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REVIEW 3 major objections 5 minor 64 references

Optimisation Is Not What You Need

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Pith's one-line read This paper argues that catastrophic forgetting is not a fixable flaw of neural networks but a mathematical consequence of tuning weights to minimise a loss, so weight-based optimisation cannot be the route to artificial general…

desk verdict A well-intentioned position paper whose formal no-go theorem rests on an unstated injectivity assumption; the central claim is not proved. read the letter →

arxiv 2507.03045 v1 pith:JAQVHBPA submitted 2025-07-03 cs.LG cs.AI

classification cs.LGcs.AI
keywords catastrophicforgettingoptimisationmethodsoverfittingartificialgeneralintelligenceworld-modellingweightedmappingscontinuallearning
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

This paper tries to prove that catastrophic forgetting is inherent to any learning method that solves problems by adjusting weights to minimise a loss function, not a bug that better architectures can remove. It formalises an optimiser as a weighted mapping $M_w(i)=i\circ w$ and shows that two different tasks demand different weights, so retraining on a new task must change the outputs on the old task. The same formal setup is used to argue that overfitting is also a necessary consequence of the optimisation framing. To make the point constructive, the paper reports small experiments with a representation-based world-modelling algorithm that keeps its accuracy after sequential training on two datasets, and concludes that general intelligence requires moving outside optimisation.

What carries the argument

The load-bearing object is the weighted mapping $M_w: I \to_w O$ with $M_w(i)=i\circ w$, where $\circ$ is any operation that combines an input $i$ with a weight vector $w$. Theorems 1 and 2 establish an injectivity principle for this object: for two different optimisation problems, distinct weights imply distinct outputs on every input, and equal outputs across both training sets force equal weights. Theorem 3 applies that principle to sequential training, and Definition 6 defines catastrophic forgetting as a change in previously learned input assignments; the proof of Theorem 3 is what converts 'the weights must change' into 'the old outputs must change'.

What would settle it

Train a linear model $f(x)=w^\top x$ on task A whose inputs span a subspace $S$, then train on a disjoint task B while projecting every gradient step onto the orthogonal complement of $S$, so outputs on task A's inputs never change. If the model learns B without losing accuracy on A, a weight-based optimisation method has avoided catastrophic forgetting, contradicting Theorem 3; the same test can be done on an overparameterised neural network by projecting updates onto the null space of the Jacobian of task-A outputs.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 3: any optimisation method that produces a weighted mapping $M_w$ and is trained on a new task $Q'\neq Q$ on a disjoint input set will suffer catastrophic forgetting as defined in Definition 6. The proof runs through Theorem 1, which says that two weighted solutions to two different problems must have different weights, and therefore that a weight update forced by the new task must alter the mapping on the old task's inputs. The same machinery yields Theorem 5, which states that a model that optimally solves one training set cannot simultaneously solve a disjoint but similar test set, making overfitting inherent too. The paper presents these results as the formal reason why mitigation strategies, from elastic weight consolidation to rehearsal to architectural changes, have only reduced but never eliminated the problem, and as evidence that the field needs non-optimisation, world-modelling approaches.

Load-bearing premise

The proof depends on the assumption that two different weight vectors always give different outputs on every single input, but real overparameterised networks routinely have many weight vectors that implement the same function on a given finite set of inputs.

Editorial extensions

If this is right

  • A single weight-based model trained by loss minimisation cannot learn a second, disjoint task without degrading its performance on the first, so continual learning with one network is impossible in principle.
  • Existing mitigation strategies such as elastic weight consolidation, rehearsal, and architecture changes can only reduce the damage; if the proof is right, none of them can eliminate it.
  • Overfitting is not a training pathology but a consequence of the same setup: the better a model solves its training problem, the further it sits from the slightly different test problem.
  • Any AGI built from a single loss-guided weight-tuning learner is blocked; the viable direction is modular or representation-based systems where each component handles one task.
  • A set of single-task optimisation methods can still solve multiple tasks, as long as no single model is retrained on a different problem.

