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Traversing the solution space of neural networks with Hessian Null Space Continuation

Ann Huang, Mitchell Ostrow, Zhouyang Lu, William T. Redman, Leo Kozachkov, Kanaka Rajan

arXiv:2609.38081v1cs.LGq-bio.NCstat.ML

TL;DR

It is unknown whether the connected low-loss region around one trained network contains genuinely different internal mechanisms. HNC traverses function-preserving alternatives using local curvature and can steer the search toward specified properties. Across RNNs, Vision Transformers, and reinforcement-learning agents, it finds diverse representations, dynamics, and strategies, including reward-hacking policies.

  • Problem

    It remains unknown whether the connected low-loss region around a single trained solution contains genuinely distinct representations and mechanisms, rather than minor modifications.

    Source: §1 · p. 2

  • Method

    HNC uses local curvature to traverse function-preserving alternatives around a trained network and can steer movement toward a differentiable objective.

    Source: §1 · p. 3, §2.2 · p. 3

  • Results

    Across RNNs, Vision Transformers, and reinforcement-learning policies, HNC found substantially different representations, dynamics, and behavioral strategies while maintaining task behavior or comparable return.

    Source: §1 · p. 3, §7 · p. 9

  • Takeaways & Limitations

    HNC quantifies local solution-set geometry, revealing how model size and task complexity shape its dimension and functional sensitivity.

    Source: §7 · p. 9

    Function preservation is approximate on a finite probe set, so low matching loss does not guarantee agreement on unseen inputs, particularly under distribution shift.

    Source: §7 · p. 9

Abstract

from arXiv · show

On a single task, deep networks can learn many solutions, depending on their optimizer, training data, architecture, and hyperparameters. Many of these solutions are mode-connected: rather than isolated points in weight space, they are connected by low-loss regions. Yet how their internal computation varies within these regions is unknown. A parallel line of work has identified the degeneracy of neural representations: many networks reach similar training loss with distinct internal structures. However, it is unclear how these solutions are related in weight space. We unify these subfields and show for the first time that many different internal mechanisms exist within a local mode-connected region in weight space. To do so, we introduce Hessian Null Space Continuation (HNC), a scalable method that uses local curvature to traverse regions of weight space that preserve network function, and can be steered toward solutions with specified properties. In RNNs trained on a memory task, HNC reaches drastically different representations and dynamics with maintained behavior. In ImageNet-trained Vision Transformers, HNC finds representations that differ more from the original network than any independently trained model with a different architecture or objective. In reinforcement-learning agents, HNC uncovers a distinct navigation strategy at comparable return and exposes reward hacking in an AI Safety Gridworld. Finally, HNC measures the local geometry of the solution set, showing how model size and task complexity shape its dimension and functional sensitivity. Our results show that a surprisingly large amount of representational diversity exists near a single trained solution, unseen by standard gradient-based optimization. HNC identifies and quantifies this diversity, opening new possibilities for mechanistic understanding of solution spaces and for model merging, editing, and fine-tuning.

1 INTRODUCTION

Although mode-connectivity work finds connected low-loss regions, it has not established whether one such region contains genuinely different internal mechanisms. HNC traverses function-preserving alternatives around a trained network, revealing substantial diversity across several settings.

1 INTRODUCTION

  • Prior mode-connectivity studies show that low-loss paths can connect minima, but do not systematically test whether one solution’s surrounding region contains distinct mechanisms.Earlier work examined pairs of trained networks or characterized regions through connectivity and loss.Source: §1 · p. 2
  • Work on solution degeneracy reports distinct representations and behaviors for similar task performance, but their relationships in parameter space remain poorly understood.The open question is how much representational diversity lies in a connected, function-preserving region around one trained solution.Source: §1 · p. 2
  • Characterizing a model’s broader solution set can separate task demands from optimizer-induced constraints and expose nearby degrees of freedom for fine-tuning, merging, and editing.The practical applications are framed as possibilities if representationally distinct solutions are reachable through local, function-preserving updates.Source: §1 · pp. 2–3
  • HNC finds diverse function-preserving solutions near one trained network, including different RNN dynamics, ViT representations, and reinforcement-learning strategies.The authors also use HNC to quantify diversity and local solution-set geometry.Source: §1 · p. 3

2 METHOD

HNC uses the function-matching loss’s local curvature to identify directions that approximately preserve a trained network’s outputs. It iteratively follows those directions, restores function matching, and can steer movement toward a differentiable objective.

