What should a neuron aim for? Designing local objective functions based on information theory

Authors: Andreas Christian Schneider, Valentin Neuhaus, David Alexander Ehrlich, Abdullah Makkeh, Alexander S Ecker, Viola Priesemann, Michael Wibral
Primary area: interpretability and explainable AI
Keywords: local learning, interpretability, neuro-inspired, information theory, partial information decomposition

Abstract:
In modern deep neural networks, the learning dynamics of individual neurons are often obscure, as the networks are trained via global optimization. Conversely, biological systems build on self-organized, local learning, achieving robustness and efficiency with limited global information. Here, we show how self-organization between individual artificial neurons can be achieved by designing abstract bio-inspired local learning goals. These goals are parameterized using a recent extension of information theory, Partial Information Decomposition (PID), which decomposes the information that a set of information sources holds about an outcome into unique, redundant and synergistic contributions. Our framework enables neurons to locally shape the integration of information from various input classes, i.e., feedforward, feedback, and lateral, by selecting which of the three inputs should contribute uniquely, redundantly or synergistically to the output. This selection is expressed as a weighted sum of PID terms, which, for a given problem, can be directly derived from intuitive reasoning or via numerical optimization, offering a window into understanding task-relevant local information processing. Achieving neuron-level interpretability while enabling strong performance using local learning, our work advances a principled information-theoretic foundation for local learning strategies.

Decision: accept (oral)
Reviewer scores: 8, 8, 8, 6

Reviews and discussion:
<thread>
creation date:
1730391579814

thread_type:
official_review

<thread_object>
creator: reviewer_wfku

<content>
summary:
This paper introduces trivariate infomorphic neuron model to develop an interpretable and biologically-inspired local objective function for training artificial neural networks. This abstract neuron model is based on Partial Information Decomposition (PID) framework, which decomposes information into unique, redundant, and synergistic components (atoms). Local objective (goal) function is constructed as a linear combination of these PID information atoms. Neural network models with one hidden layer and various configurations (e.g. sparse lateral connectivity, different context signal) are trained with the proposed loss function on MNIST and CIFAR10 classification tasks, demonstrating promising results.

strengths:
* The problem is well-stated in the introduction.

* Section 3 and Section 4 effectively introduce the PID framework and explain how it is used to develop local objective functions for training neural networks.

* The shared code is well-structured, which aids the reproducibility of the work.

* MNIST results seem promising.

weaknesses:
The paper has a few areas that could benefit from further clarification or expansion. Please see the following points and the Questions section for more details.

*  If I did not overlook it, the consistency equations for the trivariate infomorphic neuron model are not included in the paper.

* The discussion of weight updates is limited. It is mentioned that either autograd or the analytical formulation in (Makkeh et al. 2023) can be used. However, the analytical approach in (Makkeh et al. 2023) applies to  bivariate informorphic neuron model. Although the weight update derivation is potentially a similar approach, the absence of an explicit derivation raises a concern. Without this, I am uncertain if the resulting learning is biologically plausible.

* It is mentioned that (line 308) the $\gamma$ parameters can be derived from intuitive notions. However, unless I overlooked something, the explanation for determining these parameters is not provided in detail beyond a brief note in the caption of Figure 2.D.

* I am concerned that plug-in estimation of the information atoms based on empirical probability mass functions may face challenges in high-dimensional settings due to the difficulty of density estimation. While MNIST may be manageable despite being high-dimensional, this could be an issue in more complex settings.

* Lines 511-515 mention that deeper networks were trained and achieved comparable or better accuracy results; however, these results are not shown. The paper only includes experiments with a single hidden layer.

* The algorithm in Function 2 (Appendix A.1) requires quantities labeled 'isx_redundancies' and 'pid_atoms' (lines 732 and 733), but it is not specified how these quantities are computed.

* According to Appendix A.3, the proposed method requires a lot of computational power even for MNIST. This raises concerns about its scalability to more complex datasets.

* I think the abbreviation IM in Table 1 likely stands for "infomorphic networks," but this is not clarified in the paper.

**Minor Comments:**

* In line 907, the word 'function' is repeated.

* I think Figure 9 is neither referenced in the text nor is it fully interpreted beyond the caption. Given its complexity, more discussion could make it easier to understand.

questions:
* Is the activation function in line 283 commonly used? How the $\alpha_i$ and $\beta_i$ (for $i = 1, 2$) values are determined for this activation function?

* Regarding lines 293-297, which mention that lateral connections introduce recurrence, how did you determine that presenting the same input twice is sufficient? For more complex datasets, more iterations may be needed. I am curious if the output converges after two presentations.

