Stability-Plasticity Trade-off in Continual Learning

• SPT is the balance between preserving past knowledge (stability) and enabling the learning of new information (plasticity) across various domains.
• It underpins methods in continual learning, reinforcement learning, and parameter-efficient finetuning by explicitly trading off retention and adaptation.
• Recent studies model SPT via representation shifts, gradient conflicts, and architectural biases, introducing metrics such as CKA and SPR.

Searching arXiv for papers on the stability–plasticity trade-off across continual learning, reinforcement learning, and model adaptation. Searching arXiv for “stability plasticity trade-off continual learning”, “reinforcement learning”, and “PEFT”. The stability-plasticity trade-off (SPT) is the tension between preserving previously acquired knowledge and remaining adaptable enough to incorporate new knowledge. In continual learning, this tension is usually expressed as the conflict between stability, meaning retention of old tasks, classes, or representations, and plasticity, meaning effective adaptation to newly arriving data or objectives. Recent work treats SPT not as a narrow catastrophic-forgetting phenomenon, but as a broader principle that governs sequential adaptation in class-incremental learning, continual reinforcement learning, post-training of LLMs, parameter-efficient finetuning, early-exiting networks, recommender systems, and biologically motivated neural models (Kim et al., 2023, Chua et al., 25 May 2026, Qiang et al., 6 Feb 2026).

1. Conceptual formulation

In continual learning, the standard formulation is a stream of tasks or distributions, often written as $\mathcal{D}_t$ or $\mathcal{P}_t$, observed sequentially. Within this setting, stability is typically operationalized as preserving performance on old tasks or old samples, while plasticity is operationalized as learning the current task well. Replay-based formulations make this explicit through a weighted combination of current-task loss and replay loss, such as

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$

which already encodes SPT as a balance between two conflicting objectives rather than a single monolithic training criterion (Lai et al., 30 Mar 2025).

A central development in recent work is that SPT is increasingly treated as a property of representations, gradients, and architectures, not only of scalar regularization weights. In class-incremental learning, the key question is whether the feature extractor itself changes meaningfully as new classes arrive, rather than whether the final classifier remains calibrated (Kim et al., 2023). In continual reinforcement learning, the issue depends strongly on the nature of non-stationarity: abrupt task switches can make plasticity appear dominant, whereas gradual environmental drift can make stability the primary bottleneck (Chua et al., 25 May 2026). In GRPO-style post-training of LLMs, the trade-off appears as a conflict between a reward-driven plasticity term and a KL-driven stability term, and the paper explicitly argues that this is a geometric conflict problem in gradient space, not merely a matter of tuning a scalar $\beta$ (Qiang et al., 6 Feb 2026).

This broader view also implies that SPT is not necessarily symmetric across domains. In recommender systems, it concerns retraining behavior under changing interaction patterns rather than discrete tasks (Lavoura et al., 5 Aug 2025). In parameter-efficient finetuning, plasticity is target-task adaptation and stability is retention of pretrained general capabilities (Huang et al., 27 May 2026). In continual named entity recognition, the trade-off is complicated further by the semantic shift of the non-entity label, which mixes true non-entities with old and future entity types (Zhang et al., 5 Aug 2025).

2. Measurement and empirical diagnosis

A major theme in the literature is that standard end-point accuracy often fails to distinguish whether a method is genuinely balancing stability and plasticity or merely freezing useful components. In class-incremental learning, one influential diagnostic is to freeze the feature extractor learned at each stage and retrain a fresh classifier on the full ImageNet-1K dataset. If later representations have actually improved, the retrained model should exceed the base model. The representation-improvement metric is

$$\Delta M'_i := \text{Acc}(M'_i, D) - \text{Acc}(M'_0, D),$$

where a positive value indicates useful new features, $\Delta M'_i \approx 0$ indicates a stable but nearly static feature extractor, and a negative value indicates forgetting in the representation (Kim et al., 2023).

Stability of representations has been measured with Centered Kernel Alignment (CKA). Given activations $\mathbf{X}$ and $\mathbf{Y}$, the relevant formulation is

$$\text{CKA}(\mathbf{X}, \mathbf{Y}) = \frac{\text{HSIC}(\mathbf{XX^\top}, \mathbf{YY^\top})} {\sqrt{\text{HSIC}(\mathbf{XX^\top}, \mathbf{XX^\top})\, \text{HSIC}(\mathbf{YY^\top}, \mathbf{YY^\top})}},$$

which is used to compare layers even when dimensionalities differ. Together with t-SNE visualizations, this made it possible to show that many class-incremental methods are highly stable precisely because their features barely move (Kim et al., 2023).

Other subfields use domain-specific diagnostics. Architectural studies in continual learning evaluate AAN for plasticity and AF / FAF or RF / FRF for stability, yielding the empirical observation that deeper networks improve plasticity whereas wider networks improve stability under equal parameter constraints (Lu et al., 4 Jun 2025). Flashback Learning defines a stability-to-plasticity ratio (SPR),

$$\text{SPR} = \frac{\text{Forgetting on Old Classes}}{\text{Accuracy on New Classes}} \times 100,$$

with lower values indicating a better balance (Mahmoodi et al., 31 May 2025). In recommender systems, retraining-aware evaluation introduces

$\mathcal{P}_t$0

where the scores compare a legacy model and a retrained model across old and new data regimes (Lavoura et al., 5 Aug 2025).

