Goldfish Loss: Reduced Memory in LLM and Dynamics

• Goldfish Loss is a concept that intentionally reduces long-term memory in systems to prevent overfitting and mitigate privacy risks.
• It optimizes generative language models by selectively dropping tokens during loss computation, significantly reducing verbatim memorization.
• The approach extends to integrable systems, stochastic processes, and control systems, balancing memory utility against computational efficiency.

Goldfish Loss is a concept and technical tool that appears across several domains, notably in generative LLM training, integrable system theory, control systems engineering, and stochastic processes. At its core, Goldfish Loss refers to the deliberate reduction or avoidance of long-term memory—whether in statistical learning, physical dynamics, or algorithmic control—to mitigate undesirable effects such as memorization, loss of label information, adverse accumulation of error, or computational inefficiency. The name invokes the metaphor of the goldfish’s reputedly short memory: systems or models designed with "goldfish-like" memory tend to forget or never accumulate complete sequences or histories. The following survey provides a comprehensive account of Goldfish Loss, focusing on its formal definitions, implementations, practical applications, and theoretical significance in contemporary research.

1. Goldfish Loss in Generative LLMs

Goldfish Loss, as formalized in "Be like a Goldfish, Don't Memorize! Mitigating Memorization in Generative LLMs" (Hans et al., 14 Jun 2024), is a modification of the standard next-token prediction objective for training LLMs. In conventional causal LLMing, each token in a training sequence contributes to the loss, which encourages the model to reproduce long contiguous sequences if seen often enough. Goldfish Loss mitigates this risk by randomly excluding a subset of tokens from the loss calculation. Specifically, let $x = \{x_1, x_2, \dots, x_L\}$ be a training sequence and $G \in \{0,1\}^L$ a masking vector; the Goldfish Loss function is

$$L_\text{goldfish}(\theta) = -\frac{1}{|G|} \sum_{i=1}^L G_i \cdot \log P(x_i | x_{<i}; \theta),$$

where only positions with $G_i = 1$ are used for gradient calculation. The mask can be either periodic (dropping every $k$th token) or pseudo-random (using a hash over previous context). Crucially, all tokens remain in the input for conditioning, but dropped tokens are not directly learned or memorized, preventing verbatim reproduction.

Experimental evidence indicates that Goldfish Loss markedly reduces extractable memorization in billion-parameter-scale LLaMA-2 models, with almost no degradation of standard downstream task performance. In extreme scenarios (e.g., repeated training on the same short document), Goldfish Loss with $k=4$ entirely prevents exact reproduction of source passages, compared to near-perfect memorization with the conventional loss. The divergence between generated and training sequences almost perfectly aligns with dropped positions. Adversarial extraction attacks directed at verbatim memorization (e.g., by greedy or beam search) are successfully mitigated, reducing true positive rates for membership inference attacks by an order of magnitude for small $k$.

A plausible implication is that Goldfish Loss offers an effective and scalable defense against privacy and copyright risks inherent in LLM training, and, as demonstrated by validation loss curves, utility loss can be managed by appropriately scaling training iterations to account for the reduced token supervision per batch. However, some adversarial extraction metrics (such as those using compression-based membership inference) may still achieve moderate success rates, indicating that risk mitigation rather than complete protection is attainable.

2. Goldfish Loss in Integrable Systems and Dynamical Models

In mathematical physics, Goldfish Loss describes subtleties in labeling and solution structure of integrable many-body problems of goldfish type (Bihun et al., 2015, Calogero, 2012, Leyvraz, 2017). The goldfish family of models features equations of motion with nonlinearly velocity-dependent forces, often solvable via reductions to polynomial zero dynamics or matrix eigenvalue problems. For example, the goldfish equations

$$\ddot{z}_n = \sum_{\ell \ne n} \frac{2\,\dot{z}_n\,\dot{z}_\ell}{z_n - z_\ell}$$

admits solutions where the coordinates $\{z_n(t)\}$ are the roots of a monic polynomial $\psi_N(z; t) = z^N + \sum_{m=1}^N c_m(t) z^{N-m}$, with the coefficients $c_m(t)$ obeying their own autonomous dynamical systems.

Goldfish Loss arises as a loss of particle identity and trajectory label information: because solutions are constructed by determining the set of zeros of a symmetric polynomial or the eigenvalues of a time-dependent matrix, the identification of which particle (or coordinate) corresponds to each root at later times is unique only up to permutations. Without continuously tracking the evolution from the initial conditions, detailed labeling is lost—a phenomenon referred to as Goldfish Loss. This is most apparent when considering the polynomial solution in large $N$ or under isochronous dynamics, where the symmetry of the model induces an intrinsic ambiguity in particle tracking.

