Deep Distributed Optimization for Large-Scale Quadratic Programming
Authors: Augustinos D Saravanos
Hunter Kuperman
Alex Oshin
Arshiya Taj Abdul
Vincent Pacelli
Evangelos Theodorou
Venue: ICLR 2025 Poster (2025)
Keywords: Learning-to-Optimize
Distributed Optimization
Large-Scale Quadratic Programming
Primary area: optimization
TL;DR: We propose DeepDistributedQP, a new deep learning-aided distributed optimization architecture for large-scale quadratic programming.

Abstract: Quadratic programming (QP) forms a crucial foundation in optimization, appearing in a broad spectrum of domains and serving as the basis for more advanced algorithms. Consequently, as the scale and complexity of modern applications continue to grow, the development of efficient and reliable QP algorithms becomes increasingly vital. In this context, this paper introduces a novel deep learning-aided distributed optimization architecture designed for tackling large-scale QP problems. First, we combine the state-of-the-art Operator Splitting QP (OSQP) method with a consensus approach to derive **DistributedQP**, a new method tailored for network-structured problems, with convergence guarantees to optimality. Subsequently, we unfold this optimizer into a deep learning framework, leading to **DeepDistributedQP**, which leverages learned policies to accelerate reaching to desired accuracy within a restricted amount of iterations. Our approach is also theoretically grounded through Probably Approximately Correct (PAC)-Bayes theory, providing generalization bounds on the expected optimality gap for unseen problems. The proposed framework, as well as its centralized version **DeepQP**, significantly outperform their standard optimization counterparts on a variety of tasks such as randomly generated problems, optimal control, linear regression, transportation networks and others. Notably, DeepDistributedQP demonstrates strong generalization by training on small problems and scaling to solve much larger ones (up to 50K variables and 150K constraints) using the same policy. Moreover, it achieves orders-of-magnitude improvements in wall-clock time compared to OSQP. The certifiable performance guarantees of our approach are also demonstrated, ensuring higher-quality solutions over traditional optimizers.

Published as a conference paper at ICLR 2025
DEEP DISTRIBUTED OPTIMIZATION FOR LARGE-SCALE QUADRATIC PROGRAMMING
Augustinos D. Saravanos, Hunter Kuperman, Alex Oshin, Arshiya Taj Abdul, Vincent Pacelli and Evangelos A. Theodorou
Georgia Institute of Technology, USA (asaravanos, kup, alexoshin, aabdul6, vpacelli3, evangelos.theocdoruy @ gatech.edu
ABSTRACT
Quadratic programming (QP) forms a crucial foundation in optimization, appear- ing in a broad spectrum of domains and serving as the basis for more advanced al- gorithms. Consequently, as the scale and complexity of modern applications con- tinue grow, the development of efficient and reliable QP algorithms becomes increasingly vital. In this context, this paper introduces a novel deep learning- aided distributed optimization architecture designed for tackling large-scale QP problems. First, we combine the state-of-the-art Operator Splitting QP (OSQP) method with a consensus approach to derive DistributedQP, a new method tai- lored for network-structured problems, with convergence guarantees to optimality. Subsequently, we unfold this optimizer into a deep learning framework, leading to DeepDistributedQU which leverages learned policies to accelerate reaching desired accuracy within a restricted amount of iterations. Our approach is also the- oretically grounded through Probably Approximately Correct (PAC)-Bayes the- ory. providing generalization bounds on the expected optimality gap for unseen problems. The proposed framework, well its centralized version DeepQP, significantly outperform their standard optimization counterparts on a variety tasks such as randomly generated problems, optimal control, linear regression, transportation networks and others. Notably, DeepDistributedQP demonstrates strong generalization by training on small problems and scaling to solve much larger ones (up to 50K variables and 150K constraints) using the same policy. Moreover, it achieves orders-of-magee -fa-nnnittee improvements wall-clock time com- pared to OSQP. The certifiable performance guarantees of our approach are also demonstrated, ensuring higher-quality solutions over traditional optimizers.
INTRODUCTION
Quadratic programming (QP) serves as a fundamental cornerstone in optimization with wide va- riety of applications in machine learning (Cortes & Vapnik 1995 Tibshirani, 1996), control and robotics (Garcia et al.) 1989 Rawlings al. (017), signal processing (Mattingley & Boyd 2010), finance (Coruujooss et al. 2018), and transportation networks (Mota et al. 2014) among other fields. Beyond its standalone applications, QP also acts the component of many advanced non-convex optimization algorithms such as sequential quadratic programming (Nocedal & Wright 1999). trust-region methods (Conn al. (2000), augmented Lagrangian approaches (Houska al. 2016 mixed-integer optimization (Belotti et al. 2013), etc. For these reasons, the pursuit of more efficient QP algorithms remains ever-evolving area of research from active (Wolfe, 1959) and interior point methods (Nesterov & Nemirovskii, 1994) during the previous century to first-order methods such as the state-of-the Operator Splitting QP (OSQP) algorithm (Stellato al. 2020).
As the scale of modern decision making applications rapidly increases, there is an emerging in- terest developing effective optimization architectures for addressing high-dimensional problems. Given the fundamental role of QP in optimization, there is a clear demand for algorithms capable of solving large-scale QPs with thousands, and potentially much more, variables and constraints. Such problems arise in diverse applications including sparse linear regression (Mateos et al.) 2010) and support vector machines (Navia- Vazquez et al. 2006) with decentralized data, multi-agent con- trol (Van Parys & Pipeleers 2017), ), resource allocation (Huang al.) 2014), network flow (Mota) ct al. 2014). power grids (Lin et al. 2012) and image processing (Soheili & Eftekhari- Moghaaaammm
2020 Traditional centralized optimization algorithms are inadequate for solving such problems

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scale (see for example Fig. prompting the development of distributed methods that leverage the underlying network/deccntraaliee structure to parallelize computations. this context, the Alter- nating Direction Method of Multipliers (ADMM) has gained widespread popularity as effective approach for deriving distributed algorithms (Boyd et al. 2011] Mota et al., 2013). Nevertheless, scale increases, such algorithms continue to face significant challenges such as their need for meticulous tuning, the absence of generalization guarantees and restrictions on the allowed number of iterations imposed by computational or communication limitations.
Learning-to-optimizz has recently emerged as methodology for enhancing existing optimizers or de- veloping entirely new ones through training on sample
10a
DeepDistrOP (Ours)
Total Wall Clock Time (ms)
Distr@P (Ours)
053P lindirect)
problems (Chen al. 2022b; Amos al., 2023). A notable approach within this paradigm is deep unfold-
10
ing, which unrolls optimizer iterations as layers of
deep learning network and learns the optimal param- eters for improving performance (Monga et al. 2021
10a
Slimo
Shlezinger et al. 2022). Our key insight is that deep unfolding is particularly well-suited for overcoming the limitations of distributed constrained optimization, as it can eliminate the need for extensive tuning, man- age iteration restrictions and enhance generalization. However, its combination with distributed ADMM has only recently been explored in Noah & Shlezinger 2024). While this framework shows promising initial that studies unconstrained problems, assumes local updates consisting of gradient steps, focuses solely parameter tuning, and is not accompanied by any formal performance guarantees.
10a
3desh
NAA
Random 10=4024]
Coupled Pendulums Network 200)
5000)
Figure Wall-clock time comparison: DeepDistributedQR. DistributedQP (ours) and OSQP on large-scale QPs.
results, it relies on relatively simple setup
Contributions. This paper introduces novel deep learning-aides distributed optimization archi- tecture for solving large-scale constrained QP problems. Our approach relies unfolding a newly introduced distributed QP algorithm as a supervised learning framework for prescribed number of iterations. To our best knowledge, this is the first work to present a deep learning architecture for distributed constrained optimization using ADMM, despite the widespread popularity of the latter. Our framework demonstrates remarkable scalability, being trained on small problems and then ef- fectively applied to much larger ones. Furthermore, its performance is theoretically supported by establishing guarantees based on generalization bounds from statistical learning theory. We believe that this work lays the foundation for developing learned distributed optimizers capable of handling large-scale constrained optimization problems without requiring training at such scales.
Our specific contributions can be summarized as follows:
First, we introduce DistributedQP. new decentralized method that combines the well- established OSQP solver with consensus approach. We further prove that the algorithm is guaranteed to converge to optimality, even under varying local algorithm parameters.
Then, we propose DeepDistributedQP, deep learning-aided distributed architecture that unrolls the iterations of DistributedQP in a supervised manner, learning feedback policies for the underlying algorithm parameters. As a byproduct, we also present DeepQP, its centralized counterpart which corresponds to unfolding the standard OSQP solver.
To certify the performance the learned solver, we establish generalization guarantees on the optimality gap of the final solution of DeepDistributedQP for unseen problems using Probably Approximately Correct (PAC)-Bayes theory.
Finally, present extensive experimental evaluation that validates the following:
For centralized QPs, DeepQP consistently outperforms OSQP requiring 1.5-3 times fewer iterations for achieving the desired accuracy.
DeepDistributedQP successfully scales for hinh--dimensoonal problems (up to 50K variables and 150K constraints) while being trained exclusively on much smaller ones. Furthermore, both DeepDistributedQQP and DistributedQP outperform OSQP in wall- clock time by orders of magnitude as the problem dimensionality increases.
The proposed PAC bounds offer valuable guarantees on the quality of solutions pro- duced by DerpDistributedQP for unseen problems from the same class.
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2 RELATED WORK
This section provides an overview of existing related literature from the angles of distributed opti- mization and learning-to-optmmize. An extended discussion provided in Appendix A
Distributed optimization with ADMM. Distributed ADMM algorithms have emerged as scal- able approach for addressing large-scale optimization problems (Boyd et al., [01 1 Mota et al.) 2013). Despite their significant applicability to machine learning (Mateos et al. 2010), robotics (Shorinwa et al. 2024) and many other fields, their successful performance has been shown to be highly sensitive to the proper tuning of underlying parameters (Xu et al., 2017a Saravanos al. 2023a). Moreover, tuning parameters for large-scale problems is often tedious and tmm--onsummnng making it desirable develop effective learned optimizers that can be trained smaller problems instead. Furthermore, even if an distributed optimizer performs well for a specific problem instance, its generalization to new problems remains challenging to verify. These challenges constitute our main motivation for studying learning-aided distributed ADMM architectures. We also note that an ADMM-based distributed QP solver resembling simpler version of DistributedQP was presented in Pereira et al. (2022), but focusing on multi-robot control and lacking any theoretical analysis.
Learning-fo-optimizze The area of coamminngttooptmmizz methods has emerged as an effective ap- proach for enhancing existing optimizers or even deriving new algorithmic updates through training on sample problems (Chen al., 2022a (Shlezinger et al., 2022. Amos et al. 2023). A prominent technique in this paradigm is deep unfolding, which under the realistic assumption of computational budget restrictions, unrolls fixed number iterations as layers of a deep learning framework and learns the optimal parameters for improving performance 2021 Zhang et al., 2020). Nevertheless, combining deep unfolding with distributed ADMM has only been investigated recently in Noah & Shlezinger (2024). Although this framework demonstrates promising results, it is limited to an unconstrained problem formulation, assumes gradient-based local updates, focuses exclusively on parameter tuning and lacks formal performance guarantees. A reinforcement learning algorithm for accelerating OSQP was presented in Ichnowski et al.] (2021). While this approach also explores learning policies for algorithm parameters, it limited central- ized quadratic programming, lacks guarantees and its training comes at a significant computational cost. In the context of establishing generalization bounds for learned optimizers, (Sambharya & Stellato (2024a) recently explored incorporating PAC-Bayes bounds in learned optimizers, yet our approach differs fundamentally, as their method employs a binary error function, whereas ours di- rectly establishes bounds based on the optimality gap of the final solution. The works in [Suckrr & Ochs| (2023) and [Sucker et al. (2024) are also investigating generalization bounds for learned opti- mizers, considering the update function as gradient step multi-layer perceptron, respectively.
specific problem class (Monga al.)
Global variable components
3 DISTRIBUTED QUADRATIC PROGRAMMING
Wi
W2
W3
W
3.1 PROBLEM FORMULATION
convex (centralized) QP problem is expressed in general as Qx q s.t. Ax b,
Wts
1
min 2dd
(1)
where x E R is the decision vector and ( {Q E S E R", A E Rmxx E R are the problem data. As the scale of such problems increases to higher dimensions, there is often an underlying networkedddeecentraiizee structure that could be leveraged for achieving distributed computations. This work specifically aims address problems characterized by such structures. Let w E IR" be the main global variable and æ, E IR be local variables E V= [L..... ).. Then, assume a mapping (i.j) G(i.j) from all index pairs (i.j)
1
32
X3
Local variables
Figure 2: Example of consensus mapping 9 in problem (
of local variable components {m::] to indices 1 = (i. of global components w for an example see Fig. 2 We consider QP problems of the following distributed consensus form:
min
Σ TE
2 Qx:; a
+. gT., s.t. A, < b,, x. wi, iEV.
I Note that equality constraints can also he captured as pairs inequalities.
Thii formulation is adopted from the standard consensus ADMM framework (Boyd et al. local variables are typically associated with their respective computational nodes.
201 wherein
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where the problem data Im x b, RT J. The vector w; E RR defined as w = w/ (ee (= . is the selection of global variable components that correspond to the components of 2;, This form captures a wide variety of large-scale QPs found in machine learning (Mateos et al. 2010 Navia- Vazquez et al. 2006), optimal control (Van Parys & Pipeleers 2017). transportation networks, (Mota et al. 2014), power grids (Lin al. 2012),
given by Ç { with S. =Q E S E R", A, [[ hev] is the concatenation of all local variables, while
resource allocation (Huang et al. 2014) and many other fields.
3.2
DISTRIBUTEDQP: THE UNDERLYING OPTIMIZATION ALGORITHM
This section introduces a new distributed algorithm named DistributedQU for solving problems of the form The proposed method can be viewed as combination of consensus ADMM (Boyd)
al. 2011 and OSQP (Stellato et al., 2020) using local iteration steration-varying penalty parameters.
