From: ICLR.cc/2023/Conference/Program_Chairs
title: Paper Decision
decision: Accept: poster
metareview:_summary,_strengths_and_weaknesses: The paper presents a novel theoretical framework to further our understanding of predictive coding networks (PCN). The authors derive analytical results establishing connections between predictive coding, targetprop, and backprop in linear networks. They further provide experimental validation suggesting that the theory applies to nonlinear networks. Three of the four reviewers reported that the work was sound while a fourth reviewer noted some errors in the proofs (which the authors appeared to have addressed during the rebuttal but this was not confirmed by the reviewer who did not update their score/review). 
justification_for_why_not_higher_score: While the work appears to be solid, the proofs are only for linear networks and only empirical evidence is provided for nonlinear networks. 
justification_for_why_not_lower_score: NA
note_from_PC: if the above contains the word "oral" or "spotlight" please see: "oral" presentation means -> notable-top-5% and "spotlight" means -> notable-top-25%. As stated in our emails, we are disassociating presentation type from AC recommendations

From: ICLR.cc/2023/Conference/Paper5300/Authors
title: Response to reviewer c2Qw
comment: We thank the reviewer for their detailed review. With regards to the self-reference, we did not realize it may be excessive and this was not intentional on our part. We believe that we have fairly cited all relevant literature and all citations are relevant to the work.

As to the weaknesses of the paper, you highlight:

1.) While many of our arguments are for linear networks, this is because these are the only networks that can be solved analytically. We argue that the primary intuitions behind these results transfer to nonlinear models of interest and show empirical results to this effect. Also, some of our main results relating to the relationship between prospective configuration and target propagation, as well as the convergence of PC and BP apply also to nonlinear networks.

2.) We are not aware of the links the reviewer alludes to with deep gaussian networks. We would appreciate some citations to relevant work the reviewer has in mind so we can try to understand the potential connections and add it in to the related work

From: ICLR.cc/2023/Conference/Paper5300/Authors
title: Response to reviewer 3s9Y
comment: We thank the reviewer for their detailed and critical review. Overall we disagree with the assertion that our claims are dubious or trivial; we believe that this paper offers a state of the art and correct analysis of the learning and inference dynamics of PCNs including providing a full theoretical understanding of the limits in which PCNs converge to backprop, as well as a convergence proof of the prospective configuration algorithm, all of which are novel.

The reviewer, however, has highlighted some sloppiness or mistakes in notation, and we are thankful for that. These have been fixed in the updated version. Specifically:

In equation 3, we did indeed make a typo and the \epsilon_l should be \epsilon^*_l. This does not affect the validity of the proof since the whole point of the proof is taking the limit where \epsilon_f tends to 0 and the backprop recursion is identical whether $\epsilon$ or $\epsilon^*$ is used since for the output prediction error they are the same: $\epsilon = \epsilon^*$ and thus the rest of the recursion is identical. We have fixed this notational typo in the updated version.

Secondly, the reviewer is correct that we the order of the weights and error terms in Equation 2 should be transposed and we have fixed this in the updated version.

With regards to theorem 3.5, we do not assume that the loss function L depends only on x_{L-1} and hence the partial derivative is not trivial. In a feedforward architecture, L depends directly only on x_{L-1} and the penultimate layer weights, but x_{L-1} also depends on the previous activations in the network, and hence the partial derivatives are not 0. 

As for the reviewer’s comment:

> Calling the xl ‘inference’ via a local mode, not the expectation, the E-step Eq (14), and then invoking theorems about local convergence of expectation maximization seems confusing.

We do not quite understand the point the reviewer is trying to make here and would appreciate this being expanded upon. A key and explicit assumption that we make in predictive coding is that of a delta function approximation (alternatively Laplace approximation but it works out the same) around the variational posterior which is parametrized by x. With this assumption, the mode and expectation of the posterior are identical. This should not affect the probabilistic interpretation of the model as performing variational inference nor the results with expectation maximization.

Overall, given that the reviewer has, in our eyes, only brought up relatively minor typos and misunderstandings, it does not support their claim that the results are dubious or trivial nor their low score. We would be especially interested in hearing in what ways the reviewer thinks our results, especially our convergence results and relationship to target-propagation are not novel, since we have not seen them before in the literature.

