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Forgetting is Everywhere

This paper proposes an algorithm- and task-agnostic theory that defines forgetting as a lack of self-consistency in predictive distributions, demonstrating through comprehensive experiments across diverse learning settings that this phenomenon is universal in deep learning and can be precisely measured and analyzed to improve information retention.

Original authors: Ben Sanati, Thomas L. Lee, Trevor McInroe, Aidan Scannell, Esmeralda S. Whitammer, David Abel, Amos Storkey

Published 2026-07-28

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CC BY 4.0

Original authors: Ben Sanati, Thomas L. Lee, Trevor McInroe, Aidan Scannell, Esmeralda S. Whitammer, David Abel, Amos Storkey

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine your brain as a super-smart, ever-updating notebook. Every time you learn something new—like a new video game strategy or a fresh fact about dinosaurs—you write it down. But here's the tricky part: if your notebook has limited space, writing a new page might accidentally smudge or erase an old one. This isn't just a human problem; it happens to computer programs too. In the world of artificial intelligence, this is called "forgetting." It's the moment a computer learns a new trick and suddenly loses the ability to do an old one perfectly. Scientists have been studying this for decades, mostly in situations where computers have to learn a never-ending stream of new tasks, like a robot learning to walk, then run, then jump. But until now, no one had a single, clear rule for what forgetting actually is. Is it just losing a specific fact? Or is it something deeper about how the computer's "mind" changes? Understanding this is crucial because if we want to build truly smart, lifelong learning machines, we need to know exactly how and why they lose their memories.

This paper, titled "Forgetting is Everywhere," dives deep into that mystery. The authors propose a new way to look at forgetting that doesn't care about the specific task or the type of computer program. Instead of just checking if a robot gets worse at an old game, they ask a simpler, more fundamental question: "Does the computer's prediction about the future stay consistent with itself?" Imagine a weather forecaster who predicts rain for tomorrow. If they learn a new fact today, their prediction for tomorrow should only change if that new fact actually matters. If they change their prediction for no good reason, or if they change it in a way that contradicts what they already knew, that's a sign of "forgetting." The authors define forgetting as this kind of "self-inconsistency"—when a learner's internal logic gets messy and starts dropping pieces of information it should have kept.

The paper makes a few big discoveries. First, they prove mathematically that a perfect type of computer learner, called an "exact Bayesian learner," never forgets. It's like a student who can absorb new information without ever erasing a single page of their old notes, no matter how many new facts they learn. However, the paper also shows that the computers we actually use today (like the ones powering your favorite apps) are not perfect. They do forget, and this happens even when they are learning things that are very similar to what they already know. The authors ran experiments showing that forgetting is everywhere in deep learning, from simple math problems to complex video game playing.

Interestingly, the paper finds that forgetting isn't always a bad thing. In fact, for the computers we actually build, a little bit of forgetting can be helpful. It's like a gardener pruning a bush: if you don't cut away some old branches, the plant gets too crowded and can't grow new ones efficiently. The authors found that there is a "sweet spot" where a computer forgets just enough to learn new things quickly, but not so much that it loses its old skills. If it forgets too little, it gets stuck; if it forgets too much, it becomes unstable.

The paper also explains a weird phenomenon where AI models get worse when they are trained on data they created themselves (like a drawing program trying to learn from its own drawings). The authors explain this simply: when a model trains on its own output, it's essentially practicing on a version of reality that is already slightly "forgotten" or distorted. By the time it tries to learn from that, it's just forgetting more and more, leading to a rapid decline in quality.

In short, this research gives us a new, universal ruler to measure forgetting. It shows that forgetting is a natural part of how learning works, especially for the imperfect machines we build today. It's not just a bug to be fixed; it's a feature that needs to be managed. By understanding that forgetting is really about a loss of "self-consistency," scientists can now design better algorithms that know exactly when to hold on to old memories and when to let them go to make room for the new.

Technical Summary: "Forgetting is Everywhere"

Problem Statement

The paper addresses a fundamental challenge in machine learning: the tendency of learning algorithms to lose previously acquired knowledge as they adapt to new data. While "forgetting" is a well-documented phenomenon in specific domains like Continual Learning (CL) and Reinforcement Learning (RL), existing literature lacks a unified, principled definition that explains the underlying dynamics of forgetting across different learning paradigms.

Current metrics often conflate two distinct phenomena: backward transfer (where new learning improves performance on past tasks) and forgetting (where new updates erode prior knowledge). Furthermore, most existing measures are task-specific (e.g., performance degradation on previous tasks in CL) or mechanism-specific (e.g., parameter drift), failing to provide a general, algorithm-agnostic framework. The authors argue that forgetting is a fundamental property of any adaptive system updating its beliefs, occurring even in independent and identically distributed (i.i.d.) settings without task shifts.

