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REVIEW 3 major objections 6 minor 16 references

Forgetting is a learner's failure of predictive self-consistency: learning from its own imagined future changes its predictions, and this is the quantity a general definition should measure.

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2026-08-03 23:36 UTC pith:WYGOJ7RA

load-bearing objection A genuinely useful reframing of forgetting as predictive self-consistency, but the proposed estimator's dependence on an external validation distribution weakens the 'learner-only' claim and the empirical sweep is too thin to back the title. the 3 major comments →

arxiv 2511.04666 v4 pith:WYGOJ7RA submitted 2025-11-06 cs.LG stat.ML

Forgetting is Everywhere

Ben Sanati , Thomas L. Lee , Trevor McInroe , Aidan Scannell , Esmeralda S. Whitammer , David Abel , Amos Storkey This is my paper
classification cs.LG stat.ML
keywords forgettingpredictive self-consistencycontinual learningBayesian inferencedeep learningcatastrophic forgettingreinforcement learninggenerative modelling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that forgetting is not a special failure of continual learning but a general property of any learner: forgetting occurs when a learner's predictive distribution changes after it performs simulated updates on data sampled from its own predictions, with no new information from the environment. This yields a task- and algorithm-agnostic measure, the k-step propensity to forget Γ_k(t), which is zero for exact Bayesian inference (Theorem 5.1) and non-zero across regression, classification, generative modelling, and reinforcement learning in deep learning. The paper also reports a trade-off: moderate forgetting improves training efficiency, while too little slows adaptation and too much destabilises learning. A curious reader should care because it offers a unified vocabulary for a phenomenon previously measured only through task-specific performance drops.

Core claim

The central claim is that forgetting can be defined as a violation of predictive self-consistency: for a learner at state Z_{t-1} with history H_{0:t-1}, the predictive distribution q(H_{t+k:∞}|Z_{t-1},H_{0:t-1}) must equal the k-step simulated marginalisation q*_k obtained by sampling targets from the learner's own predictive distribution, sampling unmodelled inputs from an external kernel q_e, and applying the learner's update rule k times. If the divergence Γ_k(t) between the two is positive, the learner has forgotten. The paper proves that exact Bayesian updates satisfy the equality with Γ_k(t)=0 for all k,t, so parameter updates need not cause forgetting, and it demonstrates experimenta

What carries the argument

The load-bearing object is the predictive distribution q(H_{t+1:∞}|Z_t,H_{0:t}), a distribution over infinite future observation-output sequences the learner would generate by simulating its own inference-mode updates u′. On top of it, the k-step simulated marginalisation q*_k averages the predictive distribution after k updates on self-generated samples; the measure Γ_k(t) is a divergence between the original and the simulated predictive distributions. This construction isolates forgetting from backward transfer and from any change in parameters that leaves predictions intact, because it compares only predictive distributions before and after introspection-driven updates.

Load-bearing premise

The operational measure requires sampling unmodelled inputs from a held-out validation set; if the measured forgetfulness changes when the validation set changes, without any change to the learner itself, the measure is not purely a property of the learner as Desideratum 4.4 demands.

