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On the Identifiability of User Adaptation in Co-Adaptive Neural Interfaces

This paper demonstrates that in co-adaptive neural interfaces, closed-loop encoder estimates cannot uniquely identify user adaptation because they reflect the properties of the entire joint system, and it subsequently proposes conditions necessary for achieving true identifiability.

Original authors: Philip Waggoner

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Philip Waggoner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Dance Between Human and Machine

Imagine a dance floor where two partners are learning to move together: a human user and a computer program (the "decoder"). The goal is to move a cursor on a screen using muscle signals (EMG).

In the study being discussed (by Madduri et al.), both partners are learning at the same time. The computer changes how it translates muscle signals into movement, and the human changes how they move their muscles to compensate. The researchers observed that when the computer changed its rules, the human's behavior changed too. They concluded that they could figure out exactly how the human was learning and adapting just by watching this dance.

Waggoner's paper argues: While the researchers correctly saw that the dance changed, they cannot be 100% sure about the specific steps the human dancer was taking inside their own head.

The Core Problem: The "Black Box" of Adaptation

The paper focuses on a concept called Identifiability. In simple terms, this asks: "If I see the final result, can I uniquely figure out exactly what caused it?"

The author says the answer is no in this specific setup.

The Analogy: The Shifting Stage

Imagine a magician (the user) performing a trick on a stage.

The Setup: The stage lights (the computer decoder) keep changing color and angle.
The Observation: An audience member (the researcher) watches the magician and tries to guess the secret method behind the trick based on how the light hits the props.
The Confusion: Because the lights are constantly moving, the shadows change. The audience member sees the shadows change and thinks, "Aha! The magician changed their hand position!"

Waggoner's point: The magician might have kept their hand in the exact same position the whole time. The shadows changed only because the lights moved.

In the paper's terms:

The Shadows are the estimated "encoder" (what the researchers calculate the human is doing).
The Lights are the computer decoder.
The Hand Position is the human's actual internal learning process.

Because the computer decoder keeps changing, it alters the "state" of the system (where the cursor is, what the user is looking at). This change in the environment makes it look like the human is learning or changing their strategy, even if the human is actually doing the exact same thing every time.

The "Two Different Stories, Same Ending" Proof

The paper uses a mathematical proof (Theorem 1) to show that two completely different scenarios can produce the exact same data.

Scenario A: The human is a robot. They never change their strategy. They just react to the computer's changes perfectly.
Scenario B: The human is a student. They are constantly learning and changing their strategy to adapt.

The paper shows that if the computer changes its rules in a specific way, the data recorded in both Scenario A and Scenario B will look identical. The researchers cannot tell the difference between a "robot" and a "student" just by looking at the final numbers.

Why This Matters

The author isn't saying the original study was wrong about the results. The results show that the system (human + machine) behaves in a certain way.

However, the author warns against a specific conclusion: You cannot uniquely identify the human's internal learning process.

If the original study claims, "We know exactly how the human brain is learning," Waggoner says, "Not quite. You know how the team is performing, but you can't be sure if the human is changing their mind or just reacting to the machine's changes."

How to Fix It (According to the Paper)

The paper suggests that to truly understand the human's internal learning, the experiment needs to be tweaked to break this confusion.

The Solution: Introduce "random noise" or "exogenous perturbations."
The Analogy: Imagine the magician is performing, but suddenly, a random gust of wind blows the props around (independent of the magician's actions). If the magician reacts to the wind differently than they react to the stage lights, you can finally tell what their true strategy is.

The paper suggests adding random, unpredictable inputs to the system so that the human's reaction can be separated from the computer's influence. Without this extra "noise," the human's internal adaptation remains hidden behind the computer's changes.

Summary

The Claim: The experimental design used to study co-adaptive neural interfaces cannot uniquely determine how the human is learning.
The Reason: Changes in the computer's behavior alter the environment so much that it looks like the human is changing, even if they aren't.
The Takeaway: We can predict how the team will perform, but we cannot uniquely predict or identify the human's internal adaptation mechanism without adding more random variables to the experiment.

Technical Summary: On the Identifiability of User Adaptation in Co-Adaptive Neural Interfaces

Problem Statement
This paper addresses a critical theoretical gap in the analysis of co-adaptive neural interfaces, specifically building upon the experimental and theoretical framework recently proposed by Madduri et al. [2]. In co-adaptive systems, a human user and a machine decoder jointly adapt within a closed loop. While Madduri et al. demonstrated that decoder learning rates and regularization influence both system performance and observed user behavior, this paper argues that the existing experimental design suffers from a fundamental identifiability problem.

