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REVIEW 4 major objections 5 minor 59 references

Online Curvature-Aware Replay: Leveraging $\mathbf{2^{nd}}$ Order Information for Online Continual Learning

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Replay-based online continual learning can be reframed as second-order optimization with a KL-divergence stability constraint, and the resulting OCAR optimizer beats prior state-of-the-art on continual metrics in three benchmarks.

desk verdict OCAR is a serious, well-engineered OCL method with strong anytime-accuracy results, but the paper's central claim that second-order curvature drives the gains needs an ablation that isolates K-FAC from the Tikhonov schedule. read the letter →

arxiv 2502.01866 v1 pith:DDHSJGMQ submitted 2025-02-03 cs.LG cs.AI

Edoardo Urettini , Antonio Carta This is my paper · ORCID
classification cs.LGcs.AI MSC 68T0768T05
keywords onlinecontinuallearningreplaysecond-orderoptimizationFisherinformationmatrixK-FACstabilitygapstability-plasticitytradeoffnaturalgradient
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Online continual learning models must stay accurate at every moment of a nonstationary stream, yet replay-based methods typically suffer a sharp accuracy drop right after each task shift, the stability gap. This paper argues that the gap is partly an optimization failure: first-order replay treats all parameter directions alike, so when new-task gradients dominate at a task boundary, old knowledge is overwritten. The authors formalize replay as a joint second-order optimization problem with an explicit constraint that the model's predictions on buffered old data barely change, measured by the KL divergence, and enforce it by preconditioning gradients with the Fisher Information Matrix, approximated block-wise by K-FAC. The resulting method, OCAR, achieves the best average-anytime and worst-case accuracy among all compared methods on Split-CIFAR100, Split-TinyImageNet, and Online CLEAR, and its 2D trajectory analysis shows much smaller deviations at task boundaries than experience replay. If the paper is right, much of the instability in online continual learning can be fixed at the optimizer level rather than by inventing new losses or architectures.

What carries the argument

The central object is the preconditioned update rule $\delta^*_t = -\alpha(F_{N_t} + (1+\lambda)F_{B_t} + \tau I)^{-1}(\nabla_{N_t} + \nabla_{B_t})$, which unifies the paper's three ingredients: the replay gradients, the Fisher Information Matrix (FIM) as a stability constraint, and Tikhonov damping as a plasticity regulator. The FIM is the Hessian of the KL divergence between pre- and post-update predictive distributions on buffer data, so the constraint 'do not change predictions on old data' becomes the quadratic form $\delta^T F \delta \le \rho$; by the Fisher/Generalized-Gauss-Newton equivalence the same FIM replaces the Hessians of the two loss terms, so one curvature quantity plays three roles at once: metric of the parameter manifold, stabilizer for old tasks, and accelerator for new ones. Tractability comes from K-FAC: the FIM is approximated block-diagonally as per-layer Kronecker products of activation and gradient factors, updated by exponential moving average, and inverted cheaply through $(A \otimes B)^{-1} = A^{-1} \otimes B^{-1}$. Three practical mechanisms carry the method in non-iid streams: the EMA factors, with a reset of only the classifier factor when its shape changes at a class-incremental boundary; the $\lambda$ schedule, which strengthens the stability constraint as the buffer accumulates more classes; and the $\tau$ schedule, which grows damping each step because a fixed small $\tau$ dangerously amplifies low-curvature directions when new classes appear.

What would settle it

Run OCAR on Split-CIFAR100 twice: once with the default exponential-moving-average Fisher, and once with all K-FAC factors reset at every task boundary, and record the accuracy on previously seen tasks in the steps immediately after each boundary. If the EMA-carryover version does not show a consistently smaller drop than the reset version, and than plain experience replay with matched hyperparameters, then the assumption that averaged curvature stays valid across tasks is what carries the result, not the second-order update itself.

