plus-rounding principle in Fig. 2
Our method shares similar spirit with (T] while making the following significant contributions. We abandon the probabilistic guarantee in [T), because it is hard to use for general CO objectives, especially those based on proxies. Instead, design a deterministic objective relaxation principle that gives performance guarantee. We prove that if the objective relaxation is entry-wise concave w.r.t the binary optimization variables, low-cost soft solution plus deterministic sequential decoding guarantees generating a valid and low-cost integral solution. This principle significantly broadens the applicability of this unsupervised learning framework. In particular, it guides the design of model architectures to learn the objectives PCO problems. We also justify the wide applicability of the entry-wise concave principle in both theory and practice.
We evaluate our framework over three PCO applications including feature-based edge covering node matching problems, and two real-worke applications, including imprecise functional unit assignment

---

in upproximate computing (AxC) 42 46 and resource allocation in circuit design [2] In all
three applications, our framework achieves significant performance boost compared to previous RL-basedd approaches and relaxed gradient approaches based on the Gumbel-softmax trick [47 49 The datasets for these applications are also provided in our github repository.
1.1 Further Discussion on Related Works
Most previous LCO approaches are based RL 13 16 22 50 52 supervised learning
[2] as these two frameworks do not hold any special requirements on the formulation CO problems. However, they often suffer from the issues of training instability and subpar generalization. Previous works on unsupervised learning for CO have studied satisfaction problems [9 53)
Applying these approaches to general CO problems requires problem reductions. Others have considered max-cut (37] and TSP problems (20), while these works depend on carefully selected problem-specific objectives. The work most relevant to ours is (T) and we give detailed comparison in Sec. 3 Note that all previous works on unsupervised learning for CO do not apply to PCO as they need an explicit objective to manipulate. For PCO problems, previous studies focus on how to learn more generalizable proxies of the objectives. such as via Bayesian learning (5455) and adverserial training [56] Once proxies are learned, direct objective relaxation plus gradient descent (56), or RL [I] or general MIP solvers with reformulations of the objectives [58] 59) have been often adopted. Studying generalization of proxies is out of the scope of this work while we conjecture that our suggested entry-wise concave proxies seem smoother than those without constraints (See Fig. and thus have the potential to achieve better generalization.
Preliminaries and Problem Formulation
In this section, we define several useful concepts and notations.
Combinatorial Optimization (CO), Let C' E C denote a data-based configuration such as a graph with weighted edges. Let Ω be a finite set of all feasible combinatorial objects and each object has binary vector embedding X = (X:)ı<ign E {0, 1)". For example, in the node matching problem, each entry of X corresponds an edge denote whether this edge is selected not. Note that such binary embeddings are applicable even when the choice is not naturally binary: Choosing at most one element from tuple (,)3 can be represented as 3-dim binary vector (X1, X2. Xx)) with the constraint X1 + X2 + X3 < 1. W.l.o.g. we assume an algebraic form of the feasible set Ω A [X E [0, C); < 1} where g(X;C) 2 0 for all X E 10, 11 For notational simplicity, we only consider one inequality constraint while our later discussion Sec. and our case studies in Sec. 3ec.5mmay contain multiple inequalities. Given a configuration C' and a constraint Q
a combinatorial optimization (CO) is to minimize a cost f(:: C') by solving
min J(X:C), s.t. o(X:C)= 1. XE(0,1)
Proxy-based CO (PCO). In the many applications, the cost or the constraint may not be cheaply evaluated. Some proxies of the cost ff or the constraint g often need to be learned from the historical data. With some abuse of notations, we interchangably use f (g. resp.) to denote the objective (the constraint, resp.) and its proxy.
Learning for CO/PCO (LCO). A LCO problem is to learn an algorithm As(.) C + {0, 1)",, say neural network (NN) parameterized by 0 to solve CO or PCO problems. Given configuration C' C, we expect As to (a) generate a valid solution X = As(C) E lt and (b) minimize f(X; C.).
There are different approaches to learn Ag. Our focus is unsupervised learning approaches where given configuration C, ground-truth solution X is used during the training. 0 can only be optimized just based on the knowledge of the cost and the constraint, or their proxies. Note that in this work, for PCO problems, the proxies are assumed to be trained based ground-tru tuuth values of f(;; C¹) given different (X, C) pairs, which is supervised. Unsupervised learning in this work refers to the way to learn Ao (C).
Erdös' Probabilistic Method (EPM). The EPM has recently been brought for LCO I Specifi- cally, The EPM formulates Ag(C") randomized algorithm that essentially gives probabilistic
Normalization (g(:: C) 9mmi)/ to Imin) where I min ( (X;C) and Jmin e (1.1)" 9(X; C) always satisties the property. g Jmni often can be easily estimated in practice.
3

---

distribution over the solution space {0, 1}", which solves the optimization problem:
min E [[((; C)), where 1(X; C) - + X ~ Ao((C) and , > 0. (2)
Karalias & Loukas proved that with 3 > maxxER I f(X; C) and small expected loss E [(((, C)] < 8, sampling sufficiently large number of X ~ AC( guarantees the existance of a feasible X E Ω that achieves the cost f(X: C) s E [[X C'] [1]. Although this guarantee makes EPM intriguing.
applying EPM in practice is non-trivial. We will explain the challenge in Sec. [3.] which inspires our solutions and further guides the objective design for general CO and PCO problems.
3 The Relaxation Principle for Unsupervised LCO
In this section, start with the practical issues of EPM. Then, we introduce our solutions by proposing a relaxation principle of the objectives, which gives performance guarantee for general practical unsupervised LCO.
3.1 Motivation: The Practical Issues of EPM
Applying EPM in practice has two fundamental difficulties. First, optimizing 0 in Eq. is generally hard as the gradient dX (note here each cntry of X is binary) may not exist so the chain rule cannot be used. We discuss the potential solutions to this problem in Sec. 3.4 Second, EPM needs to sample a large number of X ~ AO(C') for evaluation to achieve the performance guarantee in [ This is not acceptable where the evaluation per sample is time-consuming and expensive.
do
So, in practice, Karalias & Loukas consider deterministic method. They view Ag(C) E [0. 1]" as the parameters of Bernouli distributions to generate the entries of X so E[x] AC). First, they optimize ming ((Aa(C) (C),, C) instead of ming [E[ C)), and then, sequentially round the probability Ao(C') to discrete X E {0, 1 by comparing conditional expectations, e.g., IE[((X,) C 0) v.s. E,((X, [ = 1] to decide X1. However, such conditional expectations cannot be efficiently computed unless one uses Monte-Carlo sampling or / has special structures as used in the two case studies in [ i.c., max-clique and graph-partition problems. However, what special structures are needed has not been defined, which blocks the applicability of this framework to general CO problems, especially for the PCO problems where the objectives are learned as models.
3.2 Our Approach: Relaxation plus Rounding, and Performance Guarantee
Our solution does not use the probabilistic modeling but directly adopts approach. We optimize a relaxation of the objective I, and obtain a soft solution X E [0, 1]". Then, we deterministically round the entries in X to solution X in the discrete space 10, Note that throughout the paper, we use X to denote a soft solution and X to denote a discrete solution. The question is whether the obtained solution may still achieve the guarantee EPM does. Our key observation is that such success essentially depends on how to relax the objective 1.
Therefore, our first contribution beyond | is to propose the principle (Def. to relax general costs and constraints. With this principle, the unsupervised LCO framework can deterministically yield valid and low-cost solutions (Thm. 1 as the EPM guarantees, and is applied to any objective l.
First, we introduce the pipeline. Consider a relaxation of a deterministic upper bound Eq.
min (0: A fx(:C))+ (x: C), where X Ao(C) [0, 1]" 3 >>>0 0.
Here f+(::C) [0. 11" + R is the relaxation of f(: C),, which satisfies fx:: = f(X: C) for X E 10.1)" The relation between the constraint and its relaxation 9r- is similar, i.c., gr (X; C*) = C) for X E 10". Here, we also use the fact that gr (X; C*) provides a natural upper bound (((:0)21 & CX: for X E (0. 1}" given the normalization of gXx; C*) adopted in Sec. 2
Now, suppose the parameter 0 gets optimized so that (0; C) is small. Further, we adopt the sequential rounding in Def. 1 to adjust the continuous solution X Ao(C)) to discrete solution X. Definition 1 (Rounding). Given continuous vector X E [0. 1]" and an arbitrary order of the entries, w.o.l.g., = 1, ..., n, round X; into 0 or 1 and fix all the other variables un-changed. Set X; = arg m fr fr(XX,,., X, Xit1, .... Xni C*) + (gr(X1, X,-1, X,+1, X,; C), replace X, with X; and repeat the above procedure until all the variables become discrete.