What would settle it

Train a fixed learner to the same state, then compute Γ_k(t) twice using two different held-out validation sets (e.g., MNIST-style vs. CIFAR-style inputs) while freezing the learner; if Γ changes materially despite identical learner state, the operational measure depends on the environment input distribution and fails the learner-only desideratum. Alternatively, run the same exact Bayesian posterior but replace q_e with a wildly misspecified input source and check whether Γ remains zero.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Exact Bayesian inference never forgets, by Theorem 5.1, so continual learning theories based on parameter drift cannot be the whole story.
  • Γ_k(t) gives a way to compare forgetting across classification, regression, generative modelling, and RL with the same definition, rather than task-specific accuracy drops.
  • In deep learning, forgetting is non-zero even in i.i.d. training and spikes at task boundaries in continual learning.
  • A moderate amount of forgetting is associated with faster training; maximal training efficiency occurs at non-zero Γ, suggesting forgetting is a mechanism for selective adaptation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: because Γ_k(t) depends on the external distribution q_e used to supply unmodelled inputs, the same learner can score very differently against different validation sets; one testable extension is to report Γ for multiple q_e choices and check whether rankings of learners are stable.
  • Beyond the paper: the framework suggests a direct recipe for designing non-Bayesian learners that forget less—modify the update u so that it approximately leaves the predictive distribution invariant under self-generated targets, for instance by adding a self-consistency regularisation term penalising Γ.
  • Beyond the paper: the martingale view in the appendix implies that forgetting can be monitored online as drift of predictive statistics; one could turn Γ into a training signal rather than a diagnostic.
  • Beyond the paper: model collapse on synthetic data appears as a special case of Γ>0 in generative models, so the measure could be used to predict or detect collapse before quality degrades.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript proposes a general theory of forgetting as a violation of predictive self-consistency. It introduces an interaction formalism (Definitions 3.1–3.5) in which a learner's predictive distribution is simulated forward through k introspective updates, and defines the k-step simulated marginalisation q*_k and the propensity-to-forget measure Γ_k(t) (Definitions 4.5–4.7). The central theoretical result, Theorem 5.1, states that exact Bayesian inference satisfies Γ_k(t)=0 under Assumptions B.1–B.5. The empirical sections report nonzero Γ in regression, classification, generative modelling, and DQN on CartPole, and interpret the results as showing that forgetting is ubiquitous in deep learning and that moderate forgetting is associated with training efficiency.

Significance. If the framework were fully successful, it would provide a genuinely useful unification: it separates forgetting from backward transfer and parameter drift, it is algorithm-agnostic, and it connects to predictive Bayesian and martingale ideas. The proof of Theorem 5.1 is algebraically coherent under its stated assumptions, and the paper includes a concrete operational algorithm (Algorithm 1) with experiments spanning several learning paradigms. However, the central measure is not learner-only as claimed: Γ_k(t) depends on an externally supplied input distribution q_e, which is instantiated by a validation set in Algorithm 1. This conflicts with Desideratum 4.4 and affects the interpretation of every reported value. In addition, the assumptions of Theorem 5.1 do not match the operational estimator used in the experiments. These are load-bearing issues for the paper's main conceptual and empirical claims.

