Title: Measuring and Controlling Solution Degeneracy across Task-Trained Recurrent Neural Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: recurrent neural network

dynamical systems

neuroscience

neural computations
Abstract: Task-trained recurrent neural networks (RNNs) are widely used in neuroscience and machine learning to model dynamical computations. To gain mechanistic insight into how neural systems solve tasks, prior work often reverse-engineers individual trained networks. However, different RNNs trained on the same task and achieving similar performance can exhibit strikingly different internal solutions, a phenomenon known as solution degeneracy. Here, we develop a unified framework to systematically quantify and control solution degeneracy across three levels: behavior, neural dynamics, and weight space. We apply this framework to 3,400 RNNs trained on four neuroscience-relevant tasks: flip-flop memory, sine wave generation, delayed discrimination, and path integration, while systematically varying task complexity, learning regime, network size, and regularization. We find that increased task complexity and stronger feature learning reduce degeneracy in neural dynamics but increase it in weight space, with mixed effects on behavior. In contrast, larger networks and structural regularization reduce degeneracy at all three levels. These findings empirically validate the Contravariance Principle and provide practical guidance for researchers seeking to tune the variability of RNN solutions, either to uncover shared neural mechanisms or to model the individual variability observed in biological systems. This work provides a principled framework for quantifying and controlling solution degeneracy in task-trained RNNs, offering new tools for building more interpretable and biologically grounded models of neural computation.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the solution degeneracy of recurrent neural networks (RNNs) trained on a suite of tasks, evaluating solution diversity across behavioral, dynamical, and weight-space levels. The central claim is that task complexity predictably modulates degeneracy: increasing complexity leads to more similar neural dynamics across training runs, but greater divergence in weight space. The authors use recently proposed tools such as Dynamical Similarity Analysis (DSA) and introduce the Permutation-Invariant Frobenius distance (PIF) to quantify similarity between trained RNNs. The paper proposes methods to quantify and control degeneracy (e.g., via task complexity, auxiliary loss functions, regularization).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n•\tThe paper is clearly written, with well-organized figures and a clean narrative structure\n\n•\tThe focus on degeneracy and its controllability is relevant for both interpretability and robustness in ML models\n\nWeaknesses:\n\n\n•\tThe validation of the proposed complexity and degeneracy measures is limited. For example, DSA scores in the paper in part (e.g. Fig. 3B and Fig. 6) show values on the order of 0.12—similar to what's considered \"invariant\" in previous DSA work (e.g., Ostrow et al. 2023, Fig. 4,[1]). Without a stronger argument for statistical significance, these differences appear too small to support strong claims\n\n•\tSeveral key claims (e.g., the significance of degeneracy differences, the effect of “dispersion of local minima”) are made without sufficient visual or quantitative evidence. The paper would benefit from more direct visualizations of flow fields or attractors, especially for chaotic or high-dimensional tasks, to support the dynamical claims\n\n•\tThe scale of experiments is modest—typically 2–4 data points per task variant—which limits the strength of the generalization claims (e.g. Figs. 3,6,7). Tasks are also relatively low-dimensional and may not reflect the complexity of real-world RNN applications\n\n\n\n\nReferences\n[1] Ostrow, M., Eisen, A., Kozachkov, L., & Fiete, I. (2023). Beyond geometry: Comparing the temporal structure of computation in neural circuits with dynamical similarity analysis. Advances in Neural Information Processing Systems, 36, 33824-33837.\n\nQUESTIONS:\n•\tThe paper needs stronger empirical or theoretical validation for DSA and PIF as measures of degeneracy. What threshold should count as significant difference in DSA or PIF? Why is a DSA of ~0.12 meaningful if previous work considered that range as \"invariant\"?\n\n•\tCan the authors provide flow fields, phase portraits, or other dynamical visualizations to show the difference in dynamics?\n\n•\tHow do the measures and trends extend to chaotic or high-dimensional RNNs? For instance, are the same trends visible with chaotic attractors?\n\n•\tIn Figure 3, are network sizes held constant across varying complexities? Could increasing complexity simply lead to networks using more of their capacity, thereby reducing variability?\n\n•\tThe paper does not report task performance. How well do these RNNs actually solve the tasks? Are higher-degeneracy networks lower-performing?\n\n•\tThe paper refers to dispersion of local minima but does not show empirical support. Could the observed weight variability be due to wide basins in a large solution space?\n\n•\tAre DSA comparisons based on autonomous (free-running) dynamics or input-driven trajectories? This distinction is crucial for interpreting the meaning of “similar dynamics.”\n\nLIMITATIONS:\nWhile the idea of the paper is interesting and novel, some limitations remain:\n\n•\tLack of analysis for chaotic or more realistic dynamical systems.\n\n•\tAbsence of flow field visualizations to confirm qualitative differences in neural dynamics.\n\n•\tLack of explicit model performance reporting.\n\n•\tLimited number of datapoints (e.g., 2–4 variants per task) which weakens broad generalization claims.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper proposes a unified framework to measure degeneracy in RNNs across three dimensions: behavioral, weight, and dynamic degeneracy. It quantifies the degree of degeneracy as a function of various factors, including task complexity, network size, learned features, and regularization.

