Title: Functional Equivalence and Path Connectivity of Reducible Hyperbolic Tangent Networks
Venue: NeurIPS 2023
Accepted: yes
Keywords: theory

neural network theory

structural redundancy

functional equivalence

functional equivalence class

partial identifiability

parameter canonicalisation

parameter space

piecewise-linear

connectivity
Abstract: Understanding the learning process of artificial neural networks requires clarifying the structure of the parameter space within which learning takes place. A neural network parameter's functional equivalence class is the set of parameters implementing the same input--output function. For many architectures, almost all parameters have a simple and well-documented functional equivalence class. However, there is also a vanishing minority of reducible parameters, with richer functional equivalence classes caused by redundancies among the network's units.

In this paper, we give an algorithmic characterisation of unit redundancies and reducible functional equivalence classes for a single-hidden-layer hyperbolic tangent architecture. We show that such functional equivalence classes are piecewise-linear path-connected sets, and that for parameters with a majority of redundant units, the sets have a diameter of at most 7 linear segments.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates the functional equivalence class for reducible neural network parameters and its connectivity properties. The authors focus on single-hidden-layer hyperbolic tangent networks, but the findings can be generalized to other feed-forward network components.\n\nSTRENGTHS:\nThe paper provides a comprehensive understanding of the functional equivalence class for reducible neural network parameters. This can be key to understanding the structure of the parameter space and the loss landscape on which deep learning takes place.\n\nThe authors describe a complex union of manifolds, displaying rich qualitative structure. This includes a central discrete array of reduced-form parameters, connected by a network of piecewise linear paths, and various manifolds branching away from this central network.\n\nThe paper establishes that with a majority of blank units, the diameter of this parameter network becomes a small constant number of linear segments. This can be beneficial in understanding the trade-offs between shortest path length and rank for different unit permutations.\n\nThe paper also discusses the relevance of their findings to modern architectures and deep learning, suggesting that understanding reducible functional equivalence classes may be key to understanding these topics.\n\nWEAKNESSES:\nThe paper is highly theoretical and may not provide immediate practical applications for those working with neural networks. The exact relevance of reducible parameters to these topics remains to be clarified.\n\n\nThe paper focuses on single-hidden-layer hyperbolic tangent networks, which may limit the applicability of the findings to more complex architectures.\n\nQUESTIONS:\nIn the paper, the authors focus on single-hidden-layer hyperbolic tangent networks. How do the authors think the results would change ifwe considered networks with multiple hidden layers or different activation functions?\n\nThe paper discusses the trade-offs between shortest path length and rank for different unit permutations. Could the authors provide more insights into how these trade-offs could impact the performance of neural networks in practical applications?\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces an algorithm to calculate the canonical equivalent parameter for a tanh feed-forward network.\nFurther, the full equivalent class of each parameter is given by a union of subsets constructed out of (inverse-)reductions and permutations of the canonical parameter.\nFinally, the paper shows that each functional equivalence class is path-connected and that the diameter of a functional class with rank at most the half of the number of units in the hidden layer is 7.\n\nSTRENGTHS:\nThe canonicalisation algorithm is novel for feed-forward networks with a tanh transfer function. Furthermore, this is the first result that characterizes the diameter of some of the equivalence classes of these networks.\nThe paper fully proves each theorem and includes useful visualizations for the path structure of a functional equivalence class.\nThe paper is overall very clearly written, with only a few possible minor typos.\nThe results presented here could be of importance to loss minimization problems.\n\nWEAKNESSES:\nThe scope of applications of the algorithm and the theorems seems to be rather narrow or at least is not properly motivated to be broad.\n\nThe paper has very limited evaluation of the proposed algorithm and further uses of the results.\n\nIt is for example unclear in which practical application Algorithm 4.1 could be used, because of the real valued parameters being usually all different in a trained network, see Questions.\n\nQUESTIONS:\n196: What is a branch?\n\n265: Should it be ``connect $w'$ to $u$\"?\n\nWhat are the practical applications of this algorithm?\nCould you apply it to a loss function example you mention in 354?\nIf the parameters are real values, wouldn't they (and their absolute values) all be different from each other with probability equal to 1?\n\nCould you say anything about the loss function at different parts of the functional class that belongs to zero loss? Is it steeper at certain points in parameter space than others?\n\nCould you clarify and justify the remark in 362? What is this approximation? \n\nCould you give an example of a network that has rank higher than half of $h$ that is part of a functional equivalence class with a diameter more than 7?\n\nWhy does it matter what the diameter is of a functional equivalence class?\n\nWhy are networks that are in a functional equivalence class of diameter at most 7 of particular importance?\n\nLIMITATIONS:\nThey are adequately discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on the study of functional equivalence classes in neural networks, specifically in fully-connected, single-layer networks with hyperbolic tangent activation. It  presents a Canonicalisation algorithm for reducible networks to determine canonical representative parameters for different functional equivalence classes. Finally, it explores the concept of connectivity between these classes.\n\nSTRENGTHS:\nThe paper gives a well-motivation exploration of functional equivalence classes in neural networks. It provides a well-written definition of these classes, introducing an analysis of connectivity between them. The article references foundational works on simple neural networks and incorporates recent literature in the field of deep learning, enhancing the credibility and relevance of the study.\n\nWEAKNESSES:\nThe weaknesses of this article stem from the focus on a very theoretical network type with implausible parameters. While a useful theoretical case, further discussion on the applicability of this analysis to modern architectures would benefit the article.\n\nSpecifically, a drawback of the article is its narrow focus on a specific architecture type, namely single-layer networks with tanh activation, a single input and single output. While it does discuss expanding the input and output space, it fails to address how the findings extend to more complex multi-layer networks or different activation functions commonly used in modern architectures. By limiting the scope of investigation, the article has limited application to contemporary neural network design and learning.\n\nThe proposal of the Canonicalisation algorithm also presents limitations. The algorithm relies on exact equivalence between weights or to 0, which is rarely achievable in practice, especially for weights trained through backpropagation. This raises questions about the adaptability of the Canonicalisation algorithm to networks trained through backpropagation, and whether it can provide meaningful results. The article would greatly benefit from addressing these concerns and exploring potential adaptations for networks trained through backpropagation.\n\nIt appears to me that the Canonicalisation algorithm is a unit pruning / neuron removal algorithm. While Kuditipudi et al. is cited, the article would benefit from considering and discussing other neuron removal methods, specifically the following which focus on individual neurons in fully-connected layers:\n\n+ Mao Ye, Chengyue Gong, Lizhen Nie, Denny Zhou, Adam Klivans, and Qiang Liu. Good subnetworks provably exist: Pruning via greedy forward selection. In Proceedings of the 37th International Conference on Machine Learning, pp. 10820–10830. PMLR, 2020.\n+ Xiaocong Du, Zheng Li, Yufei Ma, and Yu Cao. Efficient network construction through structural plasticity. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 9(3):453–464, 2019.\n+ Lemeng Wu, Bo Liu, Peter Stone, and Qiang Liu. Firefly Neural Architecture Descent: a General Approach for Growing Neural Networks. In Advances in Neural Information Processing Systems, volume 33, 2020.\n\nAdditionally, GradMax is a method which adds new neurons, similarly to Fukumizu and Amari, which was cited. The new neurons do not impact the function of the network upon addition by setting fan-in weights to 0. This case doesn't seem covered by remarks 5.3-5.5.\nEvci, Utku, et al. \"GradMax: Growing Neural Networks using Gradient Information.\" International Conference on Learning Representations. 2021.\n\nQUESTIONS:\nThe practical contribution of the path connectivity analysis was furthermore unclear for me. What does this analysis mean for learning or architecture design? Is it an argument for sparse training or initialization to reach an architecture with a majority of blank units?\n\nI believe the reference format is incorrect; I believe that NeurIPS uses number-based references instead of name, year.\n\nLIMITATIONS:\nThe paper does not adequately address the limitations induced by the narrow focus on a highly theoretical architecture. The sections on \"towards modern architectures\" and \"functional equivalence and deep learning\" offer insight into the potential application of the analysis to contemporary deep neural networks, but are not comprehensive.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-12"}