Watch

Extended reading notes

Core claim

The paper's central claim is that the instability of replay-based online continual learning is a first-order optimization artifact, and that a second-order formulation removes most of it. OCAR solves, at every step, a constrained joint optimization over new and replayed data: minimize the KL divergence of the model on both batches while requiring the KL divergence between the pre-update and post-update predictive distributions on buffer data to stay below a threshold $\rho$. A Taylor expansion turns this into a preconditioned update $\delta^* = -\alpha(F_{N_t} + (1+\lambda)F_{B_t} + \tau I)^{-1}(\nabla_{N_t} + \nabla_{B_t})$, where $F$ is the Fisher Information Matrix on current and buffer data, $\lambda$ is the Lagrange multiplier of the stability constraint, $\tau$ is Tikhonov damping, and $\alpha$ is the learning rate. The Fisher is the 'true' Fisher, an expectation over the model's predictive distribution rather than over observed labels, and it is made tractable with a block-diagonal K-FAC approximation whose Kronecker factors are updated by exponential moving average; at class-incremental boundaries the classifier factor is reset when its shape grows, and $\tau$ grows each step. On this basis the paper reports state-of-the-art continual metrics, average anytime accuracy and worst-case accuracy, on Split-CIFAR100 (20 tasks), Split-TinyImageNet (20 tasks), and Online CLEAR (10 tasks), with the OCAR-ACE combination best overall on the class-incremental benchmarks. The paper also identifies the ratio $\alpha/\tau$, rather than $\alpha$ or $\tau$ alone, as the control knob of the stability-plasticity tradeoff: $\tau$ caps the maximum acceleration in low-curvature directions, while $\alpha$ scales all directions uniformly.

Load-bearing premise

The load-bearing premise is that the exponential-moving-average estimate of the Fisher matrix stays trustworthy when the data distribution shifts: if the averaged curvature mixes old and new tasks in the wrong proportions, the preconditioner can amplify exactly the wrong directions and OCAR loses its stability advantage over plain replay.

Editorial extensions

If this is right

  • OCAR achieves the best average anytime accuracy and worst-case accuracy among all compared methods on Split-CIFAR100 and Split-TinyImageNet, and the largest gains on Online CLEAR (75.3 vs LPR's 65.2 final accuracy), indicating that the stability gap is not an inevitable property of replay methods.
  • Because OCAR is an optimizer rather than a new loss or architecture, it can be stacked with other replay methods: OCAR-ACE sets the best Acc, AAA, and WC-Acc overall, while OCAR-DER++ fails on TinyImageNet, which the authors attribute to DER's entropy regularization breaking the KL assumption behind the Fisher.
  • The $\alpha/\tau$ ratio, rather than the learning rate alone, controls the effective step size in low-curvature directions, so tuning the growth of $\tau$ on a short validation stream generalizes better to longer streams than tuning a fixed damping value.
  • OCAR improves linear probing accuracy on both main benchmarks, suggesting that curvature-aware updates produce better feature representations over the stream, not merely a better final classifier.
  • The method's runtime is about three times that of plain experience replay but below several prior sophisticated methods (MIR, SCR, LPR), so the stability gain does not come at extreme computational cost.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the optimizer-level story is right, a corollary the paper leaves implicit is that the stability gap should be treatable in other nonstationary settings, such as reinforcement learning or fine-tuning on drifting data, with the same damped-Fisher preconditioning even without a replay buffer.
  • The reported OCAR-DER++ failure suggests a testable boundary on the method: any replay loss whose gradients are not approximately the gradient of a KL-type objective will not inherit the stability guarantee, and one could quantify this by measuring how far the DER loss's Hessian deviates from the Fisher on buffer data.
  • The $\tau$-scheduling result hints that the FIM's conditioning degrades as the stream lengthens, so a natural extension is adapting $\tau$ online from the eigenvalue spectrum of the K-FAC factors instead of a fixed additive schedule.
  • The reset policy assumes most boundary instability lives in the classifier, yet the strong Online CLEAR results, a domain-incremental setting with a shared classifier, suggest the reset is not the main source of gains; ablating the reset there would isolate its actual contribution.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper formalizes replay-based online continual learning (OCL) as a sequence of second-order optimization problems with a KL-divergence stability constraint on buffer data, and proposes OCAR, which uses a K-FAC approximation of the Fisher Information Matrix to precondition the replay gradient. A Tikhonov regularization term with a time-increasing schedule is introduced to control the stability-plasticity tradeoff. The method is evaluated on Split-CIFAR100, Split-TinyImageNet, and Online CLEAR, reporting improvements on continual metrics (AAA, WC-Acc) over strong baselines, as well as combinations with ER-ACE and DER++. The authors provide code and follow the evaluation protocol of prior OCL surveys.