Let us introduce the auxiliary variables Z;, 8; E Ri"; such that problem can be reformulated as
min
Σ
x0 Q13; + 9 az s.t. A,x = Zi, 8; S bi. 2, 8/, a w. i V.
(3)
1EV 2
The proposed DistributedQP method is summarized below, where k 0, 1. denotes iterations:
1. Local updates for Fi, Zi. For each node E V, solve in parallel:
Q. + 1
A 11pp
k+11
+ 1 w y/ 1/p A;
(4)
A;
I
b+1
and then update in parallel:
= = s + 1/ (1/ttt x²).
A+1
(5)
2. Local updates for s; and global update for w. For each node i E V, update in parallel:
s -I, (a²2/ 2/tt (1 )+)s + X /pt)
In addition, each global variable component wi is updated through:
wi k+1 = =a E [[ + (1 (1-a ja
Σous)=
3. Local updates for dual variables do, Us For each node i V, update in parallel:
k+1 X + (a(az
2 k+1 + (1 - )+)
s s
(8)
y, k+1 v + ut (a k+1 + (1 a²) au) 1
(9)
The Lagrange multipliers v, A; and y: correspond to the equality constraints A,æ; 24, 8;
and x; = W. respectively. The penalty parameters Pi, 14: > correspond to z; = 8; and x; while a E [1, over-relaxation parameters. complete derivation is provided in Appendix B
3.3 CONVERGENCE GUARANTEES
Prior to unrolling DistributedQP into deep learning framework, it is partieularly important to estab- lish that the underlying optimization algorithm is well-behaved even for varying parameters, i.e., is expected to asymptotically converge to the optimal solution. This property is especially important in deep unfolding where parameters are expected to be distinct between different iterations.
In the simpler case of a 1, o = p. n nt, the standard convergence guarantees of two- block ADMM would apply directly (Deng & Yin 2016); for a detailed discussion, see Appendix penalty parameters pt μ well as
C Nevertheless, the introduction of local deration-varyinn the over-relaxation with varying parameters a makes proving the convergence of this algorithm non-trivial. In the following, we provide convergence guarantees to optimality for DistributedQP.
We consider the following assumption for the penalty parameters. Assumption 1. As k + xc, the parameters p p. uf 1--..for all i E V.
The following theorem states the convergence guarantees of DistributedQP to optimality. The proof, as well as necessary intermediate results, are provided in Appendix D
Theorem (Convergence guarantees for DistributedQP). If Assumption 1 holds and a E [1,2), then the tterates w converge to the optimal solution w" of problem (2) as k oc.
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Node
Local update (4+55
update
Duel updatte e3 (9)
Local updlate (4+15))
Local Problem & Inibalizations
Local rediduals
Feed reed-forvard parama
Local Aresiduals
Feed feed-forwadd parama
Feedbback
Feedback
FC layer
FC layer
Global update
Global update
Training Loss
=8.af
...4
Local update (4x5)
Local updatte (4-55
& Intializations
Problem
residuals
Feed forword params
A.
Dual updaste (8) (9)
params
Fweebback
rb.el
Local update
Feedback FC layer
FC layer
Node
N
Layer ==1
Layer & = K
Figure 3: The DeepDistributedOP architecture. The proposed framework relies unrolling the DistributedQP optimizer as a supervised deep learning framework. In particular, we interpret its iterations (-) as sequential network layers and introduce learnable components (orange blocks) to facilitate reaching the desired accuracy after a predefined number of allowed iterations.
4 THE DERPDISTTRIUTTEDOPP ARCHITECTURE
The proposed DeepDistributedQU architecture emerges from unfolding the iterations of the Dis- tributedQP optimizer into a deep learning framework. Section 4. illustrates the main architecture, key aspects of our methodology, well as the centralized variant DeepQP. Section 4.2 leverages implicit differentiation during backpropagation to facilitate the training of our framework.
4.1 MAIN ARCHITECTURE
Architecture overview. The DeepDistributedOP architecture arises from unrolling the Distribut- edQP optimizer within the supervised learning paradigm. (Fig. 3). This is accomplished through treating the updates (4) as blocks in sequential layers of deep learning network. The number of layers is equal to the predefined number of allowed iterations K. with each layer corresponding to an iteration k ...... K. The inputs of the network are the local problem data Ça and initializations ar w. s/. X and y.- These are initially passed to N parallel local blocks corresponding to (5 which output the new variables af and z/- Then, all 2 are fed into N new parallel local blocks (6). yielding the new iterates s/. In the meantime, all ar are communicated to a central node that computes the new iterate w through the weighted averaging step (7.. Subsequently, the global variable components w; are communicated back to each local node i, to perform the updates (8)- (9) which output the updated dual variables A, ys. This group of blocks then repeated K times, yielding the output of the network which is the final global variable iterate "
Learning feedback policies. Standard deep unfolding typically leverages data learn algorithm parameters tailored for specific problem (Shlezinger et al. 2022). From a control theoretic point of view, this process can be interpreted as seeking open-loop policies without the incorporating any feedback. In our setup, this is equivalent with learning the optimal parameters P.. p.. a* with
p (los 1 SoftPlus(("),,
k = ,
(10)
for all i= 1, 1..... N and l 1,...,,,, K,, where the SoftPlus(-) function is used guarantee the positivity of p 14,; and the sigmoid function Sigmoid,, (.) restricts each a to lie between (1, 2).
In the meantime, the predominant practice for online adaptation of the ADMM penalty parameters relies on observing the primal and dual residuals every few iterations (Boyd et al. (201)). The widely-used rule suggests that if the ratio of primal-to-dual residuals is high, the penalty parameter should be increased; conversely, if the ratio is low, should be decreased. Despite its heuristic nature, this approach includes notion of "feedback" since the current state of the optimizer used to adapt the parameters, and as a result, it can be interpreted as closed-loop policy. Based on this point of view, our goal to learn the optimal closed-loop policies for the local penalty parameters
p = Pofi (p. + & (r pp otp))) p.t = SoftPlus (at + TT (ri sh (lww) (11)
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First update (10-91)
Second update (93)
First update (900-91)
Problem Data
InitSalizzaion
Food-torward
Dual update (94)
Food-forward params
Training Loss
Residuals
params
poc
Residuals
p-11
Feedback
Feedback FC layer
FC
Layer à =1
Layer k K
Figure 4: The DeepQP architecture: The simplified variant for centralized QP.
where the feedback components are obtained through the policies o.. (ri... of Of..) parameterized by fully-connected neural network layers with inputs rf,,, st. and weights 0%.- The terms rf. and sf. represent the local primal and dual residuals of node at layer is and are detailed in Appendix
Solving the local updates. The most computationally demanding block in DeepDistributedOP is solving the local updates ( as this requires solving a linear system of size n, m, Similar to OSQP (Stellato et al. 2020). this can be accomplished using either direct or an indirect method. The direct method factors the KKT matrix, solving the system via forward and backward substitution. This approach is particularly efficient when penalty parameters remain fixed, as the same factorization can then be reused accross iterations. Nevertheless, at larger scales, this factorization might become impractical. In contrast, with the indirect method, we eliminate V to solve the linear system:
(Q: + M
A A,
x
qi + μ w y: A p 2; A A
(12)
Q
This new linear system is solved for क. k+1 using an iterative scheme such as the conjugate gradi- ent (CG) method. We then substitute uº* o¹ (A, 2,) A, The indirect method has three important properties that make it particularly attractive in our setup. First, its computational complexity scales better w.r.t. the dimension of the local problem, while no additional overhead is introduced by changing the penalty parameters. Second, it can be warmstarted using the solution from the previous iteration, greatly reducing the number of iterations required to converge solu- tion. The final important property, which is critical for the scalability of the DeepDistributedOOP is that training with the indirect method can be much more memory efficient shown in Section 4.2
Training loss. Let S ((Q.. q: A, b,) N w*); subject to the known mapping G of problem ( The loss we are us- ing for training is the average of the 7y-scaled distances of the global iterates Wi...., Wy from the known optimal solution of each problem instance Ç3, provided as
( be dataset consisting of H problem instances e²
K
1
((S;0)
= H
Σ
Tie Tk
Tx w ww'ww")())))) (c')
(13)
j=
where 0 corresponds to the concatenation of all learnable parameters/weights.
Centralized version. While this work primarily focuses on distributed optimization, we also intro- duce DeepQP, the centralized version of our framework, for addressing general QPs of the form (I) In the centralized case, our architecture simplifies to 1, eliminating the need for distinguishing between local and global variables. Under this simplification, the DistributedQP optimizer coin- cides with OSQP. Hence, DeepQP consists of unfolding the OSQP updates (see Appendix F) and learning policies for adapting its penalty and over-relaxation parameters. The resulting framework is illustrated in Fig. 4 Additional details on DeepQP are provided in Appendix F
4.2 IMPLICIT DIFFERNNTIAION
When solving for the local updates in using the indirect method, it is computationally intractable to backpropagate through all CG iterations. This is especially important in the context of unfolding, as it would become necessary to unroll multiple inner CG optimization loops. To address this, we leverage the implicit function theorem (IFT) to express the solution of (12) implicit function

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Published as a conference paper at ICLR 2025
of the local problem data. This allows to compute gradients in manner that avoids unrolling the CG iterations and requires solving a linear system with the same coefficient matrix, but with new RHS, achieved by rerunning the CG method. This result is formalized in the following theorem.
Theorem 2 (Implicit Differentiation of Indirect Method). Let ax+1 be the unique solution to the linear system x = = in 12). Let V=L(a, 1 be backward pass vector computed through reverse-mode automatic differentiation of some loss function L. Then, the gradient of L with respect Q and by is given by
V q: L
2 (x; k+1
@ dx
+ dx
8+1 @ x 1
VOL= dxz
where dx + the unique solution to the linear system dx + - (=L((t)))
The proof is provided in Appendix G and is a straightforward application of the IFT, similar to the results established by Amos & Kolter 2017) and Agrawal al. (2019).
GENERALIZATION BOUNDS
In this section, we establish guarantees on the expected performance of DeepDistributedQR. To achieve this, we leverage the PAC-Bayes framework (Alquier) 2024), well-known statistical learn- ing methodology for providing bounds on expected loss metrics that hold with high probability. In our case, we provide bounds on the expected progress of the final iterate w towards reaching the optimal solution w" for unseen problems drawn from the same distribution the training dataset.
Learning stochastic policies. PAC-Bayes theory is applicable to frameworks that learn weight distributions rather than fixed weights. For this reason, in order establish such guarantees, we switch to learning Gaussian distribution of weights P = N(ue, Le) based on a prior Po = N(r Eg). This choice motivated by the fact that PAC-Bayes bounds include Kullback-Leibeer (KI.) divergence terms which can be easily evaluated and optimized for Gaussian distributions.
Generalization bound for DeepDistributed(R. To facilitate the exhibition our performance guarantees, we provide necessary preliminaries PAC-Bayes theory in Appendix [H] generalization guarantee for DespDbstributeQQR a meaningful loss function must first be selected. This quamtity will be denoted q(S: 6) to differentiate from the loss used for training. To capture the progress the optimizer makes towards optimality, we propose the following progress metric:
To establish
4(::0) = min || (5:0) - w0 (C)
(14)
(() w (c)ll2
This loss function measures progress by comparing the distance between the final iterate w () and problem solution (C) with the distance between the initialization (C: f) and the solution. This choice satisfies the requirement of being bounded between 0 and I while being more informa- tive than the indicator losses used in prior work that simply determine whether the final iterate is within specified neighborhood of the optimal solution (Sambharya & Stellato 2024a). Moreover, this loss is invariant to the scale of the problem data since it is relative mcasurement.
As in Appendix H. let qp (P) be the true expected loss and (P()) the empirical expected loss. To evaluate the PAC-Bayes bounds in (101). the expectation (oop[g((C: 0)] must be computed as part of the definition of qs (P). Since no closed-form solution available, an empirical estimate using M sampled weights (0 is required to upper bound (P) with high probability. We adopt standard approach involving a sample convergence bound (Majumdar et (202T) Dziugaite & Roy
2017 Langford & Caruana (001)). Specifically, define the empirical estimate of (s(P) as:
(s(P;
1 MH
ΣΣ (G:;
(15)
Then, the following sample convergence bound provides an upper bound on (P(),,
qs (P) < (sPP; M,c) = DxL. (is(P; M) M- (2/c)).
(16)
with probability e. The following theorem summarizes the PAC-Bayes bound we use to evaluate the generalization capabilities of our framework.
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Random
Random OPs. (with Equality Constraints)
Dauble Integratir Optimal Contral
OSOP 1o foot OSof au
Ore
13"
SeepOr (0pen Loogl
13
Leesl
107
32-4
12
Oaclliating Masses Optimal Controll
Portfolie Optimisation
LASSO
13
12
33"4
10"1
33
18"4
00
2.5
50
755
10.0
12.5
15.6
37.5
20.3
5
16
15
25
25
20
0.
2.5
5.0
33.2
32.5
15.0
17.5
Figure 5: Small-scale centralized comparison of DeepQP and OSQP. Across all tested problems, DeepQP consistently outperforms OSQP (same per-iteration complexity using the indirect method).
Theorem (Generalization bound for DeepDistributedQ(P) For problems ( E 2 drawn from dis- tribution D, the true expected progress metric of DeepDistributedQ with policy P. i...
I|o (C: w" (C)ll2
.1]]
qD (P) = ECED Ey-p min
(17)
(() w (c)ll2
is bounded with probability at least 8 - bb:
(P) S DKL (Bs(P: M, s//l (PIIP))) + log(2v (2122///6)) /H)
(18)
where ãs (P;Me)) E) is the estimate of 4PP: M, e) described in (6)
We explain in detail how we train for optimizing this bound in Appendix
6 EXPERIMENTS
We conduct extensive experiments to highlight the effectiveness, scalability and generalizability of the proposed methods. Section [6.1 shows the advantageous performance of DeepQP against OSQP on a variety of centralized QPs. In Section[6.2) we address large-scale problems, showcasing the scalability of DeepDistributed(( despite being trained exclusively on much lower-dimensional instances. Additionally, we discuss the advantages of learning local policies over shared ones and evaluate the proposed generalization bounds, which provide guarantees for the performance of our framework unseen problems. An overall discussion and potential limitations are provided in Section 6.3 All experiments were performed on an system with an RTX 4090 GPU 24GB, a 13th Gen (ntel(R) Core(TM) 1390KK and 64GB of RAM.
