Detecting Evaluation Schema Shifts in AI Monitoring: A Mathematical Framework with GhostDrift Algorithms

Problem Statement: Why Conventional Drift Detection Fails in Algorithmic Audit Accountability

Traditional machine learning monitoring focuses on Data Drift (changes in $p(x)$) or Concept Drift (changes in $p(y|x)$). However, these metrics assume a static and honest evaluation framework. In real-world AI governance, a more insidious phenomenon occurs: the manipulation of the evaluation criteria themselves to mask performance degradation.

We define this as GhostDrift. Whether intentional or accidental, altering sampling weights, metric definitions, or filtering conditions (the "Evaluation Operator") renders historical comparisons meaningless. To establish true Algorithmic Audit Accountability, we need a mathematical framework that ensures Non-retrospective Audit Metrics through operator invariance.

Keyword Definitions: Understanding the GhostDrift Taxonomy

To navigate this framework, we define several core concepts:

• Drift (Data/Concept): Change in the underlying data distribution or the relationship between features and labels.

• GhostDrift: A shift in the evaluation schema (parameters, filters, or aggregation logic) that alters the perceived performance without changing the model or data.

• Evaluation Operator: A functional mapping that transforms raw data stores and model parameters into performance metrics.

• Evaluation Plan Commitment: A cryptographic anchor that fixes the state of an evaluation plan, preventing post-hoc optimization.

High-Level Overview: Operator Shift Detection in ML

The GhostDrift algorithm treats evaluation as a mathematical operator. By anchoring this operator with a cryptographic commitment and performing interval analysis, we can prove whether a change in reported accuracy is due to the environment (Data) or a "ghost" in the evaluation logic (Operator).

Chapter 1: Definition of the World and Evaluation Operators

1.1 Spaces and Records

Let $\mathcal{X}$ be the feature space, $\mathcal{Y}$ the label space, $\mathcal{M}$ the metadata space, and $\mathcal{T} \subseteq \mathbb{R}$ the time domain. A single observation record $r$ is defined as an element of the following Cartesian product space:

$$r = (x, y, m, t) \in \mathcal{X} \times \mathcal{Y} \times \mathcal{M} \times \mathcal{T}$$

Definition 1.1 (Evaluation Store): The evaluation store $S$ is defined as a finite set of records:

$$S = \{r_i\}_{i=1}^N$$

where each record has a unique ID. The store is append-only; modification of past records is defined as a "transition to a different store."

1.2 Freshness Sampling and Induced Measures

To dynamically generate evaluation sets from operation logs, we introduce time-decay weights (Freshness).

Definition 1.2 (Freshness Weight): For an audit time $t_{\text{now}} \in \mathcal{T}$, the freshness weight $w_t(r)$ of record $r$ is:

$$w_t(r) = \exp\left( - \frac{t_{\text{now}} - r.t}{\tau} \right) \quad (\tau > 0)$$

where $\tau$ is the decay parameter.

Definition 1.3 (Sampling Distribution): The probability measure (sampling distribution) $p_t$ on store $S$ is:

$$p_t(r) = \frac{w_t(r) \cdot \mathbb{I}[r \in S \land C(r)]}{\sum_{u \in S} w_t(u) \cdot \mathbb{I}[u \in S \land C(u)]}$$

where $C(r)$ is a logical function representing the evaluation filter conditions.

1.3 Evaluation Operator

Let the model be $f_\theta: \mathcal{X} \to \hat{\mathcal{Y}}$ and the loss function (metric) be $\ell: \hat{\mathcal{Y}} \times \mathcal{Y} \times \mathcal{M} \to \mathbb{R}$.

Definition 1.4 (Evaluation Operator): Given an evaluation plan $P$ and time $t$, the operator $E_{P,t}$ maps the store $S$ and model $\theta$ to a real value (or an interval):

$$E_{P,t}(S, \theta) := \text{Agg}\left( \{ \ell(f_\theta(r.x), r.y, r.m) \}_{r \sim p_t} \right)$$

where $\text{Agg}$ is an aggregation function. For expected value evaluation, the point value is:

$$\mu(P; S, t, \theta) := \sum_{r \in S} p_t(r) \cdot \ell(f_\theta(r.x), r.y, r.m)$$

Chapter 2: Non-Retroactivity and Commitments

2.1 Evaluation Plan and Commitment

Definition 2.1 (Evaluation Plan): The evaluation plan $P$ is a set of parameters uniquely determining the operator:

$$P = (C, \tau, n, \text{Method}, \ell, \text{Agg}, \text{SeedPolicy}, \text{Thresholds}, \dots)$$

Definition 2.2 (Commitment): Using a canonical mapping $\text{Canon}(\cdot)$ and a cryptographic hash function $H(\cdot)$:

$$c(P) := H(\text{Canon}(P))$$

This $c(P)$ serves as an identifier that fixes the Identity of the evaluation operator.

2.2 Definition of GhostDrift

Definition 2.3 (Binary GhostDrift): Let $P_{\text{run}}$ be the plan used at runtime and $P_{\text{committed}}$ be the pre-committed plan. The GhostDrift flag $G$ is:

$$G := \mathbb{I}[c(P_{\text{run}}) \neq c(P_{\text{committed}})]$$

Definition 2.4 (GhostDrift Distance): For continuous parameter components $\alpha \in \mathbb{R}^d$ and discrete structure $\sigma$:

$$D_{\text{ghost}}(P, P') := \begin{cases} \|\alpha - \alpha'\|_W & (\text{if } \sigma = \sigma' \land c(P) = c(P')) \\ +\infty & (\text{if } \sigma \neq \sigma' \lor c(P) \neq c(P')) \end{cases}$$

Chapter 3: Interval Analysis and Ledger Integrity

3.1 Outward Rounding and Interval Arithmetic

Let $\mathbb{IR} := \{[a, b] \subset \mathbb{R} \mid a \le b\}$ be the set of real intervals. Definition 3.1 (Outward Rounding): $\forall X \in \mathbb{IR}^n, \{ f(x) \mid x \in X \} \subseteq f^\diamond(X)$.

3.2 Ledger-based Evaluation Algorithm

Definition 3.2 (Evaluation Procedure): Procedure $\mathcal{A}_{P,t}$ outputs an interval evaluation $\widehat{E}$ and an operation ledger $L$:

$$\mathcal{A}_{P,t}(S) \to \big(\widehat{E}_{P,t}(S), \ L_{P,t}(S)\big)$$

Definition 3.3 (Verify Mapping): $\text{Verify}(L) := \bigwedge_{k=1}^K \left( \llbracket \text{op}_k \rrbracket(\text{In}_k) \subseteq \text{Out}_k \right)$.

Theorem 3.1 (Soundness of Interval Evaluation): If $(\widehat{E}, L) = \mathcal{A}_{P,t}(S)$ and $\text{Verify}(L) = \text{true}$, then:

$$E_{\text{true}} \in \widehat{E}$$

Proof: By induction on the operation steps in $L$, as each step satisfies the inclusion property. $\square$

Chapter 4: Resistance to Post-hoc Optimization

4.2 Non-anticipative Update

Definition 4.1 (Admissible Update): A sequence $\{P_t\}_t$ is valid if the update mapping is adapted to Calibration info $\mathcal{F}_t^{\text{cal}}$ and independent of Test info $\mathcal{F}_t^{\text{test}}$:

$$P_{t+1} = \text{Upd}(P_t, \mathcal{F}_t^{\text{cal}})$$

Chapter 5: Geometry of Minimal Ghost-Drift

5.1 Decision Function and Boundary

$\text{Dec}(P) = \text{OK} \iff E^+(P) \le \vartheta$. Let $g(P) := E^+(P) - \vartheta$.

