Causal Adjustment for Feedback Loops (CAFL)

• Causal Adjustment for Feedback Loops is a framework that defines techniques to debias causal estimates in systems with cyclic feedback, such as control settings and recommender environments.
• It leverages structural equation models and generalized adjustment formulas to recover interventional effects, even when traditional methods fail due to bidirectional dependencies and latent confounders.
• Empirical applications in process control and online recommendation systems demonstrate CAFL’s capability to retrieve true causal parameters from biased observational data.

Causal Adjustment for Feedback Loops (CAFL) is a class of statistical and algorithmic techniques designed to identify and estimate causal effects in systems characterized by feedback loops—settings in which the action of an agent (controller, algorithm, recommender, or participant) influences outcomes, and those outcomes in turn influence subsequent actions. CAFL provides principled ways to de-bias causal effect estimates in observational and operational data from such cyclical or closed-loop systems, ensuring that interventions can be accurately evaluated or optimized in the presence of feedback and confounding. The methodology is foundational to modern causal inference in fields as diverse as control theory, recommender systems, time-series analysis, and structural equation modeling, especially as deployed in complex, interconnected, or engineered environments.

1. Structural Foundations and Problem Motivation

Feedback loops present a fundamental challenge for causal learning because naive statistical models trained on closed-loop (operational) data often produce biased effect estimates. In control settings, operational data is confounded by the feedback action of the controller, which adapts to disturbances or reference variables; in recommender systems, iterative retraining on data generated by past recommendations leads to “rich-get-richer” effects and loss of diversity; in biological, economic, and social sciences, cyclic causal structures and unmeasured confounders are prevalent (Løvland et al., 2022, Krauth et al., 2022, Lorbeer et al., 18 Nov 2024, Xu et al., 2022, Li et al., 16 Jul 2024).

Standard causal frameworks built solely on directed acyclic graphs (DAGs) and back-door criteria often fail to identify causal parameters in the presence of cycles, bidirectionality, or latent confounders. CAFL generalizes adjustment formulas and identification techniques to settings involving instantaneous cyclic relations, control-induced dependencies, and unobserved confounding.

The general aim is to estimate interventional (do-operator) quantities such as $p(Y|do(U))$, or to identify total effect coefficients $\Theta_{j,i}$ in structural systems, from observed steady-state or time series data acquired under feedback.

2. Formal Models and Assumptions

CAFL analysis is grounded in structural equation models (SEMs) and structural dynamical causal models (SDCMs), which encode both instantaneous and dynamic relationships between system variables, actions, and disturbances.

Continuous-Time and Steady-State Models

A typical SDCM for process control has state $x(t)\in\mathbb{R}^n$, control input $U(t)\in\mathbb{R}^m$, disturbance $D(t)$, and output $Y(t)=g(x(t))$. The dynamics obey

$$\dot{x}(t) = f(x(t), U(t), D(t)), \quad Y(t) = g(x(t)).$$

Data is collected under a fixed control law, either feedforward ($U = k_{ff}(Y_{ref}, D)$) or feedback ($U = k_{fb}(Y_{ref}, Y)$). The exogenous processes $D(t)$ and $Y_{ref}(t)$ are assumed statistically independent and time-invariant during data acquisition. At steady-state, the system admits globally asymptotic equilibria $x_{ss}(U, D)$, and the resulting structural equations for $U, Y$ are acyclic in the feedforward representation (Løvland et al., 2022).

Instantaneous Cyclic SEM with Hidden Confounders

In systems with instantaneous feedback, the model is

$X = B X + \Gamma H + \varepsilon$

where $X$ is the $d$-dimensional vector of measured variables, $B$ is the direct-effect matrix (zero diagonal, possibly cyclic), $H$ is a vector of hidden confounders, $\Gamma$ specifies loadings, and $\varepsilon$ is noise. The system is solvable under a "weak stability" condition:

$\det(I - U B) \neq 0$

for any intervention mask $U$; multiply-intervention data is essential for identifiability (Lorbeer et al., 18 Nov 2024).

Dynamic Systems with Feedback in Recommender Systems

In collaborative filtering or online recommendation, feedback loops are formalized via temporal graphs:

• At each $t$, recommendations $A_t$, user outcomes $R_t$, and retrained parameters $\hat\Theta_t$ evolve as

$$A_t \rightarrow R_t \rightarrow \hat\Theta_t \rightarrow A_{t+1} \rightarrow \cdots$$

The causal model includes latent preference parameters $\Theta$, observed ratings, and sequential retraining, resulting in data whose marginal distribution is heavily biased by the feedback structure (Krauth et al., 2022, Xu et al., 2022).

Bi-directional Models with Invalid Instruments

For trait pairs $(X,Y)$, the model allows both $X\to Y$ and $Y\to X$ causality:

$$Y = \beta_{X\to Y}\,X + \pi_Y^T Z + \xi_Y(U) + \zeta\ X = \beta_{Y\to X}\,Y + \pi_X^T Z + \xi_X(U) + \eta$$

Unmeasured scalar confounder $U$, $p$-vector of candidate instruments $Z$ (possibly invalid), and arbitrary noise structures are supported. Valid identification in such models requires special conditions (enhanced plurality or covariance-heterogeneity), and the functional structure parallels the classical equilibrium of cyclic SEMs (Li et al., 16 Jul 2024).

3. Adjustment Formulas and Identification Strategies

Back-Door Adjustment in Feedback Systems

When the true causal quantity of interest is $p(Y|do(U=x))$, direct estimation from observational $p(Y|U=x)$ is biased due to feedback via unmeasured disturbances $D$ or retrained parameters. Under Pearl’s back-door criterion, adjustment for $D$ is sufficient

$p(Y|do(U=x)) = \int p(Y|U=x, D=d) p(D=d) dd$

This formula holds both for feedforward and feedback control, provided the feedback controller is time-invariant and the system equilibrates (Løvland et al., 2022).

Linear Cyclic SEM Adjustments

When cycles and confounders are present, the total effect of $X_i$ on $X_j$ is given by the $(j,i)$ element of the resolvent:

$\Theta = (I-B)^{-1}$

More generally, for classical settings (possibly cyclic, with confounders), the adjustment becomes:

$$\Theta_{j,i} = \frac{\Sigma_{ji} - \Sigma_{jZ}\Sigma_{ZZ}^{-1}\Sigma_{Zi}}{\Sigma_{ii} - \Sigma_{iZ}\Sigma_{ZZ}^{-1}\Sigma_{Zi}}$$

with $Z$ a sufficient conditioning set (often all variables except $i, j$ in cyclic models) (Lorbeer et al., 18 Nov 2024).

Recommender Systems: Breaking Feedback by Conditioning

In collaborative filtering, the intervention distribution for outcomes $R_s$ under recommendations $A_s$ is identified as:

$P(R_s|\do(A_s=a_s)) = \int P(R_s|A_s=a_s,\hat\Theta_{s-1}=\theta) P(\hat\Theta_{s-1}=\theta) d\theta$

which can be estimated using importance sampling weights based on propensity

$\frac{P(A_s)}{P(A_s|\hat\Theta_{s-1})}$

and by partitioning cells with positivity violations (Krauth et al., 2022, Xu et al., 2022).

Bidirectional Causal Discovery with Invalid IVs

Identification in bidirectional (possibly cyclic) models with invalid IVs uses:

• Mode-based identification: $$\beta_{X\to Y} = \mathrm{mode}\{\gamma_{Y,j}/\gamma_{X,j}\}$$
• Covariance-heterogeneity-based identification:

$$\beta_{Y\to X} = \frac{\theta_X^T \Sigma \theta_Y}{\theta_Y^T \Sigma \theta_Y}$$

with adaptive diagnostics to select the supporting direction (Li et al., 16 Jul 2024).

4. Robustness, Consistency, and Practical Estimation

Robustness to Outliers and Confounders

In estimation procedures based on covariance matrices (e.g., for $B$, $\Theta$), the vanilla sample covariance estimator has breakdown point zero; contamination in small sample batches can make effect estimates unbounded. CAFL incorporates robust M-estimators such as Minimum Covariance Determinant (MCD, breakdown $\sim0.5$), and gamma-divergence estimators, which are drop-in substitutions in the covariance estimation step. However, even robust covariances do not guarantee overall robustness after downstream linear system inversion, though empirical sensitivity is substantially reduced (Lorbeer et al., 18 Nov 2024).

Statistical Guarantees

Under requisite identifiability and regularity assumptions:

• Direct-effect matrices ($B$) and confounder covariances ($\Sigma_e$) are estimated at rate $O_p(n^{-1/2})$ in linear Gaussian models.
• Causal parameters such as $\beta_{X\to Y}$ in bidirectional SEMs are asymptotically normal, with bootstrapped or plug-in variance estimates providing valid confidence intervals.
• In recommender feedback adjustment, empirical inverse propensity weighted (IPW) estimates converge to the causal objective, subject to positivity and consistency conditions (Løvland et al., 2022, Lorbeer et al., 18 Nov 2024, Krauth et al., 2022, Li et al., 16 Jul 2024).

