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arxiv: 2606.10406 · v1 · pith:JDQFEVCSnew · submitted 2026-06-09 · 💻 cs.LG · cs.AI

FOGO: Forgetting-aware Orthogonalization Optimizer

Toan Nguyen , Yang Liu , Trung Le , Celso de Melo , Flora D. Salim This is my paper

Pith reviewed 2026-06-27 14:08 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords forgettingoptimizerorthogonalizationcontinual learninggradient interferencerandom projectionmomentum updatescodebook memory
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The pith

Dominant mini-batch gradients suppress rare directions during every training step, and FOGO resolves the resulting short-term and long-term forgetting through spectral orthogonalization plus codebook correction.

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

The paper argues that forgetting is not limited to continual learning but arises in ordinary training whenever dominant gradients in a mini-batch crowd out rarer but useful update directions. These suppressed directions cause immediate short-term forgetting that accumulates into long-term loss when the affected knowledge is never revisited. FOGO counters this by spectrally orthogonalizing momentum updates so no single direction monopolizes the step, then maintains a compact codebook of representative past directions via random projection that preserves distances. Conflicts with the current update are fixed by lightweight orthogonal correction and a proximal step, all without storing raw data. Experiments across class-imbalanced classification, continual visual learning, LLaVA-7B fine-tuning, and GPT-2 pretraining show faster convergence and higher retention than Adam or Muon.

Core claim

Forgetting occurs as a general optimization phenomenon because dominant mini-batch gradients suppress rare update directions at each step; when such knowledge is never revisited these losses compound. FOGO continuously detects and resolves gradient interference by spectrally orthogonalizing momentum updates and storing representative past directions in a random-projection codebook whose pairwise distances are provably preserved, then applies lightweight orthogonal correction and a proximal step to resolve conflicts with minimal overhead.

What carries the argument

Spectral orthogonalization of momentum updates combined with random-projection codebook memory and proximal correction to resolve gradient conflicts.

If this is right

  • Faster convergence on class-imbalanced classification problems.
  • Improved retention under domain and class shifts in continual visual learning.
  • Higher performance during continual fine-tuning of models such as LLaVA-7B.
  • Better pretraining outcomes for models such as GPT-2 compared with Adam and Muon.

Where Pith is reading between the lines

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

  • If the interference-resolution view holds, standard optimizers may need systematic rethinking for any training regime where rare directions matter.
  • The internal codebook approach could reduce reliance on external replay buffers in continual learning settings.
  • The method might extend naturally to other optimization interference problems such as multi-task training or federated settings.

Load-bearing premise

Dominant mini-batch gradients are the primary cause of both short-term and long-term forgetting, and spectral orthogonalization plus random-projection codebook correction can remove the interference without creating new instabilities or harming non-forgetting tasks.

What would settle it

Run the same set of tasks with a version of FOGO that disables the spectral orthogonalization and codebook correction steps and check whether the reported gains over Adam disappear.

Figures

Figures reproduced from arXiv: 2606.10406 by Celso de Melo, Flora D. Salim, Toan Nguyen, Trung Le, Yang Liu.

Figure 1
Figure 1. Figure 1: Short-term forgetting under class imbalance (CIFAR-10, ResNet-18, 5 rare classes at 10% data). (a) Dominant and rare gradients maintain negative cosine similarity throughout training. (b) The combined update initially aligns with the dominant group; after dominant loss saturates (∼step 200), the bias reverses. (c) Per-step validation loss change on a held-out probe: the suppressed group’s loss increases af… view at source ↗
Figure 2
Figure 2. Figure 2: Class-imbalanced CIFAR-10. Top: FOGO outperforms Adam and Muon under class￾imbalanced training, improving rare-class accuracy and reducing rare-class forgetting while preserv￾ing non-rare accuracy. Bottom: Removing slow–fast fusion (w/o FS) and short-term filtering (w/o FS+ST) progressively increases rare-class forgetting. Curves show the mean over ten seeds, with shaded regions denoting one standard devia… view at source ↗
Figure 3
Figure 3. Figure 3: Class-incremental learning on CIFAR-10. Left: comparison with Adam, Muon, and UPGD. Right: ablations on long-term protection strength γlt and the number of frozen centroids per layer. Results are averaged over five random seeds. Results & Ablation Study [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Training loss (left) and validation loss (right) during GPT-2 pretraining. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Training loss curves across four continual learning settings. From left to right: class [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Gradient structure preservation under random projection (CIFAR-10, ResNet-18, d=128). (a) Pearson correlation between pairwise cosine similarities in the original and projected spaces remains above 0.9 for both groups. (b) Sign preservation rate stays around 89%, well above the 50% random baseline. (c) Scatter of individual cosine pairs at the final training step; both groups cluster tightly around the y=x… view at source ↗
Figure 7
Figure 7. Figure 7: t-SNE visualization of expert activations in continual VisualQA. Both plots use the same number of sampled points per expert, isolating representation geometry from routing-load differences. (a) FOGO maintains balanced within-expert variance and clear expert separation, suggesting a healthier distribution of task-specific representations. (b) AdamW leads to representation imbalance, where one expert collap… view at source ↗
Figure 8
Figure 8. Figure 8: Qualitative comparison between LoRA-MoE and FOGO. FOGO consistently provides [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
read the original abstract

