Algorithmic drift in generative adaptive learning

Firuz Kamalov; Hana Sulieman; Ahmed El Sayed; Sivaraj R.; Nitesh Sughnani

cs.AIOct 22, 2026

Algorithmic drift in generative adaptive learning

Authors: Firuz Kamalov, Hana Sulieman, Ahmed El Sayed, Sivaraj R., Nitesh Sughnani

Organizations: Department of Computational Sciences, Canadian University Dubai, Dubai, UAE · Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE · Department of Mathematics, Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, India · Knowledge and Human Development Authority, Dubai, UAE

Abstract

Generative artificial intelligence promises to democratize personalized tutoring, yet the longitudinal stability of adaptive systems remains under-theorized. Current ethical discourse focuses on static bias. This paper argues that a distinct threat lies in algorithmic drift: a feedback loop in which minor, demographically asymmetric observation errors compound over time into systemic inequities. We make three contributions. First, we formalize algorithmic drift as a coupled learner--tutor dynamical system. Second, we corroborate this theory with an agent-based Monte Carlo simulation in which either a transparent online tracker or an LSTM deep-knowledge-tracing network drives an IRT-based curriculum policy. Third, we introduce the temporal fairness stabilizer (TFS), a targeted intervention that periodically re-probes learner capability, and benchmark it against standard exploration baselines with explicit group-fairness metrics and replication-level inference. Within the modeled environment, TFS closes the achievement gap where generic exploration does not, and we characterize the conditions under which it succeeds. We separate what is established analytically, what is demonstrated in simulation, and what remains to be validated with real learners. Our results reframe algorithmic fairness in education as a temporal property of a closed loop rather than a deployment-time checkbox.

Figures & tables

Symbol	Meaning	Default	Status
$N$ , $T$ , $K$	learners, time steps, knowledge components	$10{,}000$ , $100$ , $50$	design choice
$\|S_{prot}\|/N$	protected-group share	$0.20$	assumed (stylized)
$\theta_{i}(0)$	initial latent ability	$\mathcal{N}(0.5,0.1)$ , clipped	assumed; varied in Sec. 4.7
$\alpha_{i}$	learning rate	$\mathcal{N}(0.02,0.005)$ , $>0$	calibrated (reference ceiling $\approx 0.92$ )
$\lambda$	forgetting rate	$0.002$	assumed
$\omega$	ZPD kernel width	$0.12$	assumed; sets $\epsilon^{\star}$

Table 1: Simulation parameters. “Calibrated” values were set so that the noiseless reference group reaches the documented mastery ceiling within the horizon; “assumed” values are modeling choices whose influence is examined in the sensitivity and robustness analyses.

Panel of Figure 2 — Fig 1: Dynamical system representation of algorithmic drift in the adaptive learning environment. The recursive loop, illustrates how asymmetric observation noise ( $\epsilon$ ) generated by linguistic friction depresses DKT mastery estimations ( $\hat{p}_{t}$ ). This leads the generative policy to issue a remedial curriculum ( $\beta_{t+1}$ ), which in turn stalls the learner’s actual cognitive state ( $\theta_{t+1}$ ). The TFS serves as a regulatory layer that interrupts this structural trajectory decoupling by forcing periodic exploration probes.

Panel of Figure 3 — Fig 2: Flowchart of the TFS algorithm. Operating as a targeted regularization layer over the standard generative policy, TFS tracks a longitudinal pessimism counter ( $\kappa$ ) to detect sustained periods of remedial curriculum assignment ( $\beta<\mu_{threshold}$ ). Once the consecutive threshold ( $\tau$ ) is reached, it overrides the depressed estimations of the DKT, forcing a diagnostic exploration probe to verify actual cognitive ceiling and break the algorithmic drift loop. For visual clarity the figure depicts the absolute-threshold form of the pessimism trigger; the deployed policy uses the cohort-relative criterion and capability ratchet of Algorithm 1 .

