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Path Independent Equilibrium Models Can Better Exploit Test-Time Computation

Cem Anil, Ashwini Pokle, Kaiqu Liang, Johannes Treutlein, Yuhuai Wu, Shaojie Bai, Zico Kolter, Roger Grosse

arXiv:2211.09961v1cs.LGstat.ML

TL;DR

Neural networks that use more test-time computation may generalize better to harder instances, but how to design them to exploit that computation remains an open question. The paper studies path independence in equilibrium models, measuring it with Asymptotic Alignment and testing its relation to accuracy. It reports that alignment tracks performance with increased inference depth, and that interventions promoting or penalizing path independence shift accuracy in corresponding directions.

  • Problem

    The paper asks how neural architectures can reliably exploit additional test-time computation to improve accuracy on harder instances, a form of upwards generalization.

    Source: §1 · p. 1

  • Method

    The authors study path independence in equilibrium models, quantify it with Asymptotic Alignment, and test its relation to accuracy and to interventions that promote or penalize it.

    Source: §1 · p. 2, §4.2 · p. 6

  • Results

    Asymptotic Alignment strongly correlates with performance when test-time inference uses far more iterations than training, and promoting or penalizing path independence shifts accuracy in the corresponding direction.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Path independence is a useful indicator of whether equilibrium models can scale test-time computation to improve accuracy on harder instances.

    Source: §1 · p. 2, §4.2 · p. 6

    The analyses exclude non-input-injected models that represent constant, input-independent functions.

    Source: §4.1 · p. 5

Abstract

from arXiv · show

Designing networks capable of attaining better performance with an increased inference budget is important to facilitate generalization to harder problem instances. Recent efforts have shown promising results in this direction by making use of depth-wise recurrent networks. We show that a broad class of architectures named equilibrium models display strong upwards generalization, and find that stronger performance on harder examples (which require more iterations of inference to get correct) strongly correlates with the path independence of the system -- its tendency to converge to the same steady-state behaviour regardless of initialization, given enough computation. Experimental interventions made to promote path independence result in improved generalization on harder problem instances, while those that penalize it degrade this ability. Path independence analyses are also useful on a per-example basis: for equilibrium models that have good in-distribution performance, path independence on out-of-distribution samples strongly correlates with accuracy. Our results help explain why equilibrium models are capable of strong upwards generalization and motivates future work that harnesses path independence as a general modelling principle to facilitate scalable test-time usage.

1 Introduction

The paper asks how neural networks can use more test-time computation to generalize to harder instances, and studies equilibrium models as a candidate architecture. It proposes that path independence is a key determinant of this ability and reports that it tracks performance under increased inference depth.

1 Introduction

  • The study asks when equilibrium models’ input-dependent ability to scale test-time computation can yield upwards generalization.Equilibrium models compute outputs as fixed points of learned dynamical systems.Source: §1 · p. 1
  • Path independence means hidden activations converge to the same limiting behavior for a given input regardless of initialization.The limiting behavior may be a fixed point or a limit cycle.Source: §1 · p. 2, Figure 1 · p. 2
  • The paper hypothesizes that models fitting the training distribution with a path-independent function exploit increased test-time iterations better on harder instances than path-dependent models.The intuition is that additional iterations can compensate for harder instances in path-independent systems, unlike in path-dependent systems.Source: §1 · p. 2
  • Across several task types, Asymptotic Alignment strongly correlates with performance when test-time inference uses far more iterations than training.Path-independent models generally improve monotonically with more iterations, while path-dependent models degrade beyond their training-time iteration count; interventions that promote or penalize path independence shift accuracy in the corresponding direction.Source: §1 · p. 2

2 Background

Equilibrium models explicitly seek a fixed-point representation, unlike fixed-depth weight-tied networks, which may not approach one. Their forward computation and training can use iterative solvers and implicit differentiation, while convergence depends on properties of the model cell.