6.1
SMALL-SCALE CENTRALIZED EXPERIMENTS: DEEPQP vs OSQP
Setup. We begin with comparing DeepQP against OSQP for solving centralized QPs 1I. The following problems are considered: i,ii) random QPs without/with equality constraints, iii, iv) opti- mal control for double integrator and oscillating masses, v) portfolio optimization, and vi) LASSO regression. For all problems, we set a maximum allowed amount of iterations K for DeepQP within 30] and examine how many iterations OSQP requires to reach the same accuracy. We train DeepQP using both open-loop and closed-loop policies and with a dataset of size H E [500, 2000]. For OSQP, consider both constant and adaptive penalty parameters p and we set to be either 1.0 or 1.6. Additional details on DeepQP, OSQP and the problems can be found in Appendix [
Performance comparison. The comparison between DeepQP and OSQP is illustrated in Fig. 5 Note that both methods share the same per-iteration complexity from solving (22). We evaluate their performance by comparing the (normalized) optimality gap ||a - x l|2/ Vn. n. For all tested problems, DeepQP provides a consistent improvement over OSQP, requiring 1.5 3 times fewer iterations to reach the desired accuracy. Furthermore, the advantage of incorporating feedback in the policies is shown, as closed-loop policies outperform open-loop ones in all cases.
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Randoms OPc 1824)
Coupled Onc. Masses (H 200)
Randon
Coupled Masses
Optiastt Cep
1:0%
L. Axtioc
10
10
20
10
10
10"
60
10"
10"
10
1024
M=100
N00
Random 0P% 1624)
Nottookk Flow 5000)
Randon
Network
Optinalit Cap
100
10
10"
10
20-1
10
30
10
Steal
1124
110000
5000
Coupled Pondulano (M-
Distriluted LASSO (N- 5000
(mn)
Coupled Pondulumo
Distributed LASSo
Optinaity Gap
10e
10-1
107
10
20
10
10"
10
10
10
207
10
1000
1.00
500
ADHN Iteration
ADHN Itaration
Figure 6: Scaling DeepDistributedQOP high-dimensional problems. Left: Comparison be- tween DeepDistributedQP and its traditional optimization counterpart DistributedQP (same per- iteration complexity). Right: Total wall-clock time required by DeepDistrrbutedQP, DistributedQP and OSQP (using indirect or direct method) to achieve the same accuracy.
6.2 LARGE-SCALF DISTRIBUTED EXPERIMENTS: SCALING DERPDISTRIBUTEDDP
Setup. The purpose of the following analysis is compare the performance and scalability of DeepDistributedQP (ours), DistributedQP (ours) and OSQP for large-scale QPs of the form (2). We consider the following six problems: i.ii) random networked QPs without/with equality constraints, iii, iv) multi-agent optimal control for coupled pendulums and oscillating masses, v) network flow, and vi) distributed LASSO. We select a maximum allowed number of iterations K for DeepDis- tributedQP within [20, 50] and examine what is the computational effort required by DistributedQP and OSQP to achieve the same accuracy measured by the optimality gap ||w - 1|2/ VT. More details about our experimental setup are provided in Appendix J
Training on low-dimensional problems. One of the key advantages of DeepDistributed@P1 is that it only requires using small-scal scale problems for training. The training dimensions for each problem are detailed in Table T Both open-loop and closed-loop versions are trained using shared policies on datasets of size E [500, 1000]. We employ the shared policies version of DeepDDstribuuee(P to enable the same policies to be applied to larger problems during testing.
Scaling to high- high-dimensionnll problems. Subsequently, we evaluate DeepDistributedQP on prob- lems with significantly larger scale than those used during training. The maximum problem dimen- sions tested are shown in Table 1 On the left side of Fig. o we highlight the superior performance of DeepDistributed(U over its standard optimization counterpart DistributedQP (same per- per-iteration complexity). In all cases, the learned algorithm achieves the same level of accuracy while requiring 1.5-3.5 times fewer iterations. Additionally, the right side of Fig. compares the total wall-clock time between DeepDisttriutedQQe DistributedQP and OSQP (using indirect or direct method). For complete illustration, we refer the reader to Table Appendix p.5) The provided results emphasize the superior scalability of the two proposed distributed methods against OSQP for large-scale QPs, as well as the advantage of our deep learning-aided approach over traditional optimization.
Local vs shared policies. When applying a policy to a problem with the same dimensions as used during training, leveraging local policies instead of shared ones can be advantageous for better ex- ploiting the structure of the problem. On the left side of Fig. 7 we compare the performance of local and shared policies random QPs (N = 16) and coupled pendulums (N 10). For the cou- pled pendulums problem, which exhibits significant underlying structure, local policies demonstrate clear superiority. For the random QPs problem, where structural patterns are less pronounced, the advantage of local policies is smaller but still significant.
Performance guarantees. Next, we verify the guarantees of our framework for generalizing on unseen random QPs (N = 10) and coupled pendulums (N = 5) problems. We switch from learning deterministic weights to learning stochastic ones and follow the procedure described in Appendix ! with H 15000 training samples, M = 30000 sampled weights for the bounds evaluation, = 0.009 and e = 0.001. The resulting generalization bounds, illustrated in Fig. | (right), are expressed in terms of the the expected final relative optimality gap the progress metric used for

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Table Training and maximum testing dimensions for DeepDistributeeQQP The metric nnz (nz(Q,A) A) denotes the total number of non-zero elements in Q and A.
Training
Max Testing
Problem Class
12
nn2 A)
NV
12
m
nna Q A)
Random QPs
16
160
120
4,000
1,024
10,240
9,920
300,800
Random QPs w/ Eq. Constr.
16
160
168
4,960
1.024
10,240
9,920
300,800
Coupled Pendulums
10
470
640
3,690
1,000
47,000
64,000
380,880
Coupled Ose. Masses
10
470
1,580
4,590
300
28,200
47,400
141,180
Network Flow
100
5,000
25,000
35,000
150,000
Distributed LASSO
10
1,100
3,000
29,000
500
50,100
150,000
1,450,000
gPs
Coupled Penduluns
Coupled
Lucall
Cytteralty Cae
10"
Optieultyy Cre
deyrep Palidee)
10-1
10-1
10'
10
Desodkrrip 10 Locg)
80
3.20 3.00
10
20
30
10
30
35
10
35
Figure 7: Left: Local vs shared policies. We showcase the advantage of learning local policies over shared ones. Right: Performance guarantees. The obtained generalization bounds guarantee the performance of DeepDistributed(O and its improvements over DistributedQP.
deriving bounds in Section 5 implying that with 99% probability the average performance of our framework will be bounded by this threshold. The bounds are observed to be tight compared to actual performance, underscoring their significance. Moreover, they outperform the standard opti- mizers, providing a strong guarantee of improved performance for DeepDistribute:((.
6.3 DISCUSSION
In which cases can we use the direct method? As illustrated in Fig. 6 and Table 6 and further discussed in Stellato et al. (2020), the indirect method is generally preferred for solving systems of the form
or (90) for DeepQP/OSQP once the problem reaches a certain scale. In this work, adopt this approach both for training, due the memory and computational advantages out- lined in Section [.2 and evaluating DeepDistrbuttee(///eeep//
However, it is worth considering
whether the direct method might be advantageous during evaluation, choice that depends on the problem scale and capabilities of the available hardware. Overall, the results of this work show that learning policies for the algorithm parameters is significantly beneficial in the context of both dis- tributed and centralized QP assuming the indirect method is used. In future work, we wish to also explore schemes that adapt the parameters less frequently using the direct method and/or designing mechanisms to dynamically switch between the two approaches.
Limitations. One limitation of the proposed framework is its reliance on supervised training loss, requiring dataset of pre-solved problems. In future work, we aim to explore training through directly minimizing the problem residuals rather than the optimality gaps. Furthermore, while PAC- Bayes theory provides an important probabilistic bound on average performance, stronger guaran- tees may be necessary for cafetclcritical applications to ensure reliability and robustness.
CONCLUSION AND FUTURE WORK
In this work, introduced DeepDistributedQR, a new deep learning-aided distributed optimiza- tion architecture for solving large-scale QP problems. The proposed method relies on unfolding the iterations of novel optimizer named DistributedQP layers of supervised deep learning frame- work. The expected performance of our learned optimizer on unseen problems is also theoretically established through PAC-Bayes theory. DeepDistributedQP exhibits impressive scalability in effec- tively tackling large-scale optimization problems while being trained exclusively on much smaller ones. In addition, both Detribistrb and DistributedQP significantly outperform OSQP in terms of required wall-clock time to reach the same accuracy as dimension increases. Furthermore, we showcase that the proposed PAC-Bayes bounds provide meaningful practical guarantees for the performance of the learned optimizer on new problems. In future work, we wish to extend the pro- posed framework to a semi- superviie version that relies less on pre-solved problems for training. In addition, we wish to explore incorporating more complex learnable components such LSTMs for feedback within our architecture. Finally, we wish to consider other classes of distributed con- strained optimization methods outside of quadratic programming.
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ACKNOWLEDGMENTS
This work is supported by the National Aeronautics and Space Administration under ULI Grant 80NSSC22M0070 and the ARO Award #W911NF2011151. Augustinos Saravanos acknowledges financial support by the Onassis Foundation Scholarship. The authors also thank Alec Farid for helpful discussions PAC-Bayes Theory.
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APPENDIX
CONTENTS
Extended Related Work
16
Complete Derivation of DistributedQU Algorithm
17
Standard Convergence Guarantees for Simplified Version Distributed
19
D Proof of Distributed Asymptotic Convergence
19
D.I Sketch Proof
20
D.2 Necessary Lemmas
20
D.3 Proof of Theorem
27
Details on DeepDistributed((P Feedback Policies
29
The Centralized Version: DeepQP
29
Proof of Indirect Method Implicit Differentiation
30
H Background on PAC-Bayes Theory
31
Optimizing and Evaluating Generalization Bound
31
J
Details on Experiments
32
J.1
Problem Types in Centralized Experiments
32
0.2
Problem Types in Distributed Experiments
33
J.3
Details on Training and Testing
................................
36
U.4 Details on Standard Optimizers
................................
36
J.3
Details on Wall-Clock Times
37
Additional Experiments
37
K.I
Varying Training Dataset Size
37
K.2
Can Policies Trained for Specific Problems Be Applied to Other Problems?
38
K.3
Varying the Number of Layers in Testing DeepDDistributed(11
39
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A EXTENDED RELATED WORK
Distributed optimization with ADMM. The ADMM method was first introduced in Glowinski & Marroco (1755) and Gabay& & Mercier 1976), yet it only became widely popular after the highly influential review by Boyd et al. (2011). which highlighted its suitability for deriving distributed algorithms whose computations can be parallelized. Since then, ADMM-based architectures have been introduced in a variety of setups including distributed learning (Huang et al.) 2019 Zhou &
Li 2023). multi-agent robotics (Saravanos et al.) [203a, [hhrriwaa al. 2024), signal processing Matcos et al. 2010; Zhang & Kwokj 2014). stochastic systems (Saravanos et al. 2021 2023b), sensor networks (Mota al. 2013), power grids (Erseghe) 2014) and many other areas, often problems. Nevertheless, it is well-known that the convergence speed of distributed ADMM algorithms is heavily dependent on the choice of their underlying tuning parameters. To address this difficulty, several adaptation schemes have been presented, typically relying on the residuals of the current iterates (Song et al. 2016 Xu al.) b).7a|b). Nevertheless, such approaches tend to be myopic and cannot leverage data or provide guarantees on the performance of the algorithms on unseen problems under finite iterations.
hibiting an impressive scalability for high-dimensionn
Learning-to-optimizz for distributed optimization. The concept of integrating learning-to- optimize approaches into distributed optimization is particularly compelling, as algorithms of the latter class typically rely a significant amount of designing and tuning by experts. Nevertheless, the area of distributed tooninggto-optmmize methods remains largely unexplored. For instance, al- though distributed ADMM has achieved widespread success in decentralized constrained optimiza- tion, its unfolded extension as a deep learning network has only been recently explored by Noah & Shlezinger 20224). This framework shows promising results, but it is limited to unconstrained problem formulation, assumes gradient-based local updates, focuses solely on parameter tuning and lacks performance guarantees, Biagioni et al. (2020) presented an ADMM framework which uti- lizes recurrent neural networks for predicting the converged values of the variables demonstrating substantial improvements in convergence speed. In Zeng et al. (2022). a reinforcement learning (RL) approach for learning the optimal parameters of distributed ADMM was proposed, showing promising speed improvements, but requiring substantial amount of training effort.
Beyond distributed ADMM, Wang et al. (2021) proposed unrolling two decentralized first-order op- timization algorithms (ProxDGD and PG-Extra) as graph neural networks (GNNs) for addressing the decentralized statistical inference problem. Hadou et al. (2023) presented distributed gradient descent algorithm unrolled as a GNN focusing on the federated learning problem setup. From a different point of view. He et al. (024)) recently introduced distributed gradient- based learning-to- optimize framework for unconstrained optimization which partially imposes structure on the learn- able updates instead of unrolling predefined iterations. A deep RL approach for adapting the local updates of the approximate method of multipliers was recently proposed in ZZuu et al. (2023). Fi- nally, Kishida et al. (2020) and and[gaww & Ishi (202I) have presented distributed learned optimization methods for tackling the average consensus problem.