Reading between the lines

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

  • The proof's hidden injectivity assumption means the result covers parametric families where weights are identifiable from the mapping on the training inputs; overparameterised networks, where many weights give identical outputs, are not covered by the argument as written.
  • If non-identifiability is the escape route, then overparameterisation and null-space-constrained updates are what could make continual learning work for neural networks, the opposite of the paper's dismissal of weight-based methods.
  • The empirical demonstration uses two small numeric datasets with largely non-overlapping representations; a natural extension is to run the same sequential protocol on image or language tasks where representations overlap, where the claimed immunity of world-modelling methods is not yet shown.
  • The theorem applies to loss-guided weight tuning, not to learning in general; lookup tables, non-parametric methods, and prototype-based classifiers fall outside it, which suggests the real distinction is not optimisation versus world-modelling but global weight identifiability versus other forms of memory.
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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

3 major / 5 minor

Summary. The paper claims to formally prove that catastrophic forgetting is inherent to all optimisation methods that learn by tuning weights, and that overfitting is similarly inherent. It introduces a formal framework of weighted mappings and machine-learning optimisation problems, states four theorems (Theorems 1–4 on forgetting, Theorem 5 on overfitting), and argues that world-modelling frameworks such as the author's Unsupervised Cognition avoid both problems. Two small experiments on UCI datasets (Wisconsin Breast Cancer and Pima Indians Diabetes) are reported as empirical support for the world-modelling alternative.

Significance. If the claimed formal result were correct, it would be a far-reaching negative result for continual learning and for the optimisation-based approach to artificial general intelligence. The paper also attempts a positive contribution by pointing to world-modelling alternatives, which is a direction worth exploring. However, the formal core is the load-bearing contribution, and it is invalid: Theorem 1 relies on an unstated injectivity assumption that is false for overparameterised networks, and the later theorems inherit this flaw. The empirical evidence is too limited to carry the general claims. The paper is therefore not publishable in its current form, although the framework and the discussion of alternative approaches may contain seeds for future work.

major comments (3)
  1. [Section III, Theorem 1] The proof of the 'only if' direction asserts that if w≠w' and Q≠Q', then i◦w≠i◦w' for every i∈I. This is a substantive injectivity/uniqueness assumption, not a consequence of Definitions 2–4. It fails for overparameterised networks, where many weight vectors implement the same function on a finite training set. For example, with I=R^2, M_w(x)=w·x, L={(1,0)}, L'={(2,0)}, w=(1,1), w'=(1,2), both weight vectors achieve zero loss on both tasks and give identical outputs on L∪L', yet w≠w'. This directly falsifies the claimed 'only if' direction of Theorem 1, which is then used as the foundation for Theorem 3.
  2. [Section IV, Theorem 3] The proof invokes Theorem 1 to conclude that y_L=y'_L implies w=w'. Theorem 1, however, requires equality of the two mappings on both L and L'. The proof has explicitly assumed y_{L'}≠y'_{L'}, so the condition of Theorem 1 is not met. The earlier step 'that implies that w≠w′ because i◦w≠i◦w′ ∀i∈L′' again relies on the same unstated injectivity assumption. Consequently Theorem 3 does not establish that every weight-based optimisation method must suffer catastrophic forgetting; it establishes a consequence of the paper's hidden axiom, not of the general definitions.
  3. [Section V, Theorem 5] The proof states that 'by Theorem 1, P cannot solve Q too', but Theorem 1 concerns two different mappings that solve two different problems and says nothing about a single mapping solving two optimisation problems. Moreover, Definition 7 already defines overfitting as failing to minimise on a new input set, so the theorem's conclusion is essentially built into its premise. The deduction is therefore not a proof that overfitting is inherent to optimisation methods.
minor comments (5)
  1. [Throughout] The paper writes "training set L ∈ I", but L is a subset of the input set I; it should read L ⊆ I.
  2. [Definition 1] The notation "∃o ∈ O∥M(i)=o" uses ∥ rather than a proper separator such as "such that"; the definition is also missing a colon after ∀i∈I.
  3. [Section III, Theorems 1–2] The symbol ≡ is used for equivalence of optimisation problems (Q' ≡ Q), but no definition of equivalence is given; this matters because the proofs rely on it.
  4. [Section VI(A)] There is a typo: "Hence. the learning" should read "Hence, the learning".
  5. [Section VI] The empirical section reports only two datasets and does not provide model hyperparameters, dataset split details, or measures of variability; as presented, it is too limited to support the general claim that world-modelling methods avoid catastrophic forgetting and overfitting.

Circularity Check

2 steps flagged · score 8.0 of 10

Theorem 3's inevitability proof assumes pointwise output-distinctness for different weights, which is exactly the catastrophic-forgetting conclusion, and the empirical alternative evidence rests on the author's own Unsupervised Cognition system.

  1. other [Section III, Theorem 1 proof (used in Theorem 3, Section IV)]
    "Now, as Mw solves Q, and Mw′ solves Q′, with Q ≠ Q′, then i ◦ w ≠ i ◦ w′ ∀i ∈ I, where ◦ is the operations performed by the mapping M."

