Distributionally Robust and Safe Imitation Learning

# Distributionally Robust and Safe Imitation Learning*
Source: [https://arxiv.org/html/2607.13436](https://arxiv.org/html/2607.13436)
Ahmed Aboudonia1and Naira Hovakimyan2\*This work was supported by the Swiss National Science Foundation under grant agreement P500PT\_\\\_230223, the Air Force Office of Scientific Research Grant \(AFOSR\) Grant AF FA9550\-25\-1\-0274, the National Aeronautics and Space Administration \(NASA\) under Grant 80NSSC22M0070, and the National Science Foundation \(NSF\) under Grants CMMI 2135925, CPS 2311085 and IIS 2331878\.1Ahmed Aboudonia is with the University of California Berkeley, Berkeley, CA 94720, USAaboudonia@berkeley\.edu2Naira Hovakimyan is with the University of Illinois Urbana\-Champaign, Urbana, IL 61801, USAnhovakim@illinois\.edu

###### Abstract

Imitation learning \(IL\) has achieved remarkable success in complex decision\-making tasks\. However, its performance is highly sensitive to distribution shifts, which can pose significant safety risks\. We propose a distributionally robust and safe IL framework that explicitly addresses both policy\-induced and uncertainty\-induced distribution shifts\. Our approach develops a unified framework leveraging Taylor Series Imitation Learning \(TaSIL\) to mitigate policy\-induced shifts and distributionally robust adaptive control to handle uncertainty\-induced shifts\. This architecture enables the formulation of an IL problem that optimizes performance under distributional uncertainty while systematically accounting for safety constraints\. We demonstrate the effectiveness of the proposed approach on an unmanned aerial vehicle \(UAV\) case study where the UAV performs a task in an uncertain environment while avoiding unsafe regions\.

## IINTRODUCTION

Learning\-from\-demonstrations has emerged as a compelling paradigm for enabling autonomous systems to acquire control policies from expert trajectories\[[15](https://arxiv.org/html/2607.13436#bib.bib1)\]\. In this context, imitation learning \(IL\) has shown significant success in learning policies that reproduce expert behavior\[[20](https://arxiv.org/html/2607.13436#bib.bib2)\]\. IL has been successfully applied in domains such as robotic manipulation\[[5](https://arxiv.org/html/2607.13436#bib.bib3)\], autonomous driving\[[11](https://arxiv.org/html/2607.13436#bib.bib4)\], and legged robots\[[12](https://arxiv.org/html/2607.13436#bib.bib5)\]\.

Behavioral cloning \(BC\) is considered the basic IL approach, in which expert demonstrations are collected as state\-action pairs\[[20](https://arxiv.org/html/2607.13436#bib.bib2)\]\. A supervised learning problem is then solved to learn a mapping from states to actions\. To reduce the need for expert action labeling, imitation from observation has emerged as a promising approach that enables learning from state\-only demonstrations\[[19](https://arxiv.org/html/2607.13436#bib.bib9)\]\.

While IL has shown promising results, it is limited by the intrinsic difficulty of distribution shift\. One source of this shift is policy\-induced distribution shift, where small deviations in the learned policy can accumulate over time, potentially causing cascading errors\[[20](https://arxiv.org/html/2607.13436#bib.bib2),[2](https://arxiv.org/html/2607.13436#bib.bib10)\]\. Another source is uncertainty\-induced distribution shift, in which minor variations in the environment, modeled as epistemic and aleatoric uncertainties, can similarly propagate through the system, leading to degraded performance\[[18](https://arxiv.org/html/2607.13436#bib.bib17),[6](https://arxiv.org/html/2607.13436#bib.bib19)\]\. As a result, both policy\-induced and uncertainty\-induced distribution shifts can propagate errors that pose significant safety risks\.