Learning- for (centralized) QP. Recent works have focused on accelerating QP through learning; however these efforts have soiely concentrated centralized setup. In particu- lar, Ichnowski et al. (202T) introduced an RL-based algorithm for accelerating OSQP demonstrating promising reductions in iterations, yet training this algorithm incurs significant computational costs. From a different perspective, (Sambharya et al. (2023) (024) focused on learning-to-inn to-initializz fixed- point methods including OSQP, while maintaining constant parameters in the unrolled algorithm iterations. Concurrently with the development of the present work, (ambbaayy & Stellato (2024b) presented methodology for selecting the optimal algorithm parameters for various first-order opti- mization methods. Considering OSQP as the unrolled method coincides with the open-loop version of the proposed DeepQP framework without any notion of feedback policies.
Generalization guarantees for learning-to-optimil The works in Sucker & Ochs 2023) and Sucker et al. (2024) presented generalization bounds for learned optimizers, considering the update function gradient step multi-layer perceptron, respectively. [Sambharya & Stellato (2024a) recently also explored incorporating PAC-Bayes bounds in assuming specific underlying algorithm structure. However, our approach differs fundamentally,
methods without
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their method employs binary error function, whereas in this work directly establish bounds based on the optimality gap of the final solution.
B COMPLETE DERIVATION OF DISTRIBUTEDQP ALGORITHM
Problem transformation and augmented Lagrangian. Here, we present the detailed derivation
of the DistributedQP algorithm presented in Section 3.2 We consider the over-relaxed version of ADMM (Boyd al. 2011) with a E (1. First, let us rewrite problem (2)
Q.a min Σ 2. Qaa, + q. az s.t. A, = Zi, S bi, F. W;; iEV,
(19)
SEV
where we have introduced the auxiliary variables z; for each i ,..., N. In addition, let define the variables St, i= 1....,,, N,, and rewrite problem (19 as
min
Q.a q. az s.t. A, = Zi, 8; S bi 2, 8i 2; = Wi i
(3)
iEV
The above splitting constitutes the problem suitable for being addressed with two-block ADMM scheme, where the first block variables consists of (I,, 2i).=....... and the second one contains {s; (s and w. The (scaled) augmented Lagrangian (AL) for problem (3) is given by
L =
Σ VEV
q: 3i + IA (1:;, zi) + Im Sb. (8:)
2
(20)
Pi
A
14
yi
+
2i
+
+
w.; +
2
Ps
2
μ4
First block of primal updates. The first block of variables is updated through
(a, 2 hiev arg min [(x, 2, s w A y²).
This minimization can be decoupled the following N subproblems for each E V.
1 (0; 2 arg min 50 Q,æ; q, x; 2i 8,
Pi
A
Mi
y/
+
+
2
w
+
Pi
2
2
pi
s.t. A,
24.
where have temporarily dropped the superseript iteration indices for convenience. Since these problems are conditions for each subproblem are given by
QPs, we can obtain closed-form solution. particular, the KKT
Q + qi + Hı( - -.;) - + A v; 0, 9.) + X V, 0,
(2la)
p.(z
(21b)
A, 2; = O,
(21c)
where 2; the Lagrange multiplier corresponding to the constraint A; = 2,. Eliminating z; leads to the following system of equations
[
Q:
+ A,
mil
A. 1/pil
[ q: + 1 y!
(22)
V
2; 1/p:>1
with +++ given by
z k+1 = oi + p; (v. k+1 Xt).
(23)
Second block of primal updates. The second block of updates is given by
{silie, arg min ((* k+1 8, w. A )),
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more analytically by
A
(sihiv. w arg min
Σ
Pb
az
k+
(1
aast
Pi
iEV
Pi
aa;
k+1
+ (1 ox)o
y w. +
s.l. 8i b;.
2
Note that this minimization can be decoupled w.r.t. all 8;, i E V, and w. In particular, each s; is updated parallel through
++1 sb (azę + (1 )s + //pp))
(24)
The global variable minimization can also be decoupled among its components / 1...., n, which gives
[eo;ls + [ut]] #b 2
w7 arg min Σ
||at 1 (1 a)(
2
((())=
Setting the gradient to be equal yields
[3
Σ ((,))=
us
aalo + (1 a) k+ +
0,
M4
leading to
[B/'1
Σ G(.
k+1 Σ Me
aaaltti] , + (1 aww +
p4
((.1))
which eventually gives the update rule
ly
w k+1
(1 a)
(25)
Eou
Dual updates. Finally, the dual variables are updated through dual ascent steps as follows
X k+1 x + pi(az k+1 + (1 - a² 1 +1 + (1 a auw -w y k+1 y + (aae
(26)
(27)
Simplifying the global update. the global update can be simplified to
It is important observe that after the first iteration (for A > 1),
Eou.n M1 + 1
w
k+I
(1 a)wi
(28)
E M4
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since the summation
Σ 1
+++1
Σ
[ull + ui(ala "15 + (1 - a0] - -12b [10t+11 '1,)
((:))=
((1.))=1
Σ
[ufls + u '13 + a)wf 1
((())=1
Σ
lu'll + Me aa/a (1 a1>w
((.))=1
E 0rpea (artt=+ + [y
(1 o
E Me
b/ll + u4 aaaa -t axt axwll lab "o + [y
Σ
Eo(un)
((())=1
Egtss=ile Mi - Jo + [3
Σ ((,))=
| + (upu(act++11
Esto,o
Σ [(,))=
| + au
Σ
applaxt 1o + [y =0
(29)
((u,))=
C STANDARD CONVERGENCE GUARANTEES FOR SIMPLIFIED VERSION OF DISTRIBUTEDQP
In the simplified case where p = p. n. = for all E V and for k; as well a* for all k it would be straightforward to apply the classical convergence guarantees of two-block ADMM for convex optimization problems (Deng & Yin 2016) to ensure the convergence of DistributedQP. In the following, we show how DistributedQP would fit under this setup.
Let us define the concatenated variables a [ar,eevi (zilee)) and z [(s)evv; w). Then, we can rewrite problem 3 as
min f(t) + g(z) s.l. Au + B==
(30)
where
((x)
- Qaa + t xi + (0,, 21),
-(2)= Ie.
(31)
LEV
1EV
and À = bdiag(I, = bdiag G) and c = 0, with G e m.E,,)) defined such that
Gw. In other words, G is the matrix that represents the local-to-global variable components mapping, formally defined as = =Gi; [G;...;:Grt with each submatrix G; E RR, xn given by
1, if (i,
[Giju,
(32)
0,
else
Given this representation, it becomes clear that our algorithm can be framed as a two-block ADMM. Now, note that G is a full column rank matrix since all global variable components gi are mapped to at least one local variable component [z..]. Then, since the functions f. g convex and the matri- ces A. B are full column rank, it follows from Deng & Yin (2016) that the algorithm is guaranteed to converge to the optimal solution problem
Nevertheless, this analysis is solely applicable to this simplified version of the DistributedQP algo- rithm. In Appendix D. we tackle the more complex case of ineration-varying relaxation and local penalty parameters.
D PROOF OF DISTRIBUTED@P ASYMPTOTIC CONVERGENCE
In this section, we prove that DistributedQP is guaranteed to converge to optimality, even in the more challenging case of iteration-varying relaxation and local penalty parameters (Theorem
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The following analysis extends the theoretical results presented in Xu al. (2017a), where the convergence of an adaptive relaxed variant two-block ADMM is provided. Nevertheless, this analysis is not directly applicable to our which involves distinct local penalty parameters per computational node.
D.1 SKETCH OF PROOF
To begin, we outline the following conventions. The points x",=", w", "0""""". X are the KKT points of problem (3). refer to the notion of a distance function at any (k+1)tth iteration to be rep- resenting a weighted squared norm of the difference between the variables gh+1, w++1, y *++1 and their corresponding optimal values s" "", yy ' indicating the distance from the optimal point.
We prove the convergence of DistributedQP in the following steps:
Necessary Lemmas (Section D.2) First, we derive the descent relation (65 which estab-
lishes relationship between the values of the distance function for consecutive iterations.
To derive this descent relation in Lemma 4 we first introduce the relations [RI-(R3 R8 in
Lemmas 33
Next, we use the derived descent relation from Lemma 4 to prove convergence to optimality (Theorem based on Assumption]] T
Convergence (Section D.3
D.2 NECESSARY LEMMAS
In this section, we present some necessary lemmas before proving the convergence of DistributedQP in Section D.3 For notational convenience, let us define
1 2 Q,a, x4, C, (8188 S bbb, iEV.
M(x,,) Lemma 1. For all i V, the following four relationships hold every iteration
G.T. k+1 =0
(R1)
1VV
k k+1
(y, ++1 - -,,) - ax )G + G.
(R2)
1
t 2
k+1
h+1 -x,) - a a=)) + k+1
(R3)
p.
X (t, t2) = 4h,t2 C,.
(R4)
Proof. Relationship RI is equivalent with the argument proved in 29). Indeed, we observe that
each matrix G. E Txx: indicates the mapping from local indices (r.j) to global indices I for particular i, then we can write
Σ Eguh ++11
VG.
0,
(33)
iEV
Σ Eo 1
which yields RI). Relationship R2 follows by rearranging the dual update 9 and replacing w. G,w. Similarly, relationship R3) tollows by rearranging the dual update In the remaining, we focus proving R4).. Let repeat the s,-update (6) as
++1 llc. (a 2,
k+1 +1 (1 a )s /
(34)
We define the closed convex cone C, {pl p < 0}. such that 34 is rewritten
++l Icc (s: (s.+1) b..
(35)
with
k+1 h z
k+1 (1 a X² 22/00 by.
(36)
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Next, let rearrange the dual update (8)
Pf /0
2; + (1 -
k+1
(1-aa))s )* - 1
(37)
which can be rewritten through (36) as
+ pi (5 + bb sf 1
(38)
Substituting (35) in the above, we get
X x+1 p (s IIc ($, (
(39)
For convenience, let also repeat the definition of polar cones.
Definition 1 (Polar cones). Two cone sets and Do are called polar cones if for any E D and d E D follows that d' d= O.
By Moreau's decomposition refer to Theorems 1.1 and 1.2 from Soltan 2019) 8/t1 can then be expressed as
IIg (s,+ + a (s(+1)..
(40)
where C is polar cone to C. Thus, using (39h and (40), we get X x
MTT (8/+1), which
implies that X / E C.. Further, since C₁ is cone, and p.t O, we get
E C₁
(41)
Now, any vector t E Cj satisfies t t-6 b, C. Since C. and Ci are polar cones, and using (41). the following relation holds true by the definition of polar cones,
+ (t b,) 0 for all E C..
Thus, for any vectors t,, t2 C; and for all k, we have
X x+1T (t) - (2) = X k+1T (4) - -b
(12-b;)) -b2 = 0,
(42)
which proves R4
Lemma 2. For all i E V. the following two relationships hold at every iteration k:
(Vi((a))) + vi) ( - x + + 23 (z) --,+=+11 2(+1)= =
(R5)
Vfi(at+) + vtt + μ/ ( (1 - a )=) - (2 a* )²) + G( x+1 )] (z -x,)
x;²)
x + p (a- -=))))++)) (2 a²) a²)s + s s/t
(x/+))
2, = 0.
(R6)
using the KKT conditions of problem (3). (x", 2", s* w²) the optimum of and only the following conditions are true:
Proof. We start with proving R5
The point
Optimality for æ;;
V[i(a;) A v + yy =0,
(43a)
Optimality for z;:
- - + +x; = 0,
(43b)
Optimality for 8;:
N E Ne,e((8))
& x (s: 8,) 20V E CC,
(43c)
Optimality for we
Σ G. vi = =0,
(43d)
KEV
Constraints feasibility:
& = =
(43c)
x; G.w",
(43f)
A 2i:
(43g)
s E C..
(43h)
From 43ah, we have
(V](())) + A oi + T (m) - - 0.
(44)
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and similarly from 43b). get
(-v, + +21) T (2) - -x 2x+1) = O.
(45)
Adding
44 and 45 get
(Vi((a)) + A, Vi + yE) (ar - a,; + ( vjj + );) (z) - 2x+1 = 0,
which yields
(VI((z))+))) (ar - +1 2;
(z)
2 + +u, (A,(a)
- 2)+ = 0. (46)
Using 43g) and the fact that A A
x
k+1
0, can then simplify (46 R5
Subsequently, we proceed with proving R6 given by
The KKT conditions for the (A: + 1)-th update of æ;
Optimality for ay:
fi k+ ) + A Noat + M (x) G, + +vt/u47 0,
(47a)
Optimality for z,: Constraints feasibility:
tht + p (2, 1/tt1 - 2t + +x11000) = =0,
(47b)
A k+1 = k+1
(47c)
From (47 we have
Vf((att)) ( + A.v. v + (x( h+t1 G.w + + /1/t) /
(a -x,)= = O.
(48)
We rewrite the term (x( (x,t1 G.w + yt /uf) using 9 as follows
1 (xr
G /Z///()
u ar/t1 G +/ttt//1/ //1 (ata (1 at G Gra
t + / [r.; (n,tt1 G.w aa a=attt a G + G:wh+1)
+1 + μنب (1 at ja) (2- 12-a*tea G. G.
such that 488 is given as
Vfl + + A. h+! + y. k+1
+ 1 (1- - )=) k+l (2-a (2-a+)G + G,untt) ]
(1-04)04
(x k+1 xx;) 0. (49)
Similarly, from 47bb. we get
k+1 + p (z) k+1 /p (2/t1 2,)
-,,) 00
(50)
We rewrite the term (2 k++ - o + \ / using 8 as follows
p 1 8/ + x = /.¹ xt+ 2 + x k+1 / (a" x + a - s s/t11)
x + p.¹ (z.² k+1 2/ a xx k+1 - (1 - a )*) + s/+11
x + p.c (1 a )z x+1 (2 a ) s st:tt).
such that (50 is given as
ut + X + p (1-04)24 - (2 -2-akk)s )s + + k+1 s
(2/it1 2;¹) = 0. (51)
Combining (43g). 49
51
and the fact that A,
2 x+1 0. we obtain (R6)
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Lemma 3. For all i E V. the following two relationships hold at every iteration k:
u u yi + pa ((1 - at jattt (2- at ) + +Giohtt)) (e xx]
(110, +tt v211 - 11o, - 11113) + (2 a*) (2-a4)
11 y1112
2
+ (2 - a )ue |G; - w² +
(1l Gi (w ~000 1|2
2(at)2
2
|G - =) + a (, x+1 - G
1
(i
- (1
- ca"))e)
- a² w0)
+ (ark) (gtt1 k+1 ut²² Gi ((2 at v (1 (1- )=0 -a - at ))")), (R7)
1
1pt1 - x + p3 ((1 - )2 ++1 (2 a² ) 8 s T
(2/ 23²
1
(12 2111 1| -,113) (2 )
1 2/11
2 /
)
+ p
2 (11s - =s, || s;|??) + (2 a }p}
|| 1111
2(a+)²
+ a
(2 h+1 x-- a+ + sttl a st).