2.1 LOSS LANDSCAPE CURVATURE REVEALS FUNCTION-PRESERVING DIRECTIONS

  • The function-matching loss measures output drift from the anchor, and its Hessian identifies sharp directions with high output sensitivity and near-zero directions for approximate function-preserving movement.The loss is estimated over a fixed probe set sampled from the input distribution.Source: §2.1 · p. 3

2.2 HESSIAN NULL SPACE CONTINUATION

  • HNC alternates a small null-space step with gradient-based restoration of function matching, then recomputes the local null space when restoration loss exceeds a threshold.Restoration corrects higher-order drift; relinearization addresses the fact that the Hessian is only a local, second-order approximation.Source: §2.2 · pp. 3–4

2.2.1 FORMING THE NULL SPACE

  • The method forms the approximate null space from the smallest Hessian eigenpairs, computed with LOBPCG and Hessian-vector products rather than explicitly constructing the Hessian.A direction is treated as flat when its eigenvalue is at most µrel times the largest eigenvalue at that linearization point.Source: §2.2.1 · p. 4

2.2.2 STEERING

  • For steering, HNC softly projects a differentiable objective’s gradient onto flat directions, suppressing components along sharp directions according to their eigenvalues.The projection uses damping parameter µ, set to µrel times the largest eigenvalue so the flatness threshold also controls gradient retention.Source: §2.2.2 · p. 4

3 HNC DRIVES RNNS TO LEARN DRASTICALLY DIFFERENT DYNAMICS IN A MEMORY TASK

In the 3-bit flip-flop task, HNC reached alternative RNN solutions with maintained task behavior but substantially changed representations and dynamics. The alternatives could implement memory through ongoing hidden-state movement rather than fixed points.

3 HNC DRIVES RNNS TO LEARN DRASTICALLY DIFFERENT DYNAMICS IN A MEMORY TASK

  • The task trains 64-unit tanh RNNs to maintain three binary memories, with the anchor representing eight memory states as stable fixed points at cube vertices.The anchors had a broad near-zero Hessian bulk, with a participation ratio of about 23 across five anchors.Source: §3 · p. 4
  • All HNC walks kept task loss below 0.01, while steered walks achieved greater representational and dynamical divergence than independently trained networks with less weight movement.Undirected walks reached distances comparable to independently trained networks; steered walks altered fixed-point counts and produced visibly different state-space structure.Source: §3 · pp. 4–5
  • The undirected endpoint maintained outputs in compact readout-space regions while hidden states continued drifting mainly in readout-null space; steered endpoints showed reduced perturbation robustness or prolonged-memory drift.The CKA-steered endpoint was less robust to hidden-state perturbations, while the DSA-steered endpoint showed memory drift during prolonged memory periods.Source: §3 · p. 5
  • Across 131 accurate networks, steered walks extended farther in representation space while remaining closer in weight space than undirected walks and independent seeds.The networks all achieved accuracy above 0.999, and the solutions did not collapse into one tight representational cluster.Source: §3 · pp. 5–6

4 HNC REVEALS UNDEREXPLORED DEGREES OF FREEDOM IN VITS

The ViT experiments show that HNC can substantially reorganize representations while largely preserving the anchor’s behavior. The resulting representation differs from the anchor more than those of the trained comparison models, despite a small weight change.