major comments (3)
  1. [§3.2, Definition 4.7, Algorithm 1] The measure Γ_k(t) is not a property of the learner alone, contrary to Desideratum 4.4. In Definitions 4.5 and 4.7 the simulated marginalisation draws X_s from q_e, and Algorithm 1 (line 5) instantiates q_e by sampling inputs from a held-out validation set. For a fixed checkpoint Z_{t-1}, two different validation distributions will produce different simulated update trajectories, different q*_k, and hence different Γ_k(t). A learner can therefore be scored as forgetful under one validation set and non-forgetful under another. This is not a presentation issue: the paper's central claim that forgetting is a learner property depends on Γ being invariant to the environment, but the operational definition builds the environment in via q_e. The authors should either prove the relevant invariance, define Γ as a function of (learner, q_e), or replace Desideratum 4.4 with a conditional statement.
  2. [Theorem 5.1 vs. Algorithm 1] Theorem 5.1 does not justify the empirical estimator. Assumption B.1 requires q_e(X_t|Y_t,Z_{t-1}) = ∫ q(X_t|Y_t,θ) q_{Z_{t-1}}(θ)dθ, i.e., q_e is the model's own marginal likelihood, and the proof relies on this quantity cancelling with the Bayes denominator (Eq. 28). In Algorithm 1, unmodelled input components are sampled from a validation set, which is not the model's marginal likelihood for a discriminative learner. Consequently, the cancellation in Eq. (28) has no analogue in the experiments, and the theorem does not show that the computed Γ estimates a Bayesian self-consistency violation. The authors should either extend the theorem to general q_e (if possible) or clearly state that Algorithm 1 is a heuristic estimator whose relationship to Theorem 5.1 is an additional assumption.
  3. [Definition 4.5 vs. §3.2] There is an ambiguity about which update rule defines self-consistency. The predictive distribution in §3.2, Eq. (2), is generated by the inference-mode update u′, which is required to leave q_f invariant. Definition 4.5, however, defines simulated marginalisation by evolving Z_i through the learning-mode update u, and then evaluates q(H|Z_{t'},H_{0:t'}). It should be stated explicitly whether q*_k is meant to simulate training updates (u) or introspective updates (u′), and how the two modes interact. As written, the object being measured could be either 'sensitivity to self-training' or 'sensitivity to introspective inference,' and the two are not the same when u ≠ u′. This ambiguity affects the interpretation of all reported Γ values.
minor comments (6)
  1. [Algorithm 1] Line 5 of Algorithm 1 samples inputs from p_e, while the text and Definition 4.5 use q_e; unify the notation and clarify whether these are the same kernel.
  2. [Figure 3] The left and middle panels of Figure 3 report 'Mean Γ40' and 'Training Efficiency' on different scales/axes; the caption should define the normalization of the efficiency measure and the averaging over training steps.
  3. [Takeaway 3 / §5.3] The claim that forgetting is 'necessary for efficient adaptation' is stronger than the correlational evidence in Figure 3 and the hyperparameter sweeps. The experiments show an association, not a causal necessity; consider softening the wording.
  4. [Appendix B.4] Equations (48)–(49) state a martingale condition on the random variables Q_s^k, whereas Definition 4.7 compares the unconditional distributions q and q*_k. These two notions of self-consistency should be explicitly reconciled; as written they are not equivalent.
  5. [§5.2 and Figure 5] The empirical support for the title 'Forgetting is Everywhere' is based on small synthetic tasks (sinusoid, two-moons, CartPole) plus one CIFAR-10 experiment in the appendix. Please qualify the scope or include larger-scale experiments before making a universal claim about deep learning.
  6. [Abstract / §5] The abstract accurately calls the generative-model-collapse explanation 'tautological,' but the main text presents self-generated training as empirical validation. It would help to state in §5 that this prediction is a consistency check of the definition rather than an independent empirical discovery.

Circularity Check

3 steps flagged

The central definition makes forgetting equal to predictive self-inconsistency, and the abstract itself calls the generative-model explanation tautological; Γ_k(t) is also defined through an externally chosen q_e, so the measure is not a learner-only property.

specific steps
  1. self definitional [Abstract; Definitions 4.5–4.7]
    "Our theory naturally yields a general measure of an algorithm's propensity to forget, proves that exact Bayesian inference allows for adaptation without forgetting, and provides a tautological explanation for why generative models forget when trained on their own synthetic outputs."

    The paper defines forgetting as a violation of predictive self-consistency: Definition 4.7 sets Γ_k(t) := D(q(H_{t+k:∞}|Z_{t-1},H_{0:t-1}) ∥ q^*_k(H_{t+k:∞}|Z_{t-1},H_{0:t-1})). Therefore 'generative models forget when trained on their own synthetic outputs' is not an independent derived consequence; any training that changes the predictive distribution under self-sampling is forgetful by definition. The abstract explicitly concedes that this explanation is tautological, so the advertised finding reduces to the definition rather than being derived from it.

  2. self definitional [Definition 4.5; Algorithm 1; Desideratum 4.4]
    "for i=t, . . . , t′ the expectation is taken over Xi ∼ q_e(· |Yi, Zi−1), Yi ∼ q_f(· |Zi−1, Xi−1), Zi ∼ u(· |Zi−1, Xi, Yi). ... Sample input X (n) t+s ∼ p_e(· |H0:t+s−1, Y(n) t+s) ▷ Component not modelled. Sample from validation set."

    The operational measure Γ_k(t) depends on q_e, an externally supplied distribution, which Algorithm 1 implements as sampling X from a held-out validation set. Desideratum 4.4 states 'Forgetting is a property of the learner, not of the environment in which it operates.' But the same learner evaluated with two different validation sets yields different simulated trajectories q^*_k and hence different Γ_k(t). Thus the empirical claim that forgetting is non-zero/ubiquitous is partly manufactured by the experimenter's choice of input distribution, rather than being a learner-only property derived from the theory.