From: ICLR.cc/2023/Conference/Paper5300/Authors
title: Response to reviewer G2QQ
comment: We thank the reviewer for their enthusiastic and comprehensive review. About the precision ratio in Figure 3.C – this is a good spot and we have added a clarification of this to the figure caption.

From: ICLR.cc/2023/Conference/Paper5300/Authors
title: Response to reviewer 3qX4
comment: We thank the reviewer for their detailed and positive review.

From: ICLR.cc/2023/Conference/Paper5300/Reviewer_3qX4
confidence: 2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: The paper presents a novel theoretical framework for understanding the inference and the learning in predictive coding networks (PCN). The math derivations and proof are sound. 
strength_and_weaknesses: This is a paper full of math proof and derivations, and is theoretically sound. It is quite interesting in interpreting the PNC dynamics as performing a constrained EM algorithm, and comparing the PCN with other algorithms including target propagation and back propagation. The theoretical results seem novel based on the author's description but I am not quite sure since I am not an expert on predictive coding networks.
clarity,_quality,_novelty_and_reproducibility: - Quality and clarity
The math proof and derivations are sound and are presented clearly.

- Originality
The theoretical results seem novel but I am not fully sure since I am not an expert on PCN.
summary_of_the_review: I like the deep math analysis of the inference and learning dynamics of PCN and it greatly gain our understanding of this network. 
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 8: accept, good paper

From: ICLR.cc/2023/Conference/Paper5300/Reviewer_G2QQ
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: While backpropagation is the mainstay technique used for artificial neural networks, it is considered biologically implausible, mainly due to the so-called "weight-transport" problem, wherein neurons need to know the weights of neurons that lie upstream in order to correctly update their own weights. Two frameworks that have been proposed to be more biologically realizable are predictive coding and target prop.  While both of these methods have been found to be related to backpropagation under certain circumstances, a fuller accounting of their interconnections has remained elusive. Through both mathematical analysis of linear networks and simulations of nonlinear ones, the present work demonstrates how backprop and targetprop approximately represent two endpoints of a continuum of solutions that predictive coding network inferences lie on (i.e., the equilibrium prediction errors lie close to the gradients of backprop on one end, while the equilibrium activities lie close to targetprop target activities on the other). Moreover, the authors show that predictive coding learning results in convergence of the network to the same minima as for backpropagation.
strength_and_weaknesses: Strengths

1. The authors provide a full mathematical characterization of linear predictive networks, demonstrating analytically their relationships to backprop and targetprop networks

2. Furthermore, simulated nonlinear networks are analyzed and shown to produce the same qualitative results as found for the linear networks

3. The authors' writing is clear and well-organized

Weaknesses

Major

No major weaknesses have been 
observed.

Minor

The precision ratio is introduced after Theorem 3.4 as $\Pi_{l+1} / \Pi_l$; however, I believe the precision ratio comprising the x-axis values in Fig. 3C is instead $\Pi_{l} / \Pi_{l+1}$.  Either changing the figure to have the same ratio that is initially introduced or specifying the ratio that is used in the figure would help to avoid readers' confusion.
clarity,_quality,_novelty_and_reproducibility: The present study builds on extant work that connects predictive coding and backpropagation, further elucidating the relationship of predictive coding to backprop and target prop away from prior limits studied.  The authors describe their analysis clearly and thoroughly, including demonstrating how results with full nonlinear networks are consistent with their linear network findings.  The proofs in the body of the paper are further supplemented with further descriptions and proofs in the appendices, while the numerical experiments are detailed in the appendix and will be included in a code release, allowing others to reproduce the results in a manner that is aided by, but not dependent on, the provided code.
summary_of_the_review: The authors have analytically demonstrated the connections among predictive coding, targetprop, and backprop in linear networks. They back up the analysis with numerical experiments on nonlinear networks that demonstrate the same qualitative results.  This work represents an important and novel advance in understanding predictive coding networks, which currently represent one main direction within neuroscience for understanding cortical functionality.
correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 4: The contributions are significant, and do not exist in prior works.
empirical_novelty_and_significance: 4: The contributions are significant, and do not exist in prior works.
flag_for_ethics_review: NO.
recommendation: 10: strong accept, should be highlighted at the conference