(R8)
Proof. Let us first simplify the individual terms of the LHS of R7 For that, start by rewriting the term 2. k+1 xt follows using R2
wt) - (1 - a w + G-w
x x
at
Using 43d).
rewrite the above
1
- -1- + G, -ai--laa-Gra d=G'w") -
x
(y)
(52)
a
which can be simplified to
att -x a / (v. k+1 y + at 1o0w+t1-111-- (1- at - ww)
(53)
Let us now simplify the following term in the LHS of the relationship R7
(1-a*)es (1=a")aft ++1 (2 a²) G + G, (1 - at (xx G,w") - w²). (54)
Using
R2, we have
1 at
1 1
1 (2/tt - v.¹) (1 a G' G.antt) Giu*
Giw
which can be written as
1 or ( - vn:) + at Gi(wrtl- w
xx G
(55)
Substituting (55) in we get
(1 (1- a a+)=) - (2 Grw + G,w++1
(1 (1-a") a²)
(1, ytt) + ar
1 G((w++I _ w²
a
Using the above result, we then rewrite the following term on the LHS of R7 as
y ++1
Di + 1 ((1 a ()) (2 a G + +1)
y 1++ 4 + (1 )
at (y) k+l y + 1
at G( w²)
(56)
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For notational simplicity, let us consider the LHS of (R7 as LHS(R7). Using (56) and 53
we get
(L - )
(3 k+1 vf) + m
LHS(R7)
yi
a (w - -w0)
a
1
(ott k+1
v + a
1
Gi Giw - - a² jot a² w0))
which can be further rewritten as
LHS(R7)
a²10 (y, - [
h+1
v¹¹) + at ( - -.
Gi - -11 ar)e
a a²w")= (1 (1- at)
(2 11 w111? + (1- at)
(at)3 (y (y!t+ -y²) Gi
1
(1 aa-=a a w²)" + (at (atyz (wo - w (v, -,,)
uh (a+tz (Gi ww") T - -1 - j²) a a²"").
(57)
Let us now simplify each term on the RHS of the above equation. We start with the terms including only the variables y k+1 v and y, Using the fact that a b 2 [11a]] + ||6]12 ||a
bl(2), we get
at u (ot ((y" k+ - utt
(11,
1+1 v!11 + 11 R+1 s - 11 - 22 112)
2a p
Using the above result, we write
at (vt - y:) (v. ++ - y + (1 - )
(a+ )2 11y k++1 - 12
20 (11/+11 yfll" + fl y. 112 -1yy. -vi113)
1
(1- a +
(2
(2 a²)
= 20 (lle 22 11
y 11 + 2(a 2u
11
k+
112
(58)
Next, we consider the following terms in the RHS of 57 involving only the variables #+1 w and w"
1 (aty2 (G T GG, Gi(utti- -
a*) a²)e0* - a²w²")
(59)
(1- (L-a || - wn )²²
- (Gi(w*+t-wth
T (G - (w') w)))
(a+)2
a
Using similar approach as used to derive
58
we obtain
(1-a )= [[G - +
(G -w+)) (G, -::--i-""" w'))
(G((w(t1-2"))
(at))
a
(60)
(2 - 10-a))ut [[ - w
+ w
([] wI |[Gi(w* -w))
2(a
Now, let us consider the following terms from the RHS of (57
(1- a
(a (v: k+1 y Gi(w - aa= a²=²))
(a (at)z(w**) (wi+1 w G. (B/ - -y!")
"aatp (1,t1 - /
G. ( ( jt (1 a 2² a (1 - at )=" x + w w²)
(at)a k+1 TG ((2 a+) - (1 (1 )²))) a a a+)))) (61)
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Substituting (58.. (59). (60) and (61) into (57). we get
k+1 v² -1vx v1'l²) + (2 a
LHS(R7)
20 x (11y)
110,tt1 12
)2a4)22 pg
(2- a )²
1[G;(zottz )12 /
(1 (1, w0'1
2(aa)}
2
|G "::o==w)1l"" x²))]3)++ + 1 ++ - y ) T Gi - j=) - -a aa=w')
at (1
-11-1
-
+ (a+ (u. k+1 - y TG wo - (1 + (1 - ) a² (1 - a "(00)
(62)
which proves R7
Subsequently, we prove the relationship RK
Using similar steps as for R7
we get
A A X; + p ((1 a² )2 - (2 - a*) ) + s s
(2/ 2;)
1 (11 - xi112 - -1 - >3119) + (2 - )²)
1 - 1/
2 p
2(a+)2pp
s;11- || s; (?) + (2 a*) )p
+ 2
(11s)
||stt1 11
(63)
2(a
+ 1
(A
k+1 {;)r (s (s/+11 - (1 - a )s - a s) =;)
a
+ 1
(a:)² ( k+ X T ((2 - a ) ++1 (1 + (1 - a ) - a0 - )=?))))
Let us now simplify the last term of the RHS of the above equation as follows
(2/+1 x ((2 o* 1 (1 - )²))) a² a² )s;)
(1 (1 - a²) )3 - x (0) - ot) + a² - a (A! ++1 x²) ( - -f)) (64)
From 66 and 43h).. we have that the vectors sf s k+1 s ss E C. Using R4 the above equation
gives us
k+1 T a )s E+1 (1+ a²) ))s; a² (1 a a²)s²)
= - T - a² ) b+1 (2 + (a*)² - -2 + (-a t + (a²)))s;) = O.
It follows that (63) then simplifies to
(88
Lemma 4. The following inequality holds true at every iteration k:
Σ IEV
1 (1le k+1 - vill 11 311l)) + (1), - -w=)11 - ||G - )13)
+ p
1++ -11 1ス* 1/ + p (11s - s7112 - -1(s7 - st113)
(65)
<
(2- a
1
y
k+1 - 12 اور (( w w01) + 1
p || Xا512
a
NEV
+1 + sf1
+
Proof. We start by combining the relationships R55) and (66) to get
T
R+1 ve + 1 ((1 - a )a k+1 - (2 - k G + G,
(a x;;)
+ (2/-1 - xi + p ((1 - at )²) - (2 a ))) + 8 s(t1)
T
(2/41 27?)
(66)
-(((/((a/t)) - V vf(zi)) T (ar/t1 - x;))
=
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Since fi is convex, then we have (Vf((z)+)) Vfi(())) (x. xz) 2 0, which gives
Vi R+1
y. + u ((1 a )xxtt) (2 a² G + G(ohht))
(zl xx²)
(67)
| x p (11 - ) (2 a²)s sf s(tt))
(2/ - 22)) s 0.
Summing 67
over all i V, we get
y + p ((1 a )x + - -(2 - a + G.wntl)) T
(x -xx)
y;
SEV
(68)
(2 - a ) + of oftt)) (2ftt) -x,) s 0.
+ Σ
N+ \ + az1 k+l
iEV
Now, we use the relationships
R7
and
R8
to rewrite the above inequality
vfll²) + (2 -2-c8 a
0
(1V! h+1
v111 1l
10 k+1 yf11
2a 1
32
iEV
(2 at )pp |G w0th 201111
-w")
2(
2
1
G - w)')l?) + αx (v)
x+1 - vi) -
Gi (w - -(1 -(1-a")ue aa" - -a"w")
+ 1
(at)2 (y) k+1 )
T G, (22 * ye0 k+1 - (1 + (1 a )² -a - a yw*)
(1A k+1 - 111l - || - -;1??) + (2 - a
+
1 k+1 - 21111
2a
pf
(2 ph
/
- s7112 2 | -,,1²) + (2 at )p
2 (1
| $/112
2)
1 a (A A
- s+ + s - a² a²$t)
(69)
Let us now further simplify the terms the RHS of the above equation. For that, let us start with the last term on the RHS. We have
(A k+1 -2)) (-(1 11-1--0 a )s + + 1+1 - a² (A k+1 -);) 1;) (0 -8;)
(70)
- -1- a (2 - -2;) -8²).
Using R4). (4c and the fact that oft 8. k+1 ott1, 8t E C₁, get
(A k+1 - x #+1 8;) 2 0,
(71)
(2/+1 x;) (s s;) 0.
(72)
Thus, for h 2 1, combining (70). 71 and 722 we get
(A ++ x;) (-
) +
8 a a²s?) 2 0.
(73)
Now, the following results hold based on the relationship [1 and 43d)
Σ T. = 0.
Σ
b+1
w.) G, 0.
(74)
KEV
LEV
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By substituting 733 and 74 in 669
and by rearranging the terms, get
1
(10,
1/ 11 vfll²)
u 2
(IG 20)1 -1 -1
20 22
NV
/ (11s - $1112 || -5; s;11))
+ 2a pk
1-4 - -x11 - 1
-111 +
2
(2 a*)
(2 - a ju
S S iEV
llg
y,
[G,( 20" 201
2(a*) Fu
2(
(2 )
|| || - - + (2 - a²) jpt
|| |15 - 84:133
2(a
Since, a² 2 1, we can multiply the above equation with 2 l to obtain 66
D.3 PROOF OF THEBREM
Let us first rearrange the inequality 655 derived in Lemma 4 as
(2 )
| x+1
yt 11 n² (: w
)12
1
|
X 112 + p Al1sstt1 st(1²)
at
p.
iEV
(Ilyf
y/112 - 11 - yو 11²) + p (l - -w0)11 -IIG. (1+ -w0))3
iEV
(1 - x|1?? - - -x31= + pt (1s²² - =3 - || - s; =;11))
(75)
For convenience, let us define for each iteration k, the terms n/C i E V, and r7" such that
pt
0::
k-1
nt 1 max
ps
m max m
1
EEV
and the term V as V
ll
11 + 14, k-1 (G(( - w²1 -w=)
+
- 11 - -1;12- + p -1 -s,1(3)
1
iEV
P,
Based on the definition of r//. we can write
ll y + |||| G w w²1l + p.
| -1111= + p
s
1
$ (n. 1) ( 2? + μ k 1a= |G - w )1 + - 1 - 2mo1 + p -h 1 - 87 s;1
1
p.
By adding the above result over all E V, and using the fact that n 2 nht for all i, we get
Σ
+ + G w w0)1 1/1 || - + + - et11²)
1EV
Σ
(n
+1 1)
k-1 11 - 1 + - G w²11 +
1
- || 211s
11
pi:
p
!|
11
(76)
5 (1
+ 1) Σ
1
vt + +, "1 =00)12 +
1 - 11 1
k-I
p,-
Nev
+ 11², ||5 - s7 $p()2)
= =V - 1)
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Substituting the above result in (75). we get
Σ IEV
k+1
1/1 + ( + -20")
111
11 11
a
13
(77)
+ 1 k+1 - s/112) < (Mmix + +1 pk+1
Now that we have derived the above relationship. we need to prove that V bounded. By the definition of Vk we have that V is lower bounded by zero. Thus, prove that V is upper
bounded. From 777
we have
Vk+1 I (Amiu + 1)V²,
(78)
which leads to the following relationship
k vk+1 II (Taas + 1)V1
(79)
It should be noted that based on Assumption 1 have (7max + 1) - 1, k: Therefore, 79 implies that y is upper bounded for all k, and there exists Vmax such that
V S Viaax < 0o, for all k.
(80)
Let us now consider summing 77
over A as follows
1 x+1
wt + pil G: ++1 - x0 + 1+1 1/11
a
NVV
+ p ||lls k+1
s
s/113 1
tn + 1) Vk+1
(81)
The term on the RHS of the above equation can be further simplified as follows
1)1k - pk+1
Mma
v + Σ pk+1)= = -v-
I V
k=
km
km
Based on Assumption we have V/nxx
+ as + oc, which implies that
Mmax
< oo.
(82)
km
Using the above fact and 880
we can upper bound 30
W V as follows
Umax
v
Vmax
Vmux < co.
(83)
Using the facts that V1 is upper bounded, and 1/00 is lower bounded by zero, and using the above equation, we get
Vl
Mmnx
v S v1
mmx /b < oc.
A=
1=1
Thus, can rewrite
81
2
Σ
w k+1
21 + p G2 - 20 )112
1 || 2/112
a
k=1
IEV
(84)
+ p ||| |
s1113) < oo.
Since a E 2), we have 12-at) > 0 for all h. Further, we have 0 < uti P < for all k. Thus, 84 implies that as l + oc,
h+1 vtt) 0, G:(uo+1 - O, (2 1/) 0,
s k+1 st + (85)
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for all E V. This proves the convergence of the variables ye: X, and 8,- Further, it follows that G(w wn) + 0. Since G is full column rank, this implies that as h -+ oc,
(w wx) O,
(86)
which proves the convergence of the global variable w. Subsequently, combining R2 the convergence result (85). we also obtain that as k > xX,
R
and
(a) k+1 - x O,
- -
(87)
for all E V. Hence, we have proved the convergence of the DistributedQP algorithm.
Now that we have proved the convergence all variables, we proceed with verifying that the limit point of convergence the optimal solution problem (3). For that, we need to check if the limit
point satisfies the KKT conditions (43) for the problem (3). The convergence of the dual variables
y, and X; and the update steps verify that the limit points have constraint feasibility (43e
43hp.