Significance. If the claimed effects are real, OCAR is a valuable contribution: it brings second-order optimization, previously used mainly in stationary or task-boundary settings, to the online continual setting and reports large gains on the metrics that measure the stability gap. The paper is also careful to use established benchmarks, a shared evaluation protocol, and public code, and it does not fit constants to test data. The main weakness is causal attribution: the algorithm confounds the K-FAC preconditioner with a first-order learning-rate annealing schedule, and no ablation isolates the second-order contribution. The theoretical framing also overstates the role of the explicit KL constraint, since the implemented λ schedule is heuristic and no ρ is enforced. These issues are fixable with additional experiments and a revised derivation, and therefore a major revision is appropriate.

major comments (4)
  1. [Algorithm 1, line 10; Section 5] The central claim that second-order curvature information drives the stability-plasticity improvement is not supported by an isolation experiment. In Algorithm 1, τ is initialized to α and incremented by Δτ at every gradient step, while the update is δ = -α(F + τI)^{-1}∇L; in the low-curvature directions the effective step size is α/(σ_i + τ) ≈ α/τ, which decays over time. This is a first-order learning-rate annealing mechanism that exists independently of the K-FAC preconditioner. Since no ablation keeps the τ and λ schedules fixed while removing the curvature term (e.g., F=0) and no ER baseline is run with the same effective step decay, the empirical AAA/WC-Acc gains on Split-CIFAR100 and Split-TinyImageNet do not yet establish that the second-order information is the causal mechanism. Please add such an ablation and report the continual metrics.
  2. [Section 4.1, Eq. (2); Algorithm 1, lines 11-16] The optimization problem in Eq. (2) is posed with a hard KL constraint δ^T F_B δ ≤ 2ρ, but the implemented algorithm never chooses λ to satisfy that constraint: Algorithm 1 increases λ heuristically with the number of observed classes or with time, and no value of ρ is set or measured. As a result, the 'explicit KL-divergence constraints' described in the abstract and introduction are not actually enforced by the method used in the experiments. The authors should either derive the λ schedule from the KKT conditions of Eq. (2) and verify that the constraint is active, or reframe the method as a Tikhonov-weighted replay optimizer and provide an ablation showing what the constraint term contributes.
  3. [Section 4.1, Eq. (1)-(2)] The second-order Taylor expansion of the KL divergence is written without the standard factor 1/2 in the quadratic term: the text gives ∇^T δ + δ^T H δ instead of ∇^T δ + (1/2)δ^T H δ. The derivation then jumps from Eq. (2) to the closed-form update δ* = -(H_N + H_B + λ F_B + τI)^{-1}(∇_N + ∇_B) without writing the Lagrangian or the stationary condition. The missing 1/2 factor changes the stationary condition under a strict reading of Eq. (2), and the factor is not tracked when the Hessians are later replaced by FIMs. Please write out the expansion and the KKT system explicitly, and state which factors are exact, approximated, or absorbed into hyperparameters.
  4. [Section 4.2, Section 4.3] The replacement of both Hessians by Fisher Information Matrices is justified by the FIM/GGN equivalence only under model-and-data conditions that are not checked in the nonstationary OCL setting; the text itself notes the model is far from the optimum in OCL, which is exactly where the equivalence is weakest. In addition, the boundary reset policy in Section 4.3 reinitializes only the classifier factor G while retaining the activation factor A, relying on an unstated assumption that representations are consistent across tasks. Since the optimizer's behavior is driven by this approximated curvature, I ask for a targeted check, for example comparing the K-FAC preconditioner against an exact FIM or Hessian-vector-product preconditioner on a small-scale variant, and for an ablation of the reset policy on a setting where representations change substantially.
minor comments (5)
  1. [Section 4.1, paragraph beginning 'Stability constraint:'] There is a typo in the phrase 'Stability constraint:s approach' that should be corrected to 'Stability constraint:' or 'Stability constraint approach:'. Minor: please proofread the paragraph.
  2. [Algorithm 1 and Section 4.3] The hyperparameter Δτ is described as the 'increase of τ' but the units are ambiguous: Algorithm 1 increments τ inside the inner loop over gradient steps, so it should be stated explicitly whether Δτ is per gradient step, per batch, or per task, and how the value was selected in the grid search.
  3. [Tables 1 and 2; Appendix B] Baseline numbers in Table 1 are taken from prior papers with different numbers of seeds (LPR used 10 seeds, the other methods 5) and possibly different hyperparameter-selection procedures. A sentence in the main text acknowledging this asymmetry and its likely direction of bias, rather than only in Appendix B, would improve the fairness assessment.
  4. [Figure 2 and Appendix Figure 11] The grid-search plots over α and α/τ appear to report a single run without confidence intervals; since Section 5 uses them to support claims (1)-(4) about the stability-plasticity tradeoff, the number of seeds and any variance should be reported.
  5. [Appendix C, 'Last Layer FIM'] For one-hot categorical targets, the derivation uses y_i^2 = y_i without stating this identity; adding a short note makes the algebra in the diagonal FIM computation transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: OCAR's derivation is a self-contained constrained-optimization construction, and its empirical claims are evaluated against external baselines; self-citations are present but not load-bearing.

