A Difference-of-Convex Functions Approach to Energy-Based Iterative Reasoning
Authors: Daniel Tschernutter
David Diego Castro
Maciej Kasiński
Venue: NeurIPS 2025 poster ()
Keywords: Energy-based models
Iterative reasoning
Difference-of-convex function optimization
Primary area: optimization

Abstract: While energy-based models have recently proven to be a powerful framework for learning to reason with neural networks, their practical application is still limited by computational cost. That is, existing methods for energy-based iterative reasoning suffer from computational bottlenecks by relying on expensive optimization routines during training and especially during inference. Furthermore, these routines may not always converge to minimal energy states, potentially leading to suboptimal reasoning. To address these limitations, we propose a novel and efficient algorithm for energy-based iterative reasoning based on a difference-of-convex (DC) functions approach. Our algorithm achieves a significant speedup compared to prior methods while offering theoretical convergence guarantees ensuring locally minimal energy states. In addition, we achieve state-of-the-art or superior performance on continuous reasoning tasks, as demonstrated by our experiments on multiple benchmark datasets from continuous algorithmic reasoning. As such, our method offers a leap in computational efficiency, enabling faster inference with theoretical guarantees, and hence unlocking the potential of energy-based models for iterative reasoning applications.

A Difference-of-Convex Functions Approach to Energy-Based Iterative Reasoning
Daniel Tschernutter
David Diego-Castro Infermedica
Infermedica
Graz, Austria
Gothenburg, Sweden
daniel. tacherrruterrcinfeeeiii.a com
david. dlegg-caatroiinffemmdic...oo
Maciej Kasiáski
Infermedica
Wroclaw, Poland
maciej kas ikssinkkin @infermedica..... com
Abstract
While energy-based models have recently proven to be a powerful framework for learning to reason with neural networks, their practical application is still lim- ited by computational cost. That is, cxisting methods for energy-based iterative reasoning suffer from computational bottlenecks by relying expensive optimiza- tion routines during training and especially during inference. Furthermore, these routines may not always converge minimal energy states, potentially leading to suboptimal reasoning. To address these limitations, we propose a novel and efficient algorithm for energy-based iterative reasoning based difference-of- convex (DC) functions approach. Our algorithm achieves a significant speedup compared to prior methods while offering theoretical convergence guarantees en- suring locally minimal energy states. In addition, we achieve state-of-the art or superior performance iments on multiple benchmark datasets from continuous algorithmic reasoning. As such, our method offers a leap computational efficiency, enabling faster inference with theoretical guarantees, and hence unlocking the potential of energy-based models for iterative reasoning applications.
continuous reasoning tasks, demonstrated by our exper-
Introduction
The human thinking process is described as operating through two distinct modes 31: the rapid, automatic associations of System and the slower, more controlled symbolic reasoning of System 2. Neural networks have demonstrated remarkable ability to perform System-I-like tasks within well-defined and specific environments. However, when faced with slightly different harder tasks, neural networks often fail while humans engage in System 2 processes. The latter allows for iterative reasoning about new observations drawing upon prior experience and shared abstractions which remains difficult even for extremely large neural network architectures such as LLMs [35] 51).
There is variety of recent work that tries formalize reasoning within neural network approach, see next section. In this work, we build upon the state-of-the-an in 20 21 formalizing iterative reasoning energy minimization problem, i.e.,
argmin Fo y)
39th Conference Neural Information Processing Systems (NeurIPS 2025).

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for a given problem encoded in R" with (partial) solutions x E R". Learning to reason is defined as learning the energy landscape Es parameterized by 0 via
min Σ
argmin Ea vi)
(EMP)
x
from given problem- and volution-pairs {yr x;) E R x RD from a current (partial) solution =² to a new 2k+1 with Eg y) S. EEga y) are then considered as individual reasoning steps. has been proven empirically and theoretically that this formulation is superior to direct ferd-forward computations, recurrent approaches, and various baselines from neural reasoning in terms of generalization and parameter efficiency [20] 21]. Nevertheless, learning energy landscapes involves solving ( at training well as inference time which imposes several limitations on current approaches: (i) Due to the inherent complexity of energy landscapes, heuristics for approximating solutions used instead of directly solving (T), see next section, which can result in unstable training, (ii) Relying on gradient descent at prediction time is computationally expensive and might hinder practical applications, and (iii) Theoretically, energy-based reasoning yields a natural termination criterion during inference, i.e., an indication to terminate the computation of reasoning steps, by determining if a locally minimal energy state has been found. However, there are no theoretical guarantees that such an energy state is ever reached in previous methods as they rely on gradient descent.
Optimization steps
As a remedy, we present a general energy learning framework for continuous iterative reasoning based on differennee-o-cconvex function (DC) optimization. Our main contributions are
1. We introduce a tailored form of energy functions and present a ciffereence-o--coove--iie algorithm (DCA), see 4 for powering our novel energy learning algorithm.
2. We derive theoretical convergence guarantees of our DCA routine to local solutions of [[.
3. We show that our DCA routine converges in finitely many steps and, hence, offers clear termination criterion.
4. Under additional assumptions, show how energy learning algorithm can be scaled for batch optimization and present theoretical approximation guarantees for our form of energy function.
2 Related Work
Neural Reasoning, There is an active area of research that tries to formalize reasoning with neural networks. One group of work builds upon the idea to formulate reasoning as optimization problems and then derives differentiable solvers, e.g.. (2 34 48 50). These approaches are however constrained to tasks of a particular kind, e.g., tasks that can formulated as quadratic programs [2]. Another group of research formalizes reasoning as iterative computations using neural networks. Following (21. this research can again be broadly subdivided into two areas: one that leverages explicit program representations [26] 36 38
10] 13 37 555 and another that uses recurrent neural 18, 40 [5]. In both areas a challenging problem to decide when to terminate the computation and has been tackled in various ways usually by learning some sort of halting probabilities [12 7 [0]. In contrast, our approach naturally imposes a termination criterion by stopping once a local minimum in the energy function has been reached.
networks [32 25 9 15
(56] 16
Energy-Based Models. Energy- based models formulate prediction tasks via energy minimization 33). That is, external observations y and possible predictions r are both processed by a so-called energy function E R" x TM + which measures how compatible x is with y. The convention is that lower energies indicate higher compatibility. A prediction for y is then defined
minimum
energy state x² given y, see (T) Energy- models have been used various ways to learn probabilistic models from data [54 53
19 52). Our work leverages such energy functions to formalize an iterative reasoning process similar to [20 (1).
22
24
Energy Based Iterative Reasoning. We are not the first to formulate iterative reasoning as an energy minimization problem. To the best of our knowledge, (20 the first to formalize reasoning through optimization steps using general trainable energy function. In particular, the authors make use of a fixed number of gradient steps T with a fixed step size during training to approximate

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within EMPP) i.e., = T-1 -V, T-1 U.). However, this can lead to unstable training processes as due to potentially complex optimization landscapes it is not guaranteed that xi is good approximation of local minimizer of (z, 21). As a remedy, [T] introduced an energy diffusion process in which the authors minimize sequence of energy landscapes by gradually increasing their complexity and using solutions on previous levels to initialize the gradient descent routine on consecutive levels. The energy landscape is then tuned via supervised approach on noise corrupted gradients and a contrastive loss component to enforce local minima in the learned energy landscape instead. Nevertheless, both approaches suffer from computational bottlenecks at inference time due to the inherent optimization procedure based on gradient descent and the need for auto-differentiation at test time.
3 DC Framework for Energy-Based Reasoning
In this section, we introduce our novel energy-based reasoning framework that builds upon a difference-of-convex functions approach.
3.1 DC Energy Landscapes
To allow for sufficient reasoning capabilities an energy function Eg should be able to represent wide range of functional dependencies while at the same time entail structural properties that allow for an efficient solution of argmin, (x, y) at training time. Based on this observation, we are making use of the following energy function Eo (z, y) = Σ (,o( xr) + b;), with a = ao(y). max(-, 0) the ReLU activation function.
W WW (y) and by(y), where Ng E and Note that Eg is single hidden layer neural network in while its parameters (a, b) are again parameterized functions in y with weights In particular, we set (as, Wo, bg) again as single hidden layer neural networks in y with N, N hidden neurons. Thus, we are using neural networks in y to represent parameters for a neural network æ. Due to single hidden layer neural networks being universal approximators [27] we can ensure sufficient representation capabilities for this form of energy function.
As the goal is to learn energy landscapes in a way that minimal-energy states represent solutions of particular reasoning problems, we also want ensure that such minimum always exists. Hence, our final definition of Eg is as follows
N₂
Eo y) [1a=11 (E,2) - w +
Σ ((w,, + bi),
(2)
with a = (v)),, W = Wo(v). b bo()), & = (o(.)) p= Po(v) > 0, and = wp(v). Note that we added a general quadratic form, so that for fixed 0 and y we have Es(x, y) - oa for ||>|||| + co. Hence, Eg is coercive and continuous in æ and thus argmin, Es(x, v) + 0.
Our next result shows that ② can be decomposed into a difference of convex functions in x for fixed weights E,, W, b), see Lemma
Lemma 1. For fixed weights (p(v), ((y), aty). (yy, b(y)), the energy function Eg is DC in i.e., Eo 3) Ez) g(z) h(x) with
g(x) f |
a b:)
(3)
a,
M(x)= Σ lasla ((w),)) + b.) + (E,x) + w,
(4)
a.<0
and g. h convex in x. For proof see Appendix A.1
The above DC representation entails desirable properties for our analysis later on. We summarize important characteristics in the following I Lemma.
Lemma 2. Let g and h be defined as in Lemma then the following holds
1. g is strongly convex in x.
3