We argue that forgetting is not confined to continual learning but is a general optimization phenomenon: during standard training, dominant mini-batch gradients suppress rare but useful update directions, causing short-term forgetting at every step. When such knowledge is never revisited, these losses compound into long-term forgetting-the classical failure mode of continual learning. We introduce FOGO, a scalable optimizer that continuously detects and resolves gradient interference across both regimes. FOGO spectrally orthogonalizes momentum updates to prevent dominant directions from monopolizing optimization, then stores representative past directions in a compact codebook memory built on random projection, where pairwise distances are provably preserved in low-dimensional space. At each step, conflicts between the current update and stored directions are resolved via lightweight orthogonal correction and lifted back through a proximal step, with minimal overhead and no data storage. Across class-imbalanced classification, continual visual learning under domain and class shifts, continual fine-tuning of LLaVA-7B, and GPT-2 pretraining, FOGO consistently improves convergence and knowledge retention, outperforming Adam and Muon.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper claims that forgetting arises as a general optimization issue when dominant mini-batch gradients suppress rare but useful directions, leading to both short-term and long-term forgetting. It introduces FOGO, which spectrally orthogonalizes momentum updates and applies lightweight orthogonal correction against a compact codebook constructed via random projection (with provably preserved pairwise distances), using a proximal step to resolve conflicts. The method is evaluated on class-imbalanced classification, continual visual learning with domain and class shifts, continual fine-tuning of LLaVA-7B, and GPT-2 pretraining, where it reportedly improves convergence and retention over Adam and Muon with minimal overhead and no data storage.

Significance. If the empirical gains and the underlying mechanism hold under scrutiny, the work could be significant by reframing forgetting as an in-training optimization phenomenon rather than a problem exclusive to continual learning, offering a scalable optimizer applicable to large models with potential for improved stability in imbalanced and sequential training regimes.

major comments (3)
  1. [Method description] The central claim that spectral orthogonalization plus random-projection codebook correction resolves gradient interference without introducing new instabilities or altering effective step sizes is load-bearing for all reported gains, yet the manuscript provides no ablation isolating the codebook's contribution or testing performance degradation when the correction is removed on non-forgetting tasks.
  2. [Introduction / opening argument] The assumption that dominant mini-batch gradients are the primary cause of forgetting (rather than other factors such as learning-rate schedules or data statistics) underpins the method's design, but no direct quantitative test or comparison to alternative explanations is presented to support this as the dominant mechanism.
  3. [Method] The claim of provable distance preservation in the low-dimensional codebook via random projection is stated but lacks an explicit statement of the projection dimension, the Johnson-Lindenstrauss parameters used, or a derivation showing that the proximal lift-back step preserves the claimed orthogonality properties.
minor comments (1)
  1. [Abstract] The abstract states 'minimal overhead' but does not quantify the additional compute or memory cost of the codebook maintenance and proximal step relative to Adam or Muon.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major point below and will incorporate revisions to strengthen the manuscript where appropriate.

read point-by-point responses

  1. Referee: The central claim that spectral orthogonalization plus random-projection codebook correction resolves gradient interference without introducing new instabilities or altering effective step sizes is load-bearing for all reported gains, yet the manuscript provides no ablation isolating the codebook's contribution or testing performance degradation when the correction is removed on non-forgetting tasks.

    Authors: We agree that an explicit ablation isolating the codebook component would strengthen the claims. In the revised manuscript we will add experiments that disable the codebook correction while retaining spectral orthogonalization, evaluating on standard balanced classification tasks (e.g., CIFAR-10/100 and ImageNet subsets) to verify that performance does not degrade relative to Adam/Muon baselines when forgetting is not a concern. revision: yes

  2. Referee: The assumption that dominant mini-batch gradients are the primary cause of forgetting (rather than other factors such as learning-rate schedules or data statistics) underpins the method's design, but no direct quantitative test or comparison to alternative explanations is presented to support this as the dominant mechanism.

    Authors: The paper motivates the mechanism through the observed interference pattern and demonstrates consistent gains across imbalanced, continual, and large-model regimes. To address the request for direct comparison, we will add a targeted analysis section that varies learning-rate schedules and data statistics independently while measuring directional suppression, providing quantitative support for the relative contribution of mini-batch gradient dominance. revision: yes

  3. Referee: The claim of provable distance preservation in the low-dimensional codebook via random projection is stated but lacks an explicit statement of the projection dimension, the Johnson-Lindenstrauss parameters used, or a derivation showing that the proximal lift-back step preserves the claimed orthogonality properties.

    Authors: We will revise the method section to state the exact projection dimension employed, the specific Johnson-Lindenstrauss lemma parameters (including target dimension and failure probability), and include a short derivation (or reference to the relevant lemma) confirming that the proximal lift-back step preserves the orthogonality guarantees after the low-dimensional correction. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain self-contained with no reductions to inputs

full rationale

The abstract and provided description introduce FOGO via a conceptual argument about gradient interference and describe its components (spectral orthogonalization of momentum, random-projection codebook, proximal correction) without any equations, fitted parameters, or self-citations that reduce the claimed improvements to quantities defined in terms of themselves. No load-bearing step matches the enumerated circularity patterns; the performance claims on downstream tasks are presented as empirical outcomes rather than algebraic identities or renamed fits. The central premise is an assumption about forgetting mechanisms, not a derivation that collapses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities can be extracted beyond the high-level description of the codebook memory and orthogonal correction. The central claim rests on the unverified premise that gradient interference is the dominant forgetting mechanism and that the proposed corrections resolve it scalably.

invented entities (1)
  • compact codebook memory via random projection no independent evidence
    purpose: store representative past directions while provably preserving pairwise distances in low-dimensional space
    Introduced in the abstract as the storage mechanism for past directions; no independent evidence or falsifiable prediction supplied in the provided text.

pith-pipeline@v0.9.1-grok · 5722 in / 1354 out tokens · 16498 ms · 2026-06-27T14:08:51.367306+00:00 · methodology

discussion (0)

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