---

Note that our rounding procedure does not need to evaluate any conditional expectations [] [)] which EPM []] requires. Instead, we ask both relaxations fr and 9r sat- isfy the principle in Def. 2 With this principle, the pipeline allows achieving a valid and low-cost We leave the proof in Appendix [A.]
solution X, as proved Theorem |
Definition (The Entry-wise Concave Principle). For any C C. hy(:; : [0. R is
entry-wise concave for any [0, and any X, X E [0, 1)th that are only different in one entry,
(x; C) + (1 - 7),, C) & hr (1 + (1 - C))
Note that entry-wise concavity is much weaker than concavity. For example, the function x2) Xel)((X X2 E R is entry-wise concave but not concave.
Theorem 1 (Performance Guarantee). Let B > maxxen f(X;C) and minxes f(X; 2 0 in 33 Suppose the relaxed cost fr and constraint 9r entry-wise concave, and the learned parameter 0 achieves (.;0C) < B. Then, rounding (Def.
[ the relaxed solution X = Ao(C)
generates valid discrete solution X such that J(X; C) (0:
When there are multiple constraints (X;C) < for 1, we may use relaxation 3 E, ( (X;C) as the penalty term in Eq.(3). where g. is relaxation of g( It can be shown that if E3 gr satisfies the condition of entry-wise concavity, the guarantee of Thm. still applies.
3.3 The Wide Applicability of Entry-wise Concave Relaxations
We have introduced the entry-wise concave principle to relax the objective to associate our framework with performance guarantee. The question is how widely applicable this principle could be.
Actually, every function with binary inputs can be relaxed as an entry-wise affine (also called multi- linear) function with the exactly same values at the discrete inputs, as shown in Theorem 2 Note that entry-wise affinity is a special case of entry-wise concavity. In Sec. [ we will provide the design of NN architecture (for PCO) and math derivation (for CO) that guarantee formulating an entry-wise concave function. Note that the objectives for max-clique and graph-partition problems used in [ are essentially entry-wise affine.
Theorem 2 (Wide Applicability). For any binary-input function h(-) 1)
1)" - R, there exists relaxation (-) : [0, 11m > IR such that (a) h, (X) = h X E {0. 11"h and (b) hy is entry-wise affine, ie., for any 7 [0, 1] and any X, X' E [0, 1|ah that are only different in one entry.
(X) + 71
= hr x + (1 (1)
Proof sketch. Set (X) = (x. ((X) I[ X3, (1- (1-X,) which satisfies (a) and (b). Note that we suppose that Xy- = 1 for any X, [0, 1]. The detailed proof is in Appendix A.2
Although Theorem shows the existence of entry-wise affine relaxations, the constructed repre- sentation in the proof depends on higher-order moments of the imput entries, which make it hard to be implemented by model, say NN architecture. However, concave functions is able to implicitly generate higher-order moments via representations based on low-order moments. For example, when n = 2, we could use the composition of -Relu-)) and affine operators (only sst-order moments) to achieve universal representation (See Prop. P and the proof in Appendix A.3). For general n. leave as future study.
claim that using entry-wiss
Proposition 1. For any binary- input function h(X1, X2), there exists parameters (wyy } such that an entry-wise concave function h, (X,, X2) Wuo (x, X2) = hxX,, any X1, E {0, 1}.
D Relu( X + W22 + wap) satisfies
3.4 Discussion: Methods to Directly Optimize the Randomized Objective in EPM Eq.
The naive way to optimize the randomized objective in Eq. 2 without worrying about the specific form of the objective is based on the policy gradient in RL via the logarithmic trick, i.c., estimating the gradient di via (f(X; B1 log P(X) by sampling X ~ Ao().. However, the policy gradient suffers from notoriously large variance [39] and makes RL hard to converge. Therefore, methods such as actor critic (60) or subtracting some baselines ((X;)) b (61) have been proposed.
5