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2. h is (up to constant) polyhedral convex in I, i.e., there exists Qh E I and ps E R for & E {I..... (.....KK K} such that
h(a)
maxx( x) Pk- kEK
(5)
For a proof see Appendix A.2
3.2 Locally Minimal Energy States
This section shows how the above defined energy function can be minimized in x via a tailored DCA. For introduction DCA, refer 4 Following [45], the DCA routine for minimizing (x(x) = g(x) - h(x) starting in an arbitrary point zo E IR" is
# E @h(xx)
Fh+1 E argmin g(x) (v,
Note that an element in the subgradient of h in (6 is given by
Σ |a Ja ((w,, xx) + b;) + + (xx))..
(8)
where (=) denotes the Heaviside function, i.e., (=)) 1 for 2 and H(z) = else. To compute + in the DCA, has to solve the minimization problem in (7). The following lemma shows that the problem can be equivalently stated as convex quadratic program.
Lemma 3. The optimization problem
min g(x) (v, x)) a ZER
(9)
can be formulated as
min 2,2 2
( T
1
90 0
0
2 (
(it) " (-w- f ()) (5)
(QP)
where t the vector of strictly positive weights the output layer, i.e., at aie for all ij e {1....,N., with oi, > W is the matrix with rows formed by the corresponding weight vectors
Wiy and by the vector formed by the corresponding bias terms bi,- For proof see Appendix A.3
One step in DCA thus simplifies to evaluating Equation (8) and then solving QPP. The next theorem shows that this simple iteration entails favorable convergence propertics in the view of supervised learning of minimal energy states.
Theorem 1. Given an arbitrary starting point To, the DCA routine ( (16) and 7 with given by
Equation (8) and Tx41 given as the solution of QPP converges in finitely many iterations to a DC critical point x² E RRn i.e.
0g(x") n Oh(x*)(())))))))))))))()))))00 + 0.
(10)
Furthermore, x² a local minimum of Fo(..y) .) if and only if
Oh(x*) C 0g(x*).
(11)
For proof see Appendix A.4
Note that [1] is always fulfilled if @h(x") or 0g(x*)) a singelton. The latter holds true in particular if (w,, x²) +b, hold we can restart the DCA routine with a point zg that yields a strict energy reduction in the first restart procedure.
Vi E {i a; > 0} or (w,,
6, * Vi E {i a, < 0}. If (II does not
DCA step following (44, see Appendix A.8 for in-depth discussion
Herre 0f stands for subgradient a convex functions f defined as 0f(y)
{u IR" V E. f(x) flv) 2 (u, 8)}

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4 Scalable Energy Learning
In theory. the DC framework presented in Section 3 can now be used learn energy landscapes by supervising the resulting minimal energy states through regression using (EMP) similar to [20]. In particular, one can make use of differentiable convex optimization layers [ or specialized batched quadratic programming solvers [ to solve (QP) and run our DCA routine in batches which will converge finitely many steps due to Theorem Nevertheless, our research shows that (i) relying on differentiable QP solvers, and (ii) the need for our restart routine to ensure local optimality, hinders the ability of our approach to scale to large-scale settings. As a remedy, we introduce additional assumptions on the energy function defined in Equation (2) and show that under these assumptions we can find analytical solutions
(QP) and can guarantee that (TT) always holds true for x=, i.e., DCA always converges to a local minimum of the energy function. In addition, in Section 4.1 we show that our energy function approximates a sub class of continuous functions that we call convexly-repe -regular arbitrarily well. We also show that for the univariate prediction case the approximation is universal.
For the remaining part of this work, we make the following additional assumption summarized in Assumption 1
Assumption 1. Let ai S for all i E [L,...,NT) No) in Equation
Note, that this can be easily accomplished by using non-negative activation functior in the neural network component (-) and use the resulting values directly in Equation (4). Under this assumption, the following lemma can be derived.
Lemma 4. Let Assumption 7 hold true. Then, (OP) can be solved analytically and Øx+1 = e. Furthermore, 0h(z") ) S 0g( [x") always holds true in this case. For proof see Appendix
In the next section we analyze how Assumption I [fffcts the approximation capabilities of our energy function, as indeed restricting shallow neural networks to only positive weights can make them loose their universal approximation guarantees (49
4.1 Approximation Guarantees
In our energy function (v)) wy()), as (v))). W o (y). bo())), are all single hidden layer neural networks in y. and hence universal approximators (27). see also (ypendix[A.6] We thus focus the following analysis on approximations of functions in and need the following definitions and results from earlier work.
the weights to parameterize the function in I, i.e.,
Definition 1. Let X S RR be convex. A function f X + R is called p-weakly-conve, or simply weakly-conver if there exists a p > 0 and convex function h such that f p/211-13 h. The function is called weakly-concuve if - f is weakly-convex. The set of all weakly-convex functions in is denoted by (X..).
It can be shown that weakll-coonve functions are universal approximators. To formalize this claim recall that (X) denotes the set of continuous functions that vanish at infinity, i.e., for all e there exists compact set K C X such that [f(x) < e if x & K. Then, the following holds.
Lemma 5 (Theorem in [43)). Let X S R" be closed (or open) convex. Then, WC(X) n (x(X) is dense in Co equipped with the infinity norm. This statement also holds for coakle functions (by switching signs).
Furthermore, weakly-convex functions can be represented in a special form, see Lemma 6 Lemma 6 (Theorem 3 in 43) ). A (closed) function f is p-weakly-conver if and only if f(x) =
supter (qt, x) Pt | for some (not necessarily finite) index set 3
Now, by combining Lemma 2 and Lemma 6 we see that our energy function is weakly- in x,
ie., it has exactly the form
2 || max x) + Pk-
(12)
In our implementation. we are using softplus activations for both as (-) and pe(-).
'Note that Theorem 3 in [43 is stating that f(x) = supter (0r, 2) + by + a [|x||| as the authors use denote the modulus of convexity.
5

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Note, however, that it is not immediately clear that also any weakly-coneave function can be ap- proximated by our energy function as the vectors q and biases Pt in Lemma follow special form. Nevertheless, if this is the case or equivalently if we are able to prove that h can approximate continuous convex functions, Lemma yields theoretical approximation guarantees for our energy function.
To derive sufficient conditions for h to be able to approximate continuous convex function, we note that we merely impose sign constraints on the output layer of the involved shallow neural network. Thus, can be seen as a single layer input convex neural network (ICNN) with weighted input skip connections in (23 Input skip connections were introduced for ICNNs in [3] to increase their expressivity by allowing identity mappings between layers as otherwise the non- non-negativity constraint would be too restrictive. Indeed, it has then been shown that ICNNs are able to approximate arbitrary continuous convex functions on compact convex domains (see, e.g., Theorem in [4] or Proposition 3 in (28). However, those approximation guarantees require deeper ICNNS while h is merely single layer ICNN. As pointed out by [23], the derivations in [4 28 are merely for theoretical purposes as they require as many layers affine pieces of the piecewise linear convex function they are trying to approximate and only a single neuron per layer. The authors, thus, derive the following result.
Lemma 7 (Corollary 4.8 and Proposition 4.9 (3]). ). convex function implemented by a single hidden layer ReLU network (with or without weighted input skip connections) can also be implemented by a single hidden layer ICNN with the same width.
The question if h can approximate a continuous convex function thus boils down to the question if it can be approximated by a convex shallow ReLU network. Hence, we define the following
Definition 2. Let X C R" be convex and compact. We call continuous convex function r X I Re if there exists a convex single layer ReLU network (with or without weighted input skip connections) NN such that [|WN -l <.
Given a function f E (X) we know from Lemma 5 that for all > 0 there exists a p-weakly- concave function Je with Selloo < make the following definition.
Then we know that r -fe + |/2 is convex and we
Definition 3. Let X C RR" be convex and compact and let I E (x)) We call f convexly-rguular, if we can always choose fe such that is e-convexly-ReLe reree-eppesen
Now, Theorem 2 summarizes the above derivations
Theorem 2. Let X IR" be convex and compuct. Under Assumption 7 every convexly-regular function f E Co (X) can be approximated arbitrarily well by Eg a function in x.
Proof. Let e > 0 be arbitrary and é = e/2. From Lemma 5 we know that there exists a weakly- concave function fe such that f felloo < Hence, there exists p 0 and a convex function r such that -f((x) + p/2[|z]| 22 r(z). Furthermore, Definition 3 ensures that we can always choose fa such that r is www.aaewwyy-...--eeeeeeetttaeeae
Thus, there exists a convex ReLU neural network ACN with ||r NV < ē From Lemma 7 we know that there exist a S 0, W, b, &. and w such that (x)(x) (l
b.) + (T,x) + which we define as h. Hence, for
Ex(x) = 1 h(x:) we have
Ilf - (x)ll = || (6 [11l M(x))
|1 < I1 - -fello + +111 (G11o12- M(xi)
(13)
< é + ||/ - -rlooo < 28 =e.
(14)
Definition is rather technical and the best of our knowledge there no general results conditions under which a convex function can be approximated by a single layer ReLU network which is itself convex. However, the following theorem shows that the class of convexly-reuular functions is sufficiently large in the univariate case.
Theorem 3. Let X C R be compact and convex. Then, every f E Co(X) is convexly-regular For a proof see Appendix A.7