The recent PEFT-Arena benchmark makes the measurement problem especially explicit by using the ordered pair

$\mathcal{P}_t$1

and comparing methods through their Pareto frontier rather than through target accuracy alone. This shifts evaluation from a single metric to a two-axis operating point and supports the argument that downstream performance alone systematically underestimates forgetting (Huang et al., 27 May 2026).

3. Continual-learning mechanisms for balancing SPT

A large family of methods tries to satisfy stability and plasticity simultaneously by changing the feasible update set or by explicitly interpolating between stable and plastic solutions. In LRFR, stability is guaranteed by projecting updates into the null space of the past-task representation matrix: $\mathcal{P}_t$2 while plasticity is improved by forcing past feature representations to be low-rank, thereby enlarging the null space via the rank-nullity relation

$\mathcal{P}_t$3

The method achieves this low-rank structure by disabling a subset of neurons selected through sparsity regularization on batch-normalization scale parameters (Liu et al., 2023).

Auxiliary Network Continual Learning (ANCL) instead introduces an auxiliary network trained only on the current task and adds a second regularizer toward this plastic solution: $\mathcal{P}_t$4 For several base methods, the paper shows that the resulting gradients target an interpolation between old and auxiliary outputs or parameters, so the balance is controlled by the relative strengths of $\mathcal{P}_t$5 and $\mathcal{P}_t$6 (Kim et al., 2023).

Flashback Learning (FL) follows a related but two-phase design. It first learns the new task aggressively to create a Plastic Knowledge Base (PKB), then restarts from the previous stable model and trains with both a stability loss and a plasticity loss: $\mathcal{P}_t$7 The key claim is that conventional continual learning regularizers are mostly one-sided, whereas FL uses bidirectional regularization through a Stable Knowledge Base (SKB) and PKB, causing optimization to interpolate between old and newly acquired knowledge rather than only anchoring to the past (Mahmoodi et al., 31 May 2025).

A more geometric variant is the simple linear connector, which separately trains a stability-oriented model via null-space projection and a plasticity-oriented model via SGD plus feature distillation, then combines them by

$\mathcal{P}_t$8

The default choice $\mathcal{P}_t$9 is motivated by an upper-bound argument showing that the centroid of task-specific optima reduces a bound on accumulated loss. This turns SPT into an explicit interpolation problem in weight space (Lin et al., 2021).

These mechanisms share a common logic. Stability is not treated as absolute immobility, and plasticity is not treated as unconstrained change. Instead, each method constructs an intermediate object: a null-space-constrained update, an auxiliary target, a bidirectionally regularized solution, or a connected pair of optima.

4. Dynamic, multi-objective, and architectural reinterpretations

A major reinterpretation of SPT is that the “right” balance is often dynamic rather than fixed. Pareto Continual Learning (ParetoCL) reformulates replay-based continual learning as a two-objective problem,

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$0

and learns a preference-conditioned model with preference vector $$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$1. At inference time, it samples multiple preferences and chooses the output with minimum entropy,

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$2

This makes the trade-off adaptive both across training stages and across test samples (Lai et al., 30 Mar 2025).

A related but more explicitly post-hoc solution is IBCL, which maintains a convex hull of posterior parameter distributions and produces preference-specific models through zero-shot convex combination: $$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$3 Its central claim is that a requested SPT operating point should not require retraining, because there are infinitely many possible preferences and training one model per preference is inefficient (Lu et al., 2023).

Architectural work argues that SPT is not only about losses or priors but also about network shape. Under equal parameter constraints, deeper, thinner networks are reported to be more plastic, while wider, shallower networks are more stable. Dual-Arch therefore uses two independent learners: Pla-Net, trained only with

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$4

and Sta-Net, trained with

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$5

The method serves as a plug-in component for iCaRL, WA, DER, Foster, and MEMO, and reports maximum improvements of +10.29% LA and +7.62% AIA, with at least 33% fewer parameters (Lu et al., 4 Jun 2025).

The same continual-learning perspective has been transferred to sequentially trained early-exiting neural networks. Here each newly added exit behaves like a new task, and previously trained exits require stability. The proposed total objective is

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$6

with EWC acting at the parameter level and LwF at the output-distribution level. The paper reports about a +4.25\% to +4.46\% boost at the 25\% budget relative to the strongest sequential baseline, indicating that SPT-aware regularization is particularly important for shallow exits (Zniber et al., 6 May 2026).

5. Reinforcement learning, language-model adaptation, and other domains

In continual reinforcement learning under naturalistic, gradual non-stationarity, recent results argue that the common intuition “more plasticity is better” can fail. In Slippery Four Rooms and MuJoCo variants with noisy sine or Ornstein–Uhlenbeck drift, methods favoring stability, such as synaptic consolidation, outperform plasticity-oriented methods such as parameter resetting. The strongest approach combines Successor Features (SFs) with multi-timescale consolidation. The canonical decomposition is

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$7

with

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$8

The paper’s conclusion is that in gradual drift, the dominant issue is often retaining stable predictive structure rather than injecting more plasticity (Chua et al., 25 May 2026).

A finer-grained reinforcement-learning solution is NBSP, which moves SPT control from the network level to the neuron level. It defines RL skill neurons through a goal-oriented score based on the alignment between neuron activation and task success, then applies gradient masking,

$$\mathcal{L}_t = \lambda \mathcal{L}_{new}(f_\theta; \mathcal{D}_t) + (1-\lambda)\mathcal{L}_{replay}(f_\theta; \mathcal{M}_t),$$9

together with experience replay. The claim is that specific neurons encode task-relevant skills, so protecting those neurons preserves stability while leaving the rest of the network plastic (Lan et al., 9 Apr 2025).

For post-training LLMs with GRPO, SPT is formalized as the conflict between a plasticity gradient $\beta$0 and a stability gradient $\beta$1. PCR models these gradients as Gaussian random variables and resolves conflicts with an uncertainty-aware soft projection. Its central update form is

$\beta$2

where $\beta$3 depends on the relative precision of the two gradient estimates. This yields a stochastic, MMSE-motivated alternative to deterministic gradient projection (Qiang et al., 6 Feb 2026).

In parameter-efficient finetuning, PEFT-Arena shows that different parameterizations occupy distinct SPT profiles under comparable trainable-parameter budgets. The benchmark analyzes retention through spectral and activation-space geometry, including the Procrustes residual

$\beta$4

linear CKA, and pairwise Gram distortion. Forgetting is linked to non-isometric representation distortion, and the paper reports that orthogonal finetuning yields the most favorable Pareto frontier under comparable parameter budgets (Huang et al., 27 May 2026).

SPT has also become domain-specific. In continual named entity recognition, the proposed SPT method balances representation-level and weight-level constraints through pooled knowledge distillation, selective weight fusion, and confidence-based pseudo-labeling to address the semantic shift of the non-entity type (Zhang et al., 5 Aug 2025). In recommender systems, a retraining-aware offline protocol profiles algorithms such as UKNN, BPRMF, and NeuMF by comparing their old-regime retention and new-regime adaptation after retraining, suggesting that different algorithm families occupy different stability-plasticity regimes (Lavoura et al., 5 Aug 2025).

6. Biological perspectives, misconceptions, and open questions

Biologically motivated work treats SPT as a question of how adaptive change can remain organized rather than destabilizing. In networks of excitatory Hodgkin–Huxley neurons, the combined action of short-term plasticity (STP) and spike-timing-dependent plasticity (STDP) reshapes initially dense connectivity into directed modular networks. The reported interpretation is that STDP alone tends to reinforce synchronization and directed pathways, whereas STP imposes a balancing constraint through resource depletion, allowing modular rather than globally synchronized organization (Lameu et al., 2019).

A related spiking-neuron line argues that plain STDP is too plastic and not stable enough. The proposed remedy is a constrained plasticity reserve or protein reserve that limits potentiation according to firing-rate homeostasis. Positive STDP updates are scaled by available dendritic reserve, while LTD can return reserve, so the neuron controls not just weights but its capacity to potentiate. The paper presents this as a natural way to filter intense noise while preserving unsupervised recognition of correlated inputs (Nikitin et al., 2021).

Several misconceptions recur across the modern literature. One is that reducing catastrophic forgetting is equivalent to solving SPT. Class-incremental analyses show that many methods are highly successful at preserving features precisely because those features hardly change, to the point that the base feature extractor can be almost as effective as the final one once the classifier is retrained on all data (Kim et al., 2023). Another is that a single fixed trade-off coefficient is sufficient. ParetoCL, IBCL, and several application papers instead treat the desired balance as time-varying, sample-dependent, or user-specified (Lai et al., 30 Mar 2025, Lu et al., 2023). A third is that more plasticity is always better; continual RL under gradual drift argues the opposite in many settings, namely that the main failure mode is instability rather than insufficient adaptation (Chua et al., 25 May 2026).

An important limitation of the current evidence base is that not every work marketed as an SPT solution is equally documented in accessible source material. For “PromptFusion”, the available source here provides bibliographic context linking the work to continual learning, catastrophic forgetting, prompt tuning, and stability-plasticity references, but does not contain the method description, equations, modules, or experiments needed to reconstruct its specific SPT mechanism (Chen et al., 2023). This underscores a broader point: SPT is now a cross-domain research program, but its empirical interpretation depends heavily on whether one measures feature growth, output preservation, gradient conflict, architectural bias, or post-retraining behavior.

Taken together, the literature suggests that SPT is not a single dilemma with a single cure. It can be treated as a constraint problem in representation space, a multi-objective problem over old and new losses, a geometry problem in gradient space, an architectural allocation problem over depth and width, a multi-timescale consolidation problem, or a functional-retention problem over outputs and general capabilities. The strongest common conclusion is not that stability or plasticity should dominate universally, but that effective sequential learning requires explicit mechanisms for deciding what must remain invariant, what may change, and on what timescale that change should occur.