---

Ramdom line
(i Label ((X
Randinnly Sumpled confituaaioon
Supervised Training
concerelaftine
entry-aiie
(((X;C)
A Testing Instance
passy
3-18;0)
Label of Ca Co
Ao
Proxy Relaxationin PCO Problems
Unsupervised Training MARCC) + (g)(&;C)
Relexed Solution
Rounding
Expliciily Write
concrvelsffirs procy 9-13:0)
cuiry-wie
[(1&,))
Proxy Relaxation problems
Discrete Solution
Training
Testing
The pipeline of empirical evaluation in Sec. and Sec. on our delaxatioo-ann-roumaee principle. For a PCO task whose cost function or constraints are unknown, we first build NNs with the entry-wise concave (CON)/ affine (AFF) structure to learn their proxies (f+(X; C) gr (X; (X;C)) via supervised learning. For traditional CO tasks, we follow our principle and explicitly write their entry-wise affine costs and constraints relaxation. Then, we learn the algorithm As to optimize the relaxed loss function f-(X; C) + (X; C') in an unsupervised manner. After training, with any unseen testing instance C", we run our Ag infer the relaxed soft solution X, and then round the soft solution into discrete solution X with performance guarantee.
Figure 2:
RL
Gumbel-sullnn
Ours
Another way to solve (q.(2) is based on repa- rameterization tricks to reduce the variance of
Objective
Limit
Limit Gumbel Trick
Entry- Concave
Optimizer
Log Trick
No Limit
gradients 662 63 Specifically, we set the en- tries of output X (C) [0, 1]"h the parameters of Bernoulli distributions to gener- ate X, i.e., X; Bern(X;), for 1 < i S n. To make dX,/dX; computable, we may use the Gumble-softme this approach suffers from two issues. First, the estimation of the gradient is biased. Second, Ao(C") is essentially a randomized algorithm, sampling sufficiently many X AC()) is needed guarantee valid and low-cost solution. How- ever, such evaluation is costly as discussed in Sec. 3.1 So, empirically, we can also compare Ac(CC) with threshold to determine X, which does not have performance guarantee. We compare different aspects of RL, Gumbel-softmax tricks and our relaxation approach in Table
Inference
Sampling
Sampling
Deter. Rounding
Tmin. Time
Slow
Fast
Fast
Convergence
Hard
Medium
Easy
Time
Slow
Slow
Fast
Table 1: The comparison among RL (policy gradi-
ent), Gumbel-softmax methods and our principled objective relaxation. Our methods are in need of much less training time and inference time.
trick (47-4). However,
4 Applying Our Relaxation Principle to Learning for CO
First, we test our relaxation principle in learn- ing for CO (max clique) task, where we can explicitly write both the cost functions and the constraints into an entry-wise affine form. In this case, our framework and EPM [1] share the same pipeline, though the relaxation principle and the deterministic performance guarantee are firstly proposed in this work. The entry-wise
Method
Twitter
RBlest
Badloss+R EPM [T] AFF (OUTs)
0.768 0.203 (0.17s/g)
0.702 0.102 (0.333/g)
0.924 0.113 (7sgg)
0.788 + 0.065 (0.23s/g)
0.926 0.113 17s/g)
0.787 = 0.065 (0.33s/g)
Table 2: Approximation Rate in the max clique. 's/g" denotes the average time cost per graph.
X X,X,. Here, we use a real-world dataset (Twitter (64)) and a synthetic dataset (RBtest (T). and show the experiment results in Table. We follow the settings in [ and use 6:2:2 dataset split for training, validating and testing, each test instance runs within 8 seeds. Our entry-wise affine pipeline achieves almost the same performance as EPM. To show the importance of the relaxation principle, we also propose 'Badloss+R'. This baseline imposes trigonometric functions to the original entry-wise affine functions [x; C) = f. (sin((9) X /2); C)2 of (X; C) gr (sin (9 X /2); C"), where sin(.) operates on each entry of the input vector. Note that the relaxed functions also match the original objectives at discrete points, i.e., CC f-(X;
affine objective relaxation of the max clique is -(3 -13+1)
f(X; C) when X E (0, 1}, while with different relaxations. The poor performance of 'Badloss+R' reveals the importance of our principle for relaxation.

---

5 Applying Our Relaxation Principle to Learning for PCO
In this section, we apply our relaxation principle three PCO applications: (I) feature-based edge covering & node matching, (II) resource allocation in circuit design, and (III) imprecise functional unit assignment in approximate computing. All the applications have graph-based configurations C. So later, we first introduce how to use graph neural networks (GNNs) to build proxies that satisfy our relaxation principle. Such GNN-based proxies will be used as the cost function relaxation fr in all the applications. Our principle can also guide the relaxation of explicit CO objectives. The constraints in applications (IIIII) are explicit and their relaxation can be written into the entry-wise affine form. The constraint in (II) needs another GNN-based entry-wise concave proxy to learn.
5.1 GNN-based Entry-wise Concave Proxies
We consider the data configuration C' as an attributed graph (V, E, Z) where V is the node set, E S V V is the edge set and Z is the node attributes. We associate each node with a binary variable and group them together X :E {0, 1] IVV where for each V. Xx, 1 indicates the choice of the node v. Note that our approach can be similarly applied to edge-level variables (see Appendix C.3). which is used in application (I). Let X still denote the relaxation of X.
To learn discrete function h: {0, 131 x C - IR, we adopt a GNN as the relaxed proxy of h. We first define latent graph representation in Rf whose entries are all entry-wise affine mappings of X.
Latent representation:
<(XCC) W + Σ U
E
QuiuXoX
(4)
VEV
(,,)V,
where Win is the graph representation, U, are node representations and Qu,u are edge representations. These representations do not contain X and are given by GNN encoding C. Here, we consider at most 2nd-order moments based on adjacent nodes as they can be easily implemented via current GNN platforms 66). Then, we use to generate entry-wise wise affine & concave proxies as follows.
Eitry-wise Affine Proxy (AFF):
C) = (w", ((x;())))) C)).
(5)
Eitry-wise Concave Proxy (CON):
h₂ ,x(xCC)= (w", ( ReLUU +b.
(6)
where wh, wf E RR,, E IR are learnt parameters and w 2 0 guarantees entry-wise concavity. Other ways to implement GNN-based entry-wise concave proxies are also introduced in Appendix
5.2 The Setting up of the Experiments
Training & Evaluation Pipeline. In all the applications, we
adopt the following training& & evaluation pipeline. First, we
Baseline
fr.9r
Av
Inference
have a set of observed configurations D, C C. Each C E Dj is
Naive
no limit
no limit
munding
paired with one X {0, 13" We use the costs f(X, C') (and
RL
no limit
RL
sampling
constraints g(X, C)) to train the relaxed proxies fx( C*) (and
GS-TH+S
no limit
sampling
TH+R
no limit
GS
rounding
g. (X, C). cannot be derived explicitly), where the relaxed
Table 3: The baselines in the paper.
proxies follow either (5 (named AFF) or Eq. (named
CON). Then, we parameterize the LCO algorithm Ao(C) E
[0, 1]" via another GNN. Based on the learned (or derived) fr and Jr. we optimize 0 by minimizing Eced, (+: C),, where I, is defined according to (3. We will split D2 into a training set and a validation set for hyperparameter-tuning of both proxies and Ag. We have another set of configurations D2 C C used for testing. For each E D2, we use the relaxation X = AC(C') plus our rounding to evaluate the learned algorithm An(-). We follow [T] and do not consider fine-tuning Ag over the testing dataset D2 to match the potential requirement of the fast inference.
We consider 4 common baselines that is made up of different learnable relaxed proxies fr. Ir. algorithms Ag and inference approaches shown in Table 3 For the proxies fr, 9, for baselines, we apply GNNs without the entry-wiss concave constraint and use X as one node attribute while keeping all other hyper- parameerrsssrss exactly the same as CON other than the way to deal with the discrete variables to make fair comparison (See details in Appendix. C); For the algorithm Ag. RL
Baselines.
we provide the Gumbel-softmax trick based methods (GS-Tr) [48 49), the actor-critic- method (60) (RL) and the naive relaxation method (Naive); For the inference approaches, we consider Monte Carlo sampling (S) and our proposed rounding (R) procedure. Although the baselines adopt proxies that are different from ours, we guarantee that their proxies approximate the ground-truth
7

