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Performance Drift Detection in Machine Learning as a Service (MLaaS) for IoT Environments

Deepak Kanneganti, Sajib Mistry, Sheik Mohammad Mostakim Fattah, Erik Elmroth, Aneesh Krishna, Monowar Bhuyan

arXiv:2608.18555v1cs.LGcs.AI

TL;DR

Black-box MLaaS in dynamic IoT environments makes it difficult to distinguish real performance drift from temporary data fluctuations. The paper proposes MPDD, which combines extracted service behavior with input-data changes, and reports a 22–25% accuracy improvement over representative baseline methods.

  • Problem

    Black-box MLaaS monitoring must distinguish real performance drift from pseudo drift as IoT input distributions and service behavior change.

  • Method

    The framework learns black-box service behavior from input–output pairs and combines feature-aware data-drift measures with MLaaS behavior changes in MPDD.

  • Results

    22–25% accuracy improvement over representative baseline methods is reported for MPDD across multiple real-world datasets.

  • Takeaways & Limitations

    MPDD provides a ground-truth-independent approach for monitoring performance drift in black-box MLaaS services deployed in IoT environments.

Abstract

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Machine Learning as a Service (MLaaS) is a powerful cloud paradigm enabling data-driven intelligent applications in Internet of Things (IoT) environments, widely adopted across healthcare, smart homes, and industry due to its cost-effectiveness. However, the dynamic nature of IoT frequently alters data distributions, affecting MLaaS stability, while periodic MLaaS updates further introduce performance drift. Unlike traditional ML systems, MLaaS clients operate as black-box users without access to internal data or parameters, making drift detection particularly challenging. To address this, we propose a novel MLaaS Performance Drift Detection framework for IoT environments. The framework first employs an MLaaS extraction model that learns service behavior from input-output pairs and identifies prediction-influenced features. Building on this, the proposed MLaaS Performance Drift Detection (MPDD) model jointly captures variations in input data and MLaaS behavior. We further design an Adaptive-Temporal Performance Drift Detection Mechanism (APDDM) that dynamically adjusts monitoring frequency based on behavioral and data variations, enabling timely drift detection for effective service management. Extensive experiments on real-world datasets demonstrate that MPDD achieves up to 22-25% accuracy improvement over baseline drift detection methods. APDDM provides an average accuracy gain of approximately 4% and reduces the miss detection rate by around 9% compared to fixed-interval monitoring.

I. INTRODUCTION … B. Black-Box, Label-Independent Drift Detection Methods

MLaaS supports intelligent IoT applications but dynamic environments and black-box access constraints make performance-drift detection difficult without continuously collected ground-truth labels. The proposed framework combines learned MLaaS behavior with input-distribution changes and adaptive monitoring to detect performance degradation.

  • I. INTRODUCTION: MLaaS provides scalable infrastructure and tools for building, training, and deploying machine-learning models through web interfaces, APIs, or SDKs.It removes the complexity of managing hardware and manually designing models.
  • I. INTRODUCTION: IoT companies use MLaaS to add intelligent capabilities without training models in-house, while evaluating functional attributes and quality-of-service metrics.The passage gives healthcare IoT as an example, including prediction of low-blood-sugar events.
  • I. INTRODUCTION: Performance drift is the gradual degradation of MLaaS predictive ability, reflected in accuracy, precision, or F1 score as the relationship between inputs and labels changes.Evolving user behavior, sensor distributions, and system requirements make reliability difficult to maintain in IoT environments.
  • I. INTRODUCTION: Traditional drift detectors monitor error-related measures and assume timely ground-truth access, whereas label-free methods instead analyze internal model properties or observable input distributions.Continuous ground-truth collection is time-consuming and often impractical in real-world IoT deployments.
  • B. Black-Box, Label-Independent Drift Detection Methods: Black-box MLaaS prevents direct application of methods requiring internal parameters or training data, while input-only methods cannot establish whether distribution changes alter predictive behavior.Security and proprietary constraints limit provider disclosure of internal parameters and training data.
  • I. INTRODUCTION: The proposed framework learns MLaaS behavior from IoT inputs and outputs, identifies key features, and combines FDDS with MDES to capture data changes and their impact on MLaaS behavior.MPDD determines when the MLaaS should be re-evaluated without access to black-box parameters or training data.
  • A. Error-Rate-Based Drift Detection: Error-rate detectors such as DDM and EDDM form a prior line of IoT stream-learning research based mainly on observable error signals.The cited passage describes these methods as classical error-monitoring detectors and mentions adaptive extensions.
  • B. Black-Box, Label-Independent Drift Detection Methods: Distribution-based methods use MMD, Kolmogorov–Smirnov, or Fréchet distance to detect input shifts, but large observation windows can limit responsiveness in black-box environments.Other label-independent approaches analyze traffic-flow behavior, classification-score distributions, anomaly monitoring, or reliability streams without true labels.