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4.2 Pseudocode
We now combine our derivations in Section and Assumption to define our algorithm for scalable energy learning via batched DCA approach named DCA ReaRonsee A pseudocode is presented in Algorithm
Algorithm 1: DCAReasoner: Scalable Energy Learning via Batched DCA
Data: (gi. xx) 1R x IR", lower and upper bounds for starting points I, E R
Result:
Es
while converged
Sample hatch of data #ylsc::
Perform forward pass for parameters (o,, W. W.b,p.)) (a, W. ((tr))))) Sample uniformly random starting points (1)))EB 24(1, u);
I: <- 0:
// Initializing a in meaningful way
while maxilar x E Ela,/ws
> wot do
+6):
1
end
Update using
Σ llat 111BB/BB1 JEh
end
Note that we used maxjen ||>)) -Iy +1 > tol as stopping criterion for the DCA routine. Empirically, observe that few DCA iterations (< 10) are enough for the whole batch to converge for tol = 10-5 Indeed, most of the time observe maxje ||a ++1 precision, and hence, the finite convergence property can also be observed empirically. See Figure in Appendix
tol approaching machine
I [B.T for illustration of how the norm differences decrease zero with DCA iterations for a batch size of 512, i.e., when solving 512 reasoning tasks in parallel. We also note that we skipped the neural network for the bias term, i.e., wyy), in Algorithm | as it does not change the local minimizer of (T) and was merely used for our theoretical derivations. Furthermore, the network for b will not be updated during training as consequence of the Heaviside function. Nevertheless, similar partial optimization routines, in which parts of the parameters are randomly initialized and then frozen, have been successfully applied in neural learning, see e.g. 29
Numerical Experiments
5.1 Experimental Setup
We first evaluate our algorithm on five continuous algorithmic reasoning benchmark datasets from earlier research 20 21 in Section 5.2 All tasks are aiming capture different aspects of reasoning. We report the mean squared errors, as well as, inference times of our DCAReasoner and two state- of-the-art baselines from energy- test problems and repeated five times to report the mean and standard error of our metrics. Having established that our algorithm is superior (or on par) with thtte-of-the-art but significantly faster, then demonstrate how our DCAReasoner might unlock reasoning capabilities in language models by learning energy landscapes in token embedding spaces in Section 5.3
iterative reasoning. The evaluation is performed
10000
5.2 Continuous Algorithmic Reasoning Benchmarks
Baselines. We consider two baselines from state-of -ftthe-rrtttttt energy based iterative reasoning: (i) En- ergy based reasoning through energy minimization (IREM): This baseline learns energy function minimizing (EMP) by approximating argmin, Eg Exx y) via a fixed number gradient steps [20]. During inference it uses again a subgradient descent method but leveraging greater number of steps than during training. (ii) Energy based reasoning through energy diffusion (IRED): Here, the idea is to minimize a sequence of energy landscapes gradually increasing their complexity and using solutions on previous levels to initialize the gradient descent routine on consecutive levels 21]. The energy

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Datasel
DCA Reasonee (Ours)
IRED
IREM
MSB Inferenoo-Time
MSE
Inference- Time
MSE
Inferens-TTime [s]
Difliculty
Matrix Inverse
0.0096.1 0.0000
2.6189110.0305
0.0097 10007_10.0000
33.7056.1088818
0.0101 1.000000
22.6199.1..6127
Matrix Completion
0.0177:1-0.000
0.0179=0.0000 1.3373=:0.0125
33.6597::099125
0.0180:10.0000
22.7441 5005226
Parity
0.0301=+-00000
0.6053=000081
0.4859=0. 0002
9.1011
0,2504+0.0001
1.9797 1463
Decompostiio
0.1438:1.0.001
2.3051=:0.026
0.2175=0.0001
48.1915:::.4199
0.1521+00.0011
36.27755:10023
Matris Multiplication
0,0480 1188011..0000 0.0000
16790010.0258
0.0919 0.0919-1000000
34.1602.10.8609
0.0903110.0000
23.6510.10.6229
Harder Difficulty
Matrix Inverse
0.2077-1.0.0000
2.6017.10.0226
0.2064
33.1662.10.64422
0.2063110.000
22.7757 1..499
Matrix Completion
0.21000::0.0001
1.3091
0.2094=0.0002
33,2705::0:22M0
0.20058:100000.
22.9266:0.5148
Parily
0.0301 0.0.003
0.0063:10.0066
0.0010
8.6229-10.0879
0.2504110.001
1.9477
Decompasition
0.8847-100.0m
2.3374:1:0,22 1298
1.0776=0.0002.
4 L0362
1.3267:10.0006
35.3764
Matrix Multiplication
0.2974:10.0003
1.6877=:0.0243
0.4506=0.0002
33,3451
0.522420.0002
23.5748
Table 1: Evaluation on continuous algorithmic reasoning tasks. Models evaluated test problems
drawn from the training distribution (same difficulty) and a harder test distribution (harder difficulty). We report the mean squared error and the inference time. We perform five evaluation runs and report the mean and standard errors.
landscape is then tuned by supervising on noise corrupted gradients and a contrastive loss component enforce local minima in the energy landscape. We scale the network size in baselines to ensure that each of them has roughly the same number of parameters. For a detailed discussion of our experimental setup see Appendix B.2
Datasets. In our experiments we evaluate all baselines based on five datasets from earlier research [1.21] Each of them is evaluated once with the same level of difficulty, i.e., test cases drawn from the training distribution, and once with a harder level of difficulty, in which test cases are drawn from problem specific harder test distribution following (20 [1]. The latter should test the algorithms ability to generalize reasoning capabilities to new unseen problem settings. In particular, use: (i) Matrix Inverse: The task is invert random 20 x 20 matrix. It aims at testing numerical reasoning. Harder problems are created by creating less well-conditioned matrices to invert. (ii) Matrix Completion: The task is recover masked out values in a random low-ranked 20 20 matrix constructed from two low-rank matrices U and V. Harder tasks are created by increasing the complexity of U and V. It aims at both structural and analogical reasoning. (iii) Parity: Given random vector in [0, 1 the task is to decide whether or not the number of entries which are greater for even 1 for odd, see also (25). Harder tasks are created
than 0.5 is odd or even, i.e., the target by increasing the magnitude of vector entries. (iv) QR Decomposition: The task is to compute the QR decomposition of random 20 x matrix with entries in [-1, 1]. Harder problems are created by changing the magnitude of matrix entries. (v) Matrix Multiplication: Given random 20 x 20 matrix M the task is to compute the square, i.e., M2, Harder problems are created by changing the magnitude of matrix entries. For an in-depth discussion of our benchmark datasets see Appendix [3.]
Results. Our results are summarized in Table In terms of mean squared error, DCAReasoner is mostly on par with IRED and IREM on the matrix inverse and the matrix completion dataset, while we see larger improvements for the remaining datasets. Noteworthy, we observe a decrease in MSE by a factor of ~ 10 on the parity dataset, which might stem from the fact that the prediction is univariate in this case, for which we have established universal approximation guarantees in Theorems and In terms of inference time, we see large improvements of factors between 14 and 27 for IRED and between 3 and 18 for IREM. Furthermore, we performed additional experiments showing that our predictions are robust to noisy input data on the example of the QR Decomposition dataset. All details are reported in Appendix B.4 B.4
8

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5.3 Energy-Basee Reasoning in Token Embedding Space
In the last section we have empirically proven that our algorithm yields of-ie-ff-hh-errr perfor- mance results but is significantly faster at in- ference time than previous energy-based mod- els for iterative reasoning. Furthermore, it of- fers theoretical convergence guarantees and per- forms well high-dimensional settings. such, we think that our algorithm might be used to improve reasoning skills language models by learning energy landscapes in token embed- ding spaces which might be used during infer- ence for energy energy-guided text generation. Note that our baselines are not well-suited for such setting as large inference times especially in high-dimensional token embedding spaces might considerably slow down token generation in practice.
Alg.
MSE
Accuracy|S8]
Time[
IREM
0.012
96%
336%
IRED
0.027
96%
1330%
DCARcasoner
0.008
96%
100%
Table 2: Test evaluation performance on text clas- sification task. We report the mean squared error of the prediction and the embedding of the target text. Accuracy is computed by using the target closest to the prediction. Inference time is reported percent, with 100% indicating the lowest time.
As fully developed approach for energy-guided text generation to improve reasoning is out the scope of this work, we demonstrate how DCAReasoner is able to learn reasonable energy-landscane spaces in simpler setting. In particular, make of the symptom- symptom-to-diagnoss to-diagnosis dataset for medical reasoning
token embedding
which is freely available Hugging Face I It provides training and test dataset consisting of short texts in which a patient describes her symptoms and corresponding diag- nosis out of a set of 22 medical diagnoses. We then use the CLS token embeddings of a finetuned uncased DistilBERT model of those texts as inputs and the embeddings of the corresponding diagnosis
Malaria
Arthribs
Uninary tract infection
ground truth x train the DCAReasoree and our baselines a continuous reasoning setting. More details of our experiments are reported in Appendix B.5
We summarize the results in Table 2 Our algorithm yields the lowest mean squared error more than three times faster than IREM and thirteen times faster than IRED. We also visualize the energy landscape in token embedding space") learned by our algorithm in Figure Uusing an example the sentence "My muscles are weak, my neck is stiff, and my joints are swollen. can't move around very well, and walking is really painful." with diagnosis arthritis from the test set. Note that arthritis has indeed the lowest energy. while embedding vectors diseases like malaria which shares symptoms like weakness and joint pain are also assigned lower energy values. Furthermore, seemingly unrelated diagnoses like urinary tract infection yield higher energy values, indicating that our algorithm is capable of learning reasonable energy landscapes in the token embedding space.
Figure 1: Visualization of Energy Land- scape in Token Embedding Space for the Input Sentence "My muscles are weak, my neck is stiff, and my joints are swollen. I can't 't move around very well, and walking is really painful.".
Conclusion
We proposed algorithm named DCAReasoner for energy-based continuous iterative reasoning, is build upon a tailored class of energy functions for which we derived theoretical approximation guarantees. In addition, presented theoretical convergence guarantees for the inherent DCA routine. That is, we showed that it converges to local minima in finitely many steps independent of
toeestttoiiieeeeiiooi agnosis, the dataset is licensed un-
der Apache 2.0
'We used t-SNE to visualize the 768 dimensional embedding vectors in the plane. Furthermore, we computed the energies for all 22 target embeddings and interpolated the results using a radial basis function interpolator.