---

47
6
3
4
3
7
6
3
9
3
17
6
3 4
1 8 4
4 7
3
7
6
3
7
2
9
3
7
4
9
1
3
7
2
3
7
2
8
3
2
4
D
3
2
4
1
0
3
3
2
D
0
3
2
4
D
7
4
4
8
7
6
5
1
4
7
9
5
4
4
8
7
6
5 4
4
(a) One Config.
(b) Optimal X
(c) AFF (Ours)
(d) GS Trick + R
Figure 3: The visualization for node matching in Application I. Our method avoids large multiplica-
tions 87* 96 and 94 82 where GS-Trick cannot, and generate solution different but close to OPT.
9 over the validation dataset least no worse than ours. In application II, we also consider two non-lcarnable algorithms to optimize the proxies without relaxation constraints, simulated annealing (SA) 335 and genetic algorithms (GA) [67 [ In application III, we put all of the required AxC units either close to the input (C-In) or close to the output (C-Out) of the approximating computing circuit as additional baselines. More details of the experiments setups and hyperparameter tuning can be found in Appendix We also obtain the optimal solutions (OPT) for applications and III via brute-force search for comparison.
5.3 Application Feature-based Edge Covering & Node Matching in Graphs
This application is inspired by [69]. Here, each configuration C' is a 4 x 4 grid graph whose node attributes are two-digit images generated by random combinations of the pictures MNIST (70). We associated each edge with variables X {0, 1] [E] The objective is the sum of edge weights f(X;C) = =EEEF W.X, where We is unknown in prior and needed to be learned. The ground truth of we is multiplication of the numbers indicated by the images the two adjacent nodes. We adopt ResNet-50 (71) (to refine node features) plus GraphS (72) to encode C. We consider using both 45 and Eq. 6 formulate the relaxed cost fx(X; C). Training and validating fr are based on 100k randomly sampled C' paired with randomly sampled X. Note that 100k is much smaller than the entire space {0. 1}|8) x of size 224 x 10016
Next, as the constraint here is explicit, we can derive the relaxation of the constraints for this application. First, the constraint relaxation of the edge covering problem can be written as
Edge Covering Constraint: 9r I
VEV E:Ve
Each production term in Eq. indicates that for each node, least one edge is selected. We can easily justify that g, is entry-wise affine and Ω {X {0, 13 CX: < 1} exactly gives the feasible solutions to the edge covering problem.
Similarly, we can derive the constraint for node matching by adding a further term to Eq.
Node Matching Constraint: gr (X:C) VEV cceEc
x.)+
II
X Xezl
(8)
Here, the second term indicates that no two edges adjacent to the same node can be selected. This is a case with constraints while combine them together. We can easily justify that g. is entry-wise affine and Ω {X 10, (X; < 1} groups exactly the feasible solutions to the node matching problem.
Note that our above derivation also generalizes the node-selection framework in [ to edge selection. With the learned fr and the derived gr. we further train and validate As over the 100k sampled
(X, CY's and test on another 500 randomly sampled C's.
Method
Edge covering
Node matching
Evaluation. Thble the evaluation results. In the GS-Tr+S method, the number of sampling is set to 120 (about 2.5 times the inference time of our deterministic rounding). Note that for node matching, GS-Tr+S could hardly sample feasible node matching solution within 120 samples. The experiment results show that our principled proxy relaxation exceeds the other baselines on both tasks. Also, we observe that AFF outperforms CON, which results from the fact that fXx; C) these two problems are naturally in entry-wise affine forms
Naive+R RL
68.52
429.12
51.29
426.97
GS-Tr+S
63.36
GS-Tr+R
46,91
429.39
CON(ours)
49.59
422.47
AFF(ours)
44.55
418.96
OPT(gt)
42.69
416.01
Table 4: Performance on application (graph optimization).
8

---

The prooictio valae (DSP)
(DSP)
The cac.ocoooo San
ground truth valuo (DSP)
The ground (DSP)
The ground (DSP)
Figure Comparing different proxies for learning DSP usage. Left, no constraint; Middle, entry-wise
affine constraint (Eq. (5); Right, entry-wise concave constraint (6)
with low-order (Ist-order) moments. One instance of node matching randomly selected from the test set is shown in Fig. 3 More visualization results can be found in Fig. G in the appendix.
5.4 Application II: Resource Allocation in Circuit Design
Resource allocation in proggramoama gate array (FPGA) design is fundamental problem which can lead to largely varied circuit quality after synthesis, such as area, timing, and latency. In this application, we follow the problem formulation in (24 where the circuit is represented as data flow graph (DFG), and each node represents an arithmetic operation such as multiplication or addition. The goal to find resource allocation for each node to either digital signal processor (DSP) or look-up table (LUT), such that the final circuit area (i.e., actual DSP and LUT usage) after synthesis minimized. Notably, different allocation solutions result in greatly varied DSP/LUT usage due to complicated synthesis process, which cannot be simply summed up over each node. To obtain precise DSP/LUT usage, one must run high-level synthesis (HLS) (73 and place-and (74) tools, which
can take up to hours 24 30
In this application, each configuration C is DFG with > 100 nodes, where each node is allocated to either DSP or LUT. Node attributes include operation type (i.e., multiplication or addition) and data bitwidth. More details about the dataset can be found in Appendix C.5 Let X {0. 1)IVI denote the mapping to DSP or LUT. Let f. and 9r denote the proxies of actual LUT and actual DSP usage, respectively. Note that given different constraints on the DSP usage, we will normalize gr as introduced Sec. [ We train and validate fr.gr, 9r. Ag on 8, 000 instances that consist of 40 DFGs (C), each DFG with 200 different mappings (X), and test As over 20 DFGs. Note that the actual LUT and DSP usages of each training instance has been collected by running HLS in prior. We also run HLS to evaluate the actual LUT and DSP usages for the testing cases given the learned mappings.
Evaluation. We rank each method's best actual LUT usage under the constraint of different percent- ages (40% 70%) of the maximum DSP usage in each testing instance, then calculate the averaged ranks. Fig. shows the results. Our entry-wise concave proxy achieves the best performance. GS-Tr+R is slightly better than RL, and both of them exceed SA and GA. We do not include our entry-wise affine proxy the ranking list, because the affine proxy could be much less accurate than the proxy without constraints and the entry-wise concave proxy. The comparison between these proxies on learning DSP usage (& LUT usage) shown in Fig. 4 & Fig. in the appendix, respectively). The gap between different proxies indicates the FPGA circuit contains high-order moments of the input optimization variables and 2-order entry-wise affine proxy cannot model well. We do not include the result of GS-Tr+S and Naive+R, because these methods perform poor and could hardly generate feasible solutions given a constraint of DSP usage. We leave their results in Table. in the appendix. Moreover, we compare the training time between different methods. To be fair, all methods run on the same server with a Quadro RTX 6000 GPU. The RL based optimizer takes 22 GB GRAM, while other optimizers only take 7 GB on average. Fig, 10 in the appendix further demonstrates that our methods and GS-T methods require much less training time than RL.
Also, to give fair comparison between learning- bease approaches and traditional approaches, we implement GA with parallel (on GPU) cost-value inference for all the populations in each generation. We set the population size as 256, which is the same as the batch size that we used to train/infer Ag. The performance of GA in Fig. is obtained under the condition that the inference time of the implemented parallel GA is about the same as that of our CON method. Fig. 3 in the appendix provides more detailed comparison on the performance and the inference time between GA with different numbers of generations and our CON method.

