How Memory in Optimization Algorithms Implicitly Modifies the Loss (2502.02132v1)

Abstract: In modern optimization methods used in deep learning, each update depends on the history of previous iterations, often referred to as memory, and this dependence decays fast as the iterates go further into the past. For example, gradient descent with momentum has exponentially decaying memory through exponentially averaged past gradients. We introduce a general technique for identifying a memoryless algorithm that approximates an optimization algorithm with memory. It is obtained by replacing all past iterates in the update by the current one, and then adding a correction term arising from memory (also a function of the current iterate). This correction term can be interpreted as a perturbation of the loss, and the nature of this perturbation can inform how memory implicitly (anti-)regularizes the optimization dynamics. As an application of our theory, we find that Lion does not have the kind of implicit anti-regularization induced by memory that AdamW does, providing a theory-based explanation for Lion's better generalization performance recently documented.

• The paper proposes a framework to approximate memoryful optimization algorithms by a memoryless version plus a correction term derived via Taylor expansion of past iterates.
• This correction term implicitly modifies the original loss function, acting as either implicit regularization or anti-regularization depending on the algorithm's memory structure.
• Applying this framework explains empirical differences in generalization, showing how Lion avoids the anti-regularization effects seen in AdamW's memory mechanism.

Alright, let's start breaking down how to approach the user's query about how memory in optimization algorithms implicitly modifies the loss. The user is asking for a detailed explanation based on the paper provided. First, I need to make sure I fully understand the core concepts presented in the paper's abstract and the step-by-step breakdown from the tool's output.

The paper introduces a technique to approximate algorithms with memory using a memoryless version. The key here is the correction term that arises from replacing past iterates with the current one. The tool's response mentions Taylor expansions to relate past iterates to the current, which I need to explain clearly. Since the user is likely familiar with optimization algorithms like momentum, Adam, or Lion, connecting the correction term to these examples would be helpful.

I should structure the response to first outline the general technique, then explore the correction term and its derivation. Explaining how this term acts as a loss perturbation is crucial. The differentiation between regularization and anti-regularization based on the correction term's effect needs to be highlighted, especially since the paper contrasts AdamW and Lion's performance here.

It's important to maintain technical accuracy without sensationalizing the findings. Using equations where necessary will add clarity. The example provided in the abstract about Lion not having the anti-regularization that AdamW does should be mentioned to ground the theory in practical results.

I need to ensure that each part of the explanation flows logically, starting from the memoryful algorithm's formulation, moving to the memoryless approximation, introducing the correction term, and then discussing the implications on the loss landscape and regularization. Closing with the example from the paper will reinforce how this theory explains observed generalization performance differences.

Check that all terms are well-defined and that the explanation doesn't assume prior knowledge beyond what a seasoned researcher would have. Avoid jargon where possible, but since the user is an expert, using terms like "implicit bias" and "Taylor expansions" is appropriate. Ensure that all references to the paper's equations are correctly contextualized even if specific equation numbers aren't provided.

Finally, summarize the main points succinctly in the closing paragraph to reinforce the key takeaways: the method of approximating memoryful algorithms, the role of the correction term as a loss perturbation, and the practical implications for understanding algorithm generalization.

The paper "How Memory in Optimization Algorithms Implicitly Modifies the Loss" (2502.02132) develops a framework to analyze how optimization algorithms with memory (e.g., momentum-based methods) implicitly modify the effective loss landscape through their iterative updates. The core technical contribution involves constructing a memoryless proxy algorithm that approximates the original algorithm by replacing historical parameter dependencies with current iterates and introducing a memory-induced correction term.

Key Mechanism

For a generic memoryful update rule: $x_{n+1} = x_n - h F_n(x_n, x_{n-1}, ..., x_0),$ the memoryless approximation becomes: $$\tilde{x}_{n+1} = \tilde{x}_n - h \left[F_n(\tilde{x}_n) + M_n(\tilde{x}_n)\right],$$ where $M_n(x)$ is derived through Taylor expansion of past iterate dependencies around the current parameter $x_n$. This term captures first-order memory effects and satisfies: $$M_r^{(n)}(x) = h \sum_{k=1}^n \frac{\partial F_r^{(n)}}{\partial x^{(n-k)}}(x) \sum_{s=n-k}^{n-1} F^{(s)}(x).$$

Loss Perturbation Interpretation

The correction term $M_n(x)$ induces an effective loss modification: $\tilde{L}(x) = L(x) + h M(x).$ This perturbation arises from memory-dependent gradient interactions across time steps. The sign and structure of $M(x)$ determine whether memory introduces implicit regularization (e.g., suppressing gradient magnitudes) or anti-regularization (amplifying certain gradient components).

Algorithm-Specific Implications

1. AdamW vs. Lion:
   • AdamW's exponential moving averages induce anti-regularization through positive feedback in gradient moment estimates, potentially destabilizing sharp minima.
   • Lion's sign-based update rule suppresses such feedback, leading to better generalization as empirically observed in LLM training (2502.02132).
2. Momentum: Exponentially decaying memory creates a velocity-dependent correction term proportional to $\sum_{k=0}^\infty \beta^k \nabla L(x_{n-k})$, which the framework shows can be approximated as a second-order gradient term modifying the effective loss curvature.

The methodology provides a systematic way to quantify implicit biases introduced by optimization algorithms, offering theoretical grounding for empirical observations about generalization performance. By analyzing the structure of $M(x)$, researchers can predict whether a given algorithm's memory mechanism will favor flat minima, suppress gradient noise, or exhibit other regularization properties.