Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The present paper studies the inference of a hidden Markov model, specifically a Kalman filter-like problem. Compared with a common Kalman filter and canonical predictive coding framework, the paper considers that the feedback input (prediction) is associated with a computational cost, and how the feedback cost affects the gain of observation and predictions.
strengths_and_weaknesses: The formulation of the problem by considering a cost associated with feedback is novel, and the whole paper is structure-wise, well-written. Nevertheless, considering the neural implementation of the Kalman filter has been widely studied before, the formulation of the paper seems a bit technically simplified, and it is not clear whether the novelty of significance of the present paper is enough even though the consideration of feedback cost is new. Also, the paper directly deals with the Gaussian variables (Eq. 1-7) while we know the neural variability is Poisson-like (multiplicative). It is not clear when considering the multiplicative noise whether the main conclusions of the paper can be held. Moreover, the previous studies of neural implementation of Kalman filters typically embed the Gaussian dynamics (Eqs. 1-7) into nonlinear, biologically plausible network dynamics (e.g., Deneve, J. Neurosci., 2007; Wilson and Finkel, NeurIPS 2009; Kutschireiter 2015), hence it might be more interesting to study how the feedback cost influences the network dynamics.
questions: Please see my other comments.
limitations: A strong requirement of a network implementation of the Kalman filter is that a network model with fixed parameters is able to infer the latent variables with different uncertainties, e.g., different observation noise or feedback noise. Not sure whether the current model could achieve this or not. It might be good to discuss this point.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper presents a theory of optimal inference in presence of noisy
sensation, prediction and feedback, under linear-quadratic-gaussian
assumptions, with separate costs for feedforward and feedback
signals. The theory is used to show how the usefulness of feedback
undergoes phase transitions depending on the structure of the
environment, architectural constraints, and computational costs.

strengths_and_weaknesses: ### Strengths
This paper tackles an important problem in theoretical
neuroscience, hitting a sweet spot between level of detail and
tractability of the system under exam. The method and the results
provide meaningful progress towards understanding efficient coding in
dynamical contexts and its relationship with predictive coding, a
research area that has seen intense effort and several high-quality
contributions in the last few years (such as those by Chalk & Tkacik
and Mlynarski & Hermundstad, which the paper prominently acknowledges).

### Weaknesses
In the central part of the paper (sections 5.1, 5.2), some
passages could be made clearer/easier to follow. Please see
"questions" for details.

questions: Here are some passages that I found harder to read than necessary:
- Line 182: I was confused by the sentence "Furthermore, we also see
  that when sending feedback is optimal, it must be worth sending some
  feedforward messages even if feedback is cut off". In that context,
  it is not clear whether the reader is supposed to "see" that fact
  from some of the equations above, or this is supposed to be
  self-evident based on the consideration made in the rest of the
  sentence. Moreover, I was surprised by the use of "cut off" -
  somehow it made me doubt that I had missed a component of the model
  that accounted for specific causal interventions on the system.
- Lines 206--209, series of sentences starting with "At the
  origin...": this same idea was much clearer when repeated below,
  around lines 235--238 (even if for a slightly different case). In my
  opinion, adopting the phrasing of 235 the first time that the
  concept is presented would make for smoother reading.
- Unless I'm missing something, lines 262--265 seem redundant with
  266--270. The two passages follow immediately one another, but they
  are in different paragraphs. Please consider merging these passages
  if they are indeed redundant, or signposting better their difference
  if they are not.

limitations: In my opinion, the paper gives a fair account of its limitations.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper formulates an interesting tradeoff in optimal coding theory, in the context of (potential) predictive coding where there are feedforward messages, and top-down predictions can be sent as feedback to reduce feedforward signal cost. Previous works considered the cost of feedforward information but ignored the cost of feedback computation; this paper considers the noise cost of the feedback pathway, as well as the feedforward, and investigates the tradeoff as the noise costs in the feedforward/feedback channels vary. Interestingly, the paper shows that there is a nontrivial transition between three regimes: where predictive coding is preferred (top-down prediction feedback is useful), where efficient coding is preferred (feedback is not worth the cost, so just send feedforward signals), and where sending any messages is not worth the cost at all. The paper also provides biological evidences to argue that it is indeed plausible to have different noise costs in the feedforward/feedback pathways.

strengths_and_weaknesses: This is an important and interesting paper that studies a fundamental tradeoff in the theory of neural computations. I believe that this paper provides a conceptual advance and will be a significant contribution to the community's understanding of optimal coding. The neurobiological discussion at the end is valuable as well. The paper is original in their consideration of the feedback noise cost, and the discovery of the nontrivial structure of the tradeoff.

I find the mathematical formulation very reasonable and the logical developments convincing, although I have not checked all steps carefully.