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the starting point. Empirically, we have proven that it yields improved performance and inference times.
Limitations, Our DeAReasoner shows promising results for neural iterative reasoning. However, there still limitations. First, our DCAReasoner, presented in this work, does not yet make of any external memory. That is, DCARcasoner cannot store intermediate results during reasoning which might be beneficial in some reasoning tasks. Second, the form of our energy function might require a large number of trainable parameters if a large number of hidden neurons in æ, i.e., N2., desirable, For instance, the mapping Wo(y) requires m N2 + Ny : Nv n trainable parameters. Empirically, we see however that smaller values for N₂ and larger values for N2 are sufficient (our experiments use N₂ and N₂ 4000). Third, in the current form DCAReasoner designed for continuous reasoning tasks and hence cannot directly handle discrete reasoning tasks. However, we note that earlier work on energy-basee -baee reasoning shows how discrete tasks, formulated a continuous setting, yield promising results [20 21.. Furthermore, we conducted preliminary experiments
discrete task, specifically solving Sudoku puzzles using the dataset provided in [41] and report our early results in Appendix C
Societal Impacts. To the best of our knowledge, there no immediate positive or negative social impacts that can be derived from our work in the current form.
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A Mathematical Appendix
A.1 Proof of Lemma 1
is clear that (2, y) can be decomposed as
Ea v) Eo(x) g(x) h(x)
(15)
with
g(x) -1 |1:|| 0: (21(( + bs)
(16)
2
a,50
h(*) Σ a (aw,, x) bi) + (E) + w.
(17)
Furthermore, (wi, + by is affine in and hence convex and is non-decreasing and convex. Thus
the composition a ((w;, .)) + bi) is always convex and the claim follows by observing that and linear combinations of convex functions with positive weights.
A.2 Proof of Lemma 2
For point one, note that the modulus of strong convexity defined as
p(g) = (up(((a)
2 |||| is convex]:
(18)
2
p00
see, e.g., 44
It is easy that g 1-112 is convex and hence p(g) p> which implies
strong convexity.
For point two, polyhedral convex functions can be characterized as max x) + Phe k [1....,K]]
(19)
see, e.g..
39
Note that
h(x))
Σ (a x)) + bi) (c)
(20)
a:50
Σ
a(laylws, x) laylb.) - (E,
(21)
a,<0
max 6,E(0.1)
8 ((ass 2) + laslb.) (5, x),
(22)
0,50
where the last equality follows from the fact that the maximum is achieved in the case where 6 = 1 (lac/ur, .r) + Jai|b; 2 and 8, else. Furthermore,
max 6;E
Σ 00<0
8, ((aas/ws, x) + lassbb)) + (E,x))
(23)
Jayldww, + E,æ))
Σ
Jagldsbi
(24)
<0
a.50
which proves the claim.
A.3 Proof of Lemma
First, note that
g(x) (v,2) = P
Σ aro((w,,2) +bb) (v,,2),
(25)
which can be equivalently stated
min 2,
0
(-:
(26)
x
21

---

and the non non-linear equality constrained 2 = a((u,, x) + bç) for all i (I, with a, > 0.
Now, the inequality constraints
(27)
2 (Wx, 2) + b;
(28)
imply that 2, 2 a(w,;,)) b.). As the only term in the objective function containing is given by (a 2) and at is elementwise strictly positive, we have that problem (26) with constraints (27)
and (28) always results 2 a o(w;, + bi) at an optimal point (a ,=,=z) ,-*). Hence, the claim follows.
A.4 Proof of Theorem
From Lemma 2 we know that g is strongly convex with p(g) 2 p. Convergence thus follows by Theorem 6 in 45. Furthermore, Lemma shows that h is polyhedral convex, which proves the finite convergence property using Theorem 9 in [45] Last, the fact that xx is local minimum Equation holds true follows from Corollary 2 in 45
A.5 Proof of Lemma 4
is easy to see that under Assumption ! the function g reduces to & ||||2 Hence QP reduces
2 [|l=l| - (ar, u),
(29)
with the global solution Ju. Furthermore, in this case g is differentiable and hence 0g(x*) (Vg(z*)) is singleton. Thus, Equation [1] always holds true under Assumption
A.6 Details on Approximation in
② the weights to parameterize the function I, i.e., are all single hidden layer neural networks in y- Since (Ea (v)(y), woy)), We(y), ba) (y)) are all using the identity as an output layer activation those are universal approximators (27). Note also that (v)) and (y) are using softplus activations in the output layer. Hence, they are still able to approximate any strictly positive continuous function arbitrarily well as shown in the following.
In our energy function
(po(y), wp()) ar (v), We (e(
Let f be an arbitrary strictly positive and continuous function. Then log(eí - 1) is a continuous func- tion mapping (-00, oo). As such it can be approximated arbitrarily well by single hidden layer network by [27 Further, if we now equip this network with softplus activation, we approximate softplus (log(e) 1)) f by using that softplus is Lipschitz. We formalize this in the following lemma.
Lemma 8. Let f: _R" + (0, oo) be continuous. Then for every e there a single hidden layer network with softplus activation, Ve such that |[ - vello0 <e
Proof. Let > 0. By (27], there is single hidden layer network N. such that ||log (ef() - 1) < Nelloo
Using that the function softplus(y) = log(1 + e*) ) is Lipschitz of constant 1 and that + log (e - 1) is its inverse, we deduce that for every E R": [f(x) - log (1 eM(( s
log (ef(a) 1) N₂ (x) and therefore
f log (1 + e Nloc = sup f(x) - llog (1 + +N(c))))) =ER=
log (e((t)= -1) -1))))(()) = |log (ef - -1) - -Neloo <e
< sup =
By setting ve(ar) = log (1 + eNNot)), the claim follows.
22

---

A.7 Proof of Theorem 3
Let f E Co(X) and 0. From Lemma we know that there exists a weakly-comee concave function fo such that |l felloe < c. Hence, there exists p > 0 and a convex function r such that r(z). We need show that is Beanvexly-ReLU--rrpeeeeettt
-f((x) + p/2
We assume first that r is Lipschitz continuous. Following the derivations in Theorem 2 in 11 we find sequence of convex piecewise linear functions yn such that In uniformly for o0. Furthermore, by T'eoorem 2.2 in [6] there exist single layer ReLU networks NN,, that can represent In exactly which proves the claim.
we relax the Lipschitz assumption, we can again follow the arguments in Theorem in | and find sequence of Lipschitz continuous and convex functions f& that converge uniformly to r. Hence for é > 0 we can find 2 such that ||r - felloo < c/2. With the arguments from above we can then find Tn such that || Talloo e/2, and hence ||r rn llooo < which proves the claim.
A.8 Restart Procedure for our DCA Routine
Following [46), let the index sets (x(x)) and L (x=*) be defined as follows
Tx(x²) (i {1, ,..,, N₂) oq < and (wi, x0 b, = =],
(30)
(x()) = {i E [1,...., N2) az > 0 and (w;, x + b, - 0}.
(31)
Note that if In (x=) = 0 or (xx) = we have that 0h(x*) S 0g(x²) as one of the two sets is a singleton and xx is DC-critical. Hence, for the rest of this derivation we assume 4n (x*) * 0 and Io (2*) * D.
Now, the subgradients of h, respectively g, are given by
[0 if In(x*)
0h(x*)
Σ
Jas
He) ((wi, x") + bije E {{}} else
(32)
[10. 1] (1)
if e
0g(x")
pxx
Σ a.50
a,W,H((w,, x²) + bi)e e E
(33)
else
where (2) denotes the Heaviside function, i.e., H(2) = 1 for 2 2 0 and H(2) = else. Further-
more, let us define the following
(x=) = E (I,....,,,)) a; < 0 and x ) + b/ > 0},
(34)
1 (x") = f (1....,, No) a > 0 and x4) ) + b. > 0},
(35)
Un = + + Σ
lailun
(36)
iE
0gg px +
Σ
O,W4
(37)
VEl, (a))
as well as, the matrix A, E [x((((("') with columns Jaylw; for i E Ix(x"), and the matrix
Ag E (=x")(((=")) with columns a,w; for i E (z().. ). The inclusion 0h(x*) C 0g(x") holds true if
te [0, 1)la(=") 36 E [0, 1]("(1)) on + Anc Vgg + Age
(38)
To check if this holds, solve the max-min problem
max min Ulu + ABE" - Ug An USe SEISSees
(39)
and observe whether or not the optimal objective value is zero. If it is, a local minimum. If not, then we have found an element 2g E ah(x*) with F & 0g(x*). Following can restart the DCA routine with 2 and yield striet energy reduction in the first step.
38 holds and we have found
44
23

