Drift-Feedback Generalization Bound

• The paper introduces a framework that quantifies nonstationary generalization via the cumulative Fisher–Rao path length, known as the reproducibility budget.
• It integrates exogenous drift and learner-induced feedback to yield a minimax-optimal error rate of Θ(T⁻¹/² + C_T/T) and establishes a reproducibility speed limit.
• The model unifies stationary, drift, adaptive data analysis, and performative regimes, offering a rigorous geometric measure for algorithmic performance under dynamic data distributions.

The drift-feedback generalization bound characterizes statistical learning under nonstationary distributions where the environment law evolves over time as a coupled function of both exogenous change and learner-induced feedback. Classical generalization theory collapses in this setting, necessitating new geometric machinery. The central primitive is the reproducibility budget $C_T$, defined as the cumulative Fisher–Rao path length traversed by the underlying data-generating process over $T$ rounds. This intrinsic metric quantifies the joint effect of exogenous drift and adaptive feedback, governing a minimax-optimal generalization error rate of $\Theta(T^{-1/2} + C_T/T)$. The framework unifies stationary, drift, adaptive data analysis, and performative regimes under a single information-geometric structure, establishing a reproducibility speed limit whereby no algorithm can perform better than the imposed average drift rate.

1. Information-Geometric Drift and the Reproducibility Budget

Let $$\{p_\theta(x,y)\}_{\theta\in\Theta\subset\mathbb{R}^d}$$ be a smooth parametric model equipped with its Fisher–Rao Riemannian metric $g_\theta$. For any trajectory $\theta_0\to\theta_1\to\cdots\to\theta_T$ induced by the interaction of learner and environment, the Fisher–Rao distance between consecutive states is: $$d_F(\theta_t, \theta_{t+1}) = \inf_{\gamma} \int_0^1 \|\dot\gamma(s)\|_{g_{\gamma(s)}} ds$$ where $\gamma$ runs over piecewise-$C^1$ curves joining $\theta_t$ and $\theta_{t+1}$, and $\|v\|^2_{g_\theta}=v^\top g(\theta) v$.

The reproducibility budget is defined as the total accumulated Fisher–Rao path length: $$C_T = \sum_{t=0}^{T-1} d_F(\theta_t, \theta_{t+1}) \approx \int_0^T \|\dot\theta_s\|_{g_{\theta_s}} ds$$ This measures the intrinsic information-geometric motion of the data-generating law as the environment evolves, encompassing both unpredictable exogenous changes and endogenous learner feedback (Zaichyk, 15 Dec 2025).

2. Regularity and Decomposition of Drift

Regularity assumptions ensure analytical control:

• Fisher information $g(\theta)$ is positive-definite, twice continuously differentiable; the score $s_\theta = \nabla_\theta \log p_\theta$ has bounded second moments.
• Loss $\ell(f(x),y)$ is $\sigma$-sub-Gaussian under each $p_\theta$.
• Environment updates $\theta_{t+1} = F(\theta_t, u_t, \eta_t)$ are smooth in control $u_t$, with bounded energy $\mathbb{E}[\|u_t\|^2] \le B$. Exogenous noise $\eta_t$ is independent of $u_t$.

One-step drift is decomposed as: $$\Delta\theta_t = [F(\theta_t, 0, \eta_t) - \theta_t] + J_u F(\theta_t, 0, \eta_t) u_t + r_t$$ with

$$d_t := \mathbb{E}[\|F(\theta_t, 0, \eta_t) - \theta_t\|_{g_{\theta_t}} \mid \theta_t] \qquad \text{(exogenous drift)}$$

$$\kappa_t := \mathbb{E}[\|J_u F(\theta_t, 0, \eta_t) u_t\|_{g_{\theta_t}} \mid \theta_t] \qquad \text{(endogenous/feedback drift)}$$

$$\mathbb{E}[\|\Delta\theta_t\|_{g_{\theta_t}}] \le d_t + \kappa_t + \varepsilon_t$$

Summing yields $C_T \approx \sum_{t=0}^{T-1}(d_t + \kappa_t)$ (Zaichyk, 15 Dec 2025).