From: ICLR.cc/2023/Conference/Paper5300/Reviewer_3s9Y
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: The paper attempts to provide a theoretical analysis computation in predictive coding networks (PCNs), relating it to the more commonly used backpropagation-based (BP) training of feedforward artificial neural nets (ANNs).
strength_and_weaknesses: Strengths: 
Understanding the relation between PCN and usual ANN with BP is definitely a worthy goal.
The observation that PCN ‘inference’ is a combination of a forward and a backward signal and that the balance between the two is decided by precision ratios. This insight may not be completely new, but is brought out well in the discussion.
Weaknesses:
The proofs of some theorems seem to be spoiled by what looks like sloppy or confusing arguments. Take for example theorem 3.1. In the proof, equation 3 ignores the difference between $\epsilon^*_{l+1}$, the equilibrium value, and $\epsilon_{l+1}$, the backward propagated error, to make the argument.
In some cases, it is not clear whether some conditions are non-trivial or not, if taken literally. For example, for theorem 3.5, if one thinks $\mathcal{L}$ depends only on $x_{L-1}$ and the target, the partial derivative $\partial \mathcal{L}/\partial x_{l}$ would naively be zero for $l<L-1$, making the inequality trivially true.
Calling the $x_l$ ‘inference’ via a local mode, not the expectation, the E-step Eq (14), and then invoking theorems about local convergence of expectation maximization seems confusing.

Minor issues:
In equation (2), it seems $\epsilon_l\cdot f’(..)$ is column vector of the same size as $x_l$. Given that, in equation (1), should the order of $W^T_{l+1}$ and $\epsilon_{l+1}\cdot f’(...)$ not be reversed?

clarity,_quality,_novelty_and_reproducibility: While this is a bold attempt at some theory, several claims seem dubious and others are often trivial.
summary_of_the_review: I wish this paper was what it promises to be. I believe far more careful work is necessary to substantiate the claims made here.
correctness: 2: Several of the paper’s claims are incorrect or not well-supported.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: Not applicable
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper5300/Reviewer_c2Qw
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: I would like to begin by saying that this paper makes repeated and relentless reference to a given set of papers that induced this reviewer to suspect that it was self reference, and it took barely a minute of google search to verify this fact. I find this very disturbing, since the other papers I have reviewed have by and large played by the rules of double blind.

That said, this is a very interesting paper. The authors consider a feedforward ANN, and analyze an algorithm called prospective configuration that falls within the perview of predictive coding. Basically, given an data point and its label, the input and output of the network are clamped to their respective input and output values, and each intermediate layer searches thru gradient descent for input values that would minimize the squared difference between its output and the current output. This allows the error at the top layer to propagate downwards. Once equilibiirm has been reached, the input outputs of each layer is clamped and a gradient descent on the weight vectos is initiated, again until convergence. This process is repeated until convergence.

What the paper shows is that this this algorithm converges to the same local optima that back propagation converges to. It also show that the algorithm builds a bridge between BP and target propagation (an alternative learning algorithm for deep networks).
It builds its results on the intuitive idea that the problem can be rephrased as a missing data problem, that is the activation of the intermediate layers) and can therefore be attacked using EM.
strength_and_weaknesses: Strength: The analysis is quite comprehensive and connections to BP and TP are shown
Weekness: Several of the results are for linear networks where the problem becomes trivial. Also, a lot of the results are similar to deep gaussian networks that came out many years back.
clarity,_quality,_novelty_and_reproducibility: The paper is very clearly written, and the results are novel.
summary_of_the_review: This is a well written paper that provides theoretical justification for yet another technique (prospective configuration) for training deep networks.
correctness: 4: All of the claims and statements are well-supported and correct.
technical_novelty_and_significance: 4: The contributions are significant, and do not exist in prior works.
empirical_novelty_and_significance: Not applicable
flag_for_ethics_review: Yes, Other reasons (please specify below)
details_of_ethics_concerns: It would have not hurt the paper if they did not make so many self references. As it stands, this paper stretches what is permissible in a double blind review.
recommendation: 8: accept, good paper