4 HNC REVEALS UNDEREXPLORED DEGREES OF FREEDOM IN VITS

  • The study steers a pretrained ViT-S/16 to minimize max-over-layers CKA to its anchor; max-over-layers kNN similarity also falls without direct steering.The representational similarity decline occurs on both probe images and images unseen during the walk.Source: §4 · p. 6
  • On ImageNet, which did not constrain HNC, top-1 accuracy drops by less than 1% while top-5 accuracy remains nearly unchanged.Source: §4 · p. 7
  • With a 1.3% weight-norm change, the HNC endpoint is less similar to the anchor than every trained vision model compared, and even an untrained ViT.The compared trained models span different sizes, architectures, objectives, and training data.Source: §4 · p. 7
  • The endpoint’s closest image pairs remain similar in semantic content and color but approach the random-pair baseline in spatial-layout similarity.The pairwise similarities and MDS embedding also show reorganized neighborhoods within and across classes.Source: §4 · p. 7

5 HNC FINDS BEHAVIORALLY DISTINCT, HIGH-REWARD POLICIES IN REINFORCEMENT LEARNING

HNC searches for policies with different actions while preserving return, finding distinct navigation behavior in Plume Tracking and reward-hacking behavior in a boat-race task. These cases show that nearby policies can behave differently at similar reward while exposing a proxy reward’s exploitability.

5 HNC FINDS BEHAVIORALLY DISTINCT, HIGH-REWARD POLICIES IN REINFORCEMENT LEARNING

  • Because scalar reward rarely fully specifies optimal action sequences, reinforcement learning can admit behaviorally distinct policies with similar reward, including unintended reward exploits.Source: §5 · p. 7
  • HNC preserves an off-policy surrogate reward from anchor rollouts and steers a null-space walk to maximize KL divergence between anchor and alternative action distributions.The state-action buffer and surrogate are periodically recomputed along the walk.Source: §5 · pp. 7–8
  • In Plume Tracking, HNC increases action divergence while maintaining return and success, finding a policy that tracks the plume edge rather than surging straight toward it.The alternative casts less and stays farther from the plume centerline; it also outperforms the anchor under sparse odor or switching wind.Source: Figure 4 · p. 7, §5 · p. 8
  • In the boat-race task (Leike et al., 2017), the three most divergent endpoints exploit the proxy reward by oscillating on one arrow tile, with zero net clockwise progress.The proxy rewards clockwise entry onto an arrow tile, while true return measures net clockwise progress and is hidden from the agent during standard training.Source: Figure 4 · p. 7, §5 · p. 8

6 HNC MEASURES LOSS LANDSCAPE GEOMETRY

HNC measurements show how model width and task complexity relate to local flatness and curvature. Although approximate flat directions remain abundant, directions followed on harder tasks become more functionally sensitive.

6 HNC MEASURES LOSS LANDSCAPE GEOMETRY

  • The effective null fraction estimates parameter directions where a small step leaves outputs approximately unchanged, using a shared output-drift tolerance across conditions.The cutoff corresponds to a 1% weight-norm step along a flat direction raising function-matching loss by at most ϵ = 0.05.Source: §6 · pp. 8–9
  • The effective null fraction grows with width and shrinks as the number of classes increases, while normalized curvature generally rises with class count and falls with width.Even the smallest network on the hardest task retains a large effective null fraction; the curvature trends persist across tested relative thresholds.Source: Figure 5 · p. 8, §6 · p. 9
  • Normalized curvature measures output sensitivity to small weight changes along HNC-stepped directions; its increase on harder tasks indicates a stiffer local landscape.The study varies CNN width and number of classes as a proxy for task complexity.Source: §6 · pp. 8–9

7 DISCUSSION

HNC explores function-preserving alternatives near a trained solution and measures local solution-space geometry. The discussion relates this local diversity to optimizer bias and notes limits from finite probes and local search; the appendices detail its scalable curvature computations.

