אנליזה דטאלית של גרדיאנט דסנט עם זיכרון

תקציר מקורי באנגליתarXiv:2509.08483v2 Announce Type: replace Abstract: We analyze gradient descent with Polyak (1964) heavy-ball momentum (HB) whose fixed momentum hyperparameter $\beta \in (0, 1)$ provides exponential decay of memory. Building on Kovachki and Stuart (2021), we prove that on an exponentially attractive invariant manifold the algorithm is exactly plain gradient descent with a modified loss, provided that the step size $h$ is small enough. Although the modified loss does not admit a closed-form expression, we describe it up to $O(h^{\mathcal{R}})$-errors for arbitrary finite order $\mathcal{R}$, and prove global (finite "time" horizon) trajectory approximation bounds $O(h^{\mathcal{R}})$. We then conduct a fine-grained analysis of the combinatorics underlying the memoryless approximations of H