---

DSP usage
40%
45%
50%
55%
60%
65%
70%
rank-avg
3.08
3.50
3.33
3.66
4.16
4.08
3.60
Accal Luag
GA
2.75
3.41
2.83
2.91
3.00
3.00
2.75
2.95
RL
3.33
3.58
3.83
3.25
2.91
2.83
2,41
3.16
GS-Tr+R
3.58
2.91
2.58
3.16
2.33
2.58
3.00
2.87
CON
1.83
2.00
2.25
2.25
3.08
2.41
2.66
2.35
Figure 5: The left table shows averaged ranks of the LUT usage given by different methods with the constraint of different percentage of DSP usage in Application II (resource allocation). The right figure shows the DSP-LUT usage amount relationship on one test configuration. The HLS baseline denotes the optimal synthesis results among 200 random mappings.
Threshold 0
C-In
C-Out
Naive
RL
GS-Tr+S
GS-Tr+R
CON
AFF
OPT
AxC units
12.42
12.44
3.62
7.68
4.87
3.24
3.18
3.10
2.77
AxC units
14.68
14.65
6.20
10.15
8.03
5.86
5.13
5,38
4.74
AxC units
17.07
17.04
11.12
12.83
12.65
10.62
10.17
10.04
8.56
Table Relative errors of different methods with the AxC unit constraint as 3,5,8 Application III.
5.5 Application III: Imprecise Functional Unit Assignment in Approximate Computing
One fundamental problem approximate computing (AxC) to assign imprecise functional units (a.k.a., AxC units) to execute operations such as multiplication or addition [4). aiming to significantly reduce eircuit energy with tolerable error. We follow the problem formulation [45), where given computation graph, each node represents either multiplication or addition. The incoming edges node represent its two operands. The goal is to assign AxC units to a certain number of nodes while minimizing the expected relative error of the output of the computation graph.
In this application, each configuration C' is computation graph with 15 nodes (either multiplication or addition) that maps a vector in R 16 to I. A fixed number 0 of nodes are assigned to AxC units with produce 10% relative error. Let X E {0, 13 denote whether a node is assigned to an AxC unit or not; the proxy of the objective fr is the expected relative error the output. We use 100k (X, C*) as the training dataset and the entire solution space is 2 x 2 For each (X, C.,, the ground-truth, i.e., expected relative error, computed by averaging Ik inputs sampled uniformly random from [0, 10] The constraint 9r is Evev X, 2 f with normalization, where A E {3, (3,5,8) 5,8]. We test the learned As on 500 unseen configurations.
Evaluation. Sable.5]showe the averaged relative errors of the assignments by different methods. The problem is far from trivial. Intuitively, assigning AxC units closed to the output, we may expect small error. However, C-Out performs bad. Our proxies AFF and CON obtain comparable best results. The MAE loss values of the two proxies reason is that the circuit is made up of 4 layers in total which leads to at most 4-order moments in the objective function, which is in medium-level complexity. Training time is also studied for this application, resulting in the same conclusion as application II (See Table 8 in the appendix).
also similar, shown in Table 9 in the appendix. The
6 Conclusion
This work introduces an unsupervised end-to-end framework to resolve LCO problems based on the relaxation- relaxation-llus-ronddi rous-ing technique. With our entry-wise concave architecture, our framework guarantees that a low objective value could lead to qualified discrete solutions. Our framework is particularly good at solving PCO problems where the objectives need be modeled and learned. Real-world applications demonstrate the superiority of our method over RL and gradient-relaxation approaches in both optimization performance and training efficiency. In the future, we aim further broaden our framework to the applications where the optimization variables allow using more complex embeddings than binary embeddings.
7
Acknowledgement
We greatly thank all the reviewers for their valuable feedback and the insightful suggestions. H. Wang and P. Li are partially supported by 2021 JPMorgan Faculty Award and the NSF award 0AC-2117997.
10

---

References
[1] N. Karalias and A. Loukas, "Erdos goes neural: an unsupervised learning framework for combinatorial optimization graphs," Advances in Neural Information Processing Systems, vol. 33, 2020.
[2] C. H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity. Courier Corporation, 1998.
[3] G. Naseri and M. A. Koffas, "Application of combinatorial optimization strategies in synthetic biology," Nature communications, vol. 11, no. 1, 2020.
[4] Y. Crama, "Combinatorial optimization models for production scheduling in automated manu- facturing systems," " European Journal of Operational Research, vol. 99, no. 1, 1997.
[5] J. Hopfield and W. Tank, ""neural" computation of decisions optimization problems," Biological Cybernetics, vol. 52, no. 3, 1985.
[6] K. A. Smith, "Neural networks for combinatorial optimization: review of more than a decade of research," " INFORMS Journal on Computing, vol. 11, no. 1, 1999.
[7] O. Vinyals, M. Fortunato, and N. Jaitly, "Pointer networks," Advances in Neural Information Processing Systems, vol. 28, 2015.
[8] D. Selsam, M. Lamm, B. Benedikt, P. LLinn, L. de Moura, D. L. Dill et al., "Learning a sat solver from single-bit bit supervision," in International Conference Learming Representations, 2018.
[9] S. Amizadeh. S. Matusevych, and M. Weimer, "Learning to solve circuit-sat: An unsupervised differentiable approach," in International Conference on Learning Representations, 2018.
[10] E. Yolcu and B. Póczos, "Learning local search heuristics for boolean satisfiability." Advances Neural Information Processing Systems, vol. 32, 2019.
[11] E. Khalil, P. Le Bodic, L. Song, G. Nemhauser, and Dilkina, "Learning to branch in mixed integer programming." in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 30, no. 2016.
[12] M. Gasse, D. Chételat, N. Ferroni, L. Charlin, and A. Lodi, "Exact combinatorial optimization with graph convolutional neural networks," Advances in Neural Information Processing Systems, vol. 32, 2019.
[13] A. Delarue, R. Anderson, and C. Tjandraatmadja, "Reinforcement learning with combinatorial actions: An application to vehicle routing," Advances in Neural Information Processing Systems, vol. 33, 2020.
[14] E. Khalil, H. Dai, Y. Zhang, B. Dilkina, and L. Song. "Learning combinatorial optimization algorithms over graphs," Advances in Neural Information Processing Systems, vol. 30, 2017.
[15] Z. Li, Q. Chen, and V. Koltun, "Combinatorial optimization with graph convolutional networks and guided tree search," Advances in Neural Information Processing Systems, vol. 31, 2018 88
[16] I. Bello, H. Pham, Q V. Le, M. Norouzi, and S. Bengio, "Neural combinatorial optimization with reinforcement learning," International Conference on Learning Representations (Workshop). 2017.
[17] X. Chen and Y. Tian, "Learning to perform local rewriting for combinatorial optimization," Advances Neural Information Processing Systems, vol. 32, 2019.
[18] W. Kool, H. Hoof, and M. Welling, "Attention, learn solve routing problems!" in International Conference on Learning Representations, 2018.
[19] C. K. Joshi, T. Laurent, and X. Bresson, "An efficient graph convolutional network technique for the travelling salesman problem,"" arXiv preprint arXiv: 01227, 2019.
[20] B. Hudson, Q. Li, M. Malencia, and A. Prorok, "Graph neural network guided local search for the traveling salesperson problem," International Conference on Learning Representations, 2022.
[21] Y. Ma, J. Li, Z. Cao, W. Song, L. Zhang, 7. Chen, and J. Tang, "Learning to iteratively solve routing problems with dual-aspect collaborative transformer," Advances Neural Information Processing Systems, vol. 34, 2021.
11