7 DISCUSSION

  • Function preservation is approximate on a finite probe set, so unseen or distribution-shifted inputs may disagree; HNC is also a local search, though global methods can broaden it.Source: §7 · p. 9
  • HNC’s local alternatives help distinguish task-demanded properties from those specific to an individual training run, while larger models may admit more alternatives even as training converges to a narrower subset.The discussion also notes that HNC reaches qualitative dynamical and behavioral changes through continuous, small weight changes beyond reparameterizations.Source: §7 · p. 9

A HESSIAN NULL-SPACE CONTINUATION PSEUDOCODE

  • HNC’s pseudocode takes a trained anchor and probe set, steps within a near-null subspace, restores function matching, rejects steps above a loss ceiling, and periodically relinearizes.An optional potential steers the step direction; otherwise, exploration is undirected.Source: A HESSIAN NULL-SPACE CONTINUATION… · p. 18

A.1 COMPUTATIONAL COST

  • For the 22-million-parameter ViT-S, a dense Hessian would require about 2 PB of memory, whereas matrix-free HNC scales as O(kP) for fixed block size and iteration count.The matrix-free route applies Hessian-vector products and LOBPCG stores a block of k vectors.Source: §A.1 · p. 18, Table 1 · p. 19

A.2 FINDING THE FLATTEST DIRECTIONS BY LOBPCG

  • LOBPCG finds the flattest directions by minimizing the Rayleigh quotient, which measures function-matching-loss curvature along a parameter direction, using Hessian-vector products without forming H.Source: §A.2 · p. 19

A.3 SOLVING THE SOFT PROJECTION BY CONJUGATE GRADIENTS

  • The steered direction solves a symmetric positive-definite linear system with conjugate gradients, using products with the Gauss–Newton operator rather than forming it.The solve is warm-started from the previous direction except in the 3BFF walks.Source: §A.3 · p. 19

B GAUSS–NEWTON APPROXIMATION OF THE HESSIAN AT LOSS MINIMA

C ABLATION STUDY ON HNC ALGORITHM

The ablations examine how function restoration, steering-gradient projection, and relinearization affect HNC’s function preservation. HNC also finds greater representational divergence than penalty-based training at every matched task-loss level.

C.1 FUNCTION-RESTORING STEP

  • At η = 0.3, twelve function-restoring steps reduce endpoint task loss from 6.6 to 3.6 times the anchor’s loss, but do not eliminate drift.Smaller null-space steps protect the function more effectively than correcting drift afterward.Source: §C.1 · p. 20, Figure 6 · p. 20

C.2 NULL-SPACE PROJECTION OF THE STEERING GRADIENT

  • Larger damping speeds representational divergence at the cost of higher task loss; at µrel = 10^-9, neither changes.A strict projection filters out sharper-direction steering, while no projection can break function within a few dozen steps.Source: §C.2 · p. 21, Figure 7 · p. 21

C.3 RELINEARIZATION

  • More frequent relinearization lowers task loss, and is especially important for preventing blow-up with large steps.Relinearization recomputes the null space at the current weights as the walk moves away from its anchor.Source: §C.3 · p. 21, Figure 8 · p. 22

C.4 COMPARISON WITH PENALIZED OPTIMIZATION

  • At every task-loss level, HNC reaches greater divergence than penalized training while staying in the low-loss region.Penalized training crosses the loss ceiling within its first 20 steps and gains nearly all its divergence during that excursion.Source: Figure 9 · p. 22, §C.4 · p. 22

D PRESERVING FUNCTION VERSUS TASK LOSS IN HNC

HNC preserves the anchor’s function rather than task loss because task-loss-flat steps can drift when the anchor’s task-loss gradient is nonzero. Experiments find that function matching better preserves outputs and task performance than weaker cross-entropy constraints, while defining a narrower solution set than task compatibility alone.