comments: SUMMARY:
This paper deals with functional equivalence problems in neural networks, i.e., the characterization
of all neural networks that lead to the same given output function. This is a problem that has been
studied since the early 1990s, including work by the fields medalist Charles Fefferman. 
The present paper considers single-layer networks with tanh-nonlinearity and puts forward
a completely new perspective by paying attention to reducible parameters.

STRENGTHS:
The paper tackles a problem that has been studied in various guises for over 3 decades, and
puts forward a completely new vantage point, by considering reducible parameters. This leads
to rich insights into the functional equivalence problem. In addition, the paper exhibits a strong algorithmic component,
specifically by providing an algorithmic characterization of redundancies and connecting the underlying
theory to the beautiful concept of piecewise-linear path connected sets.

WEAKNESSES:
could not find any

QUESTIONS:
it would be interesting to see the authors' thoughts on whether the algorithmic component of the paper can
be extended to multi-layer networks

LIMITATIONS:
none
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-24

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses fully connected feed-forward neural networks (NNs) with a single hidden layer and the hyperbolic tangent activation function. A parameter is thus a vector of weights and biases. It considers reducible parameters, which means that a NN with strictly less neurons can implement the same function. The paper provides a \"canonicalise\" procedure taking a parameter as input and yielding a canonical parameter that implements the same function. Also, if two parameters implement the same function then the output of the procedure is the same for them.\n\nBased on this procedure, the paper characterizes the set of parameters that implement the same function as a given parameter. It shows that this set is piecewise linear path-connected, and if the set is defined with respect to a ``sparse'' parameter, then the number of linear pieces of the diameter of the set is bounded by 7.\n\nSTRENGTHS:\nThe paper is well written. The figures are very clear and help understanding.\n\nThe theoretical topic is well motivated and well connected to the literature. \n\nThe theoretical results are interesting, and so are the proofs in my opinion.\n\nWEAKNESSES:\nThe main weakness is that the paper is restricted to a single hidden layer. \nAlso, the hyperbolic tangent activation function is studied, while ReLU would have been more relevant, in my opinion.\n\nFor these two weaknesses, it seems they cannot really be fixed in the frame of this paper, as considering multiple hidden layers and/or ReLU would probably change the nature of the results and the proofs.\n\nQUESTIONS:\nThe content presented is overall clear.\nThe statements of the theoretical results seem clear.\nSome proofs are a bit technical and more difficult to follow. The authors could consider adding pedagogical content/details on the proofs in the appendix, if they see fit.\n\nLIMITATIONS:\nThe authors adequately discuss the main limitation (a single hidden layer) in the discussion.\nI do not see potential negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-25"}