2 Background

  • Equilibrium models compute internal representations by finding a fixed point z* satisfying z* = f_w(x, z*).The cell can depend directly on the input, a design referred to as input injection.Source: §2 · pp. 2–3
  • Fixed-point iteration repeatedly applies the cell to hidden representations and can stop after a set number of steps or when successive iterates differ by less than a threshold.When convergence conditions hold, this procedure approaches a fixed point; gradients may be computed by backpropagating through the forward computation.Source: §2 · p. 3
  • Implicit training uses root finders to solve for fixed points and implicit differentiation to compute gradients without differentiating through the root-finding procedure.The inverse-Jacobian term can be replaced with an identity or approximate inverse-Jacobian to make the backward pass less expensive.Source: §2 · p. 3
  • Unlike fixed-depth weight-tied networks, equilibrium models explicitly aim to find a fixed point, and their forward pass can use black-box solvers.Fixed-depth networks compute a set number of tied layers and may not approach a fixed point; equilibrium models may be trained using implicit differentiation.Source: §2 · p. 3
  • Contractive cells suffice to guarantee convergence to a unique fixed point, whereas unrestricted equilibrium models are not guaranteed to converge.The paper focuses its investigation on unrestricted architectures because it is unclear whether enforcing contraction limits expressiveness.Source: §2 · pp. 3–4

3 Upwards Generalization with Equilibrium Models

The experiments test equilibrium models on algorithmic tasks with explicit difficulty parameters, using longer sequences and larger mazes to assess upwards generalization. The reported results show strong upward generalization and improved performance with greater inference depth, especially on harder instances.

3 Upwards Generalization with Equilibrium Models

  • The evaluation spans prefix sum, mazes, blurry MNIST, matrix inversion, and edge copy, covering sequence prediction, visual reasoning, image classification, continuous optimization, and graph regression.Detailed analyses focus on prefix sum and mazes; results for the other tasks are provided in the supplementary material.Source: §3 · p. 4
  • Prefix sum models train on 32-bit strings and are tested on other lengths, while maze models train on 9×9 mazes and are tested on larger ones.Prefix sum predicts the parity of bits received so far at each position; mazes require finding an optimal path from start to finish without crossing walls.Source: §3 · p. 4
  • Equilibrium models show strong upward generalization compared with non-weight-tied fixed-depth models, and greater inference depth consistently improves performance, especially on harder instances.The detailed analysis focuses on prefix sum and mazes, with complementary results for other tasks in the supplementary material.Source: §3 · p. 4

4 Path Independence

Path independence describes convergence to the same limiting behavior regardless of the starting state, and the section argues it can support upward generalization. The authors identify architectural requirements, define a metric for path independence, and test that metric against adversarial initializations.

4 Path Independence

  • Path independence means convergence to the same limiting behavior regardless of the current state; the definition also permits limit cycles.For convergent computations, this is equivalent to a unique fixed point for any initialization.Source: §4 · p. 4
  • PI models show strong upward generalization on mazes, and the Path Independence Hypothesis predicts they can better exploit additional test-time iterations.The hypothesis contrasts models fitting training data with path-independent versus path-dependent functions.Source: Figure 2 · p. 5, §4 · p. 5

4.1 Architectural Components Necessary for Path Independence

  • Weight tying makes limiting behavior meaningful, while input injection lets the equilibrium depend on the input; successful non-input-injected models must otherwise be path dependent.The analyses exclude the constant-function exception, which can be path independent without input injection.Source: §4.1 · p. 5
  • Weight tying and input injection also appear in deep equilibrium models, where implicit-gradient estimation is motivated by dependence on the final hidden state rather than the path.The authors note that the two research lines arrive at these motifs for different reasons.Source: §4.1 · p. 5

4.2 Quantifying Path Independence

  • The Asymptotic Alignment (AA) score averages cosine similarity between fixed points for the same input from different initializations; higher scores indicate greater path independence.The score is described as cheap, reliable, and unitless, enabling comparisons across training runs.Source: §4.2 · p. 6
  • Adversarial-initialization tests support AA as a path-independence measure: high-AA networks resist steering away from their fixed point, while low-AA networks can be steered away.The attack searches for initializations producing distinct fixed points using L-BFGS.Source: §4.2 · p. 6

5 Path Independence Correlates with Upward Generalization

The section tests whether AA scores predict upward generalization across varied model configurations. It reports that AA scores correlate with accuracy at sufficiently large inference depth, while non-PI networks degrade under deeper inference.