* Is there any particular heuristic or rationale for using 20 equally sized bins in the algorithms in Appendix A.1?
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</thread>
<thread>
creation date:
1730505375349

thread_type:
official_review

<thread_object>
creator: reviewer_dmeh

<content>
summary:
The paper presents a local learning framework for neural networks by introducing a local training objective inspired by partial information decomposition (PID) in information theory.
It trains each neuron individually by a local goal function that is a linear combination of PID atoms, which decomposes the information that three input signals 
(feedforward, context, and lateral) provide about the neuron's output. The weights of the combination can be chosen heuristically or optimized on validation data. The weights represent 
how information in the inputs contribute to the neuron's output and are human-interpretable. Experiments on a single-hidden-layer network show that the proposed learning method 
can achieve similar performance to conventional training by cross-entropy loss, yet providing interpretable learning process of individual neurons.

strengths:
The idea of using information-theoretic loss functions to train individual neurons is interesting, with nice connection to biological neural networks and neuroscience. 

The novelty of adding a third input class representing lateral connections between neurons is intuitive and interesting and demonstrates better empirical performance than the bivariate model.

The overall method and motivation are clearly presented.

The experiments are well designed. Nice ablation study on the goal parameters. Good interpretation of "important" PID atoms identified by goal parameter optimization.

weaknesses:
Some parts of the methodology are not clear. Please see questions below.

The PID-based goal functions and infomorphic neurons were originally proposed by Makkeh et al. 2023, and the contribution of this work is to introduce lateral connections as a third input class.
However, in the Introduction, the authors claim the PID goal function to be one of the main contributions of this paper. The authors should explain more clearly the difference between this work and Makkeh et al. 2023, 
and define their contributions more accurately.

Experiments were performed on neural nets with single hidden layer, which limits the scope of the paper. It is unclear whether the observations and insights can generalize to deeper neural networks.

questions:
Questions related to numerical computation of PID atoms:
- To compute the PID atoms during training, the authors empirically evaluated the joint probability mass function of the aggregated inputs and output of each neuron. Since the input is high-dimensional, is a large batch size 
needed for such numerical estimation to be accurate and stable? For example, the input dimension is at least 28^2 = 784 for MNIST, and each dimension is discretized to 20 levels. Is the batch size of 1024 sufficient?

- The authors mentioned that the shared-exclusion redundancy and thus the PID atoms are differentiable with respect to the probability distribution. However, in my understanding, the empirical probability mass function 
of a discrete random variable is not differentiable w.r.t. its samples. Then the goal function will not be differentiable w.r.t. a neuron's weights, which is needed for training. Could the authors clarify this point?
In particular, how is the empirical probability mass function differentiated w.r.t. the output (and subsequently the weights) of the neuron?

There seems to be a gap between the claimed "neuron-level interpretability" and the proposed learning framework. 
Usually, interpretability means understanding how a trained network makes predictions and what the network has learned.
However, the method proposed in this paper only provides interpretation for the learning objective, rather than insight of what the model and each neuron has learned. 
Furthermore, the interpretability is still global, not neuron-level, as the goal parameters are shared across all neurons, i.e., the interpretation for all neurons are the same. 
Finally, the cross-entropy loss and the backpropagation process are quite human interpretable, in my opinion.
Could the authors comment on this?

"For the supervised classification task at hand, the function ... has been chosen ... This ensures that the network performs similarly during training, when context and lateral inputs are provided, and for evaluation, 
where the context signal is withheld."
It seems that the given activation function is for training, since it uses the context (label) as input. Can the authors define clearly what the activation function during testing is?

"One promising path towards constructing deeper networks is using stacked hidden layers that receive feedback from the next layer, similar to setup 3."
It is a bit unconvincing to replace label feedback by feedback connection from the next layer. In the latter, the neuron's learning goal is to capture the part of the feedforward signal that agrees with the next layer's output, which is less intuitive than 
capturing the part that agrees with the label.

Figure 8, comparision with bivariate model: Since the goal is to compare trivariate with bivariate, it might be better to put trivariate and bivariate on the same axis. If it'd be too crowded, 
maybe group the results by "Heuristic"/"Optimized".

In "Goal parameters" paragraph in Experiments, it might be better to state that "heuristic goal function" is $\Pi_{ \{ F\} \{C\}}$. Although it can be inferred from Fig. 4, it's better to define it in the text as well.