---

Figure 2: Illustration of Batched DCA for batch size 512
Energy Values
Norm differences || - 24
1800
12
1230
10
1000
500
DCA Berations
DCA lterations
Note. Illustration of batched DCA routine computing locally minimal energy states for 512 reasoning tasks in parallel.
What remains to be discussed, is how to solve (39) efficiently. To do note that we can reformulate the problem as
max min (ri, 1) + (r2, 2rri
(40)
s.t. Uh + Agc
1/ Age = r2
(41)
c e 20
(42)
c e <
(43)
This problem can be solved using branch and bound strategy for bilevel linear programming (see, e.g., section 6.2 in [8). Furthermore, since x² is DC-critical, we know that for c" = 1 there exists an el E [0, 11 (x")) with 2n + A, = Ug + Age.. Hence, we can warm start the branch and bound routine with (e", e ee,TI T2) = (1, el, O, 0). If the optimal objective value is zero, we already start with an optimal solution. If not, we can stop the branch and bound routine as soon we have found the first (c", e²) with + crch & Uig + Age². Thus, the restart procedure can be implemented efficiently.
B Details on Numerical Experiments
B.1 Visualization of Convergence
We visualize how our batched DCA routine converges for 512 reasoning tasks in parallel in Figure 2 In particular used the matrix inversion dataset, see Section [5] and for fixed visualized the energy values (x y) and norm differences | 2g. after training the parameters 0, i.e. at inference time.
B.2 Details on Experimental Setup
For our baselines we closely follow the code releases of 20 and
Model Specifications. For DCAReasoner we set the number of hidden units N₂ = 8 and N, = 4000 for all benchmark datasets. For our baselines we scale the network size depending on the dataset ensure that all models have roughly the same number of parameters. We set the convergence tolerance 10-5 and set maximum of 30 DCA iterations. However, in our experiments
in Algorithm we see that the batched DCA algorithm consistently converges to machine precision in less than 10 iterations. Hence, we also observe finite convergence guaranteed by Theorem [ Starting points x are sampled uniformly random as stated in Algorithm [ We set / = -1 and u = 1 for all benchmark datasets except for QR and Matrix Multiplication for which we l -5 and Hyperparameters are summarized in Table
code release MIT License
btttpe://githu..comm/
coleare
release
License
24

---

pyperaaraeeeer
Value
DCARcasoner
IREM
IRED
Common Hyperparameteess
Learning Rate Batch
10
10
10
Number of Gradient Steps Starting Point Sampling
10.000
10,000
10,000
uniform
uniform
normal
DCAReusoner Specilic Hyperparameeeess
Number of Neurons N=
Number of Neurons Ny
4000
DCA Convergence Tolerance tol
10
Maximum DCA lerations
30
Table 3: Hyperparameters settings for our experiments. Note that for our baselines we scale the network size depending on the dataset to ensure that all models have roughly the same number of parameters.
Training. For training. we use the Adam optimizer with learning rate of 10 suggested in [1) for all models. We set the batch size to 512 and train each model for a fixed number of 10000 iterations, i... gradient steps. Hyperparameters are summarized in Table
Evaluation. For evaluation, we are using again batch size of 512 and perform 20 test iterations, summing up to roughly 10000 test problems per difficulty level, i.e., once in the easy setting and once in the hard setting.
Hardware. All experiments performed on nl nl-standard-2 Google cloud instance with 7.5GB RAM and two NVIDIA GPUs.
B.3
Details on Benchmark Datasets
B.3.1 Motivation
In general, neural algorithmic reasoning constitutes an unsolved problem in machine learning. For an argumentation on the complexity of neural algorithmic reasoning see e.g. [47). Matrix Completion, QR Decomposition, and Matrix Multiplication represent algorithmic reasoning tasks. Du et. al. argue that effective algorithmic reasoning requires repetitive application of underlying algorithmic computations, dependent problem complexity, and thus serves as natural benchmark for iterative reasoning (20]. Learning parities a well-known reasoning benchmark and well-studied in learning theory general. Shoshani and Shamir argue that there is strong empirical evidence that suggests that parities cannot be learned using more standard general purpose learning methods, and in particular gradient methods, once the dimension is even moderately large [42]. This also goes in line with the fact that our baselines struggled particularly on this task, see also conclusion and limitations section in (20). For matrix inversion, Ji et. al. argue that despite significant progress in deep learning, there exists no universal neural-based method for approximating matrix inversion 30 showing that this is non-trivial task for neural reasoning.
B.3.2
Technical Details
In the following we give more details on our benchmark datasets, We followed the code releases of 20 and 21 closely.
Matrix Inverse. We construct well-conditioned random 20 20-matrices A = 2M +0.5 1, with M being random matrix with entries in [-1, The task is then to compute the inverse A-1, Harder tasks are created by making the matrices less-well conditioned by setting = 2M M + 0.1 I.
Matrix Completion. We randomly construct low-rank 20 x 20-matrices A = 0.1 N + 20 UVT where is standard normally distributed noise matrix and U and V are again standard normally distributed random 20 x 10-matrices. Then, random mask is created by rounding a randomly generated uniformly distributed matrix M. The model is then presented with the masked matrix A and asked to recover it. Harder tasks are created by setting A = 0.1 N + LVVT
25

---

Figure 3: Empirical Analysis of Robustenss Noise the QR Decomposition Dataset
DCAReasoner
10
IREM
IRED
MSE
Test
10
10
10"0
10
10
10
Noise Level
Note. We evaluated the trained models on the QR henchmark datnset adding different Gaussian noise levels tn the input data. IREM hocomes unstable large while TRED DCAReasoner appear to robust noisy inputs.
Parity. Similar to 25 we create uniformly random vectors in [0, 1,20 and then the target to 0 the number of values strictly greater than 0.5 is even and otherwise. Harder tasks are created by drawing random vectors from 1,,2
QR Decomposition. We create uniformly random matrices A with entries in [-1, 1] and then compute the QR decomposition, i.e., A QR. The models are then given the matrix A and asked to reconstruct both Q and R. Harder tasks are created by creating uniformly random matrices with entries in 122.5,22 2.5].
Matrix Multiplication. We create uniformly random matrices with entries in [-1, 1] and then compute the square, i.., A², The models are then given the matrix A and asked to perform the matrix multiplication A2. Harder tasks are created by creating uniformly random matrices A with entries in [-1.5, 1.5].
B.4 Further Experiments Analyzing Robustness to Noisy Data
We evaluated the trained models on the QR benchmark dataset adding different Gaussian noise levels to the imput matrix before processing with scale varying in {10-4,10-1 10-3 1 Results are visualized in Figure It appears that IREM becomes unstable for large noise levels while IRED and DCAReasoner mostly robust to noise.
1001,111 1 10).
B.5 Details on Energy-Based Reasoning in Token Embedding Space
For our experiments, we make use the comptom-to-diagnost dataset for medical reasoning which is freely available on huggingface It provides training (853 examples) and test (212 examples) dataset consisting of short texts which a patient describes her symptoms and a corresponding diagnosis out of set of 22 medical diagnoses summarized in Table 4
@
We first finetune an uncased DistilBERT model for text classification, splitting the training set again into training and validation using a 80/20 split, and using the huggingface trainer for sequence classification. In particular, we use batch size of 16, learning rate of * 10-5, weight decay of 0.01, and 10 evaluation steps saving the best performing model in terms of accuracy (which is also 96% for the energy-bau based models).
We then use the CLS token embeddings of the texts in the training dataset this finetuned model inputs y and the embeddings of the corresponding diagnosis ground truth to train the DCAReasonee and our baselines in a continuous reasoning setting. The model specifications for DCA are the same as in our main experiments, i.c., we N 8 and N, 4000,
diagnosis, the dataset licensed un-
der Apache 2.0
26

---

Diagnosis
Training Examples
Test examples
drug reaction
40
8
allergy
40
10
chicken pox
40
diabetes
40
10
psoriasis
40
10
hypertension
40
10
cervical spondylosis
40
10
bronchial asthma
40
10
varicose veins
40
10
malaria
40
dengue
40
10
arthritis
40
impetigo
40
10
fungal infection
39
common cold
39
10
gastroesophageal reflux disease
39
urinary tract infection
39
9
typhoid
38
9
pneumonia
37
10
peptic ulcer disease jaundice
37
33
7
migraine
32
Table 4: Number of examples per split and diagnosis in the Symptom-to-Diagnosii dataset.
Figure 4: Training and Validation Curves for Sudoku Experiment
(a) Training Loss (Cross-Entropy)
(b) Validation Performance (Cell Accuracies)
Vicrionee
1.10
Note, Cross-Entropy loss during training (left) and cell sccuracics on validation data (right). Validation stops are performed every 1000 training steps.
scaling the network sizes in baselines ensure that all models have roughly the same number of parameters.
For training, batch size of 64 and learning rate of 10 We train each model for 10000 iterations. For evaluation, we use the test set and process it again in batches of size 64.
C Preliminary Experiments Discrete Setting
As mentioned in the main paper, DCAReasoner is designed for continuous reasoning tasks. Thus, our method cannot be applied as-is to experiments in discrete spaces. Nevertheless, we performed
27

---

preliminary experiments using the Sudoku dataset in 41 The authors provide training dataset consisting of 1.8 million Sudoku puzzles as well validation set with 0.1 million Sudoku puzzles.
For reasoning in a discrete setting our energy landscape is defined in the logit space, i.e., a 729 dimensional setting and replace the MSE with the cross entropy loss. Processing the inputs, i.e., Sudoku puzzles, can be done in multiple ways. We decided for a convolutional neural network with the residual connection design as [2]] to process the Sudoku puzzles before feeding them to our neural network components. Using the same settings as in the main paper, i.e., N₂ 8, Ny = 4000 and a learning rate of 10 but a batch size of 64, we train DCAReasoner for a total of 10 epochs. Validation steps, i.e., computing the cell accuracy defined the percentage of unfilled cells whose values are correctly predicted on the validation data [41] are performed every 1000 training steps. The training process is visualized in Figure Note that the validation performance stagnates after roughly 6 epochs. At this time, we observe cell accuracy of 65.09% i.c., 65. 55.09% of all unfilled cells in all of the 100K validation puzzles are filled correctly. For context, the baseline from [41] with random order of input cells achieves 52%, while the one with fixed order achieves 58.64% Our results only fall short from the accuracy achieved by the model receiving additional context information about solving strategies during training (94.23%). However, we point out that the specific architecture of our model does not allow such information be provided.
28