5 Path Independence Correlates with Upward Generalization

  • The evaluation inspected trained networks’ average AA scores on in- and out-of-distribution splits across varied prefix-sum and maze configurations.Configurations varied depth, weight normalization, solvers, learning rates in prefix sum, and gradient estimators.Source: §5 · pp. 6–7
  • Adversarial initializations readily steer low-AA networks away from their original fixed-point estimate, but have difficulty doing so for high-AA networks.The stress test covers 13 × 13 mazes and 64-bit prefix sum.Source: Table 1 · p. 7
  • AA score strongly correlates with accuracy when inference depth is sufficiently large; non-PI networks degrade with deeper inference and generalize poorly on harder instances.The reported Figure 4 values combine in- and out-of-distribution validation data; further task results are in the supplementary material.Source: Figure 4 · p. 7, §5 · p. 7

6 Experimental Manipulations of Path Independence

The authors intervene on path independence because prior correlations alone could reflect unobserved confounders. Promoting path independence improved validation accuracy, whereas penalizing it pushed AA scores down.

6 Experimental Manipulations of Path Independence

  • Interventions designed to promote path independence improved generalization, while interventions designed to reduce it hurt generalization.The intervention strategy addresses the possibility that earlier correlations reflected unobserved confounding.Source: §6 · p. 8

6.1 Promoting Path Independence via Randomized Forward Passes

  • Mixed initialization combines zero states on half the training examples with standard Gaussian noise on the rest; randomized compute budgets provide another way to promote path independence.Zero-state examples match test-time initialization and avoid a training-time distribution shift.Source: §6.1 · p. 8
  • Mixed initialization and randomized solver depth increased test-time path independence and reliably improved in- and out-of-distribution validation accuracy.The experiments retrained 12 prefix-sum networks using fixed-point iterations and backpropagation without changing other conditions.Source: §6.1 · p. 8

6.2 Penalizing Path Independence via the Fixed Point Alignment Penalty

  • The fixed-point alignment penalty lowered AA scores while keeping accuracy on the same trend line.The penalty targets dot products between fixed points for the same input under different Gaussian initializations.Source: §6.2 · p. 8

7 Disambiguating Convergence and Path Independence

Different solvers and training setups show that path independence does not require convergence to a single fixed point or convergence during training. The reported results instead emphasize convergence to the same limiting behavior regardless of initialization.

7 Disambiguating Convergence and Path Independence

  • Path independence requires convergence to the same limiting behavior regardless of initialization, not training-time or test-time convergence or a shared fixed point.The solvers converge to different limit cycles yet still show good upwards generalization.Source: §7 · pp. 8–9
  • Phantom-gradient DEQs can have higher in-distribution residuals yet remain path independent and show good maze upward generalization.Table 2 compares residual, AA score, and accuracy for IFT- and phantom-gradient DEQs on 9 × 9 in-distribution and 25 × 25 OOD mazes.Source: §7 · p. 8, Table 2 · p. 9
  • Five-layer unrolled networks far from convergence during training can converge and achieve perfect performance after many more test-time iterations.The networks in this intervention also had poor in-distribution generalization.Source: §7 · p. 8
  • 99.98% (fixed point iterations) and 99.97% (Broyden solver) maze accuracy accompany an adversarial AA score of 0.99 despite differing solver asymptotic behavior.Both solvers are described as showing good upwards generalization.Source: Figure 5 · p. 9

8 Path Independence on a Per-Example Level

For prefix-sum networks trained with mixed initialization, per-instance path independence is associated with prediction correctness. The authors suggest it can serve as a label-free correctness sanity check on in- and out-of-distribution examples.

8 Path Independence on a Per-Example Level

  • Per-instance fixed-point alignment scores are distributed differently for correct and incorrect predictions in mixed-initialization prefix-sum networks.The analysis uses the most performant group from the intervention experiments.Source: §8 · p. 9
  • Per-instance path independence largely tracks prediction correctness for the prefix-sum networks.The authors describe the relationship as holding on a per-instance basis.Source: §8 · p. 9
  • Path independence may serve as a label-free sanity check of prediction correctness on both in- and out-of-distribution examples.The proposed check does not require label data.Source: §8 · p. 9

9 Related Work

Related work explores adaptive inference computation through recurrent halting, early exits, and algorithmic scratchpads. This paper distinguishes its focus on understanding scalable generalization from methods that optimize the number of computational steps.

