Einstellung Rigidity Index in Continual Learning

• Einstellung Rigidity Index (ERI) is a diagnostic tool that quantifies a continual learning model's reliance on shortcut features using a triplet of metrics (AD, PD, SFR_rel).
• It distinguishes genuine transfer from spurious cue exploitation by comparing scratch models with continual learners through patched and masked evaluation scenarios.
• Empirical application on benchmarks like CIFAR-100 reveals patterns that help adjust regularization and replay mechanisms to mitigate shortcut-induced rigidity.

The Einstellung Rigidity Index (ERI) is a lightweight, three-part diagnostic for continual learning (CL) that detects when a model is “locked in” to suboptimal, shortcut features inherited from earlier tasks, much as the human Einstellung effect describes how prior solutions inhibit finding better ones. It is designed for a two-phase setting in which apparent transfer can be inflated by incidental correlations between inputs and labels without causal meaning, and it represents rigidity as the ordered triplet $$(\mathrm{AD}, \mathrm{PD}, \mathrm{SFR}_{\mathrm{rel}})$$, intended to disentangle genuine transfer from cue-inflated performance (Gu et al., 1 Oct 2025).

1. Conceptual basis

Shortcut features are defined as incidental correlations between inputs and labels without causal meaning. In deep neural networks, such features undermine robustness and reduce reliability under distribution shifts. In continual learning, the consequences of shortcut exploitation can persist and intensify: weights inherited from earlier tasks bias representation reuse toward whatever features most easily satisfied prior labels, mirroring the cognitive Einstellung effect, in which past habits block optimal solutions.

Within this framing, shortcut-induced rigidity is distinct from catastrophic forgetting. Catastrophic forgetting measures loss of old skills, whereas rigidity measures over-reliance on old, possibly spurious, features when learning new tasks. A model can therefore adapt quickly on a new task simply by exploiting a shortcut, giving an illusion of fast transfer while the resulting behavior remains brittle and cue-driven. ERI is introduced precisely to separate two cases that can otherwise look similar in ordinary accuracy curves: genuine transfer based on semantic features that truly generalize, and shortcut reliance in which performance is inflated by suboptimal, spurious cues.

This distinction is central to the intended use of ERI. The diagnostic does not reduce continual-learning behavior to forgetting alone; instead, it asks whether prior learning constrains the acquisition of new discriminative structure, especially when regularizers or replay mechanisms protect weights that already encode an easy but misleading cue.

2. Formal construction

ERI is defined in a two-phase CL setting. Phase 2 has a label set $\mathcal{Y}_{T2}$, of which a subset $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$ is augmented with a fixed spurious cue such as a magenta patch. For any model $M$, the relevant performance trace is the macro-averaged top-1 accuracy on the patched Phase 2 test classes after $e$ effective Phase 2 epochs, where replay or generated samples do not count toward $e$:

$$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$

The comparison is between a Scratch-T2 model $M_S$, trained from random initialization on Phase 2 only, and a continual model $M_{CL}$, finetuned from a Phase 1 pretrained checkpoint. The three ERI facets are defined as follows (Gu et al., 1 Oct 2025).

Facet	Definition	Sign interpretation
Adaptation Delay (AD)	$\mathrm{AD}=E_{CL}(\tau)-E_S(\tau)$	Negative AD means the continual learner crosses the threshold faster than scratch
Performance Deficit (PD)	$\mathcal{Y}_{T2}$0	Positive PD indicates scratch ultimately achieves higher patched accuracy
Relative Suboptimal Feature Reliance ($\mathcal{Y}_{T2}$1)	$\mathcal{Y}_{T2}$2	Interpreted relative to whether the cue helps or harms scratch

For Adaptation Delay, fix a threshold $\mathcal{Y}_{T2}$3, for example $\mathcal{Y}_{T2}$4, and define

$\mathcal{Y}_{T2}$5

Then

$\mathcal{Y}_{T2}$6

For Performance Deficit, let $\mathcal{Y}_{T2}$7 denote final patched accuracy at the best Phase 2 validation checkpoint. Then

$\mathcal{Y}_{T2}$8

For Relative Suboptimal Feature Reliance, define the accuracy change induced by masking the patch:

$\mathcal{Y}_{T2}$9

Then

$$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$0

If $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$1, the scratch baseline benefits from the cue, and $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$2 means the continual learner relies on it even more. If $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$3, so that the cue is harmful, then $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$4 signals that the continual model is less harmed by the cue than scratch is. ERI is therefore not a scalar summary but the ordered triplet

$$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$5

3. Diagnostic protocol

The reported diagnostic protocol uses a two-phase CIFAR-100 benchmark. In Phase 1 (T1), a ResNet-18 is trained on 8 “semantic” superclasses with no patch. In Phase 2 (T2), 4 new superclasses are introduced; two of these, denoted “SC,” receive a $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$6 magenta patch in the top-left corner of every training and test image, while two “NSC” classes do not. At test time, each model is evaluated twice: once with the patch present (“patched”) and once with the same patch region replaced by black (“masked”).

The benchmark compares one Phase-2-only baseline and five continual learners. The baseline is Scratch_T2, trained from random initialization on Phase 2 only. The continual learners are Naïve SGD fine-tuning, Online EWC (EWC_on), Dark Experience Replay (DER++), Gradient Projection Memory (GPM), and Deep Generative Replay (DGR).