Reviews:
[{"review_id": "9Ze8FJwImJ", "reviewer": "Reviewer_zLaS", "summary": "The authors proposed a new algorithm named DCAReasoner for energy-based continuous iterative reasoning. It is build upon a tailored class of energy functions for which derived theoretical approximation guarantees. In addition, the authors presented theoretical convergence guarantees for the inherent DCA routine. That is, the authors showed that it converges to local minima in finitely many steps independent of the starting point. Empirically, the authors have proven that it yields improved performance and inference times.", "questions": "see Weaknesses", "limitations": "YES", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n1. The authors introduce a tailored form of energy functions and present a DCA routine for powering novel energy learning algorithm.\n\n2. The authors derive theoretical convergence guarantees of our DCA routine to local solutions.\n\n3. The authors show that our DCA routine converges in finitely many steps and, hence, offers a clear termination criterion.\n\n4. Under additional assumptions, the authors show how the energy learning algorithm can be scaled for batch optimization and present theoretical approximation guarantees for our form of energy function.\n\nWeaknesses\n\n1. Restrictive Assumption Limiting Expressiveness\nThe paper's core practical contribution relies on Assumption 1 (requiring all αᵢ ≤ 0), which significantly constrains the model's expressiveness. While the authors provide theoretical approximation guarantees for \"convexly-regular\" functions, this is a somewhat artificial class they define. The restriction to non-positive weights in the energy function may limit the model's ability to learn complex reasoning patterns that require both positive and negative activations. The authors acknowledge this could affect universal approximation properties but don't thoroughly investigate the practical implications of this limitation.\n\n2. Limited Experimental Scope and Generalization. The experimental evaluation is quite narrow:\n\n\t•\tSmall-scale problems: All continuous reasoning tasks involve only 20×20 matrices, which doesn't demonstrate scalability to realistic problem sizes\n\n\t•\t Artificial benchmarks: The tasks (matrix inversion, completion, etc.) are relatively simple mathematical operations that may not reflect the complexity of real-world reasoning scenarios\n\n\t•\tToken embedding experiment is preliminary: The medical diagnosis task, while interesting, is quite simple (22 classes) and doesn't convincingly demonstrate the method's potential for complex language reasoning applications", "was_revised": true, "final_justification": "The authors' rebuttal addressed my concerns!", "rebuttal": "In the following we comment on the points raised by reviewer zLaS.\n\n**\"Restrictive Assumption Limiting Expressiveness: The authors acknowledge this could affect universal approximation properties but don't thoroughly investigate the practical implications of this limitation.\"**\n\nWe would like to point out that our sign constraints are only applied to the output layer, thus making $h$ an input convex neural network (ICNN) in $x$ which are widely used to approximate convex functions in practice. In addition, our experiments show that we outperform or are on par with state-of-the-art energy based reasoning algorithms on multiple benchmark datasets. Given that those baselines offer universal approximation guarantees in theory, we think our results -- and the wide application of ICNNs to approximate convex functions in general -- rule out concerns of practical implications.\n\n**\"Small-scale problems: All continuous reasoning tasks involve only 20×20 matrices, which doesn't demonstrate scalability to realistic problem sizes\"**\n\nWe want to point out that the dimensions were chosen following **[1]** and **[2]** and do not constitute an upper bound. That said, our energy learning framework can potentially scale to even larger settings. Nevertheless, in our opinion reasoning in the space of $20\\times20$ matrices which constitutes a 400 dimensional reasoning space is non-trivial. Furthermore, our experiments in token embedding space which are 768 dimensional and the QR decomposition benchmark dataset which is 800 dimensional (two $20\\times20$ matrices as targets) show that our approach scales well.\n\n**\"Artificial benchmarks: The tasks (matrix inversion, completion, etc.) are relatively simple mathematical operations that may not reflect the complexity of real-world reasoning scenarios\"**\n\nWe acknowledge that the motivation behind the choice of our benchmark datasets was maybe not elaborated enough in our manuscript so far.\n\nWe specifically chose our benchmarks to (i) show how our energy learning framework performs on tasks on which IREM from **[1]** and IRED from **[2]** struggled most, and (ii) to test different aspects of reasoning on well-established benchmark tasks.\n\nIn general, neural algorithmic reasoning constitutes an unsolved problem in machine learning. For an argumentation on the complexity of neural algorithmic reasoning see e.g. **[3]**. Matrix Completion, QR Decomposition, and Matrix Multiplication represent algorithmic reasoning tasks. Du et. al. argue that effective algorithmic reasoning requires repetitive application of underlying algorithmic computations, dependent on problem complexity, and thus serves as  a natural benchmark for iterative reasoning **[1]**. Learning parities is a well-known reasoning benchmark and well-studied in learning theory in general. Itamar and Shamir argue that there is strong empirical evidence that suggests that parities cannot be learned using more standard general purpose learning methods, and in particular gradient methods, once the dimension is even moderately large **[4]**. This also goes in line with the fact that our baselines struggled particularly on this task, see conclusion and limitations section in **[1]**. For matrix inversion, Yuliang et. al. argue that despite significant progress in deep learning, there exists no universal neural-based method for approximating matrix inversion **[5]**, showing that this is by no means a trivial task for neural reasoning.\n\nWe will add a more elaborate justification for our benchmarks to our Appendix B.3 in a potential camera-ready version.\n\n**\"Token embedding experiment is preliminary: The medical diagnosis task, while interesting, is quite simple (22 classes) and doesn't convincingly demonstrate the method's potential for complex language reasoning applications\"**\n\nWe agree with the reviewer that those experiments should not be seen as a fully developed energy-guided language reasoning application, which is also stated in Section 5.3 as this is not the scope of this work. Our experiments were merely chosen to demonstrate that DCAReasoner is able to learn meaningful energy landscapes in token embedding space, which we believe the outcome suggests.\n\n**References**\n\n**[1]** Y. Du, S. Li, J. Tenenbaum, and I. Mordatch. Learning iterative reasoning through energy minimization. International Conference on Machine Learning (ICML), pages 5570–5582, 2022\n\n**[2]** Y. Du, J. Mao, and J. Tenenbaum. Learning iterative reasoning through energy diffusion. International Conference on Machine Learning (ICML), pages 11764–11776, 2024.\n\n**[3]** P. Veličković, C. Blundell. Neural algorithmic reasoning, Patterns, Volume 2, Issue 7, 100273, 2021\n\n**[4]** S., Itamar, and O. Shamir. Hardness of learning fixed parities with neural networks. arXiv preprint arXiv:2501.00817 (2025).\n\n**[5]** J., Yuliang, J. Wu, and Y. Xi. Rethinking Neural-based Matrix Inversion: Why can't, and Where can. The 28th International Conference on Artificial Intelligence and Statistics (AISTATS), 2025"}, {"review_id": "s8tvgL8cCZ", "reviewer": "Reviewer_GYtm", "summary": "This paper introduces DCAReasoner, an energy-based iterative reasoning algorithm leveraging a Difference-of-Convex decomposition. It provides finite-step convergence guarantees and universal function approximation, and achieves 3–27× inference speedups on tasks including polynomial root finding, matrix inversion, and quadratic programming, while matching or exceeding state-of-the-art accuracy.", "questions": "* The acronym “DCA” appears before it is explicitly defined. Could you please clarify and provide its full form at first use?\n* What criteria guided your choice of the number of DCA iterations K in each experiment? How sensitive are your results—both in terms of accuracy and convergence speed—to different values of K and to the initialization x_0?\n* The paper does not fully specify the hyperparameters and configurations used for the baseline methods. Could you include detailed settings for all baselines to ensure reproducibility and a fair comparison?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## Strengths\n\n* Theoretical guarantees. Finite-step convergence to a local minimum and universal approximation results under mild sign-constraints ensure both reliability and expressivity.\n* End-to-end trainability. By deriving closed-form updates for each DC subproblem, the entire K-step inference can be unrolled into a differentiable computation graph, enabling direct gradient-based optimization of the energy model parameters.\n* Fast training. Closed-form DC updates replace inner gradient-descent loops, reducing both per-epoch compute and memory overhead—training runs substantially faster than methods that nest gradient-descent in the forward pass.\n* Significant inference speedups. Achieves 3–27× faster inference on polynomial root finding, matrix inversion, and QP benchmarks while matching or exceeding prior accuracy.\n* Broad applicability. Demonstrated across continuous algorithmic reasoning, high-dimensional embedding-space classification, and convex/non-convex energy landscapes.\n\n## Weakness \n* Generalization Beyond Training Distribution: Because the energy function $E_\\theta(x;y) $ is learned to minimize MSE on the training pairs $(y_i,x_i)$, it may overfit to those specific problem instances. It’s unclear how well it generalizes to out‐of‐distribution y (e.g.\\ different matrix condition numbers or QP formulations).\n* High-Dimensional Scalability: Solving the DC subproblem still requires inverting a Hessian-like operator or solving a fairly large convex system at each step. While tractable up to a few hundred dimensions, it may become prohibitive in, say, $\\geq10^4$-dimensional settings without further approximations.\n* Robustness to Noise and Model Mis-specification: There is no evaluation of how DCAReasoner handles noisy or approximate y (e.g.