  3. other [Assumption B.1; Algorithm 1; §D.1]
    "Assumption B.1: qe(Xt |Y t, Zt−1) = ∫_Θ q(Xt |Y t, θ)q_{Zt−1}(θ) dθ. ... Algorithm 1: 'Sample from validation set.' ... To approximate future inputs Xt:∞, we sample uniformly over the empirical distribution of inputs observed thus far."

    The Bayesian non-forgetting theorem (Theorem 5.1) is proved under Assumption B.1, where q_e is the learner's own Bayesian marginal likelihood. But the experiments for discriminative models set q_e to a validation-set empirical distribution, which is not generally the model's marginal likelihood. Consequently, the Γ_k(t) measured in the experiments is not the same quantity for which Theorem 5.1 gives Γ_k(t)=0; the theorem's conclusion and the operational estimator are connected only by sharing the symbol q_e. This makes the empirical 'forgetting is everywhere' claim rely on a definitional choice of q_e that the theory does not justify.

full rationale

The formal core is not entirely circular: Theorem 5.1 is a genuine derivation from Assumptions B.1–B.5 using Bayes-rule algebra and the martingale property, and the experiments compute a well-defined divergence with stated hyperparameters. Self-citations (e.g., Lee & Storkey 2023, Abel et al. 2023a/b) are contextual rather than load-bearing, and no uniqueness theorem is imported from the authors. However, two load-bearing steps do reduce to definitions or inputs. First, the abstract explicitly promises a 'tautological explanation' of generative-model forgetting; given Definition 4.7, that explanation is just the definition of forgetting as self-inconsistency. Second, Γ_k(t) is defined through q_e and implemented with a held-out validation set, contradicting Desideratum 4.4's learner-only requirement; the same learner can be scored more or less forgetful by changing the validation distribution. Additionally, Assumption B.1's q_e differs from Algorithm 1's validation-set q_e, so the non-forgetting theorem and the experiments are not about the same object. These are partial circularities: the Bayesian proof and the efficiency/forgetting measurements retain independent mathematical and empirical content, but the headline conceptual claims are substantially definitional.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The paper introduces no physical entities. Its new mathematical objects—the simulated marginalisation q*_k and the propensity-to-forget Γ_k(t)—are definitions rather than postulated entities with independent falsifiable handles. The main free parameters are the estimator settings (k, M, divergence choice) and the external validation-set distribution used to supply unmodelled inputs. The theoretical result rests on the definitional postulate of forgetting-as-self-inconsistency plus the Bayesian assumptions B.1-B.5.

free parameters (5)
  • consistency horizon k = 40
    All experiments use k=40 simulated updates to define Γ_k(t); reported forgetfulness values vary with k (Figure 8), so this hand-chosen horizon is a free parameter of the measurements.
  • number of Monte Carlo particles M = 1000
    The estimate of q*_k uses M=1000 cloned learners (Table 3); the approximation error decreases with M, but the reported Γ values are tied to this choice.
  • divergence measure D = KL for classification/regression; MMD with RBF kernel for generative
    Definition 4.7 leaves D unspecified, and the paper uses different divergences for different tasks, making absolute Γ values not comparable across settings.
  • external input distribution q_e for unmodelled components = held-out validation set (100 or 1000 samples)
    Algorithm 1 samples unmodelled inputs X from a validation set, so the resulting Γ depends on this hand-chosen environmental distribution rather than on the learner alone.
  • predictive variance for regression likelihood = estimated from held-out residual errors
    For regression, the Gaussian predictive distribution's variance is estimated empirically from validation residuals, affecting the closed-form KL values.
axioms (5)
  • ad hoc to paper Forgetting is defined as violation of predictive self-consistency (Definition 4.6); no independent justification beyond desiderata and thought experiments.
    The central definition is a postulate of the paper. It is motivated but not derived from prior theory.
  • domain assumption The learner's predictive distribution q(H_{t+1:∞}|Z_t,H_{0:t}) faithfully represents the learner's state.
    Stated in the 'Scope of theory' paragraph after Definition 4.7; if false (e.g., buffer reinitialisation or target-network lag), Γ is undefined or incomplete.
  • domain assumption Assumption B.4 (mixture representation): the predictive future equals the posterior-weighted mixture of θ-conditional futures.
    Used directly in the proof of Theorem 5.1; restricts the non-forgetting result to learners whose predictive distribution has this exact mixture form.
  • standard math Assumptions B.1, B.2, B.3, B.5: likelihood existence, deterministic Bayes updates, predictive conditional independence, and Fubini/Tonelli regularity.
    Background measure-theoretic and regularity conditions for the Bayesian non-forgetting proof in Appendix B.3.
  • ad hoc to paper Unmodelled components of X are generated by an external data source (validation set) during simulated marginalisation.
    Algorithm 1 samples inputs from a held-out validation set; no principled method is given for selecting this distribution, and the resulting Γ depends on it.