The constraint feasibility of the limit points and the optimality conditions of (A: + 1)-th update of
2t, 2, (47) imply that the limit points satisfy the optimality conditions (43a
43bb. Further, using
relations (RT) and (R4), we can prove that the limit points also satisfy 443c
43d Consequently,
DistributedQP algorithm converges to the optimal point of problem which equivalent to problem
E DETAILS ON DEEPDISTRIBUTEDQP FEEDBACK POLICIES
In DeepDistributedQP, the penalty parameters are given by
p = SSfttPlus p + Th.p(rh 1,00 s s (.) n = Soft n + (
(ri sh ol,w)) (88)
where p. n are learnable feed-forware parameters and Tt, (ri,, st. 0 are learnable feedback policies parameterized by fully-connected neural network layers with inputs r st and weights 0h. The analytical expressions for rf.,, sf. are provided as follows:
|| - s 1l2 1 - s 112
|| s k-11 lle
Tp =
st.p
(89a)
Q aat
+ qi
A,
X
M ||a - 1la.
" 112.
(89b)
being motivated by the primal and dual residuals of ADMM (Boyd et al. 2011 Section 3) and the ones used in the OSQP algorithm (ttellato et al. 2020
F THE CENTRALIZED VERSION: DEEPQP
The centralized version of DeepDistributedQP boils down to simply unfolding the iterates of the standard OSQP algorithm for solving centralized QPs (T) while applying the same principles as in Section for DeepDistributed((.
For convenience, we repeat the OSQP updates from Stellato et al. (2020) here:
1. Update for (x, z): Solve linear system
Q + I
T - 11p²11
k+1 xx+l
ot gh
(90)
A
1/D A
and update
k+1 * + 1/ (u Xt).
(91)
As explained in Stellato al. 2020). the scale of the system preferable to solve the following system instead,
90 increases, it is often
(Q - + p A A)z = ox A (p yd),
(92)
using method such conjugate gradient.
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2. Update for (t, s):
gh+ a + (11 )t
(93a)
8 *+ = Ile (a x xhl + +11 (1-a")s² a²) + 1²100))
(93b)
3. Dual update for A:
A -ー k + (a h+1 + -11- a )* s 2
(94)
The DeepQP framework then emerges through unfolding the OSQP updates following the same methodology as in DeepDistributedOP In particular, its iterations are unrolled for a prescribed amount of K iterations as shown in Fig.
Similar to DeepDistributedQP. the penalty and relaxation parameters are given by
p = SoftPlus (1+ (x" "="""r"*"000) :0)) a 2
(95)
where 9" and at are learnable feed-forward parameters. The feedback components of the parame- ters p are obtained through the learnable policies x² (r* sh: 0h) parameterized by fully-connected neural network layers with inputs r² 5 and weights 0h. The expressions for rd s are also moti- vated by the ADMM and OSQP residuals, given by
1| 1 Ax
sk 1|2 l|s
(96)
st1l2
110 A 1112
G PROOF OF INDIRECT METHOD IMPLICIT DIFFERENTIATION
We start by restating the implicit function theorem, whose proof is given in Krantz & Parks 20022)
Lemma 5 (Implicit Function Theorem), Let r R" x R + R² be a continuously differentiable function. Let (Zo. Ou) be point such that r( 0,) O. If the Jacobian matrix or (æ 0.) invertible, then there exists a function x-)) defined in neighborhood of do such that x. (00) = Zu. and
Ox
(Ør (x (0). dx
@rr
-1 Ör
(0)
x ()), 00
(97)
00
Proof of Theorem Let = (Q), bl)) be the concatenation of all the parameters in Eq. 12
Q is always positive definite since Q: is positive definite and the penalty parameters are al-
ways non-negative. Therefore, Eq. 12) has unique solution x; k+1 satisfying r(x, 0) =
a 6+1 b = 0. Applying Lemma to this residual function yields the relationship *
(0)
-(Q))
3g (a (0),
8r
Now, for any downstream loss function Lax; 1+t1 (0)), we have that
0x.-11 k+1
VoL(x
(L) + (0)
00
(a)
k+1 (0), 0) -Q L (0))
00
dr
(a!
k+ (0) 0) da
(98)
00
where d is the unique solution to the linear system
da = - VoL(a krt (B)).
Expanding the matrix multiplication in 988 yields
VoLL
k+1
@ dx
dx d+t @ s
Vo. L = ddx
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H BACKGROUND ON PAC-BAYES THEORY
Here, we provide a brief overview of PAC-Bayes theory (Alquier) 2024). Consider a bounded loss function 0)) Without loss of generality, we assume that this loss is uniformly bounded between 0 and 1. PAC-Bayes theory aims to providing probabilistic bound for the true expected loss
E (P) BEWD Eg-p [e(C:
(99)
where is the data distribution - in our case, this is the distribution optimization problems are drawn from. The empirical expected loss is given by,
1 N
(s( = Es-p
Σ ((()))
(100)
H
where S = (()) is the training dataset consisting of H problem instances.
The PAC-Bayes framework operates by forming a bound that holds in high probability on the true loss (P) in terms of the empirical loss and the deviation between the learned policy and prior policy Po used to as an initial guess for P. This deviation is measured using the KL divergence. Importantly, Po need not be Bayesian prior but can be any distribution independent of the data used to train P and evaluate the sample loss. Moreover, E({: 0) need not be the loss used to train P. but can be any bounded function. This observation is useful because, both in the literature and in the sequel, it is common to use a loss function modified for practicality during training before evaluating the bound using the loss function of interest.
Specifically, the following PAC-Bayes bounds hold with probability -
(P(P) S Dii. [s(P)) Kx (P((o) + log 2VTI
< <ls(P) +1 DKL. (PI(PP) + log T/77 5
(101)
2H
where the DKL (pllc) is the inverse of the KL divergence for Bernoulli random variables B(p), B(q):
(el())))) supig [0, Dx (B(P)) sc..
(102)
The probability 8 captures the failure case that the data set S is not sufficiently representative of the data distribution D. In the sequel, both of the above inequalities will be used. As the first bound is tighter, it is used to evaluate the generalization capabilities of the learned optimizer. The benefit of the second, looser, bound is that its form convenient to use during training as a regularizer. Using both bounds in this manner is common technique in the PAC-Bayes literature (Majumdar et al.
2021 Dziugaite & Roy 2017).
I OPTIMIZING AND EVALUATING GENERALIZATION BOUND
Two important requirements for establishing a tight PAC-Bayes bound are selecting an informative prior and optimizing the PAC-Bayes bounds in (OI) instead of simply minimizing the loss function. The choice of prior Po is particularly important because the KL divergence is unbounded and can produce vacuous result (Dziugaite al. 202T). While the distribution Po need not be a Bayesian prior, it must be selected independently from the data used to optimize P and evaluate the bound. To select Po, we follow common approach in the literature and split our training S into two disjoint subsets Sn, Sj. The prior Po is first trained using the data set So and the loss E(D; +) discussed in Section
Subsequently, the posterior P is trained by minimizing the looser (i.e., rightmost) PAC-Bayes bound in (IOT). This bound used for training because it is straightforward to evaluate in comparison to computing the inverse of the KL divergence, and this objective is easily interpreted as minimizing an expected loss function with a regularizer. To evaluate the loss function in the PAC-Bayes bound, parameters sampled from P using the current network weights and an empirical average is used. Once training is complete, the PAC-Bayes bound is evaluated as described in Theorem i.c., by using the tighter PAC-Bayes bound in (10T) and the sample convergence bound in (6).
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J DETAILS ON EXPERIMENTS
This section provides further details on the problems considered in the experiments, the training of the learned optimizers, as well as the evaluation of both learned and traditional methods.
J.1 PROBLEM TYPES CENTRALIZED EXPERIMENTS
Random QPs.
We consider randomly generated problems of the following form
1 min x Qx q-x s.t. Ax S Cx d.
(103)
For each generated problem, the cost Hessian is given by Q = F TI, where each element of F TRxx is sampled through F.j ((0, and sampled as qi ~ (O, The elements of the inequality constraints matrix E m are given by Au (O, 1). while b AO, where each element of 0 IR" is sampled through Similarly, the elements of the equality constraints matrix C' E Rex= are given by Cij ~ No0, 1), while Cg. where each element of R" is & NV 1).
1.0. The coefficients of q are also
NOO, 1).
For random QPs without equality constraints, we set n 50, m 40 and with equality constraints, we set 50, m 25 and p = 20.
0. For random QPs
Optimal control.
We consider linear optimal control problems of the following form
T-1
min
Σ
0 T Qu: + u, Ru, + a. QTæT
(104a)
1=0
s.t. 1t+1
Asæ: + Baue, t= 0..,.., T 1,
(104b)
A U bus
A, bas t= 0, , T,
(104c)
2D To-
(104d)
where x: {mo. ........... 3 is the state trajectory. u {uo, ... -r-1 } is the control trajectory, Zo the given initial state condition, Q and R are the running state and control cost matrices, QT is the terminal state cost matrix, Aa and Ba are the dynamics matrices, and finally A,, b, and A,, b are the control and state constraints coefficients, respectively.
Both the double integrator and the mass-spring problem setups are drawn from Chen et al. (202aa). For the double integrator system, have 2,, E IR² and ue E R. with time horizon T 20. The dynamics matrices are given by
1
0.5 0.1
A
Ba
(105)
The cost matrices are Q QT = 12 and R 1.0. The state and control constraint coefficients are given by
I2 12
[ b, = [0.1 0.1]
A,
b, = 5- A,
(106)
Finally, the initial state conditions sampled from the uniform distribution 41[-- -0.3), [1:).]]
For the oscillating masses, we have I2 E R212 and i, E TR3, with time horizon 7 = 10. The discrete- time dynamics matrices are obtained from the continuous-tim ones through Euler discretization.
4 = I A,tt, Ba A.At.
(107)
The continuous-time dynamics matrices are given by
06x6
1s blg + dLg dL
06x3
A. als + cLa + CL
B.
(108)
F
with c - -.0, = 0.1, a -2c, b = -2.0. Lg is the 6 x 6 lower shift matrix and
F == (ei
-1
E2 E3
E2 e3
(109)
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where c1, Eg, C2 the standard basis vectors in R
The timestep is set as At = 0.5. The cost matrices are Q = QT = 112 and = 13. The state and control constraints are defined through
112 12
I3
- A
b₂ - 4 124.
Aa =
bu - 0.5 0.5 16-
(110)
- I3 :
The initial conditions Ty are sampled from
Portfolio optimization. We consider the same portfolio optimization problem setup as Stellato al. (2020). For completeness, we briefly repeat here,
max J x 7 Ex)
s.t.
In 1, 0,
(111)
where R is the assets allocation vector, l Rn is the expected returns vector, E R is the risk covariance matrix and y > 0 is the risk aversion parameter. The matrix Σ is of the form E FF D with F R the factors matrix and e R=x= is a diagonal matrix involving individual asset risks. Using an auxiliary variable t = F æ, then problem equation 111 is rewritten as
min x' Dæ + + - s.t. t= F a, 1 = 1, 1220 0.
(112)
For the problems we are generating, we use n = 250, k = 25 and 7 1.0. Each element of the expected return vector μ is sampled through μ4; (O, 1). The matrix F consists of 50% non-zero elements sampled through Fiş ~ 1), Finally, the diagonal elements are sampled with Di ~ U(00, Vkl.
LASSO. The least absolute shrinkage and selection operator (LASSO) is linear regression tech- nique with added foomr regularization term to promote sparsity in the parameters (Tibshirani) 1996). We again consider the same problem setup as in Steellaoo et al. (2020), where the initial optimization problem
min |
b12 + A||2/11
(113)
is rewritten as
min (Ax b) (Ax b) + A1 s.t. -tsost, 2,t
(114)
where E R" is the vector of parameters, A E R is the data matrix, À is the weighting parameter, and E IRn are newly introduced variables. The matrix consists of 15% non-zero elements sampled through A; ~ (0, 1). The true sparse vector v E R" to be learned consists of 50% non-zero elements sampled through v, ~ (0 1/n). We then construct E: (0, 1) represents noise in the data. Finally, we set À (1/5)]lA blloo. For the problems are generating, we set n 100 and m = 10a.
Av + & where
J.2 PROBLEM TYPES IN DISTRIBUTED EXPERIMENTS
Random Networked QPs. In this family of problems, we generate random QPs with under- lying network structure. Consider an undirected graph G(V,E), where V and are the nodes and edges sets, respectively. Each node i associated with decision variable æ; E R Then, we generate problems of the following form
Σ VEV
T Qi:: + q. x
min (w:
(115a)
2
x; [j
s.t.
Asj
bij-
Cjj
dij. (i,5) E.
(115b)
For each generated problem, cost Hessian constructed as Q: F.T F, + where each element of F. Xn, sampled through F. coefficients vectors qi are also sampled through g¹ NoO0, 1). The elements of the inequality
1) and ς 1.0. The elements of the cost
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constraints matrix A, E my ((e+t))) are given by A (0, 1). The vectors bij R obtained through b.y Oij where each element of i, E IR*, sampled through 0 ~ (0, 1). In similar manner, the elements of the equality constraints matrices Cij E RP (m)))))) are generated through C ~ AOO,, 1), while the vectors dı E 12P are acquired through di CiyEe, where each element of Eij IR"+ is generated with + (O,,
is straightforward to observe that problems of the form 115) can be cast in the form (2) by introducing the augmented node variables 2; [æ;, {x) IeEN T. The problem data then be augmented based on this new 2 to yield the desired problem structure. Most notably, the constraints can be rewritten as A, A.u0 aua S 6et0 and C, xr; dh respectively.
x
In our experiments, the underlying graph structure is square grid. For random QPs without equality constraints, we n4 10, ma 5, and Pij 0. For random QPs with equality constraints, set = 10, may = 3. and Ptj = 2 for the N = 16 training experiment and Pij = 1 for the rest of the testing experiments until N 1, 024.
Multi-agent optimal control. We adapt the distributed MPC problem from (Conte et al. 2012a a which generalizes different systems based on the choice of dynamics matrices, as described below. The optimization problem is given as
min 2.8
ΣΣ
(x))
Q + (u
R,u
+ (xf)) P.
(1 (116a)
1EV 0=0
s.t.
xt+1 A + Buu + Σ Aijayj t= O,...,T-11, i ieV
(1 166b)
jEN.