D PRESERVING FUNCTION VERSUS TASK LOSS IN HNC

  • A task-loss-flat step can still raise loss by η g⊤v when the anchor’s task-loss gradient is nonzero; function matching has zero gradient at the anchor.At HNC’s small step sizes, the first-order term can dominate the curvature term, and drift accumulates across steps.Source: D PRESERVING FUNCTION VERSUS TASK LOSS… · p. 23
  • Function-matching walks keep outputs pinned and hold task loss within 10% of its anchor value, while task-loss walks multiply task loss roughly tenfold.On the tested 3BFF anchors, first-order task-loss drift per step exceeds its second-order term by two to three orders of magnitude.Source: D PRESERVING FUNCTION VERSUS TASK LOSS… · pp. 23–24, Figure 10 · p. 23
  • On 3BFF, task MSE is nearly function matching to a fixed target because the task target is the full output trajectory and the anchor reaches it with a small residual.For a broader task-compatible set, the passages describe replacing the function-matching constraint with task loss or tightening constraints on probes and per-input outputs.Source: D PRESERVING FUNCTION VERSUS TASK LOSS… · pp. 24–25
  • On the 512-image ViT-S probe, full-logit matching preserves almost every anchor prediction, while per-image and mean cross-entropy walks lose predictions immediately.The cross-entropy walks also lose more held-out accuracy; their constraint counts are one per image and one for the whole probe, respectively, versus a thousand logits per image.Source: D PRESERVING FUNCTION VERSUS TASK LOSS… · pp. 24–25, Figure 11 · p. 24, Table 2 · p. 25
  • Function matching needs neither labels nor a task and can be applied to unlabeled probes, shifted inputs, policies, and next-token distributions.Because task loss is a function of outputs, preserving the function preserves every task loss; the function-preserving set is a subset of the task-compatible set.Source: D PRESERVING FUNCTION VERSUS TASK LOSS… · p. 25

E COMBINING HNC WITH GLOBAL SEARCH FOR ALTERNATIVE SOLUTIONS

The authors combine local, function-preserving HNC walks with fresh retraining to search for increasingly different solutions. In the 3BFF task, the resulting chain preserves accuracy while spanning distinct dynamical mechanisms and greater diversity than independent retraining alone.

E COMBINING HNC WITH GLOBAL SEARCH FOR ALTERNATIVE SOLUTIONS

  • Alternating a DSA-steered HNC walk with retraining from scratch, using summed dynamical distance as a repulsive regularizer, creates a chain of solutions that each diverges from those found so far.The walk moves the current network along its loss level set; each fresh network is trained to land far from the existing set.Source: E COMBINING HNC WITH GLOBAL SEARCH FOR… · p. 26
  • Each new 3BFF network lies farther from every predecessor in DSA distance than two independently trained seeds lie from each other, while all networks retain the anchor’s accuracy.Retraining moves about as far in weight space as an unrelated seed, and each subsequent HNC walk moves less than one anchor weight norm while remaining below the loss ceiling.Source: E COMBINING HNC WITH GLOBAL SEARCH FOR… · pp. 26–27, Figure 12 · p. 26
  • The chain spans attractor-based memories, drift-based memories without stable fixed points, and a solution with more attractors than memory states.These mechanisms have correspondingly different recurrent dynamical spectra.Source: E COMBINING HNC WITH GLOBAL SEARCH FOR… · p. 27, Figure 13 · p. 27, Figure 14 · p. 28
  • Broader search could allow higher loss ceilings followed by filtering, restart walks from perturbed endpoints, or repel solutions across architectures and optimizers; these extensions remain future work.The authors also propose steering toward a target network to bridge independently found solutions with function-preserving intermediates.Source: E COMBINING HNC WITH GLOBAL SEARCH FOR… · p. 27

F.1 ARCHITECTURE AND TRAINING OF THE 3BFF RNN

These appendices detail the 3BFF RNN experiments and additional HNC analyses, including flat-subspace measurement, structured pruning, optimization stability, and steering objectives for Vision Transformers. Across these tests, HNC finds function-preserving alternatives, though behavior can vary under distribution shift.