STRENGTHS AND WEAKNESSES:
Strengths: 
- The authors propose three metrics to quantify degeneracy across behavioral, weight, and dynamic dimensions.
- The paper analyzes multiple factors and establishes qualitative relationships between these factors and different types of degeneracy.

Weaknesses:
- The theoretical (even qualitative) rationale behind the observed relationships is limited.
- The degeneracy measurements assume that different networks achieve comparable training losses, yet no explicit justification for this assumption is provided.

QUESTIONS:
- Can the authors demonstrate that all models of interest converge to similar losses (e.g., within 2% of the average)? How robust are the results if this threshold is varied (e.g., within 5% of the average)?

- The paper defines network size in terms of width, but depth also contributes to model capacity. Do the relationships reported in Table 1 hold for networks of varying depth?

We may consider increasing the score if these questions are addressed.

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies solution degeneracy in vanilla recurrent neural networks (RNNs) across three levels: behavior, neural dynamics, and weight space. The paper runs a systematic empirical analysis by training ~50 networks on each of four neuroscience inspired tasks, varying: task complexity, feature learning, network size, and regularization. Previous work has mostly studied trained networks after fixing these variables, so this systematic study is quite timely.\n\nThe paper finds interesting relationships between the behavioral, dynamical, and weight space similarities of trained networks as a function of the task complexity, network size, and regularization. In particular, it finds that for some features (such as task complexity), the dynamical degeneracy and the weight degeneracy move in opposite directions (\"contravariant\") while for other features (such as regularization), they move in the same direction (\"covariant\").\n\nOverall, the paper provides a clear summary of the effect of key architectural choices on RNN solution degeneracy, which both can serve as a practical guide for neuroscientists using RNNs as well as opening the door for future studies to better explore the solution space of RNNs on nonlinear tasks.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n- The paper studies a critical and (as pointed out in the introduction) relatively understudied problem of solution degeneracy in RNNs.\n- The paper is really well written, with clear motivation and thorough descriptions of the experiments and results.\n- The inclusion of both code and the trained model weights will be a valuable resource for the community.\n- The study examines networks trained across four disparate (but all strongly motivated in neuroscience) tasks. The consistency of the results across these tasks really strengthens the likely generalizability of the findings.\n\n# Weaknesses\n- Narrow definition of behavioral degeneracy: The behavioral degeneracy is defined as out-of-distribution generalization to longer time sequences (temporal generalization). However, this represents only a narrow slice of possible generalization we might expect networks to exhibit. Calling this category \"behavioral degeneracy\" slightly overstates the claims, in my opinion. I think being explicit about this and referring to this category as \"temporal generalization\" instead of \"behavioral degeneracy\" would be a more accurate description. (Or, run additional experiments to examine other types of generalization: such as generalization to noise, perturbations, or other types of task variability).\n- Limitation to vanilla RNNs. The study exclusively studies vanilla RNNs with tanh nonlinearities, which lack the gating mechanisms of LSTMs or GRUs. It is unclear whether the findings would generalize to gated networks.\n- Potential confound in the weight change norm: When quantifying the weight-change norm (Section 3.2.1), do you control for the number of training steps required for each task? I would imagine that due to the stochastic nature of SGD, the change in weight norm trivially grows with the number of steps (e.g. as discussed in https://arxiv.org/pdf/1806.08805). So if the number of steps varies across tasks, the weight change norm will also, in a way that has nothing to do with weight degeneracy.