\\ measurement error in the input matrix) or errors in the DC decomposition. Such perturbations could cause the DCA iterates to converge to spurious local minima.", "was_revised": true, "final_justification": "The authors had addressed my concerns, so I keep my positive rating.", "rebuttal": "We want to thank the reviewer for the positive feedback on our work. In the following we comment on the points raised by reviewer GYtm.\n\n**\"Generalization Beyond Training Distribution: Because the energy function $E_\\theta(x;y)$ is learned to minimize MSE on the training pairs $(y_i, x_i)$, it may overfit to those specific problem instances. It’s unclear how well it generalizes to out‐of‐distribution $y$ (e.g.\\ different matrix condition numbers or QP formulations).\"**\n\nTo test the generalization on out-of-distribution samples our evaluation in Table 1 includes not only the performance on the test set of the same distribution but also on a task-specific harder test set following **[1]** and **[2]**. Details on how we increase the difficulty for each task in this test set are reported in Appendix B.3. For instance, the matrix inverse task follows exactly the reviewer's suggestion of increasing the difficulty by making the input matrix less well-conditioned.\n\n**\"High-Dimensional Scalability: Solving the DC subproblem still requires inverting a Hessian-like operator or solving a fairly large convex system at each step. While tractable up to a few hundred dimensions, it may become prohibitive in, say, $\\ge 10^4$-dimensional settings without further approximations.\"**\n\nWhile it is true that the solution of the convex system requires inverting a matrix, Lemma 4 shows that this matrix is merely a scaled identity, i.e., $\\rho I$, where $\\rho$ is a positive scalar. Thus, the inverse can be computed analytically. That is, in line 7 of Algorithm 1 the solution of the mentioned convex system is already encoded by the factor $1/\\rho$ which allows our algorithm to scale well in high-dimensional settings. We will include further explanations on Algorithm 1 in section 4.2 in a potential camera-ready version.\n\n**\"Robustness to Noise and Model Mis-specification: There is no evaluation of how DCAReasoner handles noisy or approximate y (e.g.\\ measurement error in the input matrix) or errors in the DC decomposition. Such perturbations could cause the DCA iterates to converge to spurious local minima.\"**\n\nWe thank the reviewer for raising this important point. We evaluated the trained models on the QR benchmark dataset adding different Gaussian noise levels to the input matrix before processing with a scale varying in $\\\\{10^{-4},10^{-3},10^{-2},10^{-1},1,10\\\\}$. Due to restrictions in the NeurIPS review process we can merely present the outcome of our experiments in a markdown table, but we would include proper visualizations in the appendix of a potential camera-ready version. The table below shows the MSEs for different noise levels. It appears that IREM becomes unstable for larger noise levels, while IRED and DCAReasoner are mostly robust to noise.\n\n|             | $10^{-4}$   | $10^{-3}$   | $10^{-2}$   | $10^{-1}$   | $1$    | $10$   |\n| ----------- | ----------- | ----------- | ----------- | ----------- | ------ | ------ |\n| DCAReasoner | 0.1439      | 0.1436      | 0.1439      | 0.1453      | 0.1980 | 0.7833 |\n| IRED        | 0.2176      | 0.2173      | 0.2174      | 0.2181      | 0.2646 | 0.9196 |\n| IREM        | 0.152       | 0.1521      | 0.1522      | 0.1533      | 0.444  | 15.9426|\n\n**\"The acronym “DCA” appears before it is explicitly defined. Could you please clarify and provide its full form at first use?\"**\n\nWe thank the reviewer for pointing this out. We will clarify and provide the full meaning of DCA, Difference-of-Convex function algorithm **[3]**, at it's first appearance in a potential camera ready version.\n\n**\"What criteria guided your choice of the number of DCA iterations K in each experiment? How sensitive are your results—both in terms of accuracy and convergence speed—to different values of K and to the initialization x_0?\"**\n\nIn all our experiments the DCA subroutine converged to machine precision in less than or around 10 iterations. Thus, setting the maximum number of DCA iterations to 30 is merely a safety net if for example numerical issues lead to convergence issues in one of the gradient steps and hence would end up in an infinite loop if there is no strict upper bound. Nevertheless, we did not experience such issues in any of our experiments. Thus, our results are independent of the specific setting of the maximal number of iterations. Of course if one would choose an extremely low number, e.g. 2, it would not allow the DCA routine to converge to something meaningful but this does not represent an actual limitation in our opinion. Initializations of $x_0$ are done uniformly random at training, as well as test time, as our DCA routine converges independently of the actual starting point. There might be an issue, though,  if the targets are very far from the bounds of the uniform initialization, e.g., if the target is 10 and one initializes randomly between -1 and 1. In such a case the DCA routine might need many iterations to reach the desired target. However, in a well-posed machine learning problem with scaled inputs and outputs such a situation can be ruled out.\n\n**\"The paper does not fully specify the hyperparameters and configurations used for the baseline methods. Could you include detailed settings for all baselines to ensure reproducibility and a fair comparison?\"**\n\nAll hyperparameters including the learning rate, the batch size, the number of gradient steps, and lower and upper bounds for random initialization are given in Appendix B.2. The size of the networks for our baselines is scaled to make sure that each of the networks has roughly the same number of parameters as the DCAReasoner with a size determined by $N_x=8$ and $N_y=4000$. The architecture follows exactly those in **[1]** and **[2]**. However, we agree with the reviewer that having all hyperparameters only in text might not give enough visibility. Hence, we will further include a table with all the necessary information in a potential camera-ready submission.\n\n**References**\n\n**[1]** Y. Du, S. Li, J. Tenenbaum, and I. Mordatch. Learning iterative reasoning through energy minimization. International Conference on Machine Learning (ICML), pages 5570–5582, 2022\n\n**[2]** Y. Du, J. Mao, and J. Tenenbaum. Learning iterative reasoning through energy diffusion. International Conference on Machine Learning (ICML), pages 11764–11776, 2024.\n\n**[3]** L. T. H. An and P. D. Tao. The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Annals of Operations Research, 133:23–46, 2005."}, {"review_id": "71Wbyn8k3x", "reviewer": "Reviewer_P8YW", "summary": "The paper presents a novel approach for iterative reasoning that formulates the problem of reasoning as energy minimization, which, with appropriate neural network parameterization, is transformed into a difference of convex functions. The authors provide theoretical convergence guarantees as well as expressivity results for the parameterization used. Finally, the DCA routine is scaled to perform batched optimization. The experiments are performed on matrix-based tasks (completion, inversion, decomposition, etc) and energy landscape optimization in the embedding space of language models to show the efficacy of the proposed method compared to previous approaches for energy-based iterative reasoning.", "questions": "1. What is the motivation behind selecting the task in section 5.3? The paper can be made significantly more impactful if a more realistic task, ideally in discrete space (for example, Sudoku or Zebra Puzzles), is chosen to demonstrate the effectiveness of the proposed method. In that setting, it would be nice to compare the performance of DCAReasoner with iterative feed-forward approaches like discrete diffusion and search-based auto-regressive methods (for example, [1,2])\n\n2. As I understand from the description, the DistilBERT was fine-tuned on the symptom-to-diagnosis dataset (please correct me if I'm misinterpreting here). If so, what was the accuracy of the DistilBERT classifier?\n\n# References\n[1] [[2409.10502] Causal Language Modeling Can Elicit Search and Reasoning Capabilities on Logic Puzzles](https://arxiv.org/abs/2409.10502)\n\n[2] [[2502.06768] Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions](https://arxiv.org/abs/2502.06768)", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Quality**: The authors provide theoretical convergence guarantees for DCAReasoner as well as expressivity results for the parameterization used. The experiments demonstrate the improvement in efficiency compared to previous energy-based reasoning approaches. However, the choice of tasks used in the experiments and the lack of comparison with other kinds of baselines (see question 1 below) fall short in providing a comprehensive picture.\n\n**Significance**: While the general idea of energy-based iterative reasoning is quite promising. The choice of experiments and baselines could be expanded to make the paper more impactful.\n\n**Clarity**: The paper is well written and easy to follow. \n\n**Originality**: The use of DC optimization for energy-based iterative reasoning and its proof of convergence both incorporate original ideas.", "was_revised": true, "final_justification": "Given how toyish the other tasks in the experiments are, my concern regarding the paper's significance still remains. However, given the paper's other strengths and the preliminary results on the Sudoku task, I'm updating my score.", "rebuttal": "In the following we comment on the points raised by reviewer P8YW.\n\n**\"What is the motivation behind selecting the task in section 5.3? The paper can be made significantly more impactful if a more realistic task, ideally in discrete space (for example, Sudoku or Zebra Puzzles), is chosen to demonstrate the effectiveness of the proposed method. In that setting, it would be nice to compare the performance of DCAReasoner with iterative feed-forward approaches like discrete diffusion and search-based auto-regressive methods (for example, [1,2])\"**\n\nSection 5.3 shows that our method is able to learn reasonable energy landscapes in token embedding space. That is, our energy landscape is able to capture fine-grained textual information when added on top of a transfomer based architecture. We think that DCAReasoner might be used to improve reasoning skills of LLMs by adding it as a corrective component (in a continuous sense) between the last hidden state and the LLM head. Of course the details of such an energy guided text generation approach are out of scope for our current work. Thus, section 5.3 should merely motivate such an approach and show that our baselines would not be suitable due to much higher inference times.