F.1 ARCHITECTURE AND TRAINING OF THE 3BFF RNN

  • The 3BFF experiments use a 64-hidden-unit tanh RNN to retain the latest nonzero input on each of three independent channels over 100-step trials.Training uses MSE with Adam; it stops after two consecutive epochs below 5 × 10^-5 loss.Source: §F.1 · p. 29

F.2 SIZE OF THE FLAT SUBSPACE UNDER DIFFERENT THRESHOLDS

  • The Hessian’s effective spectral dimensionality is 23.2 ± 1.6 of 4,611 parameters, while its smoothly decaying eigenvalues leave flat-subspace size dependent on the chosen threshold.The estimate is the participation ratio averaged over five anchors; Figure 15 shows the Hessian spectra and threshold-dependent null fraction.Source: §F.2 · p. 29, Figure 15 · p. 29, Table 3 · p. 29

F.5 MDS EMBEDDINGS OF THE SOLUTIONS REACHABLE FROM TEN ANCHORS

  • Across ten anchors, HNC walks spread representations several times farther than independent training, while the corresponding weight-space groups remain centered on their own anchors.The solutions separate at the representational level but remain associated with their starting anchor in weight space.Source: §F.5 · p. 32, Figure 17 · p. 32

F.6 FUNCTION-PRESERVING STRUCTURED PRUNING

  • HNC prunes the 64-unit 3BFF RNN to 11 units while retaining 100% task accuracy, leaving about 17% of the hidden units and 5% of the original parameters.The pruned model has a small increase in task MSE; truncation keeps the units with the largest outgoing group norms.Source: §F.6 · p. 33, Figure 18 · p. 33

F.7 STEERED SOLUTIONS ARE SGD-STABLE

  • After 8,000 resumed SGD steps, steered endpoints remain near their starting solutions, retaining the targeted representation, dynamics, or sparsity property rather than returning toward the anchor.Retraining moves weights far less than training from random initialization.Source: §F.7 · p. 34, Figure 19 · p. 34, Figure 20 · p. 34

F.8 OUT-OF-DISTRIBUTION PROBES OF THE WALK ENDPOINTS

  • Under out-of-distribution inputs, weak pulses degrade the DSA-steered and undirected endpoints before the anchor, while strong pulses degrade only the undirected endpoint.With rare pulses, the DSA-steered endpoint loses accuracy first, consistent with weaker memory retention.Source: §F.8 · p. 35, Figure 21 · p. 35

G.1 SMOOTH SURROGATE FOR THE MAX-OVER-LAYERS CKA

  • For ViT CKA steering, a log-sum-exp surrogate smooths the maximum over layer-pair similarities, weighting gradients toward the pairs currently closest to that maximum.The reported score remains the exact hard maximum.Source: §G.1 · p. 36

G.2 SCORING THE VISUAL ATTRIBUTES OF THE MOST SIMILAR IMAGE PAIRS

  • The mutual-kNN steering objective smooths neighborhood membership and uses a log-sum-exp over layer pairs, while probe-image similarity pairs are scored by semantic content, color, and spatial layout.The kNN score measures shared nearest neighbors across representations; the visual-attribute scores are reported as percentiles among image pairs.Source: §G.3 · pp. 36–37, §G.2 · p. 36

G.5 FUNCTIONAL PRESERVATION BEYOND THE PROBE SET FOR CKA-STEERED HNC

The CKA-steered HNC walks preserve much of the anchor’s behavior on held-out and distribution-shifted images despite substantial representational change. Their image-level agreement with the anchor generally matches or exceeds that of independently trained references, but preservation is not exact.

G.5 FUNCTIONAL PRESERVATION BEYOND THE PROBE SET FOR CKA-STEERED HNC

  • CKA-steered endpoints track the anchor’s top-k accuracy across four test sets and retain error consistency near 0.8, even as exact prediction agreement declines on Rendition and Sketch.The evaluation covers clean ImageNet, ImageNet-V2, ImageNet-Rendition, and ImageNet-Sketch; Figure 23 reports accuracy, prediction match, and error consistency.Source: §G.5 · p. 39, Figure 23 · p. 40
  • Endpoints have higher prediction match and error consistency than ViT-S/in1k on every dataset, and image-level agreement comparable to or generally higher than diff-seed networks despite greater representational movement.Accuracy loss and prediction disagreement remain, so preservation is approximate rather than exact.Source: §G.5 · p. 39, Figure 23 · p. 40
  • Random weight noise matched to the endpoint’s accuracy loss leaves CKA above 0.98, whereas the HNC walk falls below the untrained-network reference after losing less than 1% accuracy.This comparison indicates that HNC’s representational displacement is not explained by generic performance degradation.Source: §G.5 · p. 39, Figure 24 · p. 39