G G S fü G v fúr t= iEV
(1 (66cc)
x = iEV,
(116d)
where æt and ut are the state and control for agent time t. Eq. 116b) describes the dynamics and the coupling between the agents, Eq. (166c) describe local inequality constraints, and Eq. (16d describes the initial condition for each of the agents.
For the coupled pendulums, the individual state xf E R²2 for each agent consists of the angle and angular velocity of the pendulum and the control ui R! is the torque. The dynamics matrices are given as
1
dt an(s) dt m
0
0 Lmez dt
A (2 1
an(i)k m
dt
A.
k
dt
E dt
B,
where dt 0.1 is the discretization step size, 9 9.81 is the gravitational constant, mass of each pendulum, l = 0.5 is the length of each pendulum, nn(i) is the number of neighbors of agent i, k: 0.1 is the spring constant between each pendulum, and = 0.1 is the damping constant between each pendulum. We have used the small angle assumption sin so O so the dynamics are linear and therefore the optimization is convex. There are no inequality constraints for the coupled pendulums. The initial states are sampled uniformly from U[[ [[[ m]. Finally, we considered N = 10 and T 30.
1.0 is the
For the coupled oscillating masses, we adapt the same benchmark system from Chen et al. (022a) used in the non-distributed experiments. The individual state x IR² for each agent consists of the displacement and velocity of the mass and the control ue E RR. is the force acting on the mass. The dynamics matrices are
dt
0
A
A.j
B
dt
2c dt m
dt
c
dt
dt
m
m
m
where 0.5 is the discretization step size, m = 1.0 the mass, k 0.4 is the spring constant between each mass, and 0.1 is the damping constant between each mass. The initial states are sampled uniformly from 4 [ 2.0, 2.0]. Inequality constraints -4 < o < 4 and 0.5 < u < 0.5 represented as
1g -12
f == 4. 14, G -
G
fú 0.5 12
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For both the distributed MPC problems described above, the cost matrices are taken to be identity matrices: Q; = Ig. R. = I1, and P. = I2. for all i V.
The optimization (16) can be expressed in the form of (2) by defining augmented vector consist- ing of the individual agent's 's states and controls, as well as the states and controls of its neighbors. Letting 2; [a. u. "!!TT the augmented optimization vector for each agent is given as 2; mng I {z, jem] T The cost, dynamics, and constraint matrices can be augmented straightfor-
wardly based on this new "h For all problems, we considered 7 15.
Network flow. The network flow problem is adapted from Mota 2013): Mota et al. (2014). We consider a directed regular graph with 200 nodes and 1000 directed edges Tij E E. Each edge has an associated quadratic cost function Φi (1ay) [1.0, 2.0, 3.0, 4.0, 5.0, 10.0] with probabilities [0.2, 0.2, 0.2, 0.2, 0.1, 0.1). The objective to opti- mize the flow through the graph subject to equality constraints on the flow into and out of each node. Namely, the flow into each node should be equal to the flow out of the node. For node the flow conservation constraint is ΣjEE, Iiji = Exeet Tik, where &,- is the set of all incoming edges
_ (x - azz)², where dzj is sampled from
to node i, and similarly E is the set of all outgoing edges from node i. 100 nodes are randomly selected and injected with an external flow f& sampled identically aij. For each of these nodes, a reachable descendant is randomly selected and an equivalent amount of flow fk is removed from those nodes.
This problem is straightforward to express in the form 2 by considering each node as an individual agent and defining the local state vector for each agent as
{m JEE,
JEE, {rak keet
(117)
consisting of all the incoming and outgoing edges for node i. Each agent is responsible for its own flow constraint defined by
(1) fiee A, (-1]]ee;
{-1]REet] b, 0,
(118)
{1lkeet
where b, might instead contain the external injected or removed flow f. for that node i. The aug- mented cost matrix Q; is zero for all incoming edges and has entries 1/2 the diagonal of the outgoing edges. The augmented cost vector q: contains each of the quadratic cost offsets ax!
Q: {O)
{0]jee,
(2)NEET]
q:
{ Saakkheee; keet
(119)
Finally, we impose the constraint -mma S x; fmax on the maximum allowed flow of all edges, with fmax 5.
Distributed LASSO. Distributed LASSO (Mateos al.) 2010) extends LASSO to situations where the training data are distributed across different agents and agents cannot share training data with each other. It can be formulated as
min
Σ IA
ba
x
s.t.
E. W.
N
(120)
(ce, where u E IR global vector of regression coefficients, 2; R. is a local copy of w, A, E xm; and b E Rm are the training data available to agent and X is the weighting parameter. Similarly to non-distributed LASSO, this formulation is rewritten as
1=1
min Σ AA (A,x. b.)
1tt N
(121a)
S.L. 1, 5 < 44 x; w, 4 g. i 1= N where t; R newly-introduced variables and g is the global copy of t,-
(121b)
The matrix A, consists of 15% non-zero elements sampled through A ~ NO0, 1). The true sparse vector R be learned consists of 50% non-zero elements sampled through v, [0, 1/n). We then construct b = Av + & where & ~ (0,, 1) represents noise in the data.
Finally, we set À (1/5) max: ( by )l For the problems, have n, 50 and m, 5 103.
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Table 2: Training and testing details for DeepQP.
Problem Class
No of layers K
Training dalaset size
Epuchs
Training time
Test dataset size
Random QPs
30
2,000
500
21min
1,000
Random QPs with Eq. Construints
30
2,000
500
23min
1,000
Double Inlegrator
30
500
600
28min
1,000
Osc, Masses
15
500
600
48min
1,000
Portfolio Optimization
30
500
600
1h 14min
1,000
LAASSO
10
500
600
20min
1,000
Table 3: Training and testing details for DeepDistributedQP.
Problem Class
No of layers K
Training dataset size
Epochs
Training time
Test dataset size
Random QPs
50
1,000
300
3h 27min
500
Random QPs with Eq. Constraints
50
500
600
3h 29min
500
Coupled Peadulums
20
500
400
1h 49min
500
Coupled Osc. Masses
20
29miin
Network Flow
30
500
600
2b Smin
500
Distributed LASSO
20
500
600
56min
500
J.3 DETAILS ON TRAINING AND TESTING
Here, we discuss details regarding the training and testing of DeepQP and DeepDistributedQP in the presented experiments.
Centralized experiments. Table 2 shows the number of layers K, training dataset size, number of epochs, total training time and testing dataset size for DeepQP in every centralized problem. The increased dataset size and number of epochs for RandomQPs is motivated by the fact that the structure in these problems is less clear; learning policies that exploit this structure therefore requires more examples and takes longer. In all experiments, DeepQP was trained with batch size of 50 using the Adam optimizer with learning rate 10-3, The feedback layers are set as x 16 MLPs. DeepQP and OSQP always start with zero initializations in all comparisons. The weights of the training loss were set to 7 = exp ((k K) /5) in all experiments. Both the training and testing datasets contructed after letting OSQP running until optimality.
Distributed experiments. Table 3 shows the number of layers K. training dataset size, number of epochs, total training time and testing dataset size for DeepDistributedQ@ in every distributed problem. In all experiments, DeepDistributed(QP was trained with a batch size of 50 using the Adam optimizer with learning rate 10 3, The feedback layers are set as 2 x 16 MLPs. DeepDistributedQQ and DistributedQP always start with zero initializations in all comparisons. In all experiments, the weights of the training loss were set to % = exp ((k- K) /5). For the dim-dimensional testing datasets, these datasets are constructed using OSQP. For larger scales, the testing dataset con- structed with DistributedQP instead as it is much faster (see Table after ensuring convergence to optimality.
Generalization bounds experiments.
These experiments were performed on a networked random QPs problem with N = 10, 10, m.j 5, Ptj = 0 and on coupled pendulums problem with N = and the same parameters as described in the previous section. The prior was obtained through training on a small separate dataset of 500 problems for 50 epochs. The posterior was then acquired through optimizing for the generalization bound with dataset of 15, 000 problems for 50 epochs.
J.4 DETAILS ON STANDARD OPTIMIZERS
When comparing with OSQP using fixed penalty parameters, we selected the best-performing subsequence of (..., 0.1, 0. 0.3, 0.5, 1.0,3.0, 5.0,...) ) as the penalty parameters to plot against. Table 4 shows these parameters for every centralized problem in our experiments. For equality constraints, we scaled by 103, as in Stellato et al. 2020)). For the adaptive version, we preferred the standard heuristic adaptation rule shown in Boyd et al. (2011) with
Details on OSQP.
2.0 and = 10.0, instead of the OSQP adaptation scheme (Stellato et al. 2020). as it performed better in our problem instances. We hypothesize that this might be due to the fact that scale increases the infinity norm is ignoring more information that the n-norm. The initial p0 was initialized as the median of the range of fixed penalty parameters.
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Table List of OSQP penalty parameters used in centralized experiments.
Prublem Class
List of penalty paraineerrs
Random QPs
0.3. ...3, 10
Random QPs with Eq. Construints
0.1. ............ 10
Double Inlegrator
3,5,
Osc. Masses
0.1, 10
Portfolio Optimization
3,5. 100, 300
LASSO
30, 50. 1000, 3000
Table List of DistributedQP penalty parameters used distributed experiments.
Problem Class
List of penalty parameters and P
Random QPs
O.T.
Random QPs with Eq. Constraints
0.1. 0.3 3, 10
Coupled Pendulums
................................
Coupled Ose. Masses
0.1, 10
Network Flow
0.3, ....
Distributed LASSO
30, 50, 1000, 3000
Details on DistributedQP. The range of fixed penalty parameters to compare with was chosen using the same methodology with OSQP. Table 5 shows these parameters for every distributed problem in our experiments. For the adaptive version, we used the standard heuristic adaptation rule shown in Boyd et al. (201T) with 2.0 and μ 10.0. The initial value again always chosen as the median value of the above lists.
J.5 DETAILS ON WALL-CLOCK TIMES
In T list the observed wall-clock times for DeepDistributedQP (ours). DistributedQP (ours) and OSQP using either the indirect or the direct method. The table presents all six studied problems with an increasing dimension. As clearly observed, DeepDistriuutedQP and DistributedQP demon- strate a substantially more favorable scalability than OSQP. In fact, the two algorithms can efficiently solve problems that OSQP cannot handle due memory overflow on our system. Finally, Deep- DistributedQP also maintains clear advantage over its standard optimization counterpart Distribut- edQP across all experiments which signifies the importance of learning policies for the algorithm parameters.
K ADDITIONAL EXPERIMENTS
The following experiments are dedicated into providing additional insight on exploring the perfor- mance of DeepDistributedOP and DeepQP various testing scenarios.
K.1 VARYING TRAINING DATASET SIZE
This section provides additional insight on the amount of training data required for the proposed learned optimizers to perform well.
Centralized experiments. In Fig. 3 compare the performance of DeepQP on the centralized problems using a varying training dataset size 500, 1000 or 2000. To ensure an "equivalent total training effort",
train these three cases for 4e, 2e and e epochs, respectively, where random QPs and e = 150 for all other problems. This comparison highlights the robust performance of DeepQP even with limited amount of training data. Interestingly, we also observe that training with less data but over more epochs had beneficial effect on two out of six problems. We hypoth- esize that this could be attributed to the non-convex nature of training in deep learning, as well as the possibility that additional epochs might have allowed for further improvements in cases where the training the model had not yet fully converged. Overall, we conclude that DeepQP maintains reliable performance even when training data is limited.
500 for
Distributed experiments. For the training of DeepDistributeedOP a limited training dataset of 500 sample problems used for all problems except for the random QPs without equality constraints where we used 1000 problems (see Table 3). For completeness, Fig. 9 presents a performance comparison of the learned optimizer when trained with 500 sample problems (600 epochs) and 1000
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Published as a conference paper at ICLR 2025
Table 6: Wall-clock times and iterations for DeepDistributedQRP, DistributedQP, OSQP (in- direct) and OSQP (direct). This comparison shows the total wall-clock times for DistributedQP and OSQP (indirect direct method) required to reach the same accuracy as DeepDistriuutedQR. For OSQP with direct method, we only report the time for the first iteration, assuming the best- case scenario which the factorized KKT matrix can reused for subsequent iterations. Both DeepDistributeQPP and DistributedQP demonstrate orders-of -farnniudde improvements compared to OSQP as scale increases. In addition, DeepDistributedQP maintains a significant advantage over its standard optimization counterpart in all cases.
DeepDistrQP (ours)
DistrQP (ours)
OSQP (Indirect)
OSQP (Direct)
Networked Random QPs
N
7
17
nnz[Q, A)
Time
Iters
Time
liers
Time
Iters
Time (ist)
Iters
160
4,000
33.05 ms
141.9 ms
208
46. ms
29
ms
64
640
560
17,600
39.11
50
124.2 ms
192
185.1
28
23.8 ms
28
256
2,560
2,400
73,600
50.21 ms
50
128.8 ms
168
514 ms
23
703.5 ms
23
1,024
10,240
9,920
300,800
62.68 ms
50
158.9 ms
165
3.03s
23
8.20
Networked Random QPs with Equality Constraints
N
12
171
Qn,(Q.A)
Time
Iters
Time
Iters
Time
Iters
Time (1st iter.)
Iters
4,960
37.21 ms
138.9 ms
170
36.52 ms
19
ms
19
64
640
560
17,600
57.76 ms
50
238.1 ms
172
109.0
17
26.9 ms
17
256
2,560
2,400
73,600
74.54 ms
50
239.5 ms
164
692.5 ms
17
956.0 ms
17
1,024
10,240
9,920
300,800
82.55 ms
371.0 ms
172
11,60
Coupled Pendulums Optimal Control
N
1
371
nn A)
Time
Iters
Time
Iters
Time
Iters
Time (1st iter.)
