Shallow Learning In Materio.
Authors: Celestine Preetham Lawrence
Venue: Submitted to ICLR 2023 (2023)

Abstract: We introduce Shallow Learning In Materio (SLIM) as a resource-efficient method to realize  closed-loop higher-order perceptrons. Our SLIM method provides a rebuttal to the Minsky school's disputes with the Rosenblatt school about the efficacy of learning representations in shallow perceptrons. As a proof-of-concept, here we devise a physically-scalable realization of the parity function. Our findings are relevant to artificial intelligence engineers, as well as neuroscientists and biologists. 

Under review as a conference paper at ICLR 2023
SHALLOW LEARNING IN MATERIO.
Anonymous authors Paper under double-blind review
ABSTRACT
We introduce Shallow Learning In Materio (SLIM) as resource-efficient method realize closed-loop higher-order perceptrons. Our SLIM method provides rebuttal to the Minsky school's disputes with the Rosenblatt school about the ef- fieacy of learning representations in shallow perceptrons. As proof-of-concept, here we devise a physically-scalate realization of the parity function. Our find- ings are relevant artificial intelligence engineers, well as neuroscientists and biologists.
Introduction
How do we best learn representations? We do not fully understand how cognition is manifested in any brain, not even in those of a worm (Rankin 2004). It is an open question the shallow brain of worm is capable of working memory, but were then certainly must depart from the mechanistic models of large-scale brains (Eliasmith et al. 2012). Nevertheless, worm-brain inspired learning combined with "scalable" deep learning architectures have been employed in self-driving Lechner et al. 2020). At present, by scalable we refer to TPU-based architectures (Jouppi al. 2017) trained by gradient-descent (Rumelhart et al. 1986). However, one could envision super-scalable future that is less synthetic and based on self self-organized nanomaterial systems (Bose) al. 2015 Chen al., 2020 Mirigliano et al. 202T) that natively realize higher-order (Lawrence) 2022a) and recurrent neural networks. In this short communication, we shall lay yet another brick towards such future by providing theoretical arguments.
cars
Deep learning
Ashby homeostat
Synthelic
Rosenblatt perceptron
Denwinian
Hoétien
Evolution in materio
Shallow learning in materio
Figure 1: A typology cognitive material systems.

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Under review as a conference paper at ICLR 2023
Our perspective cognitive material systems is illuminated in Figure I Deep learning owes its success to our technological capacity to synthesize maasively-paralled and programmable electronic circuits. It is yet to fully exploit Darwinian and Hebbian learning methods that pioneers of the cybernetics movement experimented with by training homeostats (Ashby) 1952) and perceptrons [Rosenblati] [961]. The spirit of Darwinian (Stanley al., (0T9) and Hebbian (Scellier & Bengio, 2017) learning continues to be alive, though. Here, we add fuel to that fire by advocating for an in-materio approach.
Employing physical systems in their native form for solving computational tasks had gained atten- tion due to the efforts of the eevolution materio' community (Miller & Downing [2002). The ear- liest result was by Pask (1960) who grew dendritic metallic threads in ferrous sulphate solution to function as a sound-frequency discriminator (which called an ear, quite romantically). Now, more recent efforts are under the banner of physical reservoir computing (Tanaka et al. (019)) for realizing sequential functionality. Here, we will commit to combinational functionality by equilibrium-pon logic (Lawrence, 2022b) in material systems realizing closed-loop higher-order perceptrons.
2
Theory
Perceptrons were developed by Rosenblatt and his team, and were trained by a Hebbian learning rule cenror-cottoolld reinforcement) with proven guarantees for convergence. Unfortunately, they started recieving a bad rap after Minsky Papert (1888) published proof that 2 association neurons are required to learn the N-bit parity function. However, this analysis is only applicable if all neurons are threshold logic gates, what Rosenblatt called simple units. Physical neural networks, on the other hand, can natively realize complex units. Hence, we introduce shallow learning in materio (SLIM) perceptron as depicted in Figure
Minsky-Papert perceptron
SLIM perceptron
w11]
w[1]
x(M+ 3]
xF21
x
w[N+1]
w[2]
y
W[M]
M
x|M]
a|M
422
Figure 2: To learn the -biit parity function, the required number of synaptic weights (depicted as dashed lines) for our SLIM perceptron scales as N instead of the 2 required for the Minsky-Papen perceptron. This gain in resource-efficiency is possible because the hidden states 81-V of the SLIM perceptron can compute deep-feedforward functionality by equilibrium-poin control.
For proof-of-concepy we commit to a minimally connected recurrent network with physical states from = 1 :NN, yielding a state-space model of the form
à; 24 Ft(8;-1: 81, 81+1);
2