Grader	$\mathrm{FN}_{ref}$	$\mathrm{FN}_{prot}$	$\hat{\epsilon}$	McNemar $p$	Cohen’s $h$	$\bar{P}(\mathrm{C})_{ref}$	$\bar{P}(\mathrm{C})_{prot}$	$\Delta P$ ( $p$ )	items $\downarrow$	$\mathrm{FP}_{ref}$ / $\mathrm{FP}_{prot}$
Qwen2.5-7B-Instruct	0.000	0.050	$+0.050$	0.5	$+0.45$	0.993	0.939	$+0.055$ (0.043)	95%	0.00 / 0.00
Phi-4-mini-instruct	0.000	0.000	$+0.000$	1	$+0.00$	0.995	0.993	$+0.001$ (0.64)	98%	0.10 / 0.05
Mistral-7B-Instruct-v0.3	0.000	0.000	$+0.000$	1	$+0.00$	1.000	0.999	$+0.001$ (0.31)	62%	0.30 / 0.25
Pooled (3 graders)			$+0.017$ (mean)	0.5				$+0.019$ (mean)

Table 2: Matched-pair LLM grader measurement (40 correct and 20 incorrect items, three instruction phrasings, modal verdicts).

\mathrm{FN}

: false-negative rate on conceptually correct answers;

\hat{\epsilon}=\mathrm{FN}_{prot}-\mathrm{FN}_{ref}

; McNemar: exact two-sided

p

on the paired verdicts;

\bar{P}(\mathrm{C})

: mean grader probability of CORRECT;

\Delta P

: paired difference

\bar{P}(\mathrm{C})_{ref}-\bar{P}(\mathrm{C})_{prot}

with its paired-

t

p

-value; items

\downarrow

: share of items on which

P(\mathrm{C})

is lower for the second-language variant;

\mathrm{FP}

: false-positive rate on incorrect answers. The last row pools the discordant pairs over graders.

Fig 3: Algorithmic drift ( $\epsilon=0.05$ , online tracker, mean trajectories over $R=50$ replications). Scenario A (left) illustrates the drift divergence, where the AI’s estimation of the protected group (dashed) decouples from their true latent ability (blue solid line), suppressing their learning trajectory compared to the reference group (green). Scenario B (right) shows that the TFS mitigates this estimation error, substantially reducing the achievement gap under modeled conditions.

Policy	Final ability ( $S_{prot}$ )	Gap $\Delta$	Drift $D$ ( $S_{prot}$ )	Learner-level $d$ vs. control
Control (no exploration)	$0.884$ [0.883, 0.885]	$0.038$ [0.037, 0.039]	$0.268$	—
$\epsilon$ -greedy	$0.888$ [0.887, 0.889]	$0.035$ [0.034, 0.036]	$0.270$	$+0.03$
UCB	$0.901$ [0.900, 0.901]	$0.018$ [0.017, 0.019]	$0.283$	$+0.13$
Thompson sampling	$0.836$ [0.835, 0.837]	$0.019$ [0.019, 0.020]	$0.263$	$-0.34$
TFS (ours)	$\mathbf{0.972}$ [0.971, 0.972]	$\mathbf{0.003}$ [0.002, 0.004]	$\mathbf{0.203}$	$+0.83$

Table 3: Comparison of TFS against standard exploration baselines under

\epsilon=0.05

. Values are means over

R=50

seeds with replication-level

95\%

percentile-bootstrap CIs. The last column is Cohen’s

d

on protected-group final ability relative to the control policy, computed at the learner level and pooled over replications (positive favors the policy); the paired replication-level contrast for TFS is reported in Section 4.10 .