full rationale

The derivation chain is not circular. Section 4.1 starts from a replay objective, takes a second-order Taylor expansion, and solves the KKT system of the constrained problem to obtain the damped preconditioned update δ*_t = −α(F_N + (1+λ)F_B + τI)^{-1}(∇_N+∇_B). The FIM appears both as the Hessian of the KL stability constraint and as the curvature preconditioner; that is the method's defining construction rather than a hidden reuse of the conclusion. The Hessians are approximated by FIM/GGN following Martens and Kunstner, and the Fisher is computed from the model's predictive distribution, which is standard for natural-gradient methods and does not presuppose the empirical results. The experimental comparisons use external baselines (ER, DER++, ER-ACE, MIR, RAR, SCR, LPR, etc.), with numbers taken from the shared OCL survey code and the ICML24 LPR paper; the author's co-authorship of the survey and Avalanche is a self-citation, but it is not load-bearing because the baseline numbers are independent, externally published results rather than OCAR outputs. Hyperparameters are selected on a validation prefix (first four experiences) and all reported metrics are computed on the held-out test stream, so no fitted parameter is relabeled as a prediction. The lack of an ablation that removes the K-FAC curvature term while keeping the τ/λ schedules is a real attribution/confound concern for the causal claim that second-order information drives the gains, but a missing ablation is a correctness risk, not a circular reduction; no quoted equation reduces by construction to its inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central algorithm depends on several heuristics (EMA, reset rules, tau and lambda schedules) and on the domain assumption that a K-FAC Fisher estimate remains meaningful under nonstationarity. No new physical entities are introduced.

free parameters (4)
  • Learning rate alpha
    Controls step size; selected by validation on the first four experiences.
  • Tikhonov increase Delta_tau per step
    tau starts at alpha and is increased by Delta_tau each step; Delta_tau chosen by validation to balance stability and plasticity.
  • EMA coefficient alpha_EMA
    Controls exponential moving average of K-FAC Kronecker factors; tuned by validation.
  • Buffer-class weighting factor n/k
    The FIM on buffer data is weighted by the ratio of classes in the buffer to classes in the current batch; a heuristic choice to strengthen stability as more classes are seen.
assumptions (5)
  • domain assumption The FIM is equivalent to the Hessian of the KL divergence when the model is at the optimum, and this equivalence is assumed to hold approximately near optima and in nonstationary settings.
    Used in Section 4.2 to replace Hessians H_N and H_B with FIMs F_N and F_B.
  • domain assumption Cross-entropy loss is treated as a KL divergence, so the gradient of the loss is the same as the gradient of the KL; losses deviating from KL break the method (as observed with DER++).
    Section 6.3 and Appendix B discuss this limitation.
  • domain assumption The K-FAC block-diagonal approximation and the EMA of its Kronecker factors provide a sufficient estimate of the FIM for gradient preconditioning in online continual learning.
    Section 4.2 and Algorithm 1, where the preconditioner F_INV is built from EMA factors computed only on the first inner step s=1.
  • ad hoc to paper The model representations are consistent across tasks, so resetting only the classifier K-FAC factor when its shape changes preserves the validity of the other factor.
    Section 4.3, 'Estimate of K-FAC factors at boundaries'.
  • standard math Standard properties of the Kronecker product allow efficient inversion of block-diagonal curvature matrices.
    Section 4.2, K-FAC paragraph.