III. MOTIVATION SCENARIO · IV. KEY DEFINITIONS AND PROBLEM STATEMENT

The paper motivates performance-drift detection for black-box MLaaS used in dynamic IoT settings, where changing conditions can degrade service quality and complicate monitoring. It defines real and pseudo drift to frame the MPDD problem as distinguishing harmful predictive-behavior changes from benign input fluctuations.

  • III. MOTIVATION SCENARIO: IoT healthcare environments can use inference-based MLaaS for Human Activity Recognition when they lack expertise and infrastructure to develop, train, and maintain models.Sensor data are submitted periodically through batch inference channels to obtain predictions.
  • III. MOTIVATION SCENARIO: Changes in user behavior, device configurations, environmental factors, or patient demographics can make current IoT data differ from deployment conditions and degrade accuracy, precision, or recall.The passage identifies healthcare demographic variation as an example of changing operating conditions.
  • III. MOTIVATION SCENARIO: The central challenge is detecting performance drift without ground truth because input-distribution changes may be tolerated by robust MLaaS models and therefore constitute pseudo drift.Input changes alone cannot establish real performance drift.
  • III. MOTIVATION SCENARIO: Monitoring frequency creates a trade-off: minute-level checks may repeatedly trigger pseudo-drift notifications, whereas hourly checks may delay real-drift detection.The evaluation interval must support accurate and prompt detection of real performance drift.
  • IV. KEY DEFINITIONS AND PROBLEM STATEMENT: MLaaS is modeled as ⟨F, QoS⟩, combining functional specifications such as model and data properties with non-functional specifications such as evaluation metrics, quality factors, and response time.The functional data specification includes data volume and data features.
  • IV. KEY DEFINITIONS AND PROBLEM STATEMENT: A real drift represents a fundamental change in the relationship between input data and MLaaS predictions that reflects degraded predictive behavior.The formal setup considers the joint distribution of input data and corresponding predictions over time.
  • IV. KEY DEFINITIONS AND PROBLEM STATEMENT: A pseudo drift changes the input distribution while preserving the predictive mapping, so MLaaS performance does not undergo sustained or long-term degradation.Pseudo drift is described as a temporary input fluctuation that does not alter the service’s conditional predictive behavior.
  • IV. KEY DEFINITIONS AND PROBLEM STATEMENT: The MPDD problem is to automatically distinguish real drift from pseudo drift using recent input–prediction pairs, since real drift requires intervention while pseudo drift permits continued operation.Examples of intervention include model retraining or service reconfiguration.

V. PROPOSED MLAAS PERFORMANCE DRIFT DETECTION FRAMEWORK FOR IOT ENVIRONMENTS · A. MLaaS Extraction model

The proposed MPDD framework organizes MLaaS drift detection for IoT environments into five components, including edge/database, MLaaS, and extraction layers. Its extraction model approximates black-box service behavior from unlabeled inputs and returned class labels, then derives prediction-influencing feature importance.

  • V. PROPOSED MLAAS PERFORMANCE DRIFT DETECTION FRAMEWORK FOR IOT ENVIRONMENTS: The MPDD framework comprises five components for identifying performance drift in MLaaS within IoT environments.The framework is presented as an integrated architecture for IoT-based MLaaS monitoring.
  • V. PROPOSED MLAAS PERFORMANCE DRIFT DETECTION FRAMEWORK FOR IOT ENVIRONMENTS: IoT devices generate real-time data streams that are stored in a centralized database within the edge and database layer.Examples include smartwatches, fitness trackers, cameras, and smart rings.
  • V. PROPOSED MLAAS PERFORMANCE DRIFT DETECTION FRAMEWORK FOR IOT ENVIRONMENTS: The MLaaS layer integrates a provider’s pre-trained model into the IoT edge to produce local predictions.This layer is identified as the framework’s second component.
  • V. PROPOSED MLAAS PERFORMANCE DRIFT DETECTION FRAMEWORK FOR IOT ENVIRONMENTS: The MLaaS extraction layer trains an extraction model to capture latent factors influencing service predictions.This layer forms the basis for observing MLaaS behavior without direct access to the service internals.
  • A. MLaaS Extraction model: The MLaaS Extraction Model approximates black-box MLaaS behavior using unlabeled inputs and corresponding predictions returned by the service.It assumes the IoT provider maintains historical unlabeled inputs collected during normal operation.
  • A. MLaaS Extraction model: The model relies exclusively on predicted class labels, forming input–label pairs that capture observable MLaaS decision behavior.This avoids dependence on prediction probabilities that may be altered by rounding, quantization, or probability perturbation.
  • A. MLaaS Extraction model: Under bounded fidelity error, the extracted feature-importance vector consistently approximates the black-box service’s true behavioral importance.The stated condition is E[|M(X)−M′(X)|] ≤ ϵ.
  • A. MLaaS Extraction model: The extraction procedure trains a lightweight decision-tree model on input–label pairs to approximate the mapping X → ŷ.The algorithm outputs a feature-importance vector after training the extraction model.

B. Fr´echet Data Drift Score (FDDS)

The Fréchet Data Drift Score (FDDS) measures statistical differences between baseline data and new stream windows using mean and covariance shifts. Higher FDDS values indicate greater divergence and likelihood of data drift, while remaining low for minor deviations.

  • Definition: FDDS applies Fréchet distance to quantify statistical differences between historical baseline data and new data windows in the stream.The framework uses preprocessing and feature transformation before measuring this difference.
  • Decomposition: FDDS decomposes distributional difference into mean shift and covariance shift components.The mean term captures changes in distribution centers, while the covariance term captures changes in spread.
  • Interpretation: A higher FDDS indicates larger distributional divergence and a higher likelihood of data drift.Because covariance shift is non-negative, FDDS is lower bounded by the mean-shift component.
  • Drift sensitivity: FDDS remains stable under small mean perturbations when covariance structures are approximately unchanged.Larger mean and/or covariance changes increase FDDS, distinguishing minor deviations from significant drift.

C. MLaaS-Aware Drift Exposure Score (MDES)

MDES measures MLaaS exposure to performance drift by combining extraction-model feature importance with FDDS, thereby accounting for both input changes and inferred service-behavior variations. It classifies drift using an adaptive threshold and distinguishes real drift from pseudo drift based on weighted exposure.

  • MDES definition: MDES combines MLaaS extraction feature importance with FDDS to quantify service exposure to performance drift.This integration captures changes in input-data characteristics alongside MLaaS behavioral variations inferred by the extraction model.
  • Drift classification: Real drift is identified when the weighted combination of feature importance and data drift exceeds θm = µ + βσ.The sensitivity coefficient β controls the threshold level and is empirically determined.
  • Drift classification: Pseudo drift occurs when statistical shift is primarily concentrated on features with low importance weights, keeping cumulative weighted exposure bounded.The lemma defines b as the baseline window, w as the current window, and FDDS(b, w) as their statistical shift.
  • Detection procedure: The detection procedure trains a lightweight extraction model, computes feature importance, calculates FDDS, and forms a feature-weighted exposure score.The algorithm approximates M with M′(X) ≈ M(X), computes SFDDS from baseline and current statistics, and applies the automated threshold.

D. MLaaS Performance Drift Detection Model · E. Adaptive-Temporal Performance Drift Detection Mechanism (APDDM)

The MPDD model detects real and pseudo drift by jointly monitoring dynamic input-data properties and concurrent MLaaS behavioral changes. APDDM adaptively changes its monitoring window according to recent drift patterns to support stable performance-drift detection.

  • D. MLaaS Performance Drift Detection Model: MPDD detects both real and pseudo drift in IoT environments while addressing changes in input-data properties and MLaaS behavior.The model processes a baseline window and a current window to produce an MLaaS-Aware Drift Exposure Score, SMDES.
  • E. Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): APDDM adaptively adjusts the window size Ws according to recent drift patterns to support stable performance-drift detection.The mechanism is designed for MLaaS environments and adapts monitoring based on observed drift behavior.
  • E. Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): The procedure applies MPDD to each data block, records binary drift results in a recent buffer, and computes the drift ratio r when the buffer reaches Ws.A value of 1 indicates real drift.
  • E. Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): When the recent drift ratio exceeds θr, APDDM triggers drift and increments the consecutive-drift counter cu; otherwise, it increments the consecutive-non-drift counter co.The opposite counter is reset in each case, allowing the mechanism to track consecutive drift or non-drift behavior.
  • E. Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): After L consecutive drift outcomes, APDDM decreases Ws by ∆W but not below Wmin.This is the mechanism’s response to sustained drift.
  • E. Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): After L consecutive non-drift outcomes, APDDM increases Ws by ∆W but not above Wmax.The adapted window size is returned together with drift triggers after processing the block stream.
  • E. Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): APDDM takes a stream of blocks, buffer length b, initial window size Ws, drift-ratio threshold θr, trend length L, adaptation step ∆W, and bounds [Wmin, Wmax] as inputs.It outputs detected drift triggers and the final adapted window size.

VI. COMPUTATIONAL COMPLEXITY ANALYSIS · VII. EXPERIMENTS AND RESULTS

The paper analyzes APDDM’s online computational and memory requirements for continuous MLaaS monitoring, then evaluates MPDD against drift-detection baselines and APDDM’s adaptive monitoring timeliness across datasets.

  • VI. COMPUTATIONAL COMPLEXITY ANALYSIS: APDDM’s complexity is governed mainly by per-block drift-score computation and adaptive-window management.For each incoming block X_t of size n with d features, APDDM invokes MPDD scoring once before lightweight buffer and ratio operations.
  • VI. COMPUTATIONAL COMPLEXITY ANALYSIS: Assuming offline-precomputed extraction models and baseline statistics, online MPDD computation avoids retraining overhead.The complexity analysis explicitly assumes no online retraining.
  • VI. COMPUTATIONAL COMPLEXITY ANALYSIS: O(Tnd) is the typical streaming-time complexity when W_s ≪ n.This simplification applies under the stated streaming condition.
  • VI. COMPUTATIONAL COMPLEXITY ANALYSIS: APDDM requires O(W_s) memory for its adaptive drift-score buffer and O(d), or O(d^2) for full covariance, for baseline summaries.These requirements support continuous monitoring without retraining overhead.
  • VII. EXPERIMENTS AND RESULTS: The experiments assess MPDD’s ability to distinguish real drift from pseudo drift using accuracy, precision, and recall.MPDD is compared with ADWIN, MMD, D3, STUDD, and SCSD.
  • VII. EXPERIMENTS AND RESULTS: The evaluation also investigates APDDM’s timeliness and adaptive temporal monitoring across multiple datasets.This experiment examines whether adaptive monitoring improves drift responsiveness.

A. Experiment Setup and Dataset … B. Experiment 1: Evaluation of MLaaS Performance Drift Detection approach

The experiments evaluate MPDD in IoT settings using diverse datasets, controlled drift scenarios, benchmark detectors, and an MLaaS extraction model. MPDD achieves the highest overall detection accuracy across datasets while maintaining balanced precision and recall for pseudo and real drift.

  • A. Experiment Setup and Dataset: The evaluation uses datasets, drift setups, and baseline techniques to assess the proposed framework.
  • 1) Dataset:: PAMAP2 HAR contains 3,850,505 samples across 52 sensor channels, while Electricity and NOAA Weather contain 45,325 and 96,454 records, respectively.
  • 2) Drift Data Generation:: The study generates sudden, gradual, incremental, and recurrent drift scenarios with the SEA generator to test diverse controlled distribution changes.
  • 3) Baseline Techniques:: MPDD is compared with ADWIN and MMD as established error-based and statistical data-drift baselines for black-box MLaaS evaluation.
  • 4) Configuration of the MLaaS Extraction Model:: The extraction model M′ approximates black-box MLaaS behavior and provides feature-importance information for performance drift detection.
  • B. Experiment 1: Evaluation of MLaaS Performance Drift Detection approach: MPDD detects real and pseudo drift using accuracy, precision, and recall, and is evaluated against ADWIN, MMD, STUDD, SCSD, and D3.
  • B. Experiment 1: Evaluation of MLaaS Performance Drift Detection approach: 0.91 pseudo-drift accuracy is achieved on HAR, followed by 0.90 on Electricity, 0.89 on Weather, 0.88 on Poker, and 0.90 on Airline.Pseudo-drift precision ranges from 0.85 to 0.89, while recall remains consistently above 0.94.
  • B. Experiment 1: Evaluation of MLaaS Performance Drift Detection approach: MPDD consistently achieves the highest detection accuracy across all evaluated datasets and maintains balanced performance relative to existing approaches.The comparison attributes this effectiveness to incorporating feature-importance variations and data-drift characteristics.

C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM) · D. Experiment 3: Comprehensive Evaluation of the MLaaS Extraction Model

Experiment 2 evaluates APDDM’s timeliness and detection quality across drift streams, showing more stable accuracy, faster recovery, and fewer missed events in key datasets. Experiment 3 evaluates how accurately the MLaaS extraction model approximates the black-box service and how extraction error affects MPDD and APDDM.

  • C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): APDDM’s adaptive window adjustment is evaluated against MPDD across multiple datasets and approximately 250-sample evaluation chunks.Figure 4 uses shaded accuracy-gain regions to show APDDM’s benefit under changing drift conditions.
  • C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): During HAR drift events, MPDD accuracy drops to approximately 0.20, while APDDM maintains around 0.75 and recovers more rapidly.The comparison highlights adaptive monitoring under sudden and recurring drift.
  • C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): On Poker, MPDD drops to approximately 0.65 during drift intervals, whereas APDDM remains above approximately 0.85 for most of the stream and recovers faster.The stream exhibits recurring and incremental drift characteristics.
  • C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): APDDM and MPDD achieve relatively low false-negative counts overall, although ADWIN, MMD, and D3 record lower counts on some datasets.False negatives measure missed real performance-drift events, so lower values indicate fewer missed detections.
  • C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): Table V compares APDDM with fixed-interval MPDD and ADWIN, MMD, D3, STUDD, and SCSD using Accuracy↑, Accuracy Gain↑, MDR↓, and FN↓.The metrics assess correctness, relative improvement, miss-detection ratio, and false negatives across five datasets.
  • C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): 0.92 is APDDM’s highest accuracy on Electricity, while false negatives fall from 355 under MPDD to 280, a reduction of approximately 21.1%.MDR decreases from 0.285 to 0.23.
  • C. Experiment 2: Timeliness Evaluation of Adaptive-Temporal Performance Drift Detection Mechanism (APDDM): 0.97 is APDDM’s highest accuracy on Poker, a 9.81% improvement over MPDD, while false negatives decrease from 640 to 160, a 75.0% reduction.MDR decreases from 0.25 to 0.06.
  • D. Experiment 3: Comprehensive Evaluation of the MLaaS Extraction Model: The MLaaS extraction model approximates the original black-box service, and Experiment 3 evaluates its fidelity, prediction behavior, and error effects on MPDD and APDDM.The framework’s effectiveness depends on the quality of this approximation.

1) Fidelity and Predictive Behavior Analysis of the MLaaS extraction Model: … IX. CONCLUDING REMARKS

The paper validates the MLaaS extraction model’s agreement with black-box services, assesses how extraction errors affect MPDD and APDDM, and discusses practical service-management implications and framework contributions.

  • 1) Fidelity and Predictive Behavior Analysis of the MLaaS extraction Model:: Fidelity measures agreement between the extracted model’s predictions and the inaccessible MLaaS service, rather than accuracy against ground-truth labels.The analysis evaluates whether the extraction model approximates the service’s decision patterns.
  • 1) Fidelity and Predictive Behavior Analysis of the MLaaS extraction Model:: The four-category predictive-behavior analysis distinguishes both models correct, MLaaS-only correct, extraction-only correct, and both models incorrect.This provides more detail about extracted-service relationships than overall accuracy alone.
  • 1) Fidelity and Predictive Behavior Analysis of the MLaaS extraction Model:: Both-correct predictions dominate across datasets, reaching approximately 0.80 for Weather, 0.78 for Poker, 0.70 for Electricity, 0.63 for Airline, and 0.59 for HAR.The MLaaS-correct/extraction-wrong and MLaaS-wrong/extraction-correct categories generally remain below 5%.
  • 2) Impact of MLaaS Extraction Model Errors on MPDD and APDDM Performance:: Extraction-model approximation errors can propagate into final MPDD and APDDM drift-detection outcomes, especially when the underlying MLaaS model is complex and non-linear.The paper explicitly quantifies this propagation by comparing detection accuracy using extracted and original MLaaS models.
  • 2) Impact of MLaaS Extraction Model Errors on MPDD and APDDM Performance:: APDDM deviations range from -2.58 to +0.42 percentage points and usually show no measurable difference under extraction-model approximation errors.Its adaptive parameter selection uses recent drift-score trends rather than any single extraction prediction.
  • VIII. DISCUSSIONS AND IMPLICATIONS: The framework enables MLaaS clients to identify potential drifts, notify providers, choose alternative services, or perform service composition.Repeated random drift-point generation and aggregated metrics were used to improve robustness against sensitivity to drift placement.
  • VIII. DISCUSSIONS AND IMPLICATIONS: Proactive drift awareness supports timely service switching or composition in cost-sensitive and safety-critical domains, where undetected drift can cause incorrect decisions, operational risks, and financial losses.The paper highlights smart healthcare, industrial IoT, and intelligent transportation as especially relevant domains.
  • IX. CONCLUDING REMARKS: MPDD addresses inaccessible ground-truth labels and internal parameters by extracting observable behavioral patterns from black-box MLaaS services to monitor feature-preference variations.The framework also introduces a Fréchet-based mechanism for quantifying distributional changes in evolving data streams.

APPENDIX A PROOF OF LEMMAS · Proof of Lemma 1

Under bounded fidelity, the extraction model preserves the black-box MLaaS’s feature-importance behavior while its deviation is bounded by fidelity error and perturbation scale. The proof establishes this by comparing perturbed model outputs through triangle inequality, expectation, and sample averaging.

  • Proof of Lemma 1: The proof models M′ as an extraction model trained from query logs to approximate the black-box MLaaS M under a bounded-fidelity assumption.
  • Proof of Lemma 1: Feature importance for xi is operationally defined as the average output change caused by a controlled perturbation of magnitude δ.
  • Proof of Lemma 1: The proof applies the triangle inequality to relate importance contributions from the extraction model and the black-box MLaaS.
  • Proof of Lemma 1: Taking expectations and applying the fidelity assumption to both X and X+i yields a bounded difference between their importance contributions.
  • Proof of Lemma 1: Averaging the per-sample bound over Algorithm 1’s samples gives the corresponding bound for the estimated importance of xi.
  • Proof of Lemma 1: The extracted importance vector IM′ therefore deviates from the behavioral importance vector IM by a bounded amount.
  • Proof of Lemma 1: Because importance is measured using perturbations of magnitude δ, estimator sensitivity depends on δ and the selected norm.
  • Proof of Lemma 1: Thus, under bounded fidelity, M′ preserves the MLaaS feature-importance behavior up to a bounded deviation f(ϵ, δ), establishing Lemma 1.

Proof of Lemma 2

Lemma 2 establishes that FDDS is lower bounded by the mean shift, increases under substantial mean or covariance drift, and remains small under minor distributional changes.

  • Proof of Lemma 2: FDDS is defined between a baseline window b and a current window w using mean-shift and covariance-shift components.The proof begins from the FDDS definition in Eq. (5).
  • Proof of Lemma 2: Because the covariance discrepancy is non-negative, it cannot reduce FDDS below the mean-shift component.This follows from the symmetric positive semi-definiteness of Σb and Σw.
  • Under significant covariance change: Substantial covariance drift increases FDDS even when the baseline and current means are approximately equal.A large ∥Σw − Σb∥ enlarges the trace-based covariance gap, increasing C(Σb, Σw).
  • Under minor distributional change: Under minor distributional change, small mean shift and similar covariances make FDDS remain small.When ∥µb − µw∥ is small and Σw ≈ Σb, the covariance discrepancy is approximately zero.

Proof of Lemma 3 · APPENDIX B DRIFT GENERATION · 1) Drift Data Generation:

Lemma 3 shows that MDES flags drift when its score exceeds threshold θm, especially when statistical shifts affect highly important features. The appendix generates four drifted streams per dataset using the SEA generator, covering sudden, incremental, gradual, and recurrent drift.

  • Proof of Lemma 3: MDES flags drift when its score exceeds the threshold θm.This activates the condition MDES = 1 and indicates drift.
  • Proof of Lemma 3: MDES jointly weights feature importance and feature-wise statistical shifts between baseline and current windows.Its cumulative exposure combines I(x_i) with FDDS(b_i, w_i).
  • Proof of Lemma 3: Drift concentrated on highly important features substantially increases cumulative weighted exposure.These weighted terms are more likely to exceed θm, triggering MDES = 1.
  • APPENDIX B DRIFT GENERATION: Figure 8 depicts drift generation for HAR, Airline, Weather, Electricity, and Poker datasets.The figure organizes the datasets into panels A through E.
  • 1) Drift Data Generation:: The SEA generator creates drifted data streams across the experimental datasets.The generated streams model four common drift categories.
  • 1) Drift Data Generation:: Each dataset receives four full-length drifted streams representing sudden, incremental, gradual, and recurrent drift.One stream is generated for each drift type.
  • APPENDIX B DRIFT GENERATION: The combined generated stream contains four times as many samples as the original dataset.Table 6 summarizes stream sizes and drift statistics across datasets.
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