---

[22] Y.-D. Kwon, J. Choo, I. Yoon, M. Park, D. Park, and Y. Gwon, "Matrix encoding networks for neural combinatorial optimization." Advvances in Neural Information Processing Systems, vol. 34, 2021.
[23] M. Kim, J. Park et al., "Learning collaborative policies to solve np-hard routing problems," Advances in Neural Information Processing Systems, vol. 34, 2021.
[24] N. Wu, H. Yang, Y. Xie, p. Li, and C. Hao, "High-level synthesis performance prediction using gnns: Benchmarking, modeling, and advancing," in Proceedings of IEEE/ACM Design Automation Conference (DAC), 2022., 2022.
(25] C. Mendis, A. Renda, S. Amarasinghe, and M. Carbin, "Ithemal: Accurate, portable and fast basic block throughput estimation using deep neural networks," in International Conference on Machine Learning, 2019.
[26] G. Zhang, H. He, and D. Katabi, "Circuit-gnn: Graph neural networks for distributed circuit design," in International Conference on Machine Learning. PMLR, 2019.
[27] S. Vasudevan, W. Jiang, D. Bieber, R. Singh, C. R. Ho, C. Sutton et al., "Learning semantic representations to verify hardware designs," Advances in Neural Information Processing Systems, vol. 34, 2021.
[28] N. Mishra, C. Imes, J. D. Lafferty, and H. Hoffmann, "Caloree: Learning control for predictable latency and low energy," in Proceedings of the Twenty-Third International Conference on Architectural Support for Programming Languages and Operating Systems, 2018.
[29] A. Renda, Y. Chen, C. Mendis, and M. Carbin, "Difftune: Optimizing epu simulator param- eters with learned differentiable surrogates," in 2020 53rd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO). IEEE, 2020.
[30] N. Wu, Y. Xie, and C. Hao, "Ironman: Gnn-assisted design space exploration in high-level synthesis via reinforcement learning," in Proceedings of the 2021 Great Lakes Symposium on VLSI, 2021.
[31] A. Mirhoseini, A. Goldie, M. Yazgan, W. Jiang, E. Songhori, S. Wang, Y.J. Lee, E. Johnson, O. Pathak, A. Nazi et al., "A graph placement methodology for fast chip design," Nature, vol. 594, no. 7862, 2021.
[32] K. Chen and F. H. Arnold, "Enzyme engineering for nonaqueous solvents: random mutagenesis to enhance activity of subtilisin e in polar organic media," Bio/lechnology, vol. 9, no. 11, 1991.
[33] C. Angermueller, D. Dohan, D. Belanger, R. Deshpande, K. Murphy, and L. Colwell, "Model- based reinforcement learning for biological sequence design," in International Conference on Learning Representattions, 2019.
[34] R. Gómez--ombarelli, J. N. Wei, D. Duvenaud, J. M. Hernández-Lobato, B. Sánchez-Lengeling D. Sheberla, aagullera--prrraurrre T. D. Hirzel, R. P. Adams, and A. Aspuru- Guzik, "Au- tomatic chemical design using a data-driven continuous representation of molecules," ACS Central Science, vol. 4, no. 2, 2018.
[35] D. Bertsimas and Tsitsiklis, "Simulated annealing." " Statistical Science, vol. 8, no. 1, 1993.
[36] J. Toenshoff, M. Ritzert, H. Wolf, and M. Grohe, "Run-csp: unsupervised learning of message passing networks for binary constraint satisfaction problems," 2019.
[37] W. Yao, A. S. Bandeira, and S. Villar, "Experimental performance of graph neural networks on random instances of max-cut," in Wavelets and Sparsity XVIIII vol. 11 138. International Society for Optics and Photonics, 2019.
[38] G. Yehuda, M. Gabel, and A. Schuster, "It's not what machines can learn, it's what we cannot teach," in International Conference on Machine Learning. PMLR, 2020.
[39] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski et al., "Human-level control through deep rein- forcement learning." Nature, vol. 518, no. 7540, 2015.
[40] R. Gandhi, S. Khuller, S. Parthasarathy, and A. Srinivasan, "Dependent rounding and its applications to approximation algorithms," " Journal of the ACM (JACM), vol. 53, no. 3, 2006.
[41] Byrka, F. Grandoni, RothvoB, and I.. Sanità, "Steiner tree approximation via iterative randomized rounding," Journal of the ACM (JACM), vol. 60, no. 1, 2013.
12

---

[42] J. Han and M. Orshansky. "Approximate computing: An emerging paradigm for energy merryy-ffficiet design," " in 2013 18th IEEE European Test Symposium (ETS). IEEE, 2013.
[43] S. Mittal, "A survey of techniques for approximate computing," ACM Computing Surveys (CSUR), vol. 48, no. 4, 2016.
[44] R. Venkatesan, A. Agarwal, K. Roy, and A. Raghunathan, "Macaco: Modeling and analysis of circuits for approximate computing," in 2011 IEEE/ACM International Conference on Computer-Aided Design (ICCAD). IEEE, 2011.
[45] D. Ma, R. Thapa, X. Wang, X. Jiao, and C. Hao, "Workload-aware approximate computing configuration," in 2021 Design, Automation & Test in Europe Conference Exhibition (DATE).
IEEE, 2021.
[46] C. Li, W. Luo, S. S. Sapatnekkr, and J. Hu, "Joint precision optimization and high level synthesis for approximate computing." in Proceedings of the 52nd Annual Design Automation Conference, 2015.
[47] Y. Bengio, N. Léonard, and Courville, "Estimating or propagating gradients through stochas- tic neurons for conditional computation." arXiv preprini
2013.
[48] E. Jang, S. Gu, and B. Poole, "Categorical reparameterization with gumbel-softmax"" Interna- tional Conference on Learning Representations, 2017.
[49] C. J. Maddison, A. Mnih, and Y. W. Teh, "The concrete distribution: A continuous relaxation of discrete random variables," International Conference on Learning Representations, 2017.
[50] Y.-D. Kwon, J. Choo, B. Kim, I. Yoon, Y. Gwon, and S. Min, "Pomo: Policy optimization with multiple optima for reinforcement learning," " Advances in Neural Information Processing Systems, vol. 33, 2020.
[51] R. Wang, Z. Hua, G. Liu, J. Zhang, J. Yan, F. Qi, S. Yang, J. Zhou, and X. Yang, "A bi-level framework for learning to solve combinatorial optimization on graphs," Advances in Neural Information Processing Systems, vol. 34, 2021.
[52] Y. Nandwani, D. Jindal, P. Singla et al., "Neural learning of one-of -any solutions for com- binatorial problems in structured output spaces." " in International Conference on Learning Representations, 2021.
[53] H. Duan, P. Vaezipoor, M. B. Paulus, Y. Ruan, and C. Maddison, "Augment with care: Con- trastive learning for combinatorial problems," in International Conference on Machine Learning. PMLR, 2022, pp. 5627-5642.
[54] A. Kumar and S. Levine, "Model inversion networks for model-based optimization," Advances in Neural Information Processing Systems, vol. 33, 2020.
[55] D. Brookes, H. Park, and J. Listgarten, "Conditioning by adaptive sampling for robust design."
in International Conference on Machine Learning. PMLR, 2019.
[56] B. Trabucco, A. Kumar, X. Geng, and S. Levine, "Conservative objective models for effective offline model-based optimization," in International Conference on Machine Learning. PMLR, 2021.
[57] A. Kumar, A. Yazdanbakhshh, M. Hashemi, K. Swersky, and S. Levine, "Data-driven offline optimization for architecting hardware accelerators," 2022.
[58) R. Anderson, J. Huchette, W. Ma, C. Tjandraatmadja, and J. P. Vielma, "Strong mixed-integer programming formulations for trained neural networks," Mathematical Programming, vol. 183,
1, pp. 3-39, 2020.
[59] T. P. Papalexopoulos, C. Tjandraatmadja, R. Anderson, J. P. Vielma, and D. Belanger, "Con- strained discrete black-box optimization using mixed-integer programming." in International Conference on Machine Learning. PMLR, 2022, pp. 17 297--117 322.
[60] V. Konda and J. Tsitsiklis, "Actor-critic algorithms," Advances in Neural Information Processing Systems, vol. 12, 1999.
[61] V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. Lillicrap, T. Harley, D. Silver, and K. Kavukcuoglu, "Asynchronous methods for deep reinforcement learning," in International Conference on Machine Learning. PMLR, 2016.
[62] M. Paulus, Choi, D. Tarlow, A. Krause, and C. J. Maddison, "Gradient estimation with stochastic softmax tricks," Advances in Neural Information Processing Systems, vol. 33, 2020.
13