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Cite this review

Pith. "Pith review of Online Curvature-Aware Replay: Leveraging $\mathbf{2^{nd}}$ Order Information for Online Continual Learning." pith.science (2026). https://pith.science/paper/DDHSJGMQ

@misc{pith2026250201866,
  author       = {Pith},
  title        = {Pith review of: Online Curvature-Aware Replay: Leveraging $\mathbf2^nd$ Order Information for Online Continual Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DDHSJGMQ}},
  note         = {Machine review of arXiv:2502.01866}
}
read the original abstract

Online Continual Learning (OCL) models continuously adapt to nonstationary data streams, usually without task information. These settings are complex and many traditional CL methods fail, while online methods (mainly replay-based) suffer from instabilities after the task shift. To address this issue, we formalize replay-based OCL as a second-order online joint optimization with explicit KL-divergence constraints on replay data. We propose Online Curvature-Aware Replay (OCAR) to solve the problem: a method that leverages second-order information of the loss using a K-FAC approximation of the Fisher Information Matrix (FIM) to precondition the gradient. The FIM acts as a stabilizer to prevent forgetting while also accelerating the optimization in non-interfering directions. We show how to adapt the estimation of the FIM to a continual setting stabilizing second-order optimization for non-iid data, uncovering the role of the Tikhonov regularization in the stability-plasticity tradeoff. Empirical results show that OCAR outperforms state-of-the-art methods in continual metrics achieving higher average accuracy throughout the training process in three different benchmarks.

Figures

Figures reproduced from arXiv: 2502.01866 by the authors.

Figure 1
Figure 1. 2D projections of the training trajectories for ER and OCAR on Split MNIST (5 Tasks). Loss surface on the first task (left), second task (middle), and the average loss on all the 5 tasks (right). The black stars highlight the task boundaries. More details on the 2D projections and additional plots are available in the Appendix. dients. To avoid keeping track of errors, if Gl,l changes shape, the EMA of this factor i… view at source ↗
Figure 2
Figure 2. Grid search over α and α τ : (left) forgetting on the first task, (right) plasticity measured as the accuracy on the final task. Metrics are computed on the test stream at the end of training. 0.0 Time 0.5 1.0 L p ×10 5 0.0 Time 0.5 1.0 Ls ×10 5 ER EWC NGD OCAR [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Left: Lp Cumulative loss of single batches. Right: Ls Cumulative loss measured on all previous data of the stream. Stability-Plasticity tradeoff of OCAR hyperparameters: An interpretation of OCAR hyperparameters in the stability￾plasticity tradeoff can be given. While higher λ gives more importance to the FIM of the buffer (stability) and higher values of α allow larger learning steps (plasticity), the role of τ and… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Ratio between the norm of the gradient after being transformed with OCAR and the norm of the original gradient when a small τ is used In class-incremental settings, at task boundaries, we can assume the probability predicted for the new classes will be pretty low due t…
Figure 5
Figure 5. Figure 5: 2D projections of the training trajectories for ER and OCAR on Split MNIST (5 Tasks). The black stars highlight the task boundaries, the red star the final model. We also show learning curves on each task separately. plasticity measures. Then, on the same data, using t…
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Accuracy over time on the validation set for Split-TinyImagenet experiment. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Accuracy over time on the validation set for Online CLEAR experiment. 10 4 10 2 10 0 10 2 Eigenvalue of F ( i) 10 6 10 4 10 2 10 0 10 2 10 4 Effectiv e Ste p Size ( i) Effective Step Size vs. Eigenvalues = 10 1 , = 10 1 = 10 1 , = 10 3 = 10 1 , = 10 5 = 10 3 , = 10 1 =…
Figure 9
Figure 9. Figure 9: Effective step size against the FIM eigenvalues. The step size in the directions with small eigenvalues is regularized via τ , while large eigenvalues are unaffected by it. The learning rate α affects all the directions equally. 0 2000 4000 6000 step 0.0 0.5 1.0 ACC ER…
Figure 10
Figure 10. Figure 10: Learning curves on the first task (10b) and the average accuracy for all tasks (10a). 19 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Grid search over α and α τ : (left) average accuracy, (middle) forgetting on the first task, (right) plasticity measured as the accuracy on the final task. Metrics are computed on the test stream at the end of training. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.