G.6 FUNCTIONAL PRESERVATION BEYOND THE PROBE SET FOR MUTUAL-kNN-STEERED HNC

The additional experiments test mutual-kNN steering, layer-specific divergence, convergence toward another model, probe-size effects, and the geometry of reachable ViT solutions. They show substantial representational freedom, with functional preservation depending on the steering configuration and probe size.

G.6 FUNCTIONAL PRESERVATION BEYOND THE PROBE SET FOR MUTUAL-kNN-STEERED HNC

  • Mutual-kNN-steered endpoints retain substantial behavior but preserve it less faithfully than CKA-steered walks or diff-seed networks; error consistency remains around 0.75 across datasets.Their accuracy deficit is larger than for CKA steering, though they agree with the anchor more strongly than ViT-S/in1k.Source: §G.6 · p. 41, Figure 25 · p. 42
  • The less favorable preservation–displacement tradeoff reflects the tested walk configuration as well as the metric, not an intrinsic limitation of mutual-kNN steering.The probe-size experiment indicates that larger probes improve off-probe preservation.Source: §G.6 · p. 41

G.7 STEERING TO MAXIMIZE REPRESENTATIONAL DIVERGENCE AT DIFFERENT LAYERS

  • Both pre-logits and block-1 class-token representations can be moved away from the anchor while preserving predictions, but moving the early layer costs more function per unit of similarity lost.The block-1 experiment is shown over a shorter weight-distance range.Source: §G.7 · p. 43, Figure 27 · p. 43, Figure 28 · p. 43

G.8 STEERED CONVERGENCE BETWEEN VIT-S AND VIT-B

  • ViT-S’s mean depth-wise CKA similarity to frozen ViT-B rises from 0.78 to 0.96 while ViT-S retains 96% of its predictions on constrained images.On 5,000 held-out ImageNet validation images, it keeps nine in ten anchor predictions and loses a little over one point of top-1 accuracy.Source: §G.8 · p. 44, Figure 29 · p. 44

G.9 SCALING WITH THE SIZE OF THE PROBE SET

  • Increasing the probe set allows more mutual-kNN metric displacement and better off-probe function preservation, with compute growing proportionally and per-step cost growing about linearly in probe size.A larger probe constrains function more firmly, letting the walk stay under its loss ceiling for more steps.Source: §G.9 · p. 45, Figure 30 · p. 45

G.10 MDS EMBEDDINGS OF THE VIT SOLUTIONS REACHED BY HNC

  • CKA- and kNN-steered walks leave the anchor in different representational directions, and apparent displacement depends on the metric: kNN walks cluster under CKA but spread under kNN distance.Walks on different probe sets fan out rather than converging on one destination; each moves weights by a small fraction of the anchor norm.Source: §G.10 · p. 46, Figure 31 · p. 46

H STEERING BRAIN-SCORE AT FIXED FUNCTION

HNC steers neural predictivity while largely preserving model behavior, and the remaining passages describe reinforcement-learning steering measurements and setup.

H STEERING BRAIN-SCORE AT FIXED FUNCTION

  • HNC holds ImageNet-pretrained models’ logits fixed while steering their features to raise or lower neural predictivity.The differentiable steering surrogate and official benchmark metric use different regressions and evaluation data; the official score is not the optimization target.Source: H STEERING BRAIN-SCORE AT FIXED FUNCTION · pp. 46–47
  • 0.618 to 0.719 for the relative-position ViT-B and 0.546 to 0.627 for ViT-S/16: within-network IT predictivity ranges rival the 0.074 spread across seven architectures.Top-1 decisions remained unchanged on at least 94% of images, with logit KL between 10^-5 and 10^-3.Source: H STEERING BRAIN-SCORE AT FIXED FUNCTION · p. 47, Figure 32 · p. 48