The core issue is whether the sequence of estimated "encoders" (user models derived via linear regression) uniquely identifies the underlying latent user adaptation mechanism. The authors posit that while the current methodology successfully recovers the joint closed-loop dynamics, it fails to uniquely isolate the user-side adaptation policy from the effects of the evolving decoder and state distributions.

Methodology and Theoretical Framework
The paper formalizes the co-adaptive system using a compact empirical structure:

System Dynamics: Let $x_t$ represent task variables (e.g., target and cursor state), $u_t$ the user's motor output (EMG), and $y_t$ the controlled output. The decoder $D_t$ maps motor output to motion ( $y_t = D_t u_t$ ). The user implements an unknown control policy $U_t$ , abstractly defined as $u_t = U_t(x_t, \xi_t)$ , where $\xi_t$ represents internal noise or latent states.
Estimation Procedure: The empirical method estimates an "encoder" $\hat{E}_t$ over a finite window $W_t$ using linear regression: $\hat{E}_t = \arg \min_E \sum_{s \in W_t} \|u_s - E x_s\|^2$ . The decoder $D_t$ remains fixed within a window but updates across windows.
Identifiability Analysis: The authors define a mapping $M$ that takes the latent policy sequence $\{U_t\}$ , decoder trajectory $\{D_t\}$ , and windowing scheme to the sequence of estimated encoders $\{\hat{E}_t\}$ . The central question is whether $M$ is injective with respect to $\{U_t\}$ .

Key Results: Non-Identifiability Theorem
The paper presents Theorem 1, which establishes the non-identifiability of user adaptation under closed-loop regression. The theorem states that in a general closed-loop setting where the decoder is time-varying and state dynamics depend on the decoder, the mapping from user policies to estimated encoders is not injective.

The proof proceeds by construction, demonstrating two observationally equivalent systems:

System A (Fixed User): A user with a time-invariant policy $u_t = E^* x_t$ . Due to the time-varying decoder $D_t$ , the distribution of task states $x_t$ evolves.
System B (Time-Varying User): A user with a perturbed, time-varying policy $u_t = (E^* + \Delta_t) x_t$ .

The authors show that a sequence of perturbations $\{\Delta_t\}$ can be constructed such that the joint empirical moments within each window satisfy specific conditions (Equation 6). Under these conditions, the regression estimator yields the identical sequence $\hat{E}_t = E^*$ for both systems. Consequently, distinct user policy sequences (one fixed, one varying) generate identical encoder estimates. This proves that changes in $\hat{E}_t$ cannot be uniquely attributed to changes in the user's internal policy without additional assumptions or excitation conditions.

Discussion and Interpretation
The paper clarifies that the experimental results from Madduri et al. [2] correctly demonstrate that manipulations of the decoder trajectory affect the joint system behavior and the sequence of estimated encoders. However, due to the non-identifiability result, these observations do not uniquely determine the user's internal objective, learning dynamics, or cost functions.

Multiple distinct user models—including fixed policies reacting to shifting state distributions, delayed adaptation rules, or alternative regularization structures—can reproduce the same empirical signatures observed in $\hat{E}_t$ . Therefore, the framework identifies properties of the joint closed-loop system, but it does not uniquely identify the underlying user adaptation dynamics. Claims regarding the prediction or shaping of human behavior must be interpreted at the level of the joint system rather than the user in isolation.

Significance and Proposed Extensions
The authors emphasize that this limitation is not intrinsic to the co-adaptive framework itself but is a consequence of the current experimental design. The paper suggests that identifying user adaptation requires additional structure to break the symmetry between user policy changes and state distribution changes.

Proposed approaches to achieve identifiability include:

Exogenous Perturbations: Introducing independent noise (e.g., randomized probe inputs $\tilde{D}_t = D_t + \delta_t$ ) to generate persistent excitation independent of the closed-loop dynamics, thereby separating user response from state distribution effects.
Latent State Formulations: Adopting a state-space model (e.g., $z_{t+1} = f(z_t, x_t, D_t)$ ) with constraints on the dynamics functions, permitting identification under standard observability conditions.
Longer Horizon Experiments: Conducting experiments without frequent decoder reinitialization to allow for the separation of transient and persistent adaptation components.

In conclusion, the paper serves as a necessary clarification for interpreting co-adaptive neural interface studies, urging a shift from attributing encoder variations solely to user learning toward a more rigorous identification of the joint system dynamics.