Proximal State Nudging Overview

• PSN is a nudging framework that leverages local, data-informed adjustments to enhance state estimation and human-AI shared control.
• It integrates proximal operators and dynamic learnability estimators to balance immediate assisted performance with long-term skill retention.
• Empirical results show PSN achieves notable improvements in unassisted outcomes while reducing errors and collisions in safety-critical tasks.

Searching arXiv for the cited PSN and related nudging papers to ground the article in current literature. Proximal State Nudging (PSN) denotes a class of nudging schemes in which a state, estimate, or shared-control action is adjusted by a local, data-informed operation rather than by wholesale replacement. In the robust state-estimation literature, PSN refers to the template in which, at each step, the observer state is nudged toward measurement-consistency by a proximal operator associated with a convex, possibly non-smooth loss (Bako et al., 2024). In shared autonomy, PSN denotes an algorithm that jointly optimizes for skill development and task performance by nudging users toward states estimated to be most learnable, motivated by the problem of skill atrophy under AI assistance (Srivastava et al., 19 May 2026). Taken together, these usages suggest a common structure: controlled intervention that preserves short-term performance while shaping longer-term estimation or learning behavior.

1. Terminological scope and conceptual setting

In the shared-autonomy formulation, PSN is introduced to address skill atrophy, defined as the gradual decline of human capability under AI assistance, particularly in shared-control of semi-autonomous systems where operators may be unable to distinguish their own inputs from autonomous corrections (Srivastava et al., 19 May 2026). The algorithm is framed as a response to the trade-off between immediate assisted performance and long-term human skill retention.

In the state-estimation formulation, the same term is used for a general template in which an observer prediction is nudged toward measurement consistency through a proximal update. The setting is discrete-time, potentially nonlinear state estimation under impulsive measurement noise, where the measurement corruption may be sparse and arbitrarily large (Bako et al., 2024). In adjacent Bayesian filtering work, nudging is formalized as a likelihood-increasing operation on state-space models, yielding observation-dependent transition kernels and a new “nudged” state-space model with higher marginal likelihood for a fixed observation sequence under suitable conditions (Gonzalez et al., 2024).

These formulations share the term “nudging,” but they target different objects. One targets human-AI shared control and learnability; another targets observer robustness to attacks and outliers; a third targets robustness to model misspecification in Bayesian filtering. The commonality is methodological rather than domain-specific.

2. Shared-autonomy PSN: objective and mathematical formulation

The shared-autonomy version of PSN is motivated by Vygotsky’s Zone of Proximal Development (ZPD) and defines assistance in terms of “learnable” states: situations that are challenging, but within reach with help (Srivastava et al., 19 May 2026). The formal setting uses a state space $S$, an action space $A$, a student policy $\pi_{\mathrm{student}, t}(a \mid s)$, an expert policy $\pi_{\mathrm{expert}}(a \mid s)$, a shared policy $\pi_{\mathrm{shared}}(a \mid s)$, a task reward $R(\tau)$, and a learnability estimator $\phi_{\mathrm{zpd}, t}(s)$.

Standard shared autonomy is expressed as a convex combination of user and expert behavior,

$$\pi_{\mathrm{shared}}(a \mid s) = \alpha \pi_{\mathrm{expert}}(a \mid s) + (1-\alpha)\pi_{\mathrm{student}, t}(a \mid s),$$

where $\alpha \in [0,1]$ controls assistance level (Srivastava et al., 19 May 2026).

PSN replaces fixed, myopic blending with trajectory-level planning. For a beam of length-$T$ sampled trajectories $A$0 starting from the current state $A$1, it scores trajectories using

$A$2

where $A$3 and $A$4 are the states along $A$5 (Srivastava et al., 19 May 2026). The action executed at time $A$6 is the first action of a maximizing trajectory,

$A$7

The learnability term is estimated from reward differences between assisted and unassisted rollouts,

$A$8

The intended interpretation is explicit in the formulation: low $A$9 corresponds to states that are either already mastered or too difficult to be educationally productive, whereas high $\pi_{\mathrm{student}, t}(a \mid s)$0 indicates a state within the student’s ZPD (Srivastava et al., 19 May 2026).

3. Operational mechanism and adaptive assistance

The algorithmic workflow begins by initializing a student policy, an expert policy, a base shared-autonomy policy, datasets of student and shared experience, and a learnability estimator $\pi_{\mathrm{student}, t}(a \mid s)$1 (Srivastava et al., 19 May 2026). During each learning episode, the controller processes each encountered state $\pi_{\mathrm{student}, t}(a \mid s)$2 by adjusting $\pi_{\mathrm{student}, t}(a \mid s)$3 based on $\pi_{\mathrm{student}, t}(a \mid s)$4, sampling trajectory candidates via beam search, evaluating $\pi_{\mathrm{student}, t}(a \mid s)$5, selecting the highest-scoring trajectory, executing its first action, and recording experience. The learnability estimator is then updated using new simulated or real rollouts, and the student policy is updated from accumulated experiences.

A central mechanism is adaptive assistance. The shared-autonomy PSN formulation states that $\pi_{\mathrm{student}, t}(a \mid s)$6 is adjusted dynamically, with less assistance in states deemed highly learnable, so that the user acts more independently exactly where practice is expected to be most beneficial (Srivastava et al., 19 May 2026). This differs from fixed-weight blending, which prioritizes immediate correction but does not plan around downstream learning opportunities.

The planner is therefore not only reactive to current error. It evaluates future trajectories through a joint criterion that averages estimated learnability along the trajectory and combines it with cumulative reward. This makes the intervention proximal in the sense emphasized by the paper: assistance is local and controlled, but forward-looking rather than purely instantaneous.

4. Empirical evaluation in simulation and human studies

The simulated evaluation uses the classic LunarLander environment and compares PSN with Stochastic Q-Bumpers, Blended Shared Autonomy, Reddy et al. (2018), and No Assistance (Srivastava et al., 19 May 2026). The reported metrics include assisted mean reward and collision or crash count, together with unassisted reward and percentage of successful landings as measures of skill transfer or retention. The reported result is that PSN outperforms existing shared autonomy baselines in balancing student improvement in unassisted reward with overall shared performance, and does so without increasing crashes (Srivastava et al., 19 May 2026).

The human evaluation is conducted in the CARLA simulator across two driving tasks, High Performance Racing and Parallel Parking, with $\pi_{\mathrm{student}, t}(a \mid s)$7 (Srivastava et al., 19 May 2026). The paper describes these as, to the best of its knowledge, the first human subject studies of a planner incorporating learning-compatible shared autonomy. In High Performance Racing, the metrics include change from pre- to post-practice in lap time, Dynamic Time-Warp distance to expert, control jerk, and number of collisions during practice. In Parallel Parking, the metrics include parking time, collisions, and performance under both in-distribution and novel starts.

Across these driving tasks, PSN is reported to produce up to 7x larger gains in unassisted skill than standard blended shared autonomy, while incurring 50% fewer collisions than unassisted self-practice (Srivastava et al., 19 May 2026). The detailed task-level summary also reports that, in Parallel Parking, PSN led to greater improvement in unassisted skill, especially for novel test conditions, with over 7x larger gains than baselines, and reduced collisions by approximately 60% relative to self-practice, though blending remained even safer at the cost of learning (Srivastava et al., 19 May 2026). In High Performance Racing, both PSN and blending reduced collisions compared to self-practice, while PSN yielded greater gains in unassisted metrics than blended shared autonomy.

The comparative pattern is consistent across the reported experiments. Blended shared autonomy maximizes safety and performance in the moment but leaves users dependent; unassisted self-practice can improve skill but at substantial safety cost; PSN is presented as occupying the middle ground by preserving safety while improving underlying student ability (Srivastava et al., 19 May 2026).

5. Relation to proximal observers, Bayesian nudging, and particle methods

The proximal-observer formulation provides the clearest technical antecedent for the name. For the system

$\pi_{\mathrm{student}, t}(a \mid s)$8

with $\pi_{\mathrm{student}, t}(a \mid s)$9 including bounded dense noise and sparse impulsive noise, the observer first predicts

$\pi_{\mathrm{expert}}(a \mid s)$0

then updates by solving

$\pi_{\mathrm{expert}}(a \mid s)$1

where $\pi_{\mathrm{expert}}(a \mid s)$2 and $\pi_{\mathrm{expert}}(a \mid s)$3 is convex, possibly non-smooth (Bako et al., 2024). This is a proximal operator construction; if $\pi_{\mathrm{expert}}(a \mid s)$4, then $\pi_{\mathrm{expert}}(a \mid s)$5. The resulting implicit error dynamics involve subgradients,

$\pi_{\mathrm{expert}}(a \mid s)$6

In this setting, the role of non-smooth convex losses is central. The paper discusses $\pi_{\mathrm{expert}}(a \mid s)$7, Lasso-type mixed norms, and component-wise relaxed approaches, emphasizing saturation, thresholding, shrinkage, and implicit dead-zones as mechanisms that dampen the influence of outliers or impulsive measurement noise (Bako et al., 2024). Under suitable assumptions, the estimation error converges to zero in the noise-free setting, and under sparse impulsive measurement noise the estimation error remains bounded independently of the magnitude of the measurement noise. The paper also states that if attacks become infrequent, with large dwell-time between nonzero entries of $\pi_{\mathrm{expert}}(a \mid s)$8, the error can be made arbitrarily small (Bako et al., 2024).

Bayesian filtering work recasts nudging as an operation on the transition kernel itself. For a nudging map $\pi_{\mathrm{expert}}(a \mid s)$9 satisfying $\pi_{\mathrm{shared}}(a \mid s)$0, the nudged transition kernel is

$\pi_{\mathrm{shared}}(a \mid s)$1

and gradient-based nudging may take the form

$\pi_{\mathrm{shared}}(a \mid s)$2

for sufficiently small $\pi_{\mathrm{shared}}(a \mid s)$3 (Gonzalez et al., 2024). The main result states that, for suitable step sizes, there exists a nudged model with higher marginal likelihood than the original model for a fixed observation sequence. The same paper also warns that overshooting can create degenerate models and may increase estimation error.

Particle-filter nudging provides further neighboring usage. Residual nudging monitors the residual norm in observation space and, when it exceeds a threshold $\pi_{\mathrm{shared}}(a \mid s)$4, replaces the analysis mean by a convex combination of the particle-filter mean and an observation inversion, then shifts the ensemble mean accordingly without changing particle weights (Luo et al., 2013). Nudging the particle filter modifies a subset of particles via a map $\pi_{\mathrm{shared}}(a \mid s)$5 satisfying $\pi_{\mathrm{shared}}(a \mid s)$6, but deliberately does not correct the importance weights for the nudging. This induces a controlled bias, yet the paper proves the same $\pi_{\mathrm{shared}}(a \mid s)$7 asymptotic convergence rate as conventional particle methods when the number of nudged particles is $\pi_{\mathrm{shared}}(a \mid s)$8 (Akyıldız et al., 2017).

6. Interpretation, distinctions, and broader implications

A recurrent source of confusion is the assumption that PSN is equivalent to standard blended shared autonomy. The shared-autonomy formulation explicitly distinguishes itself from fixed-weight blending by planning over future trajectories and by adapting assistance according to estimated learnability (Srivastava et al., 19 May 2026). The point of intervention is not merely to improve the current action, but to place the user in states where practice is expected to be productive.

Another source of confusion is the assumption that all nudging methods are estimation techniques of the same type. The literature described here uses “nudging” at different algorithmic levels: proximal observers nudge state estimates toward measurement consistency; state-space-model nudging modifies transition kernels to increase likelihood; particle methods nudge particles or analysis means; shared-autonomy PSN nudges human-AI interaction toward states that balance safety and learning [(Bako et al., 2024); (Gonzalez et al., 2024); (Luo et al., 2013); (Akyıldız et al., 2017); (Srivastava et al., 19 May 2026)]. The term therefore names a family resemblance rather than a single canonical algorithm.

The broader implication drawn in the shared-autonomy paper is that long-term safety and performance in AI-assisted teams require explicit planning for skill retention, not only optimization of immediate assisted reward (Srivastava et al., 19 May 2026). The paper presents PSN as a foundation for assistive systems in domains where human agency matters, including medicine, aviation, and autonomous vehicles, and states that the core idea generalizes beyond control tasks to coding assistants, education, and complex task support (Srivastava et al., 19 May 2026). In the filtering and secure-estimation literature, the parallel implication is that robustness can be obtained not by ignoring anomalous data wholesale, but by using structured proximal or likelihood-increasing updates whose influence is bounded, thresholded, or selectively applied (Bako et al., 2024, Gonzalez et al., 2024).

Within this broader landscape, PSN is best understood not as a single model class, but as a design principle: intervention should be local, data-informed, and explicitly regularized by a competing objective—measurement consistency, marginal likelihood, residual control, or human learnability—rather than by unrestricted correction alone.