I.1 ADVANTAGE ESTIMATES

  • For plume tracking and Ant, advantages use generalized advantage estimation; boat-race walks instead use discounted proxy returns to episode end.The value head remains fixed, and advantages are standardized over the buffer.Source: §I.1 · p. 49

I.2 BEHAVIORAL DIVERGENCE FOR CONTINUOUS AND DISCRETE ACTION SPACES

  • The reinforcement-learning steering potential is action-distribution KL; with shared frozen covariance, the continuous-action form reduces to a weighted mean-squared difference between mean actions.When action-dimension variances are equal, this is proportional to the unweighted mean-action difference.Source: §I.2 · pp. 49–50

I.3 HYPERPARAMETERS OF THE REINFORCEMENT-LEARNING WALKS

  • The plume-tracking walk’s settings are listed in Table 9.The accompanying success-rate plot measures the fraction of 60 fixed initial conditions reaching the odor source against relative weight distance.Source: Table 9 · p. 50, Figure 33 · p. 51

I.5 OUT-OF-DISTRIBUTION EVALUATION OF THE PLUME-TRACKING HNC WALK

The plume-tracking walk finds policies with improved homing under shifted conditions, while Ant policies retain return despite changing actions, gait, and failure patterns.

I.5 OUT-OF-DISTRIBUTION EVALUATION OF THE PLUME-TRACKING HNC WALK

  • Every shifted plume-tracking condition improves along the walk; the alternative policy homes on some sparse-plume and noisy-wind episodes and more than doubles the anchor’s homing rate under a wind switch.At step 20, the curves collapse together where the walk leaves the reward level set.Source: §I.5 · p. 51, Figure 34 · p. 51

I.6 THE BOAT RACE POLICIES

  • In the boat race, proxy reward can be repeatedly collected by stepping on and off one arrow tile without net progress, whereas the true return measures clockwise progress.The ten reported anchors were trained on true return and learned to circle the track.Source: §I.6 · pp. 51–52

I.7 MUJOCO ANT LOCOMOTION

  • On Ant-v3, HNC moves actions away from the anchor while preserving return and forward speed; matched-action-KL random directions lose more than half the return.The alternative changes diagonal leg timing and failure modes, and equal mean return masks uncorrelated per-episode returns across initial states.Source: §I.7 · pp. 52–53, Figure 35 · p. 52, Figure 36 · p. 53

J.1 ARCHITECTURE AND TRAINING OF THE CNN SWEEP

Across the CNN sweep, harder tasks increase representational dimensionality and functional sensitivity, while wider models show larger effective null fractions and lower curvature.

J.1 ARCHITECTURE AND TRAINING OF THE CNN SWEEP

  • Every CNN anchor reaches 100% training accuracy and exact zero training loss, while test accuracy falls with class count and rises with width.The sweep uses residual CNNs trained on CIFAR-100 class subsets.Source: §J.1 · pp. 53–54

J.2 SWEEPING MAX LOSS DRIFT TO DEFINE THE NULL-SPACE

  • At every flatness threshold ϵ, the null fraction rises with width and falls as the number of classes increases, so these trends do not depend on the choice of threshold.Source: §J.2 · p. 54

J.3 PER-STEP CURVATURE AT DIFFERENT RELATIVE THRESHOLDS

  • At every relative flatness cutoff, normalized curvature rises with class count and falls with width; looser cutoffs raise its overall level but preserve cell ordering.Curvature measures output sensitivity to small weight changes along a unit direction.Source: §J.3 · p. 54, Figure 38 · p. 55

J.4 REPRESENTATION DIMENSIONALITY ACROSS THE SWEEP

  • At fixed width, effective representation dimensionality grows about fivefold from the easiest to hardest task, while at fixed task it barely changes across widths.Cumulative-variance curves show the same pattern: more classes spread variance across more directions, while extra width leaves curves nearly unchanged.Source: §J.4 · p. 54, Figure 39 · p. 55

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