\n\nAs mentioned in our manuscript, DCAReasoner is designed for continuous reasoning tasks. Thus, our method cannot be applied as-is to experiments in discrete spaces. This is also why the task in section 5.3 was formulated as a continuous reasoning task instead of a classification task. Nevertheless, we performed preliminary experiments using the Sudoku dataset in **[1]** suggested by the reviewer. For reasoning in a discrete setting our energy landscape is defined in the logit space, i.e., a 729 dimensional setting and we replace the MSE with the cross entropy loss. Processing the inputs, i.e., sudoku puzzles, can be done in multiple ways. We decided for a convolutional neural network with the residual connection design as in **[2]** to process the Sudoku puzzles before feeding them to our neural network components. Using the same settings as in the main paper ($N_x=8$, $N_y=4000$, a learning rate of $10^{-4}$) but a batch size of 64, we observe a cell accuracy of 65.09% after training for 6 epochs, i.e., 65.09% of all unfilled cells in all of the 100K test puzzles are filled correctly. For context, the baseline from **[1]** with random order of input cells achieves 52%, while the one with fixed order achieves 58.64%. Our results only fall short from the accuracy achieved by the model receiving additional context information about solving strategies during training (94.23%). However, we point out that the specific architecture of our model does not allow such information to be provided.\n\nWe will include a section into the appendix on how DCAReasoner might be used in a discrete setting for future research and report our early results. Nevertheless, we want to point out again that DCAReasoner as presented in our manuscript is designed for continuous reasoning tasks.\n\n**\"As I understand from the description, the DistilBERT was fine-tuned on the symptom-to-diagnosis dataset (please correct me if I'm misinterpreting here). If so, what was the accuracy of the DistilBERT classifier?\"**\n\nYes, this is correct. This step should ensure that the CLS token embeddings that are used within our energy-based reasoning approaches are encoding the necessary information for the classification task in this dataset. The accuracy of the DistilBERT classifier is mentioned in Appendix B.4 and is 96% as for the energy-based baselines. We note again that the aim of this experiment was not to train a better classifier but to show that DCAReasoner is able to learn reasonable energy landscapes in token embedding space while improving inference time. In a potential camera-ready version we will include an additional clarifying comment on the motivation for this experiment.\n\n**References**\n\n**[1]** Shah, K., Dikkala, N., Wang, X., & Panigrahy, R. (2024). Causal language modeling can elicit search and reasoning capabilities on logic puzzles. Advances in Neural Information Processing Systems (NeurIPS), 37, 56674-56702.\n\n**[2]** Y. Du, J. Mao, and J. Tenenbaum. Learning iterative reasoning through energy diffusion. International Conference on Machine Learning (ICML), pages 11764–11776, 2024."}, {"review_id": "sHMJV4BXHW", "reviewer": "Reviewer_A3zE", "summary": "This paper proposes an approach to energy-based reasoning where a difference of convex functions is used to represent the energy function. The paper provides theoretical results illustrating how this approach can approximate arbitrarily complex functions, while also ensuring that optimization converges to valid minima. Finally, the paper illustrates the applicability of the method on several numerical tasks, with an interesting energy landscape recovered in token reasoning. Overall, this paper has a nice blend of both emperical practice as well as theoretical analysis", "questions": "- Would the method presented in the paper also work with non single layer MLP networks?\n- How robust are the energy minima recovered from this approach? Do they have less local\nminima than prior work on energy based reasoning?\n- Could the authors provide a bit more clarification on why the subgradient in Equation 8 of the paper\nis the subgradient of Equation 4? I derived a slightly different form when going through the computation.", "limitations": "The paper adequetly discusses the limitations of the approach.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- The paper is clear and well motivated\n- The paper provides theoretical justification on their approach, with the approach\nhaving a provable minimum that it can converge to.\n- The paper illustrates a set of promising results on real-world settings\n\nWeaknesses \n\n- Most of the setting shown in the paper are a little toy, but I don't think this really \ndetracts from the paper, as I think it sufficient to illustrate the promise of the method", "was_revised": true, "final_justification": "I remain positive about this paper and maintain my positive score.", "rebuttal": "We want to thank the reviewer for the positive feedback on our work. In the following we comment on the points raised by reviewer A3zE.\n\n**\"Would the method presented in the paper also work with non single layer MLP networks?\"**\n\nYes, all the networks $\\alpha$, $W$, $b$, $\\xi$, and $\\rho$ can be set to arbitrary differentiable architectures. As such, it would also be possible to train on different modalities of $y$ such as text, images, or graphs and perform continuous reasoning in such settings. See also our preliminary experiments for reviewer P8YW, in which we make use of convolutional neural networks to process Sudoku puzzles as inputs.\n\n**\"How robust are the energy minima recovered from this approach? Do they have less local minima than prior work on energy based reasoning?\"**\n\nWe thank the reviewer for this important question. As both, our baselines and DCAReasoner, are able to represent highly non-convex energy functions it is not possible to give any general statements on the number of local minima. Also projecting the energy landscapes of our trained baselines and DCAReasoner to lower dimensional spaces to compare them empirically can lead to wrong conclusions due to a loss of information in the projection. Nevertheless, we think that the convergence guarantees to local minima in our approach alone makes DCAReasoner more robust than our baselines. We also performed additional experiments for reviewer GYtm showing that our method is robust to noisy inputs and outperforms our baselines even in the presence of large noise levels. Please see our answer to reviewer GYtm for more details on the experiments.\n\n**\"Could the authors provide a bit more clarification on why the subgradient in Equation 8 of the paper is the subgradient of Equation 4? I derived a slightly different form when going through the computation.\"**\n\nThe full subgradient of $h$ is given in equation (32) in our manuscript. Equation (8) is merely a specific element of the subgradient that results from setting all $\\epsilon_i^h=1$ in equation (32). We give more details on this specific choice later on.\n\nTo derive equation (32) we make use of the following properties of the subgradient:\n* For $\\lambda \\geq 0$, we have $\\partial (\\lambda f)(x) = \\\\{\\lambda u: u \\in \\partial f(x)\\\\}$\n* Sum rule (Proposition 5.4.6 in **[1]**): If $f_1,f_2:\\mathbb{R}^n\\to (-\\infty, \\infty]$ are convex functions satisfying $\\text{ri}(\\text{dom}(f_1)) \\cap \\text{ri}(\\text{dom}(f_2)) \\neq \\emptyset$, $\\text{ri}$ denoting the relative interior, and $f_1+f_2$ is proper, then for all $x\\in\\mathbb{R}^n$ we have $\\partial(f_1+f_2)(x)=\\\\{u+v: u\\in\\partial f_1(x), v\\in\\partial f_2(x)\\\\}$.\n* Let $g(x) = f(Ax+b)$ with $f:\\mathbb{R}\\to (-\\infty, \\infty]$ then $\\partial g(x)=A^{\\ast}\\partial f(Ax+b)$ if the range of $A$ contains a point of the relative interior of $\\text{dom}(f)$, see Theorem 23.9 in **[2]**.\n\nEquation (32) then follows directly from the fact that the subgradient of the ReLU activation function is given by $\\partial(\\text{ReLU})(x)=\\\\{1\\\\}$ for $x>0$, $\\partial(\\text{ReLU})(0)=\\[0,1\\]$, and $\\partial(\\text{ReLU})(x)=\\\\{0\\\\}$ for $x<0$.\n\nTo ensure finite convergence of our DCA routine, Theorem 9 in **[3]** makes use of a so called fixed choice of subgradients which is defined in section 2.4.2 in **[3]**. This fixed choice is formalized via a mapping from the subgradient of $h$ in a specific point (set-valued) to a specific element therein. In our case, this mapping is given by always choosing the element in which all $\\epsilon_i^h=1$ resulting in equation (8), thus avoiding to check exactly which of the affine terms evaluate to zero (and would allow for $\\epsilon_i^h\\in[0,1]$) and having a rather simple form using the properties of the heaviside function, i.e., $H(z)=1$ for $z\\geq 0$ and $H(z)=0$ else.\n\n**References**\n\n**[1]** Bertsekas, Dimitri. Convex optimization theory. Vol. 1. Athena Scientific, 2009.\n\n**[2]** Rockafellar, Ralph T. Convex Analysis. Princeton Mathematical Series Princeton University Press, Princeton, N. J., (1970)\n\n**[3]** Thi Hoai An, Le, and Pham Dinh Tao. \"Solving a class of linearly constrained indefinite quadratic problems by DC algorithms.\" Journal of global optimization 11.3 (1997): 253-285"}]

Author rebuttal:

Meta-review: This paper introduces DCAReasoner, an energy-based iterative reasoning algorithm based on difference-of-convex functions. It introduces a tailored form of energy functions, derived theoretical convergence guarantees, and show that the DCA routine converges in finitely many steps. For larger-sized problems, it provides theoretical approximation guarantees under additional assumptions. Experiments on 5 datasets on the prior benchmark show that the method achieves competitive performance with 1 order-of-magnitude smaller inference time.

The reviewers universally recognize that the method is novel, provides theoretical guarantees, has significant inference-time speedups, and is well-written, which I concur. On the flip side, reviewers raised concerns about the additional assumptions and that the experiments are relatively small-scale. Reviewer GYtm also raised questions about the method's performance under noise and model mis-specification During the rebuttal, the authors added experiments demonstrating that the proposed method is robust to noise compared to baseline methods. The authors also performed preliminary experiments using the Sudoku dataset. Overall the rebuttal addresses most of the reviewers' concerns. During the camera-ready, the authors are encouraged to incorporate the added experiments and discussion into the paper. Also, additional, larger-scale experiments are recommended to demonstrate the scalability of the method.
Decision: Accept (poster)