pith-pipeline@v1.3.0-alltime-deepseek · 28775 in / 17793 out tokens · 163127 ms · 2026-08-03T23:36:27.861238+00:00 · methodology

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read the original abstract

A fundamental challenge in developing general learning algorithms is their tendency to forget past knowledge as they adapt to new data. Addressing this problem requires a principled understanding of forgetting. Yet, despite decades of study, no unified definition has emerged that offers insight into the underlying dynamics of learning. We propose an algorithm- and task-agnostic theory that characterises forgetting as a lack of self-consistency in a learner's predictive distribution, manifesting as a loss of predictive information. Our theory naturally yields a general measure of an algorithm's propensity to forget, proves that exact Bayesian inference allows for adaptation without forgetting, and provides a tautological explanation for why generative models forget when trained on their own synthetic outputs. To validate these claims, we design a comprehensive set of experiments that span classification, regression, generative modelling, and reinforcement learning. We demonstrate that forgetting is present across all deep learning settings and plays a significant role in determining learning efficiency. Together, these results establish a principled understanding of forgetting and lay the foundation for analysing and improving the information retention capabilities of general learning algorithms.

Figures

Figures reproduced from arXiv: 2511.04666 by Aidan Scannell, Amos Storkey, Ben Sanati, David Abel, Esmeralda S. Whitammer, Thomas L. Lee, Trevor McInroe.