\n\nQUESTIONS:\n1. Do you think the results would hold for gated RNNs?\n2. Do you normalize the weight-change norm for the number of steps (T) required for convergence? If not, does incorporating some normalization there affect the results (I'm not sure what the right normalization is, maybe divide by $\\sqrt(T)$?)\n3. What about $\\ell_2$ regularization? I'm assuming it has a similar effect as $\\ell_1$ and low-rank penalties. I think it would be helpful to add $\\ell_2$ regularization experiments in section 3.5 for completeness.\n4. In figure 2B, for the sine-wave generation task, are the colored dots fixed points or samples from trajectories? I would assume the latter. If so, it might be helpful to show the fixed points as points and the trajectories as lines, to be consistent with the other panels.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates solution degeneracy in task-trained RNNs, which occurs when different networks achieve similar task performance (as measured by their training loss) but vary widely in weights, neural dynamics, and behavior.  The authors identify 4 crucial factors (task complexity, learning regime, network size, regularization) that control solution degeneracy in various task-trained RNNs, providing empirical support for the “contravariance principle.” They propose novel metrics to quantify solution degeneracy and show that task complexity and learning regime have contravariant effects (i.e., affect weights and dynamics in opposite ways), while network size and regularization have covariant effects.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- Provides empirical validation for the Contravariance Principle.\n- Identifies four key factors controlling degeneracy: task complexity, learning regime, network size, and regularization.\n- Uses a solid set of degeneracy metrics.\n- Well-written and thoughtfully designed set of experiments.\n\n**Weaknesses:**\n\n- Terms like *“learning regime”* and *“feature learning”* are used interchangeably. The paper relies on the Rich vs Lazy regime literature, but this may not be familiar to all readers (especially in neuroscience), and could benefit from clearer terminology introduced earlier.\n- In Figure 2, subfigure 2B is not clearly described (presumably PCA of neural activity). It should be clarified in the caption.\n- In Section 3.3, $\\phi$ is not defined, and a new update rule is introduced without context—unclear how this relates to prior experiments.\n- Line 267–268: Inconsistent reasoning—on one hand, the text says that high γ leads to consistent behavior; on the other, it says stronger feature learning leads to less consistent behavior.\n- Minor typo on line 160: “compleity”.\n- Some sections could benefit from better citations.\n\nQUESTIONS:\n1. Definition of degeneracy: Has the definition of behavioral degeneracy you use in Section 2.3.1 appeared in prior literature? If not, please state that it is novel; if it has, cite the appropriate references.\n2. Task motivation: Could you elaborate on why these four tasks were chosen? Do they correspond to known computational functions of brain areas, or are they commonly used RNN benchmarks in neuroscience or machine learning?\n3. Path integration claim: On line 127, you say that “After training, the network forms a Euclidean map of the environment in its internal state space.” Can you back this up? Many studies involving RNNs trained on the task of path integration find toroidal neural manifolds in the internal space, similar to grid cells in the mammalian entorhinal cortex. These are non-Euclidean. \n4. Clarify the $\\phi$ model: In Section 3.3, you introduce a hidden state update involving $\\phi$ without context. Is this a new model? How does it tie into your prior experiments?\n5. Initialization and $h_0$: How is the initial hidden state $h_0$ set in your experiments?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-07"}