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Under review as a conference paper at ICLR 2023
where F; is nonlinear function. We conjecture that all possible N-bit functions may be realized if arbitrary choices of FI.N are allowed. At present there is no engineering theory to design an optimal F; (even when N = 2).
We first take an approach amicable to discrete mathematics, and demonstrate cquilibrium-poin logic in Figure 3] with Fi:2 designed piecewise-constant functions. A promising approach to obtain F; for higher dimensions is to identify an analogy with cellular automaton that are capable of cquilibrium-point parity logic in arbitrary dimensions (Betel et al. (013). Obtaining scaling laws for the volume of state-space in action during equilibrium-point logic may be another worthy problem ponder upon.
x|2
Components of vector-fleld F
x
Input-driven dynamical system
x(1]-1
/
=[1]
Figure 3: Equilibrium-poine logic a piecewise-constant vector field à; =; F Fi 82) designed to realize 81 Iz T2. The components of F; combine generate the field 21 T2 0 for the input-driven dynamical system. The field can be updated on addition of input(s) in possible directions, and the system traverses new equilibrium-point shown (traversing the gray regions).
Eigenvalue sprend
-2
8
N
Figure 4: Maximum and minimum eigenvalues over all 2 locally-linear modes of equation 2.
Imposing conditions of physical realizability on F; would affect the neuronal capacity (Baldi & Ver- [hynin||211 2018) of our SL.M perceptron. To obtain an insight into the abundance of unique functions expressable by SLIM, let us consider a unit-resistor learnable-thresher (w,) diode network of the form
(2) with so = s and s = sN- The above equation is the simplest expression that captures the non- linear synergetic interactivity found in the L.yapunovsstable resistor tunnel-diode networks studied in
(s St, 8:41) 8;-1 + Si+1 2s; Ramp(s; - w;);

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Under review as a conference paper at ICLR 2023
For each = 1 .N, depending sgn wi, there is positive negative mode of equation 2, and thus there are 2d modes of convergence to equilibrium. Each mode has N eigenvalues, thus there are N22N different timescales. The smallest and largest eigenvalues plotted in Figure and the eigenvalue spread is larger for higher N. Because the largest eigenvalue positive, while the system is Lyapunov stable, we may expect non-trivial mixing of the modes of functionality on the way to equilibrium. This was confirmed empirically for N = and 1000 random arrays of weights with w; E (0, 1). No two weight arrays yielded the same mode of equi- libriation for all 2 = 256 inputs, and thus 1000/1000 functions expressed were unique (for N = 3 this was not true and only 266/1000 unique functions were expressed). Wolfram Mathematica code to reproduce this result and investigate it for other values of N is provided in the Appendix.
Lawrence 2022b).
3 Conclusion
Our contribution here is threefold: (1) a typology of cognitive material systems that puts spotlight on yet-to-be-appreciated alternatives to deep learning, (2) mathematically tractable framework to investigate recurrent networks for deep feedforward functionality, (3) framing open problems in equilibrium-pon logic. More theory is needed to develop constructive high- tigh-dimensional examples, and a statistical analysis of SL.M's performance. Next steps could be obtain estimates on the learning duration, and check if it is superior to estimates obtained from the principal convergence theorem for perceptrons (Rosenblatt, 1961 Section 5.5, Theorem 4). Given the well established (in silico) deep learning industry, even with more practical demonstration, business economics would prevent the shift to a SLIM paradigm in the near future. Nevertheless, the SLIM concept may act as catalyst for gifted mathematicians make new connections or help neuroscientists in unravelling the mysteries of scall-scale brains.
Reviewer contributions
Reviewer sByK asks why this work should be considered novel, in comparison to earlier concepts such as predictive coding networks. The novelty here comes from using the function F; equation]] to efficiently realize nonlinear predictors in materio, an improvement over the linear weighted-sum techniques designed
predictors (Srinivasan et al. 1982) that were inspired from image-compresse for conventional computers.
Reviewer joBI kind to provide a thoughtful summary, and suggests to compare this work to two other alternatives for realizing the parity function the wenphex-weighted neuron of AAizeberrg (2088) and the translated multiplicative neuron of [yoda et al.] (2003). In both alternatives, the implementation would be less robust to noise at large N, because only a single neuron is employed (a robust implementation would require a circuit of many physical units for the neuron, making "single" only in mathematical sense). SLIM need not suffer from such crowding problems, because the state-space can grow exponentially in volume with N. Several provably convergent schemes of Hebbian learning as given by |Pineda| (1987) may engineered materio, to act as generalization of backpropagation for closed-loop higher-order perceptrons.
Based on feedback from all reviewers, the technical novelty of this work has been clarified in the conclusion (contribution no. 2).
References
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A Appendix
Snippet Wolfram Mathematica code used to estimate the abundance of in-materio functionality.
(-Define the material's function by starting from the all-0 state and updating over 100 loops*)
F[=_] (t ArrayPad 1, "Fixed"]} 2. -. =] -3]) settlelaterio 9_, x_] 1a Nest[ [# + .01 (x F[#] Ramp [# -9] &, 0 100];
funMaterio 9_1 i= FromDigits
[UndttSteppsetttaaattteee [n, x] -9] (x, Tuples[(O, 1), m1]1, 21;
(*Count the number of distinct functions realised over 1000 random biases-)
abundance [m_] i= Countlstdinteeaaallllleeee [fuMMaterio [n, sl, (8, RandomReal [1., (1000, n}11];
abundance [3]
266
abundance
1000