---

[63] K. Struminsky, A. Gadetsky, D. Rakitin, D. Karpushkin, and D. P. Vetrov, "Leveraging recursive gumbel-max trick for approximate inference in combinatorial spaces." Advances in Neural Information Processing Systems, vol. 34, 2021.
[64] J. Leskovec and A. Krevl, "Snap datasets: Stanford large network dataset collection," 2014.
[65] M. Fey and E. Lenssen, "Fast graph representation learning with PyTTrrc Geometric," in ICLR Workshop Representation Learning on Graphs and Manifolds, 2019.
[66] M. Wang, D. Zheng, Z. Ye, Q. Gan, M. Li, X. Song. J. Zhou, C. Ma, L. Yu, Y. Gai, T. Xiao, He, G. Karypis, J. Li, and Z. Zhang, "Deep graph library: A graph-centrie, highly--erffrmant package for graph neural networks," arXiv preprint arXiv: 1909.01315, 2019.
[67] S. Mirjalili, "Genetic algorithm," in Evolutionary algorithms and neural networks. Springer, 2019.
[68] D. Whitley, "A genetic algorithm tutorial," Statistics and computing, vol. 4, no. 2, 1994.
[69] M. V. Pogančić, A. Paulus, V. Musil, G. Martius, and M. Rolinek, "Differentiation of blackbox combinatorial solvers," in International Conference on Leurning Representations, 2019.
[70] L.. Deng, "The mnist database of handwritten digit images for machine learning research," " IEEE Signal Processing Magazine, vol. 29, no. 6, 2012.
[71] K. He, X. Zhang, S. Ren, and J. Sun, "Deep residual learning for image recognition," in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016.
[72] W. Hamilton, Z. Ying, and J. Leskovec, "Inductive representation learning on large graphs," Advances Neurul Information Processing Systems, vol. 30, 2017.
[73] AMD/Xilinx, "Vitis development environment," https//wwww.xiinn.cccmmmpooocce
Besign-toolsVviiis/viittaaaiitti
[74]
"Vivado development tool," http::/wwwwwwili..cccmmoooocttcs
[75]
P. J. Huber, "Robust estimation of location parameter," in Breakthroughs statistics. Springer,
1992, pp. 492-518.
[76] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga al., "Pytorch: An imperative style, high-performance deep learning library." Advances in Neural Information Processing Systems, vol. 32, 2019.
[77] M. Fey and J. E. Lenssen, "Fast graph representation learning with pytorch geometric," arXiv preprint 100..00930222 2019.
[78] D. P. Kingma and J. Ba, "Adam: A method for stochastic optimization," in International Conference on Leurning Representations, 2015.
[79] Gilmer, S. Schoenholz, P. F. Riley, O. Vinyals, and G. Dahl, "Neural message passing for quantum chemistry," in International Conference on Machine Learning. PMLR, 2017, pp. 1663-1272.
[80] G. Corso, L. Cavalleri, D. Beaini, P. Liò, and P. Veličković, "Principal neighbourhood aggrega- tion for graph nets," Advances in Neural Information Processing Systems, vol. 33, 2020.
14

---

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Sec. 6
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See
Appendix[] [
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions all theoretical results? [Yes] See Sec and S
(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.1
A.2 and A.3
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? [Yes]
(b) Did you specify all the training details (c.g.. data splits, hyperparameters, how they were chosen)? [Yes] See the experiment details in Appendix
(c) Did you report error bars (e.g.. with respect the random seed after running experi- ments multiple times)? [Yes]
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] See Appendix [
4. If you using existing assets (e.g., code, data, models) or curating/releasing
assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] The dataset in
application II adopted from [30].
(b) Did you mention the license of the assets? [Yes] See Appendix
(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] All the assets publicly available.
(d) Did you discuss whether and how consent was obtained from people whose data you't 're using/curating? [Yes] All the assets obtained from others are publicly available. (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] See Appendix and our data contains no human information offensive content.
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid participants and the total amount spent on participant compensation? [N/A]
15


Reviews:
[{"review_id": "n0MZSBbmQR", "reviewer": "Reviewer_BQ6N", "summary": "The paper proposes an extension and variant to a previous paper by Karalias& Loukas, replacing the expectation based decoding strategy and probabilistic guarantee with another construction which relies on elementwise concavity+rounding of the function used to minimised the selfsupervised objective. The paper compares agains naive rounding, RL and Gumble softmax + sampling/rounding baselines on edge covering, node matching and 2 additional tasks (resource allocation and approximate programming) and finds their methods to be dominant across all considered methods.", "questions": "1. Most importantly: Is your method applicable to the problems studied in Karalias&Loukas? If so, can you add those experiments as well to directly compare the most relevant Literature method?\n2. I saw you ran 3 seeds, can you add std to the table (and ideally, run additional seeds and perform a statistical significance test between methods?)", "limitations": "Authors discuss binary embeddings of the optimization variables as an avenue for future work. I would like to have seen a discussion of which functions might lie outside the entry-wise convex framework, and about the computational complexity and scaling of their proxies when moving to large scale problems", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well written and provides good experimental evidence of the improvement possible with their method, evaluated on real world problems. Originality, quality/clarity and significance are all at acceptance threshold for me.\nThe only nit picks I could find are in Questions and limitations.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "kTXr-zEn7W", "reviewer": "Reviewer_odo9", "summary": "The authors propose an unsupervised learning framework for combinatorial optimization problems. They evaluate this framework by solving a synthetic graph optimization problem.", "questions": "It would be really helpful if the authors can clarify my questions above.\n\nFurther, I believe it would be productive for the rebuttal, if the authors can clarify the the following question first -\n\nGiven an instance of a minimum edge covering problem [1] G = (V,E) - how would you propose to solve this with your framework?\n\n[1] https://en.wikipedia.org/wiki/Edge_cover", "limitations": "I find the setup of finding solutions in binary encoding space to be quite restrictive and don't immediately see how this would generalize to any combinatorial optimization problem in a feasible way - for eg. network flow algorithms in which one has to assign a fractional amount of flow to a network arc.\n\n", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Overall, I found the paper quite confusing to follow .\n\nThe authors claim that they have an unsupervised learning framework, yet they seem to be using supervised and labelled data to learn proxy objectives for both cost and constraints.\n\n> This work provides the first unsupervised framework for PCO problems\n\nThe motivation behind unsupervised algorithms for PCO is unclear since you already need supervised data to train the proxy functions.\n\nKeeping the supervised vs unsupervised nitpicking aside, I don't follow how either AFF or CON are actually solving the problem or how the guarantees about rounding provided by the authors are valid - if the function being optimized is a proxy of the true loss function - how can rounding this solution have any guarantee on feasibility - since the learned loss function has no bound on its error?\n\nTaking a step back, I don't fully grasp how the authors go from an instance of an edge covering problem G = (V, E) to a valid solution X \\in {0, 1}^|E| that solves the edge covering - my best guess is that the authors try to train two GNNs - the first GNN is trained in a supervised fashion to predict the cost function f (and perhaps the constraint function g), and the second GNN is used to learn a Bernoulli distribution from which the authors will sample lots of X, and the X that minimizes the learned/proxy cost function will be rounded out using the procedure they described into the final solution.\n\nSection 4.1 - Each observed configuration is paired with a solution X  - what is the methodology for picking this X for the supervised training? Did you'll sample a solution at random and evaluate its cost and constrain function for the training data?\n\nThe notation was a bit confusing at times - \n- line 216 the data configuration is denoted by C but the GNN encoding is also denoted by C in line 224.\n- line 225 - 2nd order moments -  I am not familiar with this notation in this context of GNNs - did you mean the number of successive applications of the message passing layers. \n\nI tried going through the supplementary material but it seems that some of the READMEs are incomplete.\n\nWhat problem is being solved in section 4.3? Minimize the weighted edge covering? Why do you even have to use LCO for this problem? Don't you just have to solve MNIST on the nodes' images and then solve edge covering? Why do the edge weights have to be learned? I don't understand the motivation for picking this problem to evaluate the framework proposed in the paper.\n\nFurther, 4x4 graphs are extremely tiny. While the total search space of these graphs can be huge, it doesn't necessary imply that the family of instances in the data set has a very complex combinatorial structure or that learning this problem is hard.\n\nIf this synthetic example is just for the sake of demonstration, it would have been much easier to follow were it solving just the vanilla edge covering problem.\n\n-----\n\nMy current recommendation is a borderline reject but I remain hopeful that I fundamentally don't understand something in the paper and that the authors will clarify it.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "f4-GKsKvLBs", "reviewer": "Reviewer_kjHC", "summary": "This paper extends the relax-optimize-discretize pipeline in Erdos Goes Neural. The authors believe that the sequential decoding scheme in EGN suffers from two issues, and resolve these issues by sequentially rounding over a so-called entry-wise concave principle. The proposed scheme is claimed to require fewer samples and thus faster than EGN, while still preserving the theoretical performance guarantee in EGN. The authors mainly investigate the application of the proposed approach in proxy combinatorial optimization (PCO), where the objective function is unknown or too expansive to know.", "questions": "1. Is there any reason why pure CO problems are not evaluated?\n2. Are the proxy-GNNs learned by supervised learning?\n3. In application II, what is the proxy model for RL/SA/GA? Do they use the same entry-wise concave GNN? It seems that we can build a conventional GNN for them and I am curious whether a conventional GNN will be more accurate?", "limitations": "The authors write in the checklist that the limitations are addressed in Section 5, but it is missing.", "rating": 4, "confidence": 4, "soundness": 1, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Here is my overall judgment of this paper: the ideas/insights/experiments are novel and interesting. However, this paper reads somewhat confusing, especially since the experiments seem not to match the other parts of this paper.\n\n**Strengths**\n1. (Unsupervised) learning for combinatorial optimization is an important topic worth studying. The considered PCO problem is also practical and important.\n1. I enjoy the narrative of theoretical analysis and insights about learning CO. \n\n\n**Weakness**\n1. Experiment for learning on pure (non-proxy) CO problems is missing, and the important baseline Erdos Goes Neural is missing. In the experiment, the authors study 3 PCO problems, which I agree to be very important and novel. However, in Section 1&2&3, this paper reads to be an improved version of Erdos Goes Neural. Since Erdos Goes Neural is focused on pure CO problems, there is no reason not to evaluate such problems and compare to EGN to support the theoretical results. Besides, the authors also discussed on how to tackle the pure CO problem in detail in Section 3. It seems confused for proposing a new method for pure CO but evaluating it.\n2. This paper is titled \"unsupervised\" learning. However, it seems that the evaluated PCO problems are not \"unsupervised\", but \"supervised\" because the proxy should be learned from labels. For example, in application II, the authors write \"the actual LUT and DSP usages of each training instance has been collected by running HLS in prior\".\n3. Adding a flow chart or an algorithm block for the proposed framework will make this paper clearer and easier to reimplement.\n4. The implementation details on the application of PCO problems seem not clear. For example: In application II, what is the proxy model for RL/SA/GA? Do they use the same entry-wise concave GNN? It seems that we can build a conventional GNN for them and I am curious whether a conventional GNN will be more accurate?\n\n\n**Other Remarks**\n1. L19: Most CO problems are NP-complete -> Most CO problems are NP-complete or NP-hard.\n2. Is Erdos Goes Neural missing in the discussion in Table 1? How is EGN considering these aspects?\n3. I feel confused about L188: \"Optimizing $\\theta$ in Eq.(2) is generally hard as the gradient $dX/d\\theta$ does not generally exist so the chain rule cannot be used.\" However, $X$ should be continuous in EGN, therefore the gradient should always exist.\n4. In L128: \"The two special cases in [1] on the max-clique and graph-partition problems seem to have special structures.\" Do you have any support for this claim?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "EDiisZldr7L", "reviewer": "Reviewer_4diE", "summary": "The paper proposes an unsupervised machine learning method for combinatorial optimization (CO) problems. The focus is in particular on CO problems that have an expensive cost function, necessitating a learned surrogate model. The authors extend the work by Karalias & Loukas [1] to this so called Proxy-based CO setting, while still retaining theoretical guarantees that a feasible solution with low cost can be attained. This guarantee comes with certain assumptions about the ground-truth cost function (not too high order) and limitations on the neural network for the proxy. The method outperforms learning based baselines on three different non-standard benchmarks and appears to require less resources to train.", "questions": "It would be helpful to explicitly spell out the limitations of the current method. For example: What are the computation costs associated with n-th order proxy costs? Will it fail when the ground truth cost requires higher order moments? How well does this scale to larger problem instances (>1k nodes)?\n\nFor suggestions about the experiments, see above.", "limitations": "Potential societal impact is addressed in the appendix. Limitations should be expanded on, as for example suggested above.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n- Unsupervised machine learning for CO (especially PCO) is an interesting and important research area. In particular, this work is a significant extension of [1].\n- The paper is well written, and the main ideas are clearly presented\n- The proposed method appears sound: it is accompanied by theoretical guarantees on achieving feasible solutions with costs upper bounded by the training loss (theorem 1) and there exist relaxations fulfilling the required constraint for the proxy cost function (theorem 2)\n\nWeaknesses:\n- The experiments are extensive (multiple different baseline models and different experiments), but do not compare directly against prior work on any established benchmarks. This makes it difficult to place the work in the current literature and gauge how well it does in comparison.\n- The non-learned baselines (simulated annealing and genetic algorithm) were only applied to the 2nd experiment \"Resource Allocation in Circuit Design\", but in light of recently pointed out shortcomings of ML based approaches compared to non-learned methods, it would be interesting to see a more extensive comparison against these non-learned baselines. This also includes applying SA/GA for more iterations, to get better results and a better sense of the complexity of the investigated problems.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept