Trajectory inference via Acceleration Matching

Trajectory inference is a fundamental problem in many scientific domains: given a collection of unpaired snapshots of observations at discrete time points, the goal is to generate smooth trajectories that best resemble and interpolate the data. Existing algorithms exhibit computational challenges: they either rely on preprocessing subroutines to enforce smoothness or on simulation-based training objectives, both of which can be expensive. In order to overcome these limitations, we propose a new algorithm called Acceleration Matching (\texttt{AM}). Our approach consists of lifting the original interpolation problem to phase space and then regressing onto an explicit conditional acceleration field that induces random, smooth trajectories that agree with the prescribed marginals. Importantly, our resulting training algorithm only requires positional data, avoids trajectory simulation during training, and is devoid of expensive preprocessing. We provide ample numerical evidence suggesting that \texttt{AM} is competitive with or superior to existing algorithms on several benchmark problems from the existing literature.

High-Dimensional Interpolators Can Be Fragile: Heavy Tails and High-Dimensional Large Deviations

High-dimensional interpolation is common in modern machine learning, but its tail risk is less understood than its expected prediction risk. Existing theory shows that interpolating models can perform well in expectation, yet such guarantees do not determine the probability of rare, severe errors. In operations research and stochastic decision-making applications, rare estimation errors can have disproportionate downstream effects, so tail behavior matters alongside average performance. We study the fragility of high-dimensional linear interpolators using large-deviation methods. We focus on ridgeless regression and compare it with ridge-regularized estimators. We first show that the risk of ridgeless regression can exhibit heavy-tailed behavior: although its expected risk may remain well controlled, its upper tail can decay much more slowly than that of regularized alternatives. We then quantify this phenomenon at the level of large-deviation rates. In the regime we study, ridge regularization suppresses fixed right-tail deviations at the $n^2$ scale, whereas ridgeless regression has only $n\log n$-scale decay, where $n$ is the sample size. This gap shows that interpolation can be statistically fragile even when it is accurate on average. Thus regularization affects the frequency of rare, high-impact risk events in addition to the usual bias-variance tradeoff.

How abundant are good interpolators?

Let $S$ be the set of unit norm linear classifiers $θ\in \mathbb{R}^d$ which correctly classify every point of a labeled dataset $(X_i,y_i)_{i=1}^n$, $X_i \in \mathbb{R}^d$, $y_i \in \{-1,+1\}$, with a possibly negative margin $κ$ fixed in advance. Under two natural data-generating distributions of the $(X,y)$ pairs -- a Gaussian mixture model and a logistic model with Gaussian features -- and in the proportional regime $n/d \to α$ with small enough $α$, we establish a large deviation principle on the event that a point $θ$ chosen uniformly at random from $S$ achieves a given generalization error, with high probability over the choice of the data. The associated large deviation rate function is deterministic and describes the proportion, at the exponential scale in $d$, of interpolating classifiers having a given desired performance. As a consequence, we establish the following concentration phenomenon: all but an exponentially small fraction of interpolating classifiers have approximately the same generalization performance given by the unique maximizer of this rate function. We numerically compare this maximizer to the performance of empirical risk minimization by gradient descent and to the performance of a natural linear program, both finding a point in $S$, and deduce that in the overparametrized regime of small $α$, these efficient procedures outperform the vast majority of interpolators, pointing to their nontrivial benign overfitting in this setting.

Fast Generalization after Interpolation via Critically Damped Momentum Optimization

A central problem in machine learning is that models can achieve near-perfect training performance while generalizing substantially less well to unseen examples. This gap is especially acute in high-dimensional, low-sample regimes, where many interpolating solutions exist and optimization must implicitly select among minima with different generalization properties. Following recent theoretical advances on optimization dynamics near the interpolation threshold, we note that the two-regime structure of risk minimization, with loss minimization followed by complexity minimization, motivates a biphasic optimization schedule. We thus theoretically demonstrate that GROKtimizer, a biphasic strategy that combines rapid convergence to interpolation with Critically Damped Momentum (CDM)-based post-interpolation norm minimization, offers a natural solution for selecting low-norm interpolating solutions. Under a local quadratic model of the post-interpolation basin, GROKtimizer provides a quadratic speedup over classical gradient descent, with provable optimality among first-order optimizers. To showcase the applicability of our method, we evaluate GROKtimizer on several synthetic benchmarks common in the classical grokking literature and on various real-world datasets. Finally, we reconcile our findings with the flat-minima hypothesis, highlighting the importance of post-interpolation dynamics in the construction of high-quality, generalizing models.