Reviews:
[{"review_id": "tc_OwL0Y-F", "reviewer": "Reviewer_sByK", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "SLIM is intended to be a perceptron that involves a minimally connected recurrent network. The internal state variables have a nearest-neighbor interaction on a chain. The authors conjecture such networks could realize arbitrary N-bit Boolean functions, in contrast to the limitation of Rosenblatt perceptrons, as shown by Minsky and Papert.", "summary_of_the_review": "This paper would need much more work to rise to the level of being worthy of consideration for publication at ICLR. At this point, it is just a scheme, and not a particularly novel one.", "strength_and_weaknesses": "Strengths:\nThe paper explores alternatives to the deep learning architecture and hopes to have self-organized nanomaterial-based learning machines.\n\nWeaknesses:\nThe key idea is not so novel. From a multitude of recurrent neural networks (including the reservoir computing framework, mentioned by the authors) to predictive coding networks, many approaches utilize the idea of a network with internal dynamics, the equilibrium point of which helps perform the task.\n\nThere is very little work done to substantiate the claim for the particular architecture proposed.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "QT6oO4YgWHi", "reviewer": "Reviewer_joB1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Goals and contribution:\n* The authors try to draw attention of the community to shallow recurrent neural networks trained by Hebbian learning in the context of cognitive material systems. \n* They argue that these models could compete with deep-learning systems in the future  - thanks to upcoming advancement of in-materio technologies.\n* They demonstrate this idea on an illustrative example of the N-bit (2-bit) parity function realized by the new-proposed SLIM (Shallow learning in Materio) model.\n* The authors also point out several crucial open problems of the proposed approach as topics for future research.\n\n", "summary_of_the_review": "Although the main idea of the paper is novel and interesting, its realization is poor. The proposed model is not well-described and it is not adequately empirically or theoretically justified. ", "strength_and_weaknesses": "Strengths: \n1. I agree with the authors that the shallow recurrent higher-order neural networks are a promising area of future research - due to the progress of in-materio technologies. \n2. The illustrative example is well-chosen. It demonstrates well how the proposed SLIM model could overcame the classical problem of n-bit parity (Minsky&Pappert,1988) - thanks to the higher-order units used instead of threshold logic gates.\n3. The authors provide several open questions concerning their model as possible subjects of future research.\n\nLimitations:\n1. The paper is very brief  and vague in every aspect and its techniqual quality is imited.  \n2. There is no theoretical justification of the method (and its convergence) and the empirical evaluation is limited to a toy problem.\n\nDiscussion of the limitations:\n1. The paper is very brief  and vague in every aspect and its techniqual quality is imited.  \n  * Mainly, the proposed model should be described in more detail, including training algorithm and analysis/discussion of the choice of the F_i functions.\n  * The illustrative example also needs a more detailed description. How did you choose F_1,2 and why? Was the model trained by Hebbian learning? \n    Can you show the concrete final model (its F-functions and weights)? Figure 3 is not comprehensible - could you describe more clearly what the four images represent? \n2. There is no theoretical justification of the method (and its convergence) and the empirical evaluation is limited to a toy problem.\nThe paper addresses more limitations and problems of the proposed model than its advances.  Many important questions are not addressed: \n * Is it possible to extend the 2-bit parity model to the n-bit one? Are you able to show a solution for n=3?\n * The training process of the model is not described/analyzed.\n * The choice of F-functions for different tasks is not described/analyzed.\n * From the paper it seems like it is difficult to successfully  apply the model to different tasks. Is it right?  \n ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"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": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Ne9lIWF3G0u", "reviewer": "Reviewer_wULo", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper suggests an alternative network architecture, but lacks analytical and numerical evaluation.", "summary_of_the_review": "Ingenious, but in lack of a theory and of numerics", "strength_and_weaknesses": "Strength:\n* Out of the box\n\nWeakness: \n* Lack of box", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "2KyHTKal2GR", "reviewer": "Reviewer_h2U8", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors introduce a structural variant of single-layer perceptrons with added recurrent connections. The model seems to be mostly a standard RNN, with the different of a more general activation function.\nNo evaluation is provided, aside from a hand-designed example.", "summary_of_the_review": "N/A", "strength_and_weaknesses": "Strengths:\n- The paper is short, which makes it easy to read.\n\nWeaknesses:\n- The main problem tackled in the paper is not a particular concern since the early 80s and is largely solved.\n- The structure of the paper is unorthodox and lacks most of the required content (concerning the theory and the model) and any practical evaluation. \n- Figure 3 is difficult to understand, as no explanation is given about the notation in the text nor in the caption.\n- The proposed architecture seems to be a standard RNN with a more flexible activation function, which however the authors leave undefined.\n- Related literature is severely lacking, both to justify the significance of the problem addressed and all the work done on it since the 80s.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"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": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject