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Geometry of Forgetting: Representation Flux in Continual Learning

Maksim A. Kazanskii

arXiv:2608.15854v1cs.LGcs.CV

TL;DR

Catastrophic forgetting remains poorly understood in representation space. This paper introduces representation flux and FlowLess-R to analyze and stabilize latent representations, improving retention and reducing forgetting across evaluated replay methods and benchmarks.

  • Problem

    Existing continual-learning methods mitigate forgetting, but provide limited insight into the representation-space dynamics associated with it.

  • Method

    The paper measures sample-level representation flux and uses FlowLess-R to constrain replay representations relative to stored references during continued learning.

  • Results

    FlowLess-R improves knowledge retention and consistently reduces forgetting across evaluated replay methods and continual-learning benchmarks.

  • Takeaways & Limitations

    Representation flux is an informative geometric marker of forgetting, and stabilizing replay representations provides complementary information for continual-learning regularization.

  • Takeaways & Limitations

    The study evaluates replay-based continual learning on image-classification benchmarks, leaving effectiveness in other continual-learning settings unresolved.

Abstract

from arXiv · show

Catastrophic forgetting remains a fundamental obstacle to continual learning, where neural networks lose previously acquired knowledge while learning new tasks. Existing methods primarily mitigate forgetting through parameter regularization or experience replay, while the representation-space dynamics associated with forgetting remain less understood. We investigate latent representation evolution during sequential learning and introduce representation flux, a geometric measure of sample-level representation displacement across training. We show that representation flux is strongly associated with catastrophic forgetting across multiple benchmarks, with temporal analyses indicating that elevated flux can precede subsequent performance degradation. Representation displacement is also associated with confidence degradation, while complementary geometric properties provide additional information about sample-level forgetting. Motivated by these observations, we propose FlowLess-R, a representation-space regularization method that constrains replay representations relative to stored references while allowing continued learning. FlowLess-R is architecture-agnostic and integrates into replay-based methods through a representation-matching term. Experiments on SplitMNIST, SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet show improved final average accuracy and reduced forgetting with ER, DER++, and ER-ACE. Our results identify representation flux as an informative geometric marker of forgetting and show that stabilizing latent representations provides a simple strategy for mitigating catastrophic forgetting.

1 Introduction

The introduction reframes catastrophic forgetting through latent representation dynamics, proposing representation flux as a geometric measure of displacement during sequential learning. It links elevated representation flux and displacement to later performance degradation, forgetting, and confidence changes, motivating FlowLess-R.

  • Motivation: Catastrophic forgetting causes performance on previously learned tasks to deteriorate during sequential training, despite successful optimization of the current task.Existing methods primarily address forgetting through parameter regularization and experience replay.
  • Representation-space perspective: The paper views latent representations as an evolving geometric distribution and defines representation flux as displacement between consecutive training epochs.This perspective shifts attention from parameter updates to representation movement over time.
  • Empirical findings: Elevated representation flux can precede later performance degradation, while larger sample-level displacements are associated with higher forgetting rates and confidence degradation.These relationships are reported across multiple continual learning benchmarks.
  • Geometric characterization: The study combines temporal displacement with local-density and transition-level geometric descriptors to characterize complementary aspects of sample-level forgetting and prediction-confidence changes.These observations motivate the introduction of FlowLess-R as a representation-space regularizer.

2 Related Work

Continual learning research includes regularization, replay, optimization, and architectural approaches, alongside work stabilizing learned representations. This paper instead studies temporal, sample-level representation displacement and introduces FlowLess-R to constrain accumulated drift using persistent references.

  • Continual Learning: Continual learning methods are commonly categorized as regularization-based, replay-based, optimization-based, and architectural approaches.Regularization methods preserve parameters important for previous tasks using importance estimates such as Fisher information or path-integral measures.
  • Representation dynamics and stabilization: Representation-stabilization methods constrain current features toward previous-model features or use structured representation distillation to mitigate forgetting.Less-Forgetful Learning directly preserves penultimate-layer features by matching them to those produced by the previous model.
  • Representation dynamics and stabilization: This work studies representation displacement as a temporal, sample-level phenomenon rather than introducing representation matching solely as a regularization mechanism.Representation flux measures individual-sample displacement between successive training snapshots and is examined alongside subsequent forgetting, confidence degradation, and other geometric properties.
  • Representation dynamics and stabilization: FlowLess-R penalizes replay-sample displacement from fixed, sample-specific historical representations stored when samples enter memory.Persistent per-sample references directly constrain accumulated representation drift without requiring a frozen teacher network.

3 Representation Flux as an Early Indicator of Forgetting

Representation flux measures sample-level latent displacement between consecutive training epochs and is associated with subsequent forgetting, confidence degradation, and complementary geometric signals. Its temporal behavior and cross-benchmark trends motivate using representation-space dynamics as an early indicator of catastrophic forgetting.

  • Definition: Representation flux is the magnitude of a sample’s latent-representation displacement between consecutive training epochs.Larger flux indicates greater latent-space displacement, while smaller flux indicates greater geometric stability.
  • Temporal association: Representation flux peaks at task transitions, while hard forgetting reaches its local maximum one epoch later.Soft forgetting follows a qualitatively similar but weaker temporal pattern.
  • Sample-level forgetting: The probability that a previously correct prediction becomes incorrect increases with representation flux, while representation density alone shows a weaker association.Forgetting probability remains low at small flux and rises at larger flux.
  • Confidence dynamics: Larger representation displacements are associated with greater confidence loss, whereas confidence gains concentrate at substantially lower flux levels.These patterns characterize how representation flux relates to prediction-confidence dynamics.
  • Cross-benchmark evidence: Representation flux alone is more informative of sample-level forgetting than representation density on SplitMNIST, SplitFashionMNIST, and SplitCIFAR10, while combining both improves discrimination.On SplitTinyImageNet, neither flux nor density is informative; analogous flux-forgetting trends are observed across the benchmarks.

4 FlowLess-R

FlowLess-R stabilizes replay-sample representations by anchoring them to fixed historical references, thereby controlling accumulated representation drift. It adds a representation-matching term to existing replay-based continual learning methods without architectural modifications.

  • Representation anchoring: FlowLess-R operates directly in representation space, penalizing replay-sample deviations from fixed historical reference representations rather than constraining network weights.The method targets accumulated drift from an earlier representation, beyond displacement between consecutive training snapshots.
  • Reference construction: Each replay sample’s latent representation is computed and stored when the sample enters the replay buffer, and its reference remains fixed during storage.The reference representation is associated with the model parameters at the time of sample insertion.
  • Optimization objective: The complete objective combines the original continual learning loss with representation stabilization controlled by λ.The original objective may be ER, DER++, or ER-ACE.
  • Integration: FlowLess-R requires no architectural modifications and can be combined directly with existing replay-based continual learning methods.The formulation is described as compatible with ER, DER++, and ER-ACE objectives.
  • Computational cost: O(Brd) computation and O(Md) memory are added, with the computational cost small relative to O(BrCnet) when d ≪ Cnet.Br is replay batch size, d is representation dimension, M is replay-buffer capacity, and Cnet is per-sample network computation.

5 Results

Across replay methods and benchmarks, FlowLess-R reduces mean forgetting, often significantly, while generally improving final average accuracy. Ablations identify the last hidden layer as the strongest regularization location, whereas density weighting adds complexity without consistent gains.

  • Replay-method results: FlowLess-R reduces ER mean forgetting by 8.68, 7.68, 8.41, and 5.96 percentage points on SplitMNIST, SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet, respectively.All four reductions are statistically significant, with p < 0.0001, p < 0.0001, p = 0.0042, and p = 0.0361, respectively.
  • Replay-method results: For DER++, mean forgetting decreases from 45.09% to 26.03% on SplitTinyImageNet and from 25.78% to 12.17% on SplitCIFAR10.These correspond to reductions of 19.07 percentage points (p = 0.0001) and 13.61 points (p = 0.0021); smaller reductions on SplitMNIST and SplitFashionMNIST are not significant.
  • Replay-method results: With ER-ACE, FlowLess-R reduces mean forgetting by 6.17, 6.32, 8.78, and 4.85 points across the four benchmarks.All reductions remain statistically significant across benchmarks and with both MLP and ResNet-18 backbones.
  • Accuracy outcomes: In most evaluated settings, FlowLess-R simultaneously reduces forgetting and improves final average accuracy rather than sacrificing predictive performance.Complete final average accuracy results across all evaluated λ values are reported in Appendix D.
  • Density weighting: Representation flux and local representation density provide complementary information about sample-level forgetting, motivating an optional density-weighted FlowLess-R variant.The unweighted formulation (α = 0) achieves comparable performance across replay buffer sizes while avoiding an additional hyperparameter and computational overhead.
  • Layer ablation: Regularizing the last hidden layer yields the highest mean final average accuracy (78.30%) and lowest mean forgetting (25.90%) in the SplitMNIST layer ablation.The study uses ER with 40 replay samples per task and averages results over regularization coefficients and 10 random seeds.

6 Discussion

The discussion links catastrophic forgetting to unstable latent-representation dynamics and presents representation-space analysis as complementary to parameter-space accounts. It positions FlowLess-R as a simple, complementary stabilization strategy while identifying layer choice, scope limitations, and multivariate forgetting geometry as directions for further study.

  • Representation displacement is associated with confidence degradation and higher forgetting, while elevated representation flux can precede later performance degradation.These findings support representation-space analysis as complementary to parameter-space descriptions of forgetting.
  • FlowLess-R stabilizes replay representations against stored references and improves retention across replay methods and benchmarks through an additional representation-matching term.The method is motivated by the observed link between representation dynamics and forgetting, while allowing continued learning.
  • Regularizing only the final hidden layer is more effective than constraining earlier or multiple layers in the evaluated setting.The discussion suggests that the final hidden layer may better support decision boundaries, while earlier layers retain flexibility during adaptation.
  • FlowLess-R provides complementary stability to ER-ACE by anchoring replay-sample representations while ER-ACE reduces disruptive updates from incoming classes.The additional gains with ER-ACE suggest that directly constraining accumulated representation displacement adds stability beyond asymmetric classification.
  • The study is limited to replay-based image-classification settings, and reduced descriptor informativeness on SplitTinyImageNet suggests that forgetting geometry becomes more multivariate on complex benchmarks.Future work should examine other continual-learning approaches, sequential decision making, language models, multimodal learning, and larger-scale settings.

7 Conclusion … A.3 Density Stability

The paper frames catastrophic forgetting through latent-representation dynamics, identifying representation flux as a geometric marker and introducing FlowLess-R to stabilize replay representations. The appendix interprets these dynamics as density evolution, formalized through region transitions, transport, and density stability.

  • 7 Conclusion: Representation flux is identified as an informative geometric marker of catastrophic forgetting, motivating FlowLess-R to stabilize latent representations during training.FlowLess-R directly stabilizes replay-sample representations and improves knowledge retention.
  • A A Density Evolution Interpretation of Representation Flux: The density-evolution view explains why excessive representation flux is associated with catastrophic forgetting and motivates limiting unnecessary representation transport.The interpretation treats latent representation densities as evolving during sequential learning rather than focusing solely on parameter updates.
  • A.1 Representation Density Evolution: Network parameter changes between training snapshots induce corresponding changes in the distribution of latent representations.For an individual sample, representation displacement contributes to the collective density transition between snapshots.
  • A.2 Density Transition Matrix: A fixed partition of the latent space assigns each representation to one of K regions across consecutive snapshots, discretizing the continuous representation distribution.The partition may be induced by clustering or a Voronoi tessellation and is used for mathematical discretization.
  • A.2 Density Transition Matrix: The density vector records the probability mass contained in each latent-space region at training epoch t.Its i-th component is the probability mass in region i.
  • A.2 Density Transition Matrix: The Density Transition Matrix gives the empirical probability that representations move from region i to region j between consecutive snapshots.Rows form probability distributions; diagonal entries represent retained regional mass, while off-diagonal entries represent transport between regions.
  • A.3 Density Stability: Density stability measures the probability that a sampled representation remains in its original latent region, while density leakage measures transitions to different regions.Stability is determined by diagonal transition probabilities weighted by region probabilities; leakage is the complementary quantity.

A.4 Geometric Bound Relating Representation Flux to Density Leakage · A.5 Representation Flux and Decision Boundary Stability

The paper derives geometric bounds linking density leakage and forgetting to boundary proximity, representation flux, and classifier drift. These results motivate FlowLess-R as a method for stabilizing replay representations and suppressing displacement-driven forgetting.

  • A.4 Geometric Bound Relating Representation Flux to Density Leakage: A.4 defines representation flux as consecutive-snapshot latent displacement and boundary clearance as the minimum displacement needed to leave the current region.Region transitions therefore require displacement at least as large as the sample’s boundary clearance.
  • A.4 Geometric Bound Relating Representation Flux to Density Leakage: A.4 bounds density leakage by separating representations near region boundaries from samples undergoing sufficiently large representation displacement.The two terms respectively reflect latent-distribution geometry and learned-representation dynamics.
  • A.4 Geometric Bound Relating Representation Flux to Density Leakage: The boundary-proximity term measures mass near assigned-region boundaries, whereas the large-displacement term measures the frequency of substantial representation changes.This decomposition distinguishes geometric instability from dynamic instability in latent space.
  • A.4 Geometric Bound Relating Representation Flux to Density Leakage: FlowLess-R penalizes replay representations’ deviations from stored references to limit accumulated drift and stabilize replay-sample latent geometry.The fixed-reference displacement differs from consecutive-snapshot flux but is designed to control accumulated representation drift.
  • A.5 Representation Flux and Decision Boundary Stability: A.5 decomposes margin change into classifier drift at a fixed representation and the effect of representation displacement.This separates changes caused by the classifier from changes caused by the latent representation.
  • A.5 Representation Flux and Decision Boundary Stability: Theorem 2 states that a forgetting event requires the combined effects of representation flux and classifier drift to be sufficiently large relative to the current classification margin.Representation flux and classifier drift are complementary mechanisms associated with catastrophic forgetting.
  • A.5 Representation Flux and Decision Boundary Stability: Constraining accumulated displacement from stored replay references suppresses the representation-drift component associated with representation flux.This provides the decision-boundary-stability rationale for FlowLess-R.

B Additional Representation Flux Analyses

Additional analyses on SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet show that representation flux remains associated with confidence degradation and forgetting-related geometric instability. Temporal and joint density–flux analyses further indicate that forgetting concentrates at higher flux, while density modulates this relationship.

  • Cross-dataset geometric relationships: Across SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet, representation flux is positively associated with confidence loss, while confidence gain concentrates at lower flux.These findings qualitatively reproduce the relationships observed on SplitMNIST across increasingly complex visual benchmarks.
  • Cross-dataset geometric relationships: Forgetting probability generally increases with density leakage, indicating that unstable occupancy of latent-space regions is associated with forgetting.Density leakage measures transitions from a sample’s initial latent-space region to a different region between consecutive snapshots.
  • Temporal and joint geometric analyses: Forgetting concentrates at higher representation flux, while local representation density further modulates this relationship in joint density–flux heatmaps.Representation density is estimated using distances to the k = 10 nearest latent-space neighbors.
  • Temporal and joint geometric analyses: Across the additional datasets, elevated representation flux is associated with subsequent forgetting, although the temporal separation is less pronounced than on SplitMNIST.The temporal analyses compare flux with hard and soft forgetting across task transitions.

C Experimental Setup · D Additional FlowLess-R Results

Experiments use a standard class-incremental learning protocol across four datasets and evaluate FlowLess-R with replay-based methods under varied architectures, memory sizes, and regularization coefficients. Performance is summarized using final average accuracy and mean forgetting, with results reported across multiple random seeds.

  • C Experimental Setup: Tasks are learned sequentially with prediction over all previously observed classes, while prior-task data remain accessible only through the replay buffer.Inference does not provide task identity.
  • C Experimental Setup: SplitMNIST and SplitFashionMNIST use a 784-input multilayer perceptron, with FlowLess-R applied to its 64-dimensional pre-classifier representation.The network has hidden layers of 256, 256, and 64 units with ReLU activations.
  • C Experimental Setup: SplitCIFAR10 and SplitTinyImageNet use ResNet-18 models whose 512-dimensional pooled features feed linear classifiers with 10 and 200 output units, respectively.FlowLess-R is applied to the feature representation produced before classification.
  • C Experimental Setup: ER stores uniformly selected replay samples and their insertion-time feature representations as FlowLess-R reference representations for later replay.Samples are selected without replacement after each task and replayed uniformly without replacement during subsequent tasks.
  • C Experimental Setup: ER buffer-size experiments vary memory from 10–160 samples per task on SplitMNIST and SplitFashionMNIST, 50–1600 on SplitCIFAR10, and 200–3200 on SplitTinyImageNet.Experiments use ten seeds for the first three datasets and five seeds for SplitTinyImageNet.
  • C Experimental Setup: Final average accuracy averages task accuracies after the final task, while mean forgetting averages each previous task’s peak-to-final accuracy difference.Results are reported as mean ± standard deviation across the corresponding random seeds unless otherwise stated.
  • D Additional FlowLess-R Results: Table 4 compares final average accuracy across FlowLess-R coefficients λ, with λ = 0 representing the baseline without FlowLess-R regularization.Values are averaged over 10 seeds for SplitMNIST, SplitFashionMNIST, and SplitCIFAR10, and 5 seeds for SplitTinyImageNet; p-values use Holm–Bonferroni correction.