In Fig. 4, is the difference in validation accuracy defined as "after setting to 0 - before"?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730527543809

thread_type:
official_review

<thread_object>
creator: reviewer_mc5f

<content>
summary:
Based on the concept of partial information decomposition in the information theory, this work formulates a learning framework that allows the implementation of per-neuron goal functions and neuronwise interpretability. In an ANN setup for classification tasks, the goal function is optimized via parameters of the ‘informorphic’ neurons, with neuron-level interpretation, good performance (comparable to the same ANN trained with bp) and insights into the local computational goals. Examples with bivariate and trivariate informorphic neurons are demonstrated where the three-input classes unlock the potential of information theoretic learning, validating the abovementioned advantages in comparison with classical information composition.

strengths:
The use of partial information decomposition in a learning framework with bivariate and trivariate implementations enables interpretable information processing at the per-neuron level, which is also new. The discussion based on a comparison between heuristic and optimization approaches demonstrates the potential of the interpretability of local learning framework in a task-relevant context and without the loss of performance compared to ANN trained with bp.

weaknesses:
The title ‘What should a neuron aim for’ is broader than the scope explored in the current work, where a single-layer bivariate and trivariate local learning framework is studied. There is still a large gap between the interpretability of the model here and that of the neuron. The advantage of interpretability from an application perspective is not demonstrated or discussed, which is expected to be very limited by the single-layer simplicity here.

questions:
How do we understand the performance presented in Fig. 3B by considering that at Nhid = 100, there is a convergence for all results? Why a sparse lateral outperform a dense lateral under large Nhid conditions? What are the computational and memory costs?
What does the extraction of high-effect parameters mean, and would this be useful to construct networks with higher performance, or understand the nature of the problems under training?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730669767243

thread_type:
official_review

<thread_object>
creator: reviewer_sbrr

<content>
summary:
The authors claim that modern deep neural networks focus on global optimization and the learning objective for individual neurons are obscure, whereas real-life biological neurons grow with local learning objectives with limited global information. Based on this intuition, the authors propose a biologically inspired neural network that enhances interpretability of the learning dynamics of individual neurons. The authors leverage Partial Information Decomposition (PID) in information theory to formulate objective functions on the individual neuron level, and these neurons are termed “infomorphic neurons”. This novel formulation is able to achieve comparable performance to backpropagation on MNIST and CIFAR10 datasets, showing preliminary signs of utility. Besides, the proposed framework allows for interpretability analysis on the partial information components which is quite unique and interesting.

strengths:
1.	The topic of biologically inspired neural networks has attracted great attention in the recent years. Proposing alternative solutions to the long-dominating backpropagation method is also interesting and profoundly influential. I encourage the authors to work along these lines.
2.	Overall this is a paper very rich in content.
3.	The visualizations in Figure 1 and 2 are very helpful for understanding the partial information components in partial information decomposition.
4.	The performance on MNIST and CIFAR10 are quite promising. From what I read, the proposed method achieved beyond 98% accuracy on MNIST and 94.4% accuracy on CIFAR10.
5.	The parameter importance assessment is thoughtful and informative.

weaknesses:
1.	While I believe we should not expect super comprehensive experiments from a paper that initially introduce a novel concept or paradigm, it might be even improve the soundness of this paper if the authors can also include the performance on slightly more challenging datasets. If standard large datasets such as ImageNet are too computationally expensive, the authors could consider variants that are decently big and challenging, such as TinyImageNet.
2.	For demonstration of experimental results, while I like the richness of Figure 3B and I do notice Figure 8 in the appendix, I would personally argue it would be more straightforward to include a “less exiting” bar and whiskers plot of backprop as well as different variants of bivariate and trivariate models. I suspect the authors left the performance of the bivariate model to the appendix to avoid overcrowding the results, but this omission from the main text might trigger confusion from the audience. Again, a good old bar and whisker might be a valid solution.

questions:
1.	Please refer to Weakness 2. Would the authors consider the “less existing” representation of the quantitative results or provide some other alternative that are similarly straightforward?
2.	I still have a hard time understanding how the partial information components are realized in actual implementation. It would be good for the authors to provide a brief explanation on how that is done or point to the relevant text in the paper.
3.	It is a bit uncommon to have “Related Works” and “Limitations and Outlook” as part of the Discussion section. Is that intentional or is it just a typo?
</content>
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</thread>
<thread>
creation date:
1732188805327

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
General Response to the Reviewers

comment:
Thank you for your insightful and constructive reviews. Incorporating your suggestions into our manuscript has certainly helped to improve the clarity and precision of our work.
To avoid duplication, we will address questions raised by multiple reviewers in this general comment.