In related developments, Goldfish models appear as strong-coupling limits of Ruijsenaars–Schneider-type integrable systems (Sechin et al., 14 May 2024, Gorsky et al., 2021), such as in the duality between Toda chains on classical Lie algebras and their Goldfish-model counterparts. Under Hamiltonian reduction, the transition to a Goldfish model entails the "loss" or freezing of several degrees of freedom, with only a restricted set of conserved quantities surviving—again, this reduction is referred to as Goldfish Loss.

3. Goldfish Loss in Stochastic Processes and Volterra Dynamics

Goldfish Loss also appears in the transformation of non-Markovian stochastic processes into Markovian ones, as in "From elephant to goldfish (and back): memory in stochastic Volterra processes" (Bonesini et al., 2023). Here, a Volterra path-dependent process (the "elephant") governed by an integral over its past states,

$$X_t = \xi^0 + \int_0^t K(t-s) b(s, (\tilde{K} * X)_s) ds + \int_0^t K(t-s) \sigma(s, (\tilde{K} * X)_s) dW_s,$$

is transformed—via convolution with a pseudo-inverse kernel $\tilde{K}$—into a finite-dimensional Markovian "memory process" $\xi$ (the "goldfish"). The process $\xi_t = e^{\rho t} (\tilde{K} * X)_t$ then evolves via a standard SDE, and the transformation is reversible; the elephant’s path can be reconstructed from the goldfish's evolution. Implementing this memory loss allows efficient simulation of rough volatility models, with schemes that achieve strong convergence rate $1/2$ independent of the underlying roughness parameter—a result difficult to obtain with standard Euler schemes. This improvement in computational tractability is considered a Goldfish Loss benefit.

4. Goldfish Loss in Quantum Walks and Algorithmic Memory

In quantum information theory, Goldfish Loss denotes the loss of "quantumness" in a quantum walk induced by memory effects (Rohde et al., 2012). The goldfish walk possesses only the most recent coin value (N=1 memory), allowing full quantum interference and ballistic speedup. Introducing extended memory ("elephant walk") leads to decoherence and classical statistics, as measurement or tracing out large histories loses the interference between paths. The term thus connotes the transition from quantum to classical behavior via memory-induced loss of quantum properties.

5. Goldfish Loss in Control Systems and Adaptive Algorithms

In digital control and adaptive optics systems, Goldfish Loss is explicitly engineered through leaky integrators (Agapito et al., 2019). Goldfish-like controllers (with forgetting factor $f<1$) drop a proportion $(1-f)$ of accumulated error at every step, in contrast to pure integrators which accumulate all past error ("elephant" memory). The adoption of Goldfish Loss leads to improved stability, decreased power consumption, and reduced aliasing, particularly for high-order modes, but can potentially compromise steady-state performance in modes with high turbulence power. Empirical evidence from SOUL commissioning supports the strategic employment of Goldfish Loss for optimizing system robustness and efficiency.

6. Mathematical Implications and Applications

The consequences of Goldfish Loss extend to superintegrability and separation of variables in polynomial-integrable Hamiltonian systems (Leyvraz, 2017), dualities in integrable probability via stochastic process mappings (Gorsky et al., 2021), and algebraic properties such as closure and symmetry in Poisson bracket structures. Importantly, Goldfish Loss often entails the projection or restriction of the set of invariants or integrals of motion, with implications for both mathematical structure (e.g., loss of full spectrum in strong-coupling limits) and practical computation (e.g., simplification of dynamic equations, tractable simulation).

7. Future Directions and Open Problems

Ongoing research aims to extend Goldfish Loss methodologies to larger-scale machine learning models, further optimize token-dropping strategies, design richer privacy-preserving objectives, and exploit goldfish-type memory loss in algorithmic control for enhanced robustness. Mathematical investigation continues into the algebraic and combinatorial structures that support integrable Goldfish-type systems, dualities with quantum models, and applications in financial mathematics, stochastic control, and quantum information. Most formulations point to a fundamental trade-off: Goldfish Loss, whether in learning, control, or physical modeling, balances the utility of memory against the risks or inefficiencies of over-accumulation, underscoring its centrality in both theory and application.