9 Related Work

  • Adaptive-computation research seeks networks that adjust computational budget to task complexity.The passage introduces this research area as a longstanding line of work.Source: §9 · p. 9
  • Adaptive-compute research includes halting-based recurrent networks and PonderNet, while this work studies scalable generalization without optimizing or penalizing computational steps.PonderNet is reported to extrapolate to harder parity examples; prior work also studies architectural and training mechanisms for weight-tied networks.Source: §9 · p. 10
  • Early-exit networks select among depth-wise prediction heads, whereas equilibrium models can use root solvers to find a fixed point.The passage contrasts equilibrium models' simple architecture with the complex modules and exit policies used by most early-exit architectures.Source: §9 · p. 10
  • Scratchpads let transformers revisit and correct errors by emitting intermediate algorithmic steps, with reported OOD generalization on algorithmic tasks.The cited example concerns transformers trained to perform well on algorithmic tasks.Source: §9 · p. 10
  • The paper uses path independence for convergence to the same limiting behavior regardless of initial state, distinct from some broader uses of global stability and convergence.Those uses may require a single point or allow convergence to different points.Source: §9 · p. 10

10 Conclusion

This section summarizes how equilibrium models use additional inference-time computation for upwards generalization and how path independence relates to harder-example performance, with promotion or penalization affecting that ability. It also considers path independence at larger inference budgets and per example, including mixed-initialization networks and cases where high scores and correct predictions do not require convergence to a fixed point.

10 Conclusion

  • Path-independent equilibrium models generalize to harder instances using more test-time compute, and interventions that promote path independence improve this ability while penalties harm it.The paper presents path independence as convergence to the same limiting behavior for an input regardless of initialization.Source: §10 · p. 10

A Alternative Approaches for Quantifying Path Independence

  • Fixed Point Alignment’s cosine version is dimensionless, tests non-local convergence, and requires only two forward passes; Jacobian and adversarial-initialization checks are costly and local.The authors prefer this score among the alternatives reviewed, while noting that no considered metric verifies global path independence.Source: A Alternative Approaches for… · p. 15
  • Gaussian, Laplacian, and inverse multiquadratic kernels preserve the qualitative finding that path independence is strongly correlated with generalization.The authors report that their takeaways remain the same across these alternatives to cosine similarity.Source: A Alternative Approaches for… · pp. 15–16, Figure 6 · p. 16

C Limitations

  • The experiments cover only prefix-sum and maze tasks, and networks trained with Broyden can struggle to find test-time fixed points under fixed-point iterations.The authors call for testing more diverse tasks and studying how solver choice affects empirical path-independence scores.Source: C Limitations · p. 21

D Path Independence vs. Accuracy Plots Different Difficulty Levels

E Intervention Results on Different Difficulty Levels

  • With extra test-time compute, mixed-initialization networks reached perfect in-distribution and near-perfect OOD accuracy, especially at length 64, despite poor training-budget performance; path-independence regularization enabled very shallow models to exploit this compute.Some networks were fewer than six layers deep.Source: E Intervention Results on Different… · p. 22, Figure 8 · p. 24

F Per-Instance Path Independence Analyses - Convergence vs. Path Independence

  • Fixed-point convergence usually accompanied high FPA and accuracy, while limit cycles retained high AA and accuracy; solver divergence corresponded to high residuals, low AA, and low accuracy.The transition between full convergence and limit cycles was gradual, and divergence was usually the source of per-instance lack of path independence.Source: F Per-Instance Path Independence… · p. 22, Figure 9 · p. 25

G Test Time Convergence and Path independence

  • 0.9% for 9x9 mazes, 0.49% for 13x13 mazes, and 1.5% for 25x25 mazes were limit-cycle residuals as percentages of point Euclidean norms, despite correct predictions.Naive fixed-point iterations could enter localized limit cycles on both in-distribution and harder mazes, showing that convergence to the same test-time fixed point is not necessary for path independence.Source: G Test Time Convergence and Path… · p. 25

H Results on the Blurry MNIST Task

On BlurryMNIST, path independence correlates with accuracy when models are tested on more heavily blurred digits than those used for training. The task varies Gaussian blur across splits, with training on the four least-blurred splits and testing across all splits.

H Results on the Blurry MNIST Task

I Results on Matrix Inversion Task

On matrix inversion, the best equilibrium model matches the energy-based approach and significantly beats its baselines. Across tested configurations, lack of path independence correlates with poor generalization to matrices with worse condition numbers.

I Results on Matrix Inversion Task

J Results on the Edge Copy Task

On the edge copy task, the best equilibrium model outperforms the energy-based approach, and lack of path independence correlates with poor generalization to larger graphs. The graph cell refines edge features at each application.

J Results on the Edge Copy Task

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