Iters
470
3,690
50.99 ms
89.81 ms
35
ms
ms
20
940
1,200
7,500
66.44 ms
20
116.7 ms
35
372.0
199.7
50
2,350
3,200
18,930
75.9 ms
20
142.1 ms
34
948.8 ms
8
4.38 s
100
4,700
6,400
37,980
101.9 ms
201.9 ms
35
19,91
200
9,400
12,800
76,080
146.0 ms
20
284.8 ms
34
22.41
90.07
500
23,500
32,000
190,380
204.3 ms
20
379.8 ms
36
112.9 8
9
Out of memory memory
1,000
47,000
380,880
317.2 ms
628.2 ms
34
memory
Coupled Oscillating Masses Optimal Control
N
12
X71
nn A)
Time
Herrs
Time
Iters
Time
Iters
TIme (1st iter.)
liers
470
1,580
4,590
48.22 ms
73.58 ms
33
79.1
178. ms
20
940
3,160
9,300
67.93 ms
20
91.53 ms
33
641.9
9
2.37
50
2.350
7,900
23,430
73.92 ms
20
97.34 ms
32
1.07
8
28.
4,700
15,800
46,980
91.93 ms
148.8 ms
33
5.45
132
200
9,400
3,600
94,080
109.4 ms
20
194.4 ms
34
31.8
8
614
300
28,200
47,400
141,180
132.8 ms
20
304.8 ms
33
243
8
Out of memory
Network Flow
N
nnz [q.
Time
Ites
Time
Iters
Time
Iters
Time (1st iter.)
Iters
20
100
140
600
6.80 ms
30
10.68 ms
50
9.51 ms
15
0.59 ms
15
250
1,500
7,81
13.17 ms
48
ms
ms
200
1,000
1,400
6,000
12.08 ms
30
17.61 ms
42
208.19 19 ms
17
61.93
17
500
2,500
3,500
15,000
13.63 ms
30
19.73 ms
40
425.7 ms
17
745.2 ms
17
1,000
20.51 ms
ms
40
2,000
10,000
14,000
60,000
29.86 ms
30
47.22 ms
40
51.6
18
73.9
18
5,000
25,000
35,000
150,000
61.23 ms
30
85.99 ms
39
558 s
18
Out of memory
Distributed
LASSO
N
anz A)
Time
Iters
Time
Iters
Time
Iters
Time (1st iter.)
Iters
10
1,100
3,000
29,000
15.06 ms
20
28.57 ms
37
2.04
33
148.2 ms
33
5,500
15,000
145,000
24.92 ms
44.27 ms
38
13.74
31
100
10,100
30,000
290,000
30.51 ms
20
51.44 ms
35
85.92
32
342.9
32
200
20,100
60,000
580,000
40.88 ms
20
76.21 ms
36
418.9
32
Out of memory memory
50,100
150,000
1,450,000
69.19 ms
130.24
35
Out memory
sample problems (300 epochs). While additional training data provides some improvement, the model trained with less sample problems still significantly outperforms the standard optimization counterparts.
K.2 CAN POLICIES TRAINED FOR SPECIFIC PROBLEMS BE APPLIED OTHER PROBLEMS?
The field of learning-to-ppe to-timmize primarily focuses on improving the performance of an underlying optimizer for problems drawn from the same distribution the training data (Shlezinger et al. 2022). However, this prompts an interesting question: How would a policy perform if trained on specific problem class and then evaluated on different one?
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Published as a conference paper at ICLR 2025
Random
Random OPs. (with Equality Consbraints)
Dauble Optinal Control
Dae (Dood loen
f 2203
Ore
Deesh Lens, 2063)
10
DecoO open lenk 1003)
1071
32"5
12
Ooctliating Masses Optimal Control
Portfolie Optimisation
LASSO
17
12
1371
10"1
33:7
394
00
2.5
50
755
10.0
12.
15.6
37.5
20.0
16
25
25
20
-40
6.
3.5
5.0
30.2
32.5
15.0
17.5
Figure 8: Varying training dataset size for DeepQP. The performance of DeepQP remains robust
(for both open-loop and closed-loop policies) even as the training dataset size is reduced.
Random QPs
Random QPs (zoomed
10a
DeepDistrOP [Closed- Loop. Hous # 1000)
10a
DeepDistrGP [Open- Loop, Huue = 1000)
Optimality Gap
DeepDistrOP (Closed Loopdd-Lopp.Haa.. Loop. 500)
10-1
DeepDistrGP Loap,
10-1
10-2
10
10
10
40
80
120
160
0
10
30 ADMM Iteration
20
40
ADMM Iteration
Figure 9: Performance of DeepDistributed(P on random QPs using training dataset size 500 and 1000. While using more training data results in slight improvement in the performance of DeepDistributedOR, in both scenarios, the proposed learned optimizer consistently outperforms the traditional one. The right figure illustrates only the first 50 iterations.
At this point, we wish to emphasize the following fact:
The proposed DeepDistributedQP framework already surpasses the expected capabilities of typical learning-to- o-ooimmmiee algorithms in the literature, as it is trained on small-scale problems and then successfilly deployed much higheerdimensional ones.
For completeness, we also conduct curiosity-driven experiments, where we apply the learned poli- cies to different classes of problems than the ones used for training. In Fig. TO test policy trained on small-scale random equality-constraine QPs on large-scale random QPs without equal- ity constraints and large-scale coupled pendulums problems. In the first case, DeepDistributedQP maintains remarkable performance compared to DistributedQP due to the existing similarity be- tween the two classes. In the second setup, where the training and testing problems are entirely different, the performance is suboptimal. Overall, these results highlight that when there is degree of similarity between the training and testing setups, DeepDistributedQP is expected to still perform very well. In future work, we plan to explore extensions trained on broader variety of problem classes to improve generalization on entirely different setups.
K.3 VARYING THE NUMBER OF LAYERS IN TESTING DEEPDISTRIBUTEDOP
Another natural question that arises is how DeepDistributedOP can be adapted to run for more iterations than the number of layers it was originally trained for. A straightforward modification is to repeat the last layer of the framework for the additional iterations. In Fig. [ we add 30 extra iterations for the random QPs and 20 for the other problems. For all cases, the closed-loop policies
39

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Published as a conference paper at ICLR 2025
Random QPs
Coupled Pendulums
10"
DeepDistrOp Loop)
10a
DespDistrQP (Closed Loop)
Optimality Gap
10-1
10-1
10
10
10
10
10
40
80
10
20 Iteration
30
40
ADMM Iteration
Figure 10: Testing policies on different classes of problems. We evaluate the policy trained on scall-scale random equality-constraneet QP problems (N 16) on two large-scale scenarios of different problem types: random QPs without equality constraints (N 1, 024) and coupled pen- dulums (N = 1, 000). Notably, in the first case (left), the policy demonstrates strong performance which is attributed on the fact that there is still some similarity between the training and testing setups. In the second case (right), where the testing problems differ entirely from the training setup, the performance of the learned optimizer becomes suboptimal.
Random QPs (N 1024)
Coupled Osc. Masses (N 300)
Optimality Gap
10
DeepDistrQP (OL)
10-1
DeepDistrOP (cI)
10
DeepDutrop
DeepDbuor (CL, Ropeated Layers)
10-3
10
10
40
80
120
160
200
10
20
30
40
50
Random Eq. QPs (N 1024)
Network Flow (N=5000)
Optimality Gap
10-1
10-1
10-3
10-3
120
160
200
Coupled Pendulums 1000)
Distributed LASSO (N 500)
Optimality Gap
10-1
10-1
10-3
10-5
10-3
10-7
ADMM Iteration
ADMM Iteration
Figure 11: Varying the number of layers while testing DeepDistributedQR. If additional itera- tions are needed, DeepDistributeQQP maintains strong performance by repeating its last layer for these extra iterations. Specifically, we explore adding 30 iterations for the random QPs and 20 for the rest of the problems. In all cases, the closed-loop policies continue demonstrate superior per- formance, while in 4 out of 6 problems, the open-loop policies also remain advantageous.
continue to outperform the standard optimizers. Additionally, the open-loop policies maintain strong performance in 4 out of problems. In future work, we plan to incorporate the repetition of the last layer during training to further ensure robust performance when additional iterations are required.
40


Reviews:
[{"review_id": "X1OxhGBvIW", "reviewer": "Reviewer_fvD1", "summary": "The authors present an optimization technique for solving quadratic programming problems distributed across multiple compute nodes. The methods use a extend OSQP methodology to a distributed framework which has similarities to a two-block ADMM approach. The authors identify that selecting hyperparameters in this methodology can greatly influence convergence rates. As a result, the authors propose learning the hyperparameters through a deep learning model that has the iterations of the distributed OSQP algorithm embedded as layers.", "questions": "1.\t Can the authors provide specific details about the dataset size used in each experiment, and comment about how much data would be needed for a user to implement this strategy.\n2.\tCan the authors discuss if they have separate models for each experiment, or if a single model was used across all problems. If multiple models are used, can the authors comment on the ability of models to generalize to other problems.\n3.\tTheorem 3 depends on a set of problems drawn from a distribution. Can the authors provide any empirical evidence or discussion on how well the bounds hold for different types of problems?\n4.\tThe authors require knowing the maximum number of iterations, K, apriori. Generally, the maximum number of iterations is quite large with the intention that the optimization solver finds a solution before reaching the limit. Is choosing an appropriate value of K required to construct a deep neural network model that is memory and time efficient? \n5.\tIn many of the experiments the authors evaluate the optimality gap up to the maximum number of iterations. However, often we care about the number of iterations to reach the optimum. Does this change the methodology implementation?\n\nSome minor points are:\n1.\tThe authors should define some terms like $r$, $\\theta$ in equation (10).\n2.\tCan the authors provide more detail about the known mapping G on page 6, line 282.\n3.\tSpelling mistake on “non-distrubuted” on page 6 line 290.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Using solved QP problems to train the deep learning model, the authors are able to select optimal hyperparameters that greatly improve the convergence for solving new distributed QP problems.", "weaknesses": "A limitation of this approach seems to be the amount of training data required to train the neural network.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "P2y9YHYiQT", "reviewer": "Reviewer_Y4sR", "summary": "This paper combines the interpretability of distributed optimization with the strong empirical performance of deep neural networks to enhance optimization efficiency. The authors propose a distributed method for QPs with theoretical convergence guarantees. They then unfold this method into a deep neural network, leveraging it to accelerate QP solutions. The paper includes both theoretical proofs and numerical results to demonstrate effectiveness. The author claim that the proposed approach outperforms conventional optimization softwares.", "questions": "Questions:\nIn addtion to those mentioned in weakness, there are some specifc questions regarding this paper.\n1. What does r^k_i in Figure 1 and subsection **Learning feedback policies** represent? This notation is not defined elsewhere in the paper. Additionally, a complete update formulation would greatly aid in understanding the unfolding process.\n1. The author mentions that distributed methods are effective for large-scale decision-making; however, the tested problems are relatively small. Can the authors discuss the feasibility of extending the numerical results to larger problem sizes?\n3. The specific number of layers used in the neural network is not mentioned. From the description, I assume it corresponds to $K$. Do these layers share the same weights? Additionally, how does DEEPDISTRQP perform with varying numbers of layers?\n4. Why does Figure 6 use the relative optimality gap as a metric, while other figures utilize the absolute optimality gap? What is the rationale behind this choice?", "limitations": "", "rating": 3, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The mathematical aspects of the paper are technically correct.", "weaknesses": "The primary concern revolves around the novelty of the unfolded network. \nA similar approach can be found in [1], which also predicts parameters for ADMM in conventional OSQP using reinforcement learning, where this work seems to be a straightforward substitution of tuning efforts with basic MLPs. \nThis also raises questions about the fairness and completeness of the experiments, as the authors only compare their method against OSQP without considering other learning-based frameworks that could potentially yield better performance.\nAdditionally, the presentation in Section 6 is somewhat difficult to follow. For example, the implementation of the performance metrics is not clearly explained, and it would be helpful for the authors to provide more details on how improvements are quantified. Another minor issue is that some notations are quite confusing and not adequately introduced\n\n[1]Jeffrey Ichnowski, Paras Jain, Bartolomeo Stellato, Goran Banjac, Michael Luo, Francesco Borrelli, Joseph E. Gonzalez, Ion Stoica, and Ken Goldberg. Accelerating quadratic optimization with reinforcement learning, 2021..", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "e3sMVFerW3", "reviewer": "Reviewer_EEUg", "summary": "This paper proposes two frameworks -- Distributed Quadratic Programming (DistributedQP) and Deep Distributed Quadratic Programming (DeepDistributedQP). The former combines the state-of-the-art operator splitting method in optimization with distributed optimization, and the latter further introduces deep neural networks into the framework. The authors theoretically prove the convergence guarantees and the generalization bounds of the algorithms. Some numerical simulations are provided to empirically verify the effectiveness of the algorithms.", "questions": "I am wondering if the authors could explain the motivation and the novelty of this paper.", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "1. The organization of the paper is clear -- the authors first introduce distributed version of operator splitting, and then combine explain how to incorporate deep learning techniques into the proposed framework to solve optimization problems more effectively.\n\n2. The theoretical proof is well-structured and easy to follow.", "weaknesses": "Major:\n\n1. (Lack of Motivation) In DEEPDISTRQP framework, the authors aimed to combine distributed optimization, in particular distributed quadratic programming, with deep learning. However, it lacks motivation of why one should study this type of problems. It is true that distributed QP part has the advantage of interpretability, while the deep learning part has stronger generalization capabilities. However, the experiments are only limited to relatively small-scale problems like optimal control, portfolio optimization, etc. It is unclear if this framework can handle large-scale deep learning problems effectively, and thus a optimization journal, instead of a machine learning conference, might be a better fit for this paper.\n\n2. (Lack of Novelty) The design of the frameworks as well as the theoretical analysis are more like a combination of existing techniques, such as local and global updates, primal dual updates, the generalization bounds, etc. It is unclear if there is any novelty of simply combining these techniques, especially provided that the experiments are only conducted on some simple ones.\n\n3. (Unfair Comparison in Experiments) All figures of the experiments section have the iteration number as the horizontal axis, and the curves showcase better performance of DeepDistributedQP. However, different model architectures have different per-iteration complexities, and thus it is unfair to simply plot the optimality gap vs iteration numbers.\n\nMinor:\n\n1. The term 'OSQP' is first introduced in the abstract without additional context. It is unclear what OS is for the audience outside of optimization community.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)