The measurement procedure is explicitly stepwise. First, Scratch_T2 is trained on Phase 2 alone, and $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$7 together with final patched and masked accuracies are recorded on the shortcut classes. Second, each continual method is pretrained on Phase 1 and then continued on Phase 2, recording $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$8, $$\mathcal{C}_{\mathrm{patch}} \subseteq \mathcal{Y}_{T2}$$9, and $M$0. Third, the patched-accuracy curves are smoothed with a moving average of width $M$1, and $M$2 is used to compute $M$3 and $M$4. Fourth, AD, PD, and $M$5 are computed and reported as ERI.

An algorithmic recipe is also specified. The inputs are arrays $M$6 and $M$7 for $M$8, final paired accuracies $M$9 and $e$0, and a threshold $e$1. The computation smooths both curves with moving average window $e$2, finds the earliest threshold-crossing epochs, sets $e$3, sets $e$4, computes $e$5 and $e$6, and returns $e$7 with $e$8. If no threshold crossing occurs, AD is flagged undefined.

4. Empirical profile on the two-phase CIFAR-100 benchmark

On the patched shortcut classes with $e$9 and a 50-epoch budget, the reported ERI profile is method-dependent but not ambiguous in its aggregate pattern (Gu et al., 1 Oct 2025). For Adaptation Delay, SGD, EWC_on, DER++, and GPM all achieve the threshold earlier than Scratch_T2, yielding $e$0. DGR often never crosses the threshold within the budget, so AD is undefined, or large positive if extrapolated.

For Performance Deficit, Scratch_T2 reaches final patched shortcut-class accuracy of approximately $e$1. SGD, EWC_on, DER++, and GPM are reported at approximately $e$2–$e$3, which corresponds to $e$4–$e$5. DGR reaches approximately $e$6, corresponding to $e$7.

For Relative Suboptimal Feature Reliance, Scratch_T2 has $e$8, meaning that the scratch baseline benefits from the patch. The continual-learning methods instead exhibit $e$9, meaning that they are harmed by the patch; accordingly, $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$0. The aggregate result is therefore not a case in which continual learning exploits the spurious cue more effectively than the scratch baseline. Rather, the patch behaves as a distractor for the continual learners in this setting.

DGR is singled out as the most extreme case. It adapts very slowly, often fails to reach the threshold, underperforms greatly at convergence, and is harmed by the patch. The source interpretation is that this suggests overly broad replay constraints that lock the model out of discovering new discriminative features.

5. Interpretation and uses

The proposed “red-flag” rigidity pattern, in the case where the cue helps the baseline, is

$$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$1

This pattern indicates fast threshold crossing, no final penalty relative to scratch, and greater reliance on the shortcut than the baseline. In the reported benchmark, that pattern is not observed. Instead, the empirical configuration is $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$2, $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$3, and $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$4: continual learners adapt faster than scratch, but scratch converges slightly better on patched classes, and the patch harms continual learners more than it harms scratch. The stated interpretation is that the patch acted as a distractor for the CL methods rather than a helpful shortcut (Gu et al., 1 Oct 2025).

This interpretation helps avoid a common misreading of rapid adaptation curves. Negative AD alone is not evidence of beneficial transfer; it can coexist with lower final performance or with harmful cue sensitivity. ERI is explicitly constructed to expose this possibility by combining threshold dynamics, end-of-training performance, and a masked-versus-patched counterfactual.

The practical uses proposed for ERI are diagnostic rather than standalone. One use is screening new tasks: ERI can be computed alongside ACC, BWT, and FWT, with a triplet such as $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$5 flagging likely shortcut-induced rigidity and motivating further probes such as counterfactual patch placements or representation-drift analyses. A second use is mitigation. If ERI indicates rigidity, the suggested responses are to weaken regularizers or replay strength to boost plasticity, introduce adversarial or counterfactual augmentations that discourage shortcut pathways, or apply targeted masking-aware fine-tuning on suspected classes. A third use addresses unknown cues: when the spurious feature is not known in advance, occlusion-sensitivity maps, attribution-guided deletion, or counterfactual inpainting can be used to approximate $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$6 and still compute a conservative $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$7.

6. Scope, limitations, and terminological distinctions

ERI, as introduced for continual learning, is tied to a specific diagnostic setup: a two-phase task sequence, a comparison between scratch and continual models, a designated subset of Phase 2 classes associated with a spurious cue, and a patched-versus-masked evaluation. Its compactness comes from representing rigidity as a triplet rather than a more elaborate diagnostic suite. A plausible implication is that its interpretability depends on the quality of the counterfactual intervention used to isolate the cue, since $$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$8 is defined through the difference between patched and masked accuracies.

The term should also be distinguished from a different mathematical usage of “rigidity index” or “index of rigidity” in the theory of meromorphic connections on compact Riemann surfaces. In that setting, the rigidity index is defined as the Euler–Poincaré characteristic of the de Rham complex of the middle extension of the endomorphism connection,

$$\mathcal{A}_M(e)=\mathrm{Acc}\bigl(M,\mathcal{D}_{T2}^{\mathrm{test,patch}}\bigr).$$9

and Hiroe proves, under specific hypotheses, that this equals the Euler characteristic of the normalization of the associated irregular spectral curve (Hiroe, 2019). That object is unrelated in construction, domain, and interpretation to the continual-learning ERI. The shared acronym therefore does not indicate a shared formalism.

Within continual learning itself, the principal conceptual distinction is between forgetting and rigidity. Forgetting concerns erosion of previously acquired competencies; rigidity concerns the persistence of inherited feature preferences, including spurious ones, during acquisition of a new task. ERI is designed for the latter phenomenon. In that sense, it functions as a compact litmus test for whether apparent transfer in continual learning is truly semantic or shortcut-driven, while preserving an interpretable decomposition into adaptation speed, terminal performance, and relative cue reliance.