Robustness, Calibration, and Safe Control › Distribution Shift and Stress Testing

Distribution shift is the umbrella term for any mismatch between the distribution the world model was trained on and the distribution it is asked to act in. It is worth being precise about what can shift, because different kinds of shift demand different responses, and because a model's own uncertainty carries different amounts of information about each type.

• Input shift. The distribution of states and actions changes. In pure covariate shift, the conditional transition law given those inputs stays fixed. Observed inputs permit label-free distribution tests, but detection power depends on the change, representation, sample size, and test; visibility does not guarantee detection.
• Dynamics shift (concept shift). The conditional transition law changes, potentially at inputs that initially look familiar. The floor got wet; the payload got heavier; the actuator got slower. The model can be confidently wrong without obvious input novelty, and changed dynamics may alter the states visited later. This is the failure mode that motivates the entire chapter.
• Observation shift. The sensor or observation pathway changes: a new camera, changed calibration, lighting, or noise profile. An inaccurate state estimate can undermine an otherwise adequate transition model. Observation diagnostics and transition residuals can both signal a problem, but residuals alone need not identify its cause.
• Task shift. Goals, rewards, or constraints change. A change from docking to transit may alter the objective and also change visited states and actions. A new operating regime alone is input shift unless the task definition changes too.
• Hidden-parameter shift. A latent parameter value or its distribution changes, abruptly or gradually. Within a parameterized plant family, drifting friction is one mechanism of dynamics shift; it is not a definition covering every structural change.

The distinction affects what ensemble disagreement can reveal. The members may agree on a wrong transition when the conditional dynamics change. Conversely, disagreement can increase on novel inputs, but that response is not guaranteed: shared model bias can keep all members confident outside the data. We measure these responses in a particular synthetic example rather than assuming either is a reliable detector.

Held-out validation measures prediction on its chosen test population. A same-generator split is useful for checking generalization within that generator, but may miss deployment conditions absent from it. Safety-relevant samples can include both rare in-distribution cases and out-of-distribution conditions. A large average RMSE improvement can coexist with serious boundary errors; report conditional and closed-loop outcomes rather than assuming the aggregate captures them.

Stress testing changes the population deliberately: under which plausible conditions does performance degrade, and which consequences follow? It can falsify a claim while also producing measurements. A useful output includes quantitative outcomes and an inventory of tested mechanisms, failures, and untested conditions. Failure to find a problem is evidence about the search performed, not proof of safety or evidence that the search was necessarily inadequate.

Good stress tests are built from a shift taxonomy for the specific domain, in which each category probes a distinct mechanism by which reality can diverge from training:

• Parameter stress. Perturb parameters over an explicitly justified domain, for example reducing friction, increasing mass, or reducing actuator gain. Search for adverse values within that domain; physical constraints may make it bounded.
• Input-distribution stress. Test high velocities, large commands, or near-boundary positions with sparse training coverage. Ensemble disagreement may rise there, but sparse coverage does not guarantee high estimated uncertainty.
• Structural stress. Add an effect outside the model class, such as an unrepresented delay or coupling. More data cannot remove that representational limitation while the class remains fixed. This tests a different limitation from parameter error.
• Temporal stress. Change the control rate, introduce jitter, or hold commands longer. Whether errors remain tolerable depends on the plant, controller, and timing. Evaluate closed-loop consequences rather than assuming a universal relation between rate difference and degradation.

When a plausible stress condition breaks the model, options include broadening data, changing the model class, restricting the operating domain, redesigning the controller or hardware, or adding detection and fallback. These approaches can complement one another but none universally handles rare or unbounded disturbances. A fallback needs its own operating assumptions. Part VI: Learning World Models at Scale covers learning and adaptation; here we evaluate diagnostic and control responses.

Let us set up the running example. The plant is a mass on a spring, integrated with a semi-implicit Euler step at $\Delta t = 0.1\,$s. The state at step $t$ is the pair $(p_t, v_t)$. The update advances velocity first and then uses that new velocity to advance position.

$$\begin{aligned} a_{t+1} &= -k\,p_t - c\,v_t - f\,\mathrm{sign}(v_t) + g\,u_t \\ v_{t+1} &= v_t + \Delta t\, a_{t+1} + \epsilon_t, \qquad \epsilon_t \sim \mathcal{N}(0, \sigma_p^2) \\ p_{t+1} &= p_t + \Delta t\, v_{t+1} \end{aligned}$$

where:

• $p_t$: position at step $t$, the coordinate the safety envelope constrains
• $v_t$: velocity at step $t$
• $a_{t+1}$: the acceleration computed at step $t$ and used to advance the velocity
• $u_t$: the scalar control action applied at step $t$; it is the book's action $a_t$ under a local notation chosen to distinguish it from acceleration
• $\Delta t = 0.1\,$s: integration timestep
• $k$: stiffness, the restoring force per unit displacement; $c$: viscous damping, the drag proportional to velocity; $f$: dry friction, a constant drag opposing motion; $g$: actuator gain, how much acceleration one unit of command produces
• $\epsilon_t \sim \mathcal{N}(0, \sigma_p^2)$: process noise on velocity, the part of the dynamics no deterministic model can capture

The first equation computes acceleration; the second advances velocity and adds its innovation; the third uses that new velocity to advance position. Here $a_{t+1}$ denotes acceleration, not the action: the scalar action is $u_t$. We observe the full simulated state $s_t=(p_t,v_t)$, so no observation model or belief filter is implemented. Mass is normalized to one and action units are abstract; this oscillator is not a wheel-torque model. The simulator permits dry friction, but every experiment here sets $f=0$. Semi-implicit Euler advances velocity before position; its stability depends on the timestep and dynamics, not on the method name alone.

We build two stress sets. The input-shift generator increases initial position and velocity scales and draws abstract actions from $[-0.70,0.70]$, twice the training range, while retaining nominal plant parameter sampling. The dynamics-shift generator retains training-scale initial conditions and commands but draws damping from $[0.00,0.05]$ instead of $[0.70,1.10]$. These are distinct interventions on the generator. Later states can change under either intervention, so a dynamics change does not imply that every encountered input remains in-distribution. The measured response of the ensemble, not input visibility alone, tells us what its diagnostic does here.