Methodology

The authors propose a general, algorithm- and task-agnostic theory of forgetting grounded in predictive self-consistency.

1. General Framework

The paper formalizes learning as a stochastic interaction process between a learner and an environment, unifying supervised learning, RL, and generative modelling.

Learner State ( $Z_t$ ): Encapsulates all learner contents (parameters, buffers, etc.).
Interaction Process: The learner observes $X_t$ , emits an output $Y_t$ , and updates its state $Z_t$ via an update function $u$ .
Predictive Distribution: The learner's state induces a predictive distribution $q(H_{t+1:\infty} | Z_t, H_{0:t})$ over infinite future interaction sequences. This is derived by simulating the learner's future behavior using an inference-mode update function $u'$ .

2. Defining Forgetting via Self-Consistency

The core insight is that forgetting occurs when a learning update induces a change in the learner's predictive distribution that is not justified by new information.

Simulated Marginalisation: The authors define a process where the learner updates its state on data sampled from its own current predictive distribution (introspective data). Since this data contains no new information relative to the current state, a non-forgetting learner should remain unchanged in its behavior.
Self-Consistency Condition: A learner is $k$ -step self-consistent if its predictive distribution before and after simulated marginalisation is equivalent.
Propensity to Forget ( $\Gamma_k(t)$ ): Forgetting is quantified as the divergence (e.g., KL divergence) between the initial predictive distribution and the distribution after simulated marginalisation. If the distributions diverge, the update is forgetful.

Key Contributions

Unified Theory: A general theoretical formulation for reasoning about how learners acquire, retain, and lose capabilities, independent of specific problem settings or algorithms.
Predictive Definition: A precise definition of forgetting as a violation of predictive self-consistency, distinguishing it from mere performance changes or parameter updates.
Operational Measure: The introduction of the propensity to forget ( $\Gamma_k(t)$ ), a metric that quantifies the likelihood of forgetting at any given time step without relying on task-specific performance metrics.
Theoretical Proofs:
- Bayesian Non-Forgetting: Proof that exact Bayesian updates satisfy the self-consistency condition and thus do not forget (Theorem 5.1).
- Generative Model Degradation: A tautological explanation for why generative models degrade when trained on their own synthetic outputs (Model Collapse), framing it as an inevitable consequence of forgetting during simulated marginalisation.
Empirical Validation: Comprehensive experiments across classification, regression, generative modelling, CL, and RL demonstrating that forgetting is ubiquitous and plays a significant role in learning efficiency.

Results

Ubiquity of Forgetting: Experiments confirm that forgetting is present across all deep learning settings, including i.i.d. scenarios. It is not merely a result of task shifts but a fundamental property of gradient-based updates.
Bayesian Learners: Exact Bayesian learners exhibit zero propensity to forget, as their posterior-predictive distributions form a martingale. In contrast, approximate learners (e.g., variational inference, point estimates via gradient descent) violate self-consistency and exhibit forgetting.
Non-Monotonic Efficiency: The relationship between forgetting and training efficiency is non-monotonic. The "forgetting-efficiency" curve exhibits an "elbow," suggesting that a moderate amount of forgetting is beneficial for adaptation (stability-plasticity trade-off), while excessive forgetting destabilizes learning and insufficient forgetting slows adaptation.
Task Boundaries: In continual learning settings, the propensity to forget increases abruptly at task boundaries, correlating with shifts in the observation distribution.
RL Dynamics: In Deep Q-Networks (DQN), the forgetting curve tracks the Temporal Difference (TD) loss, indicating that forgetting is the mechanism agents use to manage the trade-off between acquiring new information and consolidating existing knowledge.

Significance and Claims

The paper claims to establish a principled understanding of forgetting that is independent of backward transfer and specific learning domains. By reframing forgetting as a lack of self-consistency in predictive distributions, the authors provide a foundation for:

Analyzing the information retention capabilities of general learning algorithms.
Explaining phenomena like generative model collapse and the stability-plasticity trade-off without relying on ad-hoc assumptions.
Designing future algorithms that explicitly manage forgetting as a mechanism for efficient adaptation.

The authors emphasize that their work does not propose a specific new algorithm to prevent forgetting but rather offers a theoretical lens to understand why forgetting occurs and how it impacts learning dynamics. They posit that "forgetting is everywhere" because it is an inherent consequence of how adaptive systems update their beliefs in the absence of exact Bayesian inference.