Cumulative Wasserstein Drift

• Cumulative Wasserstein drift is the sum of Wasserstein distances between successive probability measures, quantifying total distributional change over time.
• It underpins finite-sample guarantees, dynamic regret bounds, and concentration inequalities in stochastic processes and online optimization.
• Optimal weighting schemes leveraging cumulative drift balance bias and variance, guiding parameter choices in nonstationary and robust methods.

Cumulative Wasserstein drift quantifies the total “distance” traversed by a time-evolving sequence of probability measures, typically under the Wasserstein metric, over a given time horizon. It is a central nonstationarity measure in stochastic processes, online optimization, Markov dynamics, and empirical process theory, capturing both instantaneous and aggregated changes in distributions. Rigorous frameworks for cumulative Wasserstein drift underpin concentration inequalities, dynamic regret bounds, convergence rates for flows in the space of measures, and finite-sample guarantees in distributionally robust optimization.

1. Definition and Foundational Concepts

Let $\{P_t\}_{t=1}^T$ be a sequence of probability measures on a Polish space $\Xi$. The $p$-Wasserstein distance between $P_t$ and $P_{t+1}$ at each time $t$ is given by

$\Delta_t := W_p(P_t, P_{t+1})$

The unweighted cumulative Wasserstein drift over $T$ periods is

$D_T := \sum_{t=1}^{T-1} \Delta_t$

This sum captures the total geometric “movement” of the underlying data-generating law as measured in the Wasserstein space. In settings with weighted empirical estimators or time-decayed observations, the natural generalization is the $L_p$-norm-type drift: $$D_p(w) := \left( \sum_{t=1}^T w_t (T-t+1)^p \right)^{1/p} \rho$$ where $w \in \Delta_T$ is a vector of nonnegative weights, and $\rho$ is a uniform bound on $\Delta_t$ (Keehan et al., 21 Oct 2025).

2. Weighted Empirical Measures and Effective Sample Size

In nonstationary environments, weighted empirical measures are used to balance effective sample size against the impact of distributional drift: $\hat P_w := \sum_{t=1}^T w_t \delta_{\xi_t}$ with $\xi_t \sim P_t$. A key metric is the effective sample size

$$n_{\mathrm{eff}}(w) := \frac{1}{\sum_{t=1}^T w_t^2}$$

which quantifies the statistical reliability of $\hat P_w$ under the weighting scheme $w$. The interplay between $D_p(w)$ and $n_{\mathrm{eff}}(w)$ is critical for controlling estimation error and variance in time-evolving data (Keehan et al., 21 Oct 2025).

3. Finite-Sample Concentration and Nonstationary Robustness

A central technical result is a concentration inequality for Wasserstein distances in the nonstationary, weighted setting: $$\Pr\bigl[ W_p(\hat P_w, P_{T+1}) \geq \epsilon \bigr] \leq \exp\left( -c_1 n_{\mathrm{eff}}(w) \left( (\epsilon - D_p(w))_+^p - c_2 n_{\mathrm{eff}}(w)^{-q} \right)_+^2 \right)$$ where $c_1, c_2 > 0,$ and $q \in (0, 1/2)$ depend on the geometry of $\Xi$ and $p$ (Keehan et al., 21 Oct 2025). For sufficiently large $\epsilon$,

$$\Pr\bigl[ W_p(\hat P_w, P_{T+1}) \geq \epsilon \bigr] \leq \exp\left( -\frac{c_1}{4} n_{\mathrm{eff}}(w) (\epsilon - D_p(w))^{2p} \right)$$

This quantifies deviations of the empirical process in the presence of cumulative nonstationary drift, explicitly balancing sample variance and drift-induced bias.