5. Representative Algorithms and Implementation Steps

Equilibrium Model Estimation

For process control under feedback, practical steps include:

1. Define a parametric map $h(U,D;\theta)$ (e.g., linear $Y = cU + D$).
2. Specify priors/regularization for $\theta$ and $\{D_i\}$.
3. Maximize joint likelihood over $\theta$ and $D_i$ (in the linear-Gaussian case, convex and closed-form).
4. Output empirical $D$-distribution and fitted model.
5. Use the adjustment formula over empirical $D$ for future causal prediction (Løvland et al., 2022).

Linear Cyclic SEM Recovery

CAFL in linear cyclic systems proceeds as:

1. Compute covariances under each intervention using elected estimator (sample, MCD, or GDE).
2. Formulate and stack total-effect constraints into a linear system.
3. Solve for direct effects via least-squares with optional regularization.
4. Estimate confounder noise from observational data.
5. Compute total effects from the resolvent (Lorbeer et al., 18 Nov 2024).

Causal Adjustment in Sequential Recommender Systems

The CAFL algorithm for collaborative filtering at each time step:

1. Issue recommendations per current policy.
2. Observe outcomes.
3. Compute per-instance weights: $W_{t,ui}=1/P(A_{t,ui}=1|\hat\Theta_{t-1})$.
4. Form weighted likelihood and update model parameters.
5. Partition unobservable cells for positivity violations; apply reweighting as in Theorem 4.1. This is a direct replacement of the standard unweighted loss, agnostic to the base model (Krauth et al., 2022).

Bidirectional IV and CH-based Causal Discovery

For trait pairs with feedback and invalid instruments:

1. Compute reduced-form effects of $Z$ on both $X$ and $Y$.
2. Cluster ratio statistics to apply the enhanced plurality rule for one direction; if unique, estimate the effect.
3. In the reverse direction, check covariance-heterogeneity and compute the corresponding estimator if conditions hold.
4. Decide causal direction using combined diagnostics; output effect estimates and confidence intervals (Li et al., 16 Jul 2024).

6. Empirical Findings and Applications

Empirical validation across domains demonstrates:

• In control and steady-state system modeling, CAFL adjustment recovers the true process gain or effect even under strong closed-loop confounding; in linear cases, closed-form solutions match ground-truth causal parameters (Løvland et al., 2022).
• In recommender systems, CAFL significantly reduces rank-based and homogenization errors relative to naive feedback-trained baselines, matching the accuracy of oracle-style non-feedback models. Homogenization (collapse of recommendation diversity) is nearly eliminated in both simulated and real-world data (Krauth et al., 2022, Xu et al., 2022).
• For linear cyclic SEMs with hidden confounders, total effect matrices and confounder covariances are reliably recovered provided sufficient interventional diversity and stability, and robustification reduces sensitivity to moderate contamination (Lorbeer et al., 18 Nov 2024).
• Bidirectional causal estimation with invalid IVs demonstrates consistent directional discovery and accurate confidence interval coverage under theoretical conditions, outperforming simpler one-directional or valid-IV-only approaches (Li et al., 16 Jul 2024).

7. Limitations and Extensions

While CAFL theory is now well developed across several application types, practical limitations include:

• Necessity for sufficient variability in confounders and/or interventional actions for identifiability.
• The assumption of time-invariant control policies and unique asymptotic equilibrium in SDCMs.
• For robust estimation, the critical step is in handling breakdown through both robust covariance estimation and careful linear system design; true closed-loop robustness is not fully guaranteed by robustified sample covariance alone (Lorbeer et al., 18 Nov 2024).
• In dynamic recommender settings, specification of counterfactual histories and control of hyperparameters (personalization vs. diversity) require additional modeling or heuristics.
• In bidirectional IV models, identification can fail when neither plurality nor heterogeneity conditions obtain; model selection remains nontrivial.

Future research is oriented towards relaxing stability and exogeneity conditions, developing more general nonparametric analogs, extending robust estimation through the full causal effect estimation pipeline, and applying these frameworks to increasingly high-dimensional, multi-modal, and adaptive feedback environments.