D.1 Final Average Accuracy · D.2 Effect of Replay Buffer Size

FlowLess-R generally preserves or improves final average accuracy across replay methods, with its largest gains varying by benchmark and baseline. Across replay-buffer capacities, it is evaluated using validation-selected regularization strengths against λ = 0 baselines, with especially pronounced effects reported for larger SplitTinyImageNet buffers.

  • D.1 Final Average Accuracy: FlowLess-R generally maintains or improves final average accuracy relative to the corresponding replay baseline.The reported comparisons aggregate all evaluated values of the regularization coefficient λ.
  • D.1 Final Average Accuracy: 6.54 and 5.56 percentage points are the significant DER++ gains on SplitCIFAR10 and SplitTinyImageNet, increasing accuracy from 43.15% to 49.69% and from 13.22% to 18.78%.The corresponding p-values are 0.0082 and 0.0084; DER++ gains on SplitMNIST and SplitFashionMNIST are smaller and nonsignificant.
  • D.1 Final Average Accuracy: 4.87 and 5.05 percentage points are the significant ER-ACE gains on SplitMNIST and SplitFashionMNIST, respectively, both with p < 0.0001.ER-ACE improvements on SplitCIFAR10 (+1.90 points, p = 0.3309) and SplitTinyImageNet (+1.91 points, p = 0.0567) are not significant after correction.
  • D.1 Final Average Accuracy: Improved retention with FlowLess-R is generally not obtained at the expense of final predictive performance, and most settings show reduced forgetting alongside improved final average accuracy.The passage specifically highlights DER++ gains on SplitCIFAR10 and SplitTinyImageNet as cases where stronger retention coincides with improved final performance.
  • D.2 Effect of Replay Buffer Size: FlowLess-R is evaluated across dataset-specific replay-buffer ranges by selecting the λ with the highest validation final average accuracy, while the baseline uses λ = 0.This design compares performance across replay capacities against the corresponding baseline replay methods.
  • D.2 Effect of Replay Buffer Size: On SplitTinyImageNet, the performance gap widens substantially at larger replay capacities and is accompanied by a marked reduction in forgetting.The passage suggests that larger buffers provide more stored representation anchors, potentially enabling more effective constraints on representation drift.
  • D.2 Effect of Replay Buffer Size: Figure 5 reports final average accuracy in its first row and mean forgetting in its second row, with curves averaged over random seeds and shaded regions showing one standard deviation.The figure summarizes the effect of replay buffer size on continual learning performance.

D.3 Effect of Density Weighting · E Miscellaneous

FlowLess-R supports optional density-weighted replay regularization, but its SplitMNIST performance is largely insensitive to moderate weighting choices, motivating the unweighted formulation. The paper also provides public reproduction code and documents reviewed LLM assistance in language editing and software development.

  • D.3 Effect of Density Weighting: FlowLess-R optionally weights replay samples by their local representation density through exponent α.The density estimate for replay sample i is denoted ρ_i.
  • D.3 Effect of Density Weighting: Negative α emphasizes low-density replay samples, whereas positive α emphasizes samples from high-density representation regions.A small ε is included for numerical stability, and α controls weighting strength.
  • D.3 Effect of Density Weighting: FlowLess-R’s density-weighting sensitivity was evaluated on SplitMNIST across multiple replay memory sizes and exponent values.For each α and memory size, λ was selected to maximize final average accuracy; results are summarized in Table 5 over 10 matched random seeds.
  • D.3 Effect of Density Weighting: Across replay memories of 20 and 80 samples per task, final average accuracy and forgetting differences remained small for −1 ≤ α ≤ 1.The Friedman test detected statistically significant differences in some comparisons, but the reported practical differences were small.
  • D.3 Effect of Density Weighting: The remaining experiments use the unweighted formulation, α = 0, because it performs comparably to moderate density weighting without an additional hyperparameter.This choice follows the observed insensitivity to the density-weighting exponent.
  • E Miscellaneous: Source code for reproducing the paper’s experiments, analyses, and figures is publicly available in the FlowLess repository.The repository is hosted at github.com/maksimkazanskii/FlowLess.
  • E Miscellaneous: ChatGPT (OpenAI; GPT-5.6 Sol) assisted with language editing and software development, with all LLM-assisted outputs reviewed and verified by the author.The author states responsibility for the paper’s content.
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