Policy	$G_{\beta}$	$G_{Z}$	$G_{b}$	$\mathrm{FN}_{prot}$ / $\mathrm{FN}_{ref}$	$G_{FN}$	$\rho$
Control, $\epsilon=0$ (parity reference)	$+0.000$	$+0.000$	$+0.000$	$0.105$ / $0.105$	$-0.001$	$1.000$
Control (no exploration)	$+0.070$	$+0.059$	$-0.047$	$0.138$ / $0.105$	$+0.033$	$0.911$
$\epsilon$ -greedy	$+0.064$	$+0.052$	$-0.049$	$0.145$ / $0.115$	$+0.030$	$0.916$
UCB	$+0.059$	$+0.046$	$-0.056$	$0.098$ / $0.066$	$+0.032$	$0.957$
Thompson sampling	$+0.065$	$+0.022$	$-0.058$	$0.127$ / $0.105$	$+0.022$	$0.945$
TFS (ours)	$+0.033$	$+0.045$	$-0.026$	$0.037$ / $0.032$	$+0.006$	$0.994$

Table 4: Group-fairness metrics at

\epsilon=0.05

(means over

R=50

replications; time-averaged over

t=1,\dots,T

G_{\beta}

: exposure gap (reference minus protected mean served difficulty);

G_{Z}

: ZPD-access gap;

G_{b}

: estimation-bias gap (protected minus reference signed calibration error; negative = protected group underestimated relative to reference);

\mathrm{FN}_{prot}/\mathrm{FN}_{ref}

: remedial-misclassification rates with their gap

G_{FN}

;

\rho

: gain ratio. The first row is the control policy with a noiseless channel (

\epsilon=0

), the parity reference point. Replication-level

95\%

CI half-widths are at most

0.002

for every entry.

$a=0.7$ : LSTM-DKT held-out AUC $0.553$ (heterogeneous) / $0.553$ (deployment); oracle AUC $0.565$ / $0.559$
$\epsilon$	Control $\Delta$	Control $G_{b}$	Control $G_{\beta}$	TFS $\Delta$	TFS $G_{b}$	Tracker $\Delta$ : control / TFS
0.00	$-0.002$ [-0.003, -0.000]	$-0.000$	$+0.000$	$-0.002$ [-0.003, -0.000]	$-0.000$	$+0.001$ / $+0.001$
0.02	$-0.002$ [-0.004, -0.001]	$-0.006$	$+0.008$	$-0.003$ [-0.004, -0.002]	$-0.005$	$+0.014$ / $+0.001$
0.05	$-0.003$ [-0.004, -0.002]	$-0.015$	$+0.022$	$-0.005$ [-0.006, -0.004]	$-0.013$	$+0.039$ / $+0.003$
0.10	$-0.001$ [-0.002, -0.000]	$-0.031$	$+0.047$	$-0.008$ [-0.009, -0.007]	$-0.026$	$+0.089$ / $+0.012$
0.15	$+0.007$ [0.006, 0.008]	$-0.047$	$+0.075$	$-0.010$ [-0.011, -0.009]	$-0.038$	$+0.148$ / $+0.025$

Table 5: Main experiment with the LSTM-DKT in the loop (

N=10{,}000

T=100

R=20

replications per cell; replication-level

95\%

CIs in brackets). Held-out next-outcome AUC of the network on the heterogeneous pre-training population and on the deployment population, against the oracle AUC of the true success probability.

\Delta

: gap widening;

G_{b}

: estimation-bias gap;

G_{\beta}

: exposure gap. The last column gives the online tracker’s gap at the same

(a,\epsilon)

(

R=10

Fig 4: Online tracker versus LSTM-DKT in the loop. Left: estimation-bias gap $G_{b}$ under the control policy as a function of $\epsilon$ (negative: protected group under-estimated relative to the reference group). Right: ability gap $\Delta$ under control (solid) and TFS (dashed). The LSTM reproduces the estimation bias at every $\epsilon$ and the trapping transition at larger $\epsilon$ ; TFS holds the gap at or below zero for both estimators.

Fig 5: Algorithmic drift persists at a lower noise parameter $\epsilon=0.02$ (online tracker, mean trajectories over $R=50$ replications; gap widening $\Delta=0.013$ under control versus $0.001$ under TFS).