5.2 Minimal Alteration Problem

Theorem 5.1 (Closed-form Solution):

$$\Delta^* \approx \frac{|g(\alpha)|}{\sqrt{\nabla g(\alpha)^\top W^{-1} \nabla g(\alpha)}}$$

Chapter 6: Freshness Sensitivity and Strict Monotonicity

6.1 Gradient as Covariance

Theorem 6.1 (Covariance Gradient and Lipschitz Constant):

$$\nabla_\alpha \mu(\alpha) = \text{Cov}_{p(\alpha)}(L, \phi) = \mathbb{E}_p[L \phi] - \mathbb{E}_p[L]\mathbb{E}_p[\phi]$$

For bounded loss $L \in [0, B]$, the Lipschitz constant $L_{\text{op}}$ satisfies:

$$L_{\text{op}} \le \frac{B}{2} R_{\text{diam}} \quad \text{where} \quad R_{\text{diam}} := \sup_{i,j} \|\phi_i - \phi_j\|_{W^{-1}}$$

Proof: Follows from $|\text{Cov}(L, \phi)| \le \frac{B}{2} \sup \|\phi - \mathbb{E}\phi\|$. $\square$

6.2 Monotonicity Determination via Comonotonicity

Lemma 6.2 (Comonotonicity): If $(L_i - L_j)(\phi_i - \phi_j) \ge 0$, then $\text{Cov}_p(L, \phi) \ge 0$. Theorem 6.3 (Uniqueness of Root): Under comonotonicity, $\mu(\alpha)$ is monotonic, making the solution to $\mu(\alpha) - \vartheta = 0$ unique.

Chapter 7: Dual Drift Decomposition and the Necessity Frontier

7.1 Dual Drift Decomposition

$$\Delta_{\text{obs}} := |\mu(A_2) - \mu(A_1)| \le \underbrace{B \cdot D_{\text{data}}(S_1, S_2)}_{\Delta_{\text{data}}} + \underbrace{\frac{B}{2} R_{\text{diam}} \cdot D_{\text{op}}(P_1, P_2)}_{\Delta_{\text{op}}}$$

7.3 Ghost-Necessity Frontier

Definition 7.1 (Ghost-Necessity Lower Bound): If $c(P_1) = c(P_2)$:

$$\Omega(d) \ge \frac{2 (\Delta_{\text{obs}} - B \cdot d)^+}{B \cdot R_{\text{diam}}}$$

This asserts that if observed change exceeds data budget $d$, operator alteration is mathematically necessary.

Chapter 8: Construction of the Audit Certificate

8.1 Certificate Structure $C$

$$C = (\text{FreezeID}, S_1^{\text{ref}}, S_2^{\text{ref}}, \Delta^{\text{LB}}, d^{\text{UB}}, R^{\text{UB}}, \Omega^{\text{LB}}, \text{LedgerHash})$$

8.2 Algorithm for Verifiable Lower Bounds

1. Observation Lower Bound: $\Delta^{\text{LB}} = \max(0, l_2 - u_1, l_1 - u_2)$.

2. Geometric Upper Bound: $R^{\text{UB}} = \frac{A^{\diamond}}{\sqrt{w_a}} + \frac{S^{\diamond}}{\sqrt{w_s}} + \frac{C^{\diamond}}{\sqrt{w_c}}$.

3. Necessity Lower Bound: $\Omega^{\text{LB}}(d) = \frac{2 (\Delta^{\text{LB}} - B \cdot d)^+}{B \cdot R^{\text{UB}}}$.

8.3 Conclusion: Two-Layer Drift Verdict

$$\text{DriftVerdict} := \underbrace{\mathbb{I}[c(P_1) \neq c(P_2)]}_{\text{ID Drift (G1)}} \lor \underbrace{(\Omega^{\text{LB}}(d) > 0)}_{\text{Quantitative Drift}}$$

Use Cases: Audit Pipelines and Compliance

• Third-party Verification: Enables auditors to verify performance without model access.

• Compliance (EU AI Act): Ensures non-manipulation of performance reports.

• Non-retrospective Audit Metrics: Prevents post-hoc metric adjusting.

Note: Filed under Patent Application No. 2025-275211.