3. The Drift-Feedback Generalization Bound

At each round $t$, the learner produces $f_t$ and receives $(x_t, y_t) \sim p_{\theta_t}$. Define

$$\hat{R}_T = \frac{1}{T} \sum_{t=1}^T \ell(f_t(x_t), y_t) \qquad R_T = \frac{1}{T} \sum_{t=1}^T \mathbb{E}_{\theta_t}\left[\ell(f_t(x), y)\right]$$

Lemma (add–subtract): $$|\hat{R}_T - R_T| \leq \frac{1}{T} \left| \sum_{t=1}^T Z_t \right| + \frac{1}{T} \sum_{t=1}^{T-1} |R(\theta_{t+1}, f_t) - R(\theta_t, f_t)|$$ with $Z_t$ a martingale difference.

Under sub-Gaussianity and smoothness,

• Sampling term: $$\mathbb{E}\left| \frac{1}{T}\sum Z_t \right| \leq C_0 T^{-1/2}$$
• Drift term: $$\mathbb{E}[|R(\theta_{t+1}, f_t) - R(\theta_t, f_t)|] \leq L_p (d_t + \kappa_t)$$

Aggregated bound: $$\mathbb{E} |\hat{R}_T - R_T| \leq C_0 T^{-1/2} + \frac{L_p}{T} \sum_{t=0}^{T-1} (d_t + \kappa_t) = O(T^{-1/2} + C_T / T)$$

4. Minimax Lower Bound and Speed Limit

No estimator can outperform the average drift rate. An explicit Fano-type construction in a Fisher-arclength parameterization yields: $$\inf_{\hat{R}_T} \sup_{\text{process: } C_T \leq C} \mathbb{E} |\hat{R}_T - R_T| \geq c_1 \max(T^{-1/2},\, C/T)$$ The proof partitions time into blocks with step-size $\delta$, employing codewords and the KL/Fano argument, yielding minimax optimality. In the stationary case ($C=0$), the classical $T^{-1/2}$ rate is recovered, while for maximal drift, the $C_T / T$ term dominates.

The reproducibility speed-limit: No algorithm can achieve a worst-case generalization error below the average Fisher–Rao drift rate $C_T/T$ (Zaichyk, 15 Dec 2025).

5. Connections to Existing Drift and Variation-Budget Frameworks

The drift-feedback structure recovers well-established bounds:

• Stationary i.i.d.: $d_t \equiv \kappa_t \equiv 0 \implies O(T^{-1/2})$
• Pure drift: $O(T^{-1/2} + (1/T) \sum d_t)$
• Adaptive data analysis: $O(T^{-1/2} + (1/T) \sum \kappa_t)$
• Performative equilibrium: $d_t, \kappa_t \to 0 \implies O(T^{-1/2})$

Related work in concept drift learning (Hanneke et al., 2015) quantifies the drift by a sequence $\Delta_t$ and delivers window-adaptive error bounds: $$\text{error}_T(\hat{h}_T) \le O\left(\min_{1 \le m \le T-1} \left\{ \frac{1}{m}\sum_{i=T-m}^{T-1}\sum_{j=i+1}^T \Delta_j + \frac{d \ln(m/d) + \ln(1/\delta)}{m} \right\}\right)$$ This structure cleanly splits error into drift and complexity, paralleling the decomposition in the drift-feedback regime.

In generalized linear bandits under parameter drift (Faury et al., 2021), the variation-budget $B_T$ similarly controls regret rates:

• Orthogonal action sets: $O(d^{2/3}B_T^{1/3}T^{2/3})$
• General sets: $O(d^{9/10}B_T^{1/5}T^{4/5})$ The projection step feeds back the observed drift into the confidence set, analogous to the drift-feedback framework.

6. Unified Geometric Characterization and Model Adaptivity

The drift-feedback generalization theory unifies exogenous drift, adaptive analysis, and performative prediction by projecting complexity and stability penalties onto the intrinsic Fisher–Rao path length $C_T$. The information-geometric approach enables precise characterization and algorithm-independent lower bounds. Settings previously analyzed via extrinsic variation-budgets or stability coefficients are subcases determined by the projection of $C_T$ onto the principal mode of environmental change.

A plausible implication is that the geometric drift metric provides a canonical unifying resource for quantifying nonstationarity across statistical learning, bandit optimization, and adaptive decision processes. The reproducibility budget thus subsumes and refines previously used drift and variation notions, providing a rigorous speed-limit for adaptive generalization (Zaichyk, 15 Dec 2025, Hanneke et al., 2015, Faury et al., 2021).