**Dimensionality of inputs and estimation of PID atoms**

Two reviewers raised the question if the dimensionality of the input or the batch size of 1024 pose a problem to the estimation of the PID atoms. We believe this not to be the case, as the high-dimensional input or lateral connections are not used in the estimation themselves but only the aggregated scalar values R, C and L which are reduced to a single dimension each by a learned weighted sum. For this reason, the joint probability mass function is only four-dimensional irrespective of the dimensions of the inputs. Furthermore, the probability mass function has empirically been observed to be quite sparse, making 1024 samples sufficient to give a sufficiently good approximation of the PID atoms.
We validate this assumption, we ran MNIST classifier networks with $N=100$ neurons in the hidden layer with smaller and larger batch sizes. Despite matching the learning rates accordingly, in this first test, the runs with larger batch sizes converge significantly slower but reach approximately the same final accuracy (see new appendix section). While determining the optimal batch size remains a question for future research, we thus believe the chosen batch size of 1024 samples to be sufficient for the experiments shown.

**Scaling to multiple hidden layers for solving more complex tasks**

Furthermore, multiple reviewers have highlighted the importance of analyzing multi-layer networks using infomorphic neurons. We agree that using deeper layers is the logical next step in our research and have expanded on the preliminary results in a new appendix section. These preliminary results show that deeper infomorphic networks can train with the same goal functions as their more shallow counterparts, reaching comparable or slightly higher accuracy when matching the number of learned parameters. Nevertheless, since the goal of this paper is to introduce the framework of trivariate infomorphic networks, a thorough investigation of how optimal goal functions differ between hidden layers is left to future research.
To showcase the generality of PID goal functions, we have in the meantime applied our approach to the AudioMNIST dataset, achieving test accuracies between $94.7\%$ and $96.3\%$ for 10 training runs using the same optimized goal function from the MNIST image recognition task. While the AudioMNIST task may not be significantly more complex, this result showcases the transferability of the local goal function to classification tasks from domains other than image recognition, which likely have very different input statistics.

**On the concept of "interpretability"**

We want to emphasize that the more abstract notion of task-independent interpretability of local goals provided by infomorphic networks is  different from interpretability from an application perspective: The interpretability that infomorphic neurons offer is on the level of the information-theoretic goals of the individual neurons, revealing what information processing on the local level is sufficient to solve a particular global task. These results are expected to depend on the type of task (e.g. classification), but should be transferrable between tasks of the same nature (e.g. MNIST and CIFAR10) and between different mechanism which produce the activation values (e.g., activation function).
In future work, these tools may be used to discover which neurons do or do not contribute to the solving of a global task. Because the used redundancy measure is *local*, meaning it can be evaluated for individual samples, infomorphic neurons may in future work be used to identify particular classification labels for which a neuron does or does not contribute. 
We added a paragraph to the paper's discussion to explain this distinction.
</content>
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</thread>
<thread>
creation date:
1733152703407

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Manuscript Title Change

comment:
Following a suggestion by reviewer Mc5F, we have previously changed the title of our work to "Neuron, do your part: Self-organizing global computation from local objective functions based on partial information decomposition" to address the concern that the question "What should a neuron aim for?" in the original title  may be understood to imply more extensive analysis of larger networks. However, in communication with colleagues we found that the new title can be misunderstood as implying a global agent that "makes neurons do their part", when focus should be on their local learning and self-organization.

Should the reviewers and editors agree, we therefore suggest to revert to the original title "What should a neuron aim for? Designing local objective functions based on information theory". We believe that the first part of this title introduces the topic in a thought-inspiring manner and highlights the generality of the proposed framework in principle, with the second half clearly defining the scope of the present submission. If the reviewers have serious concerns with this original paper title, or if regulations disallow further changes of the title at this point, we are are nevertheless contented with keeping the new suggested title.

The regulations regarding title changes remain unclear to us: While the ICLR author guidelines clearly specify that the title can be augmented during the rebuttal phase, there was no option to update the official title in the rebuttal form on the OpenReview website. We thus kindly ask the editor to clarify which title should be used in the camera-ready version in accordance with the regulations in their final decision.
</content>
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</thread>


Metareview:
metareview:
This work deals with the dissonance between global and local learning rules in neural networks. It takes an information theory approach to enable neurons to steer the integration of different classes of input information to drive it's own update. The way that the different classes of information can be combined can be dictated by the researcher or fit to data. The reviewers noted that the overall conceptual framework was novel and interesting, and especially appreciated the interpretable nature of the model parameters. 

While some minor concerns were raised, these mostly focused on details, e.g., the application to harder problems and questions on the data-intensive nature of information estimation. A successful discussion with reviewers and additional preliminary results on new experiments seem to have overcome these concerns.