Figure 1
Figure 1. Figure 1: State evolution and predictive distributions. The learner’s internal state Zt evolves through training updates u given (Zt−1, Xt, Yt). Each state induces a predictive distribu￾tion q(Ht+1:∞ | Zt−1, H0:t), which is updated in inference mode u ′ . This separation illustrates how each state encodes retained capabilities, and how training vs. introspective updates interact. learning, Xt denotes (current input,… view at source ↗
Figure 2
Figure 2. Figure 2: Exact Bayesian learners can adapt without forgetting. Axes showing observations (x, y) are shaded white, and parameter axes (w0, w1) are shaded grey. Top row: The same four observations are presented to a linear regression learner in different orders. Second row: By Theorem 5.1, an exact Bayesian posterior can update its parameters without forgetting. Bottom rows: Two constrained learners – a Gaussian vari… view at source ↗
Figure 3
Figure 3. Figure 3: Approximate learners can benefit from forgetting; forgetting abruptly increases at task boundaries. Left/Middle: Training efficiency and forgetfulness across varying momentum, µ, (left) and number of parameters, s, (middle) for a regression task. Efficiency is proxied by the inverse normalised area under the training loss curve; forgetting is the mean k-step propensity to forget Γ40(t) averaged over traini… view at source ↗
Figure 4
Figure 4. Figure 4: DQN actively manages the information acquisition-retention trade-off. We show TD loss, Q-value evaluation, and the forgetting profile of a DQN learner trained on cartpole across ten seeds. Early in training, TD loss is low, rising as the agent acquires new information, then decreasing once knowledge has been consolidated. The forgetting curve follows this trajectory, highlighting that forgetting is a delib… view at source ↗
Figure 5
Figure 5. Figure 5: Forgetting occurs across all deep learning scenarios. Forgetfulness dynamics of a shallow neural network trained on re￾gression, classification, and generative modelling tasks. The mean k-step forgetfulness (k ∈ [1 − 40]) over the normalised training step, with the range shown in the shaded region. Regression and classification tasks use KL divergence, while the generative task uses the maximum mean discre… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of initial and 40-step simulated marginalisation. Axes show the input space around the two-moon binary classification task. Example datapoints are overlaid and coloured by target class (red = 0, blue = 1). The background grid is shaded according to the classifier’s logit values, from red (0) to blue (1), with white indicating uncertainty at 0.5. Left: The initial predictive distribution q(Ht+k:∞… view at source ↗
Figure 7
Figure 7. Figure 7: Optimal forgetting is not necessarily zero. We examine how a regression learner’s training efficiency and propensity to forget vary under different hyperparameter settings. Left: Increasing the batch size reduces forgetting while improving training efficiency. Forgetting plateaus once the batch size approaches the dataset size (40 datapoints), at which point training efficiency increases with minimal furth… view at source ↗
Figure 8
Figure 8. Figure 8: Impact of model size on forgetting dynamics. Plots illustrating the impact of varying numbers of hidden-layer parameters in a single-hidden-layer neural network on a regression task with 40 training datapoints (details in §F). Left: Forgetfulness as a function of the number of updates k, showing how the learner’s propensity to forget evolves over update steps. Middle: Forgetfulness throughout training, hig… view at source ↗
Figure 9
Figure 9. Figure 9: Effect of the momentum parameter on forgetting dynamics. Plots illustrating the impact of varying momentum coefficients during training of a single-hidden-layer neural network on the sinusoid regression task. Left: Forgetfulness as a function of the number of updates k. Middle: Forgetfulness over the course of training. Right: Training loss curves for each momentum value. We observe that forgetting dynamic… view at source ↗
Figure 10
Figure 10. Figure 10: Effect of batch size on forgetting dynamics. Lines and markers indicate different batch sizes used in training a single-hidden￾layer neural network on the sinusoid regression task. Left: Forgetfulness as a function of the number of updates k. Middle: Forgetfulness over the course of training. Right: Training loss curves across batch sizes. Smaller batches exhibit significant fluctuations and high variabil… view at source ↗
Figure 11
Figure 11. Figure 11: Forgetting dynamics across model architectures on a high-dimensional classification task. Different-coloured lines and marker styles denote different model types: logistic regression, MLP, CNN, and ResNet. Left: Forgetfulness over training updates. Right: Validation accuracy over training time. While CNNs and ResNets exhibit substantially higher levels of forgetting than logistic regression and MLPs (with… view at source ↗
Figure 12
Figure 12. Figure 12: Replay buffers regulate prediction support. Forgetting landscapes for different replay buffer sizes. Small buffers produce high, unstable forgetting because the learner continually overwrites supported predictions as the data distribution shifts. Larger buffers stabilise the distribution and reduce forgetting, but overly large buffers reintroduce outdated transitions that also break self-consistency, lead… view at source ↗
Figure 13
Figure 13. Figure 13: Target updates trade-off stability and adaptability. Forgetting landscapes for different target network update rates τ . Large τ yields less stable forgetting dynamics due to rapidly shifting targets; small τ produces smooth but uninformative updates that prevent learning. Intermediate τ values best maintain support for predictions and thus yield balanced forgetting and effective learning. When τ is too l… view at source ↗
Figure 14
Figure 14. Figure 14: Training frequency determines forgetting. Forgetting landscapes for different training frequencies. Updating too frequently causes highly unstable forgetting due to rapid shifts in the effective training distribution. Updating too infrequently also causes large shifts in the effective training distribution. A trade-off must be found to achieve more self-consistent updates. –appendices continue on next pag… view at source ↗

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    Rl’s razor: Why online reinforcement learning forgets less.arXiv preprint arXiv:2509.04259,

    Shenfeld, I., Pari, J., and Agrawal, P. Rl’s razor: Why online reinforcement learning forgets less.arXiv preprint arXiv:2509.04259,

    Pith/arXiv arXiv