4. Optimal Weighting: Variance–Drift Tradeoff

Optimal weights $w^*$ simultaneously control bias due to drift and estimation variance, solving

$$\max_{w \in \Delta_T} n_{\mathrm{eff}}(w) (\epsilon - D_p(w))_+^{2p}$$

The unique structure of the solution is

$w_t = (c_1' - c_2' (T - t + 1)^p)_+$

with scalars $c_1', c_2' \geq 0$ determined by simplex constraints. As $p$ grows, the optimal scheme exhibits sharper cutoff of past (older) data, reducing to pure sliding-window or exponential-decay weighting depending on parameter choices. Explicit calibrations,

$$s \approx \left\lfloor (2\epsilon/\rho - 1)/3 \right\rfloor,\quad \alpha \approx 3/(\epsilon/\rho+1)$$

arise for windowing and exponential smoothing in the $p=1$ case, providing optimal parameter choices in terms of desired accuracy $\epsilon$, drift bound $\rho$, and Wasserstein order $p$ (Keehan et al., 21 Oct 2025).

5. Cumulative Drift in Dynamic Optimization and Learning

In online convex optimization where objective distributions $\{\mathbb{P}_t\}$ evolve, the cumulative Wasserstein drift $D_T$ enters directly into dynamic regret bounds: $$D_T := \sum_{t=1}^{T-1} W_p(\mathbb{P}_t, \mathbb{P}_{t+1})$$ and the sequence of minimizers $x_t^*$ satisfies

$$\sum_{t=1}^{T-1} \| x_{t+1}^* - x_t^* \| \leq C \, D_T$$

The corresponding dynamic regret is lower-bounded by an $O(D_T)$ term—cumulative drift sets the intrinsic limit on performance in adapting to distributional changes, with all other regret contributions (noise, initialization) being controllable via algorithmic parameters (Shames et al., 2020).

6. Wasserstein Drift in PDEs, Stochastic Flows, and Markov Chains

The notion of cumulative Wasserstein drift generalizes to continuous-time measure-valued flows:

• For gradient flows in $P_2(\mathbb{R}^d)$,

$$L(0, T) := \int_0^T \left( \int \|v_t(x)\|^2 \rho_t(x) dx \right)^{1/2} dt$$

where $v_t$ is the instantaneous velocity field from the continuity equation. The total path-length controls convergence rates and is uniformly bounded in terms of the initial suboptimality of the functional $F(\rho_0) - \inf F$ (Chizat et al., 16 Jul 2025).

• In measure-valued SPDEs and diffusions, the time integral of instantaneous drift or squared gradient quantifies both cumulative displacement in Wasserstein space and the action or Fisher information over time (Delarue et al., 2024).
• In discrete-time Markov chains, geometric contractivity plus one-step non-contractive “drift” yields cumulative bounds:

$$W(\mu P^n, \nu P^n) \leq \kappa^n W(\mu, \nu) + \frac{1 - \kappa^n}{1 - \kappa} \delta$$

where $\delta$ is the per-step drift and $\kappa < 1$ is the contraction rate. The second term encodes the aggregated perturbation—the “cumulative drift” of the Markov process (Madras et al., 2011).

7. Applications and Broader Significance

Cumulative Wasserstein drift is a central concept in:

• Design and analysis of distributionally robust methods under nonstationarity, providing the basis for weighted empirical ambiguity sets and finite-sample risk guarantees (Keehan et al., 21 Oct 2025).
• Online and adaptive algorithms, where $D_T$ quantifies the inevitable penalty from time-varying environments (Shames et al., 2020).
• Empirical process concentration, governing the balance between effective sample use and tracking distributional shifts (Keehan et al., 21 Oct 2025).
• Evolution and regularity analysis in nonlinear PDEs, mean-field diffusions, and interacting particle systems, where path-length or total drift controls long-time behavior, ergodicity, and rates of mixing (Chizat et al., 16 Jul 2025, Delarue et al., 2024, Hwang et al., 2021).
• Markov chain convergence diagnostics and quantitative ergodic bounds (Madras et al., 2011).

These frameworks provide precise nonasymptotic characterizations, parameter choices for weighting schemes, and convergence rates that systematically account for nonstationarity and time-varying complexity in modern stochastic systems.