As for the weaknesses of the study, the theory is formulated in a simplified model setting, and it is still to be tested how relevant the lessons are in the actual neural systems. Also, this paper provides no direct connection to real neural data. But this is how theories should start, and the authors also discuss the limitations in the paper, with plans for future work.

Presentation-wise, the paper is nicely contextualized with respect to the existing ideas and models. Regarding writing/visualization, I eventually felt that the paper is clear enough, in that it describes all important points, and touches on most of my questions that came up over the course of reading. That said, in the initial read I had to go back and forth between the figures and the text paragraphs more than I would usually do. I will provide more comments below.

questions: i) The theory is developed in the simple LQG model setting. Can you say whether the results in this limit corresponds to any bounds to the non-linear, non-gaussian cases?

ii) One of the most interesting points in this paper is that the non-monotonic shape of the boundary curve $\Phi$ as a function of the feedforward noise cost (Eq 14, Fig 3). And as soon as I saw it, I was eager to understand why it was so. When $ \Phi $ first appears in the paper, the authors only say that it is a "complicated function", but it would be nice to provide some more insights on why it increases at first and decreases later. Looking at Appendix A7 (as referred to from Eq 14) did not help much. I eventually found relevant text later in the paper, in section 5.2 (paragraphs starting L232 and L242). I suggest that this material be moved up, so that the reader gets a clearer picture early on. It was nice that the authors explain it with a fixed, low-to-moderate feedback noise cost, and talk about small / intermediate / large feedforward noise regimes. It would help to add a horizontal line to Fig 3A and describe it right there.

iii) Minor comments:

- I felt that the text was rather dense in my initial read, and the main messages were not easily located by quick skimming. Hard to pinpoint a specific suggestion, but in general, having the main point at the beginning of each paragraph is really helpful.
- In the figures, red-green-yellow will make it difficult for color-blind readers. Suggest replacing with other color palettes.
- The 3-d plot in Fig 4 left was difficult to understand at first, because the contour lines wasn't visible in the normal pdf view. (I realized after zooming in that there were contour lines.)

limitations: The authors have addressed the limitations of their modeling assumptions and constraints.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper inspects the consequences of considering the costs associated with sending messages to represent univariate variables in the brain. The authors propose a dynamical system taking into account the inherent noise while updating the estimation of the latent world state. They subsequently derive a theory from the minimization (supposed to take place in the brain) of a cost function taking into account the accuracy of the latent state estimation as well as the feedforward and feedback costs. The main part of the paper presents a mathematical analysis of the system, that is, shows phase transitions between predictive coding (feedforward + feedback messages), efficient coding (feedforward messages only) and silence (no messages), depending on the values of parameters. Finally, the theoretical framework is put into an experimental perspective, through parallels with the properties of the biological brain and discussion about how to test the theoretical predictions.
strengths_and_weaknesses: Strengths:
1. Interesting topic, well in the scope of the NeurIPS conference
2. Very pedagogical figures and legends
3. The paper has a very clear and intuitive message
4. The paper is well-grounded mathematically
5. The link from the very theoretical framework to the biological brain is very interesting. More particularly, I enjoyed the parts about the excitatory/inhibitory processes in the brain, especially the explanation about the potential higher cost of long-range inhibitory feedback as a consequence of the anatomical constraints.

Weaknesses:
1. The authors put some effort into thinking of ways to test their theory, but the discussion remains a bit too conceptual and vague, especially the part about the comparison between brain systems or species. Add references and explicit further your predictions
2. The relation with previous theoretical work is only superficial (see section "Related work'). I would expect more developments in the discussion part, in light of the paper's results
questions: 1. Is it "just" a model of the brain, or could it be useful in another domain? I feel like this theory could be used in designing energy-efficient systems where the estimation takes place under constraints (for instance energetic). This could be acknowledged in the discussion
2. How could the abstract message-passing in the graph from Fig 1B relate to the messages sent from excitatory/inhibitory populations? 
3. Mathematical conventions: to relate more to predictive coding, shouldn't L be positive and D negative, instead of the reverse? Potentially also replace D with -D, in order to have a minus sign in Eq 5

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Comments: 
1. I think Fig 9 should be moved from the appendix to the main text
2. I didn't understand much Fig 4 with the legend. For instance, which color represents the absence of messages? None, right?
limitations: Limitations: The authors carefully address the limitations of the work, opening the way for generalizations of the theory (taking actions / multivariate variables / nonlinear tasks / learning).

Societal impact: The authors did not address any potential negative societal impact of their work, which is fine given the subject of the paper (understanding the brain)
review_title: null

Meta-review: metareview: Sound and clearly presented contribution to the field of predictive coding.
confidence: Certain
Decision: Accept