One Trajectory, Two Requirements: Accuracy and Stability

A new paper from Grafy Research Lab and author Mahindra Rautela at Los Alamos National Laboratory examines how robot AI policies choose their Euler sampling step counts. The work asks whether a step count selected for trajectory accuracy can still be too coarse when local stability matters, and it reports that the two requirements can lead to different sampling budgets.

What the study examined

The paper focuses on flow matching robot policies that generate continuous action chunks. According to the source, these policies often use a fixed Euler step count picked from offline accuracy sweeps. At inference time, each policy input, including visual observation, robot state and language instruction, plus a source noise draw, defines a different noise-to-action trajectory and local Jacobians. The researchers ask whether a step count that follows the nominal noise-to-action trajectory accurately can still be too coarse for strongly contracting directions around it.

Why the step count matters

For flow matching policies, actions come from integrating a learned ordinary differential equation from source noise to action. Each Euler step needs another action expert evaluation, so fewer steps can lower inference cost. In practice, the step count is often chosen by sweeping values on a fixed evaluation set and picking one with acceptable endpoint accuracy.

Accuracy and local stability are not the same

Trajectory accuracy asks whether the nominal Euler path ends near a refined numerical solution. Local stability asks what Euler does to nearby perturbations. The state Jacobian can show a direction that the continuous flow contracts rapidly. If the Euler step is too large, that direction can be amplified. A trajectory can be accurate at its endpoint while the local Euler update still misrepresents nearby state behavior.

What the experiments found

The source describes an exact two-coordinate system and a controlled augmentation of six trained flow models. These experiments suggest trajectory accuracy and local Euler stability can impose different step requirements even when the original learned transport is unchanged. The team then studied a frozen pi0.5-LIBERO checkpoint without modifying its weights or action dynamics. Across 30 policy inputs from 10 LIBERO-10 tasks, the Jacobian showed strong contracting directions near the action endpoint, yet no sampled mode violated the Euler non-expansion condition. On five inputs, accuracy sweeps required 4 to 9 Euler steps at 5 percent terminal error and 87 to 112 at 1 percent.

How researchers can use Grafy

For researchers reading this paper, Grafy Reader is the most direct surface. It lets users read, search and question PDFs and papers with AI. A researcher could load the arXiv paper and ask follow-up questions about flow matching, Euler sampling or stability. Grafy Reader at Reader can be used to explore the paper, and Grafy Chat at Chat can support broader questions about model behavior, though the source text does not describe such integration.

Frequently asked questions

Does the study prove that accuracy sweeps are always enough?

No. The researchers investigated whether an accuracy-selected Euler budget could still be too coarse for local stability. They found that trajectory accuracy and local stability can require different step counts, according to the source. The authors note that endpoint accuracy alone cannot rule out a separate stability requirement.

When can local stability require more Euler steps?

The paper suggests that when the Jacobian contains strong contracting directions, a large Euler step may amplify perturbations instead of preserving them. The source says the controlled experiments showed why endpoint accuracy alone cannot rule out a stability requirement. It also reports that on five paired inputs, accuracy sweeps set the step budget rather than stability.

How should researchers interpret the five paired cases?

The five cases were consistent with accuracy-limited rather than stability-limited sampling under the local criterion, the source says. The controlled experiments still showed that accuracy and stability can impose different requirements. The authors conclude that fixed offline accuracy sweeps may not fully capture inference-time stability needs.

---

Source: Grafy Research Lab — One Trajectory, Two Requirements: Accuracy and Local Stability in Euler Sampling for Flow Matching Policies