	Control				TFS
Noise $\epsilon$	Final ability ( $S_{prot}$ )	$\Delta$	$D$ ( $S_{prot}$ )	$G_{b}$	Final ability ( $S_{prot}$ )	$\Delta$	$G_{b}$
0.00	$0.922$	$0.000$	$0.235$	$+0.000$	$0.975$	$0.000$	$+0.000$
0.01	$0.916$	$0.006$	$0.242$	$-0.009$	$0.975$	$0.000$	$-0.005$
0.02	$0.909$	$0.013$	$0.248$	$-0.019$	$0.975$	$0.000$	$-0.011$
0.03	$0.901$	$0.020$	$0.255$	$-0.028$	$0.974$	$0.001$	$-0.016$
0.05	$0.884$	$0.037$	$0.268$	$-0.047$	$0.972$	$0.002$	$-0.026$

Table 6: Sensitivity of learning outcomes to observation noise (

R=25

replications per cell; reference-group final ability

0.922

under control and

0.975

under TFS throughout). Gap widening

\Delta

carries a replication-level

95\%

bootstrap CI of at most

\pm 0.001

at every row;

G_{b}

is the estimation-bias gap.

Initial-ability distribution of both groups, $\theta_{i}(0)\sim\mathcal{N}(\mu_{0},\sigma_{0})$
$\mu_{0}$	$\sigma_{0}$	Control $\Delta$	Control $D$	TFS $\Delta$
0.3	0.05	$+0.042$	$0.246$	$+0.004$
0.3	0.10	$+0.040$	$0.247$	$+0.002$
0.3	0.20	$+0.034$	$0.252$	$-0.006$
0.5	0.05	$+0.039$	$0.267$	$+0.004$
0.5	0.10	$+0.037$	$0.268$	$+0.002$

Table 7: Initial-state and group-heterogeneity sensitivity at

\epsilon=0.05

(

R=25

). Top: initial-ability distribution of both groups (control gap

\Delta

, control drift

D

, and TFS gap). Middle: initial estimate offset

\xi

of the tracker. Bottom: protected group with a different prior ability or learning rate;

\Delta_{0}

is the gap with a noiseless channel,

\Delta_{0.05}

the gap with the biased channel, and the excess

\Delta_{0.05}-\Delta_{0}

is the drift-attributable component; the last column is the TFS gap at

\epsilon=0.05

Cohort margin $m$	$\tau=3$	$\tau=5$	$\tau=8$	$\tau=12$
0.05	$-0.006$	$+0.000$	$+0.013$	$+0.023$
0.10	$-0.005$	$+0.002$	$+0.015$	$+0.025$
0.15	$-0.001$	$+0.007$	$+0.018$	$+0.026$

Table 8: TFS hyperparameter ablation at

\epsilon=0.05

: residual protected-group gap

\Delta

as a function of patience

\tau

and cohort margin

m

(

R=25

). The recovered gap is near zero across the grid.

$\epsilon=0.05$
$\phi$ \ $\psi$	$0.0$	$0.05$	$0.1$
0.00	$+0.002$ ( $0.972$ )	$+0.004$ ( $0.967$ )	$+0.004$ ( $0.964$ )
0.25	$+0.003$ ( $0.969$ )	$+0.004$ ( $0.965$ )	$+0.005$ ( $0.962$ )
0.50	$+0.003$ ( $0.965$ )	$+0.005$ ( $0.961$ )	$+0.005$ ( $0.958$ )
1.00	$+0.006$ ( $0.952$ )	$+0.007$ ( $0.949$ )	$+0.008$ ( $0.947$ )
$\epsilon=0.15$

Table 9: TFS under probe side effects (

R=25

): residual gap

\Delta

(and protected-group final ability in parentheses) as a function of the frustration penalty

\phi

and the entrenchment penalty

\psi

. The unmitigated control gap is

0.037

\epsilon=0.05

and

0.147

\epsilon=0.15

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