We analyze gradient descent with Polyak (1964) heavy-ball momentum (HB) whose fixed momentum hyperparameter $β\in (0, 1)$ 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 HB, in particular, finding a rich family of polynomials in $β$ hidden inside which include and lie coefficient-wise in between Eulerian and Narayana polynomials. We prove that these polynomials are $h$-polynomials of certain graph-associahedra. As corollaries of the main results, we derive continuous modified equations of arbitrary finite approximation order (with rigorous bounds) and the principal flow that approximates the HB dynamics, generalizing Rosca et al. (2023). Approximation theorems cover both full-batch and mini-batch HB. The results shed new light on the main features of HB and outline a roadmap for similar analysis of other optimization algorithms.

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