    Theorem 1 is meant to prove that solving two different optimisation problems forces different outputs on the old training set. Instead, the proof assumes the pointwise version of that claim: whenever w≠w′, the outputs i◦w and i◦w′ differ for every input i, including the old inputs L. Under that assumption, Mw(L)≠Mw′(L) follows immediately, so the only-if direction is the assumption itself, not a consequence of Definitions 2–4. The assumption is false in general: with M_w(x)=w·x, w=(1,1), w′=(1,2), and x=(1,0), the outputs coincide despite w≠w′. Since Theorem 3 derives the inevitability of catastrophic forgetting directly from Theorem 1, the central impossibility result reduces to this unstated injectivity condition rather than to the definitions.

  2. self citation load bearing [Section VI (World-modelling alternatives), Section VI.A, references [23]–[25]]
    "This framework has been implemented into an algorithm called Unsupervised Cognition [23], that has obtained astounding results, not only getting similar results to those obtained by optimisation methods, but actually becoming the state-of-the-art in some cases [23], [25]."

    The paper's positive conclusion—that world-modelling methods avoid catastrophic forgetting and overfitting—is supported by the author's own Unsupervised Cognition/Synthetic Cognition line of work (refs [23], [24], and [25] are by the same author) and by two small experiments run on that same implementation. No independent implementation or external benchmark is used. Thus the claim that the field should look outside optimisation rests on self-citations that are not independently verified in this paper. This is load-bearing for the overall recommendation, although it is not the basis of Theorem 3.

full rationale

The formal derivation chain collapses at Theorem 1: the only-if direction is obtained by assuming i◦w≠i◦w′ for every input whenever w≠w′. That pointwise injectivity condition is precisely a stronger form of the catastrophic-forgetting conclusion for weighted mappings, so Theorem 3's claim that any optimisation method based on weights will suffer catastrophic forgetting is not proved from the stated definitions; it is assumed. The same assumption reappears in Theorem 3's proof ('i ◦ w ≠ i ◦ w′ ∀i ∈ L′'), and Theorem 5's overfitting result also invokes Theorem 1. The empirical world-modelling evidence is additionally self-referential, since it evaluates only the author's own Unsupervised Cognition algorithm and cites the author's prior papers for its state-of-the-art claims. Because the central theorem reduces to an unstated assumption equivalent to the target result, the circularity score is high; however, the failure is also a straightforward mathematical gap, so the finding should be read as 'not proven' rather than as a demonstration that the conclusion is false.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The formal results rest on hidden assumptions rather than explicit free parameters: injectivity of the weight-to-function map, uniqueness of solutions, and the reduction of all optimisation methods to weighted mappings. No free numerical parameters are reported for the experiments; the algorithm's configuration is not given.

assumptions (3)
  • ad hoc to paper For distinct weight assignments w and w', the weighted mapping satisfies i◦w ≠ i◦w' for every input i.
    Used in Theorem 1 and Theorem 3 (Sections III and IV). Not generally true; overparameterized neural networks can have many weight vectors with identical input-output behavior.
  • ad hoc to paper If two mappings both solve the same optimisation problem Q, they agree on every input in I.
    Used in Theorem 2 (Section III). False when the loss has multiple global minima or when the training data do not identify a unique function.
  • domain assumption Every optimisation method of interest produces a single weighted mapping Mw, and learning a new task necessarily changes the weights.
    Definition 5 restricts the proof to weighted mappings, but the abstract and conclusions generalize to all optimisation methods; this scope reduction is used without justification.

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Cite this review

Pith. "Pith review of Optimisation Is Not What You Need." pith.science (2026). https://pith.science/paper/JAQVHBPA

@misc{pith2026250703045,
  author       = {Pith},
  title        = {Pith review of: Optimisation Is Not What You Need},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAQVHBPA}},
  note         = {Machine review of arXiv:2507.03045}
}
read the original abstract

The Artificial Intelligence field has focused on developing optimisation methods to solve multiple problems, specifically problems that we thought to be only solvable through cognition. The obtained results have been outstanding, being able to even surpass the Turing Test. However, we have found that these optimisation methods share some fundamental flaws that impede them to become a true artificial cognition. Specifically, the field have identified catastrophic forgetting as a fundamental problem to develop such cognition. This paper formally proves that this problem is inherent to optimisation methods, and as such it will always limit approaches that try to solve the Artificial General Intelligence problem as an optimisation problem. Additionally, it addresses the problem of overfitting and discuss about other smaller problems that optimisation methods pose. Finally, it empirically shows how world-modelling methods avoid suffering from either problem. As a conclusion, the field of Artificial Intelligence needs to look outside the machine learning field to find methods capable of developing an artificial cognition.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.