To mitigate the effects of policy\-induced distribution shifts, several IL variants have been developed\. For example, approaches such as DAgger \(Dataset Aggregation\)\[[16](https://arxiv.org/html/2607.13436#bib.bib11)\]and DART \(Disturbances for Augmenting robot trajectories\)\[[10](https://arxiv.org/html/2607.13436#bib.bib12)\]build upon BC and use supervised learning\. Other methods, such as inverse reinforcement learning\[[1](https://arxiv.org/html/2607.13436#bib.bib14)\]and generative adversarial imitation learning\[[9](https://arxiv.org/html/2607.13436#bib.bib15)\], follow different principles and have also demonstrated robustness in mitigating distribution shifts\[[20](https://arxiv.org/html/2607.13436#bib.bib2)\]\. More recently, Taylor series imitation learning \(TaSIL\) has been proposed to align expert and imitated policies along with their higher\-order derivatives\[[14](https://arxiv.org/html/2607.13436#bib.bib13)\]\.

To address uncertainty\-induced distribution shifts, several approaches have been proposed\. Robust imitation learning methods, for instance, have been developed to learn from multiple experts acting under diverse environment dynamics\[[4](https://arxiv.org/html/2607.13436#bib.bib16)\]\. Sampling augmentation methods leverage tube\-based robust MPC schemes within imitation learning frameworks such as BC and DAgger\[[18](https://arxiv.org/html/2607.13436#bib.bib17)\]\. Distributionally robust imitation learning approaches build upon BC and employ distributionally robust optimization within a Robust Markov Decision Process framework\[[13](https://arxiv.org/html/2607.13436#bib.bib18)\]\. Recently, a layered control architecture, termed DRIP, has been proposed\[[6](https://arxiv.org/html/2607.13436#bib.bib19)\]\. It integrates TaSIL to address the policy\-induced imitation gap andL1\\pazocal\{L\}\_\{1\}\-Distributionally Robust Adaptive Control \(L1\\pazocal\{L\}\_\{1\}\-DRAC\) to mitigate the uncertainty\-induced imitation gap\[[7](https://arxiv.org/html/2607.13436#bib.bib20)\]\.

In this paper, we propose a distributionally robust and safe IL framework that adopts a similar architecture to that of DRIP, where policy\-induced and uncertainty\-induced distribution shifts are handled using TaSIL andL1\\pazocal\{L\}\_\{1\}\-DRAC, respectively\. Unlike the standard empirical risk minimization over expert trajectories typically used in TaSIL, we reformulate the TaSIL problem to explicitly account for distributional robustness and safety requirements during training, inspired by the robustness certificates provided byL1\\pazocal\{L\}\_\{1\}\-DRAC\.

To this end, we first generalize the TaSIL objective to a distributionally robust formulation that optimizes performance over an ambiguity set of state distributions, thereby accounting explicitly for distribution shifts\. Additionally, we incorporate a safety term into the TaSIL objective, ensuring that the resulting policy accounts for safety requirements\. The resulting framework is designed to be robust to both policy\-induced and uncertainty\-induced distribution shifts, optimize performance under distributional uncertainty, and account for safety considerations\. Finally, we demonstrate the efficacy of the proposed framework through extensive simulations on an unmanned aerial vehicle \(UAV\) operating in uncertain environments while avoiding unsafe regions\.

The remainder of this paper is organized as follows\. Section II presents the problem setup and provides a brief overview of IL, with a focus on TaSIL\. Section III introduces our proposed distributionally robust and safe IL framework\. Specifically, we discussL1\\pazocal\{L\}\_\{1\}\-DRAC and demonstrate how it can be leveraged to reformulate a distributionally robust TaSIL problem, as well as to account for safety requirements\. Section IV evaluates the performance of the proposed framework through extensive simulations on a UAV case study\. Concluding remarks are provided in Section V\.

## IIProblem Setup

We consider the uncertain dynamical system whose dynamics are given by the Itôprocess:

d​X¯t=\[Aμ​X¯t\+Bc​\(U¯t\+Hμ​\(X¯t\)\)\]​d​t\+\[Aσ\+Bc​Hσ​\(X¯t\)\]​d​Wt,d\\bar\{X\}\_\{t\}=\\left\[A\_\{\\mu\}\\bar\{X\}\_\{t\}\+B\_\{c\}\\left\(\\bar\{U\}\_\{t\}\+H\_\{\\mu\}\(\\bar\{X\}\_\{t\}\)\\right\)\\right\]dt\+\\left\[A\_\{\\sigma\}\+B\_\{c\}H\_\{\\sigma\}\(\\bar\{X\}\_\{t\}\)\\right\]dW\_\{t\},\(1\)whereX¯t∈ℝn\\bar\{X\}\_\{t\}\\in\\mathbb\{R\}^\{n\}andU¯t∈ℝm\\bar\{U\}\_\{t\}\\in\\mathbb\{R\}^\{m\}are the system state and control input vectors, respectively, andWt∈ℝnwW\_\{t\}\\in\\mathbb\{R\}^\{n\_\{w\}\}is a Brownian motion defined on the complete probability space\(Ω,F,ℙ\)\\left\(\\Omega,\\pazocal\{F\},\\mathbb\{P\}\\right\)\. The matricesAμ∈ℝn×nA\_\{\\mu\}\\in\\mathbb\{R\}^\{n\\times n\},Aσ∈ℝn×nwA\_\{\\sigma\}\\in\\mathbb\{R\}^\{n\\times n\_\{w\}\}andBc∈ℝn×mB\_\{c\}\\in\\mathbb\{R\}^\{n\\times m\}encode the known parts of the system dynamics\. The functionsHμ:ℝn→ℝmH\_\{\\mu\}:\\mathbb\{R\}^\{n\}\\rightarrow\\mathbb\{R\}^\{m\}andHσ:ℝn→ℝm×nwH\_\{\\sigma\}:\\mathbb\{R\}^\{n\}\\rightarrow\\mathbb\{R\}^\{m\\times n\_\{w\}\}are continuously differentiable maps describing the drift and diffusion uncertainties, respectively\. Throughout the sequel, we use capital letters \(e\.g\.,XtX\_\{t\},UtU\_\{t\}\) to denote continuous\-time variables, and lowercase letters \(e\.g\.,xkx\_\{k\},uku\_\{k\}\) to denote their discrete\-time counterparts\. We refer toHμ​\(X¯t\)H\_\{\\mu\}\(\\bar\{X\}\_\{t\}\)andHσ​\(X¯t\)H\_\{\\sigma\}\(\\bar\{X\}\_\{t\}\)as epistemic uncertainties, and tod​WtdW\_\{t\}as aleatoric uncertainty\.

For completeness, we introduce the following two definitions\. A continuous functionγ​\(x\)\\gamma\(x\)is of classK\\pazocal\{K\}if it is strictly increasing inxx, and satisfiesγ​\(0\)=0\\gamma\(0\)=0\. A continuous functionβ​\(x,t\)\\beta\(x,t\)is of classK​L\\pazocal\{KL\}if, for each fixedtt,β​\(x,t\)\\beta\(x,t\)is of classK\\pazocal\{K\}, and for each fixedxx,β​\(x,t\)\\beta\(x,t\)is decreasing with respect tottand satisfieslimt→∞β​\(x,t\)=0\\lim\_\{t\\rightarrow\\infty\}\\beta\(x,t\)=0\.

### II\-AImitation Learning

We considernnrollouts of lengthLdL\_\{d\}generated by an expert policy under nominal conditions\. For a sufficiently small sampling timeΔ​T\\Delta T, these rollouts can be interpreted as state\-input trajectories of the discretized nominal deterministic dynamics corresponding to \([1](https://arxiv.org/html/2607.13436#S2.E1)\), given by:

xk\+1=A​xk\+B​uk,x\_\{k\+1\}=Ax\_\{k\}\+Bu\_\{k\},\(2\)wherek∈ℕ0k\\in\\mathbb\{N\}\_\{0\}denotes the discrete time index,xk∈ℝnx\_\{k\}\\in\\mathbb\{R\}^\{n\}is the state vector,uk∈ℝmu\_\{k\}\\in\\mathbb\{R\}^\{m\}is the input vector,InI\_\{n\}denotes an identity matrix of sizenn,A=In\+Aμ​Δ​TA=I\_\{n\}\+A\_\{\\mu\}\\Delta TandB=Bc​Δ​TB=B\_\{c\}\\Delta Tdefine the discrete\-time system matrices\. Each rollout is initialized from an independently and identically distributed initial conditionξ\\xidrawn from a distributionD\\pazocal\{D\}with a compact supportX\\pazocal\{X\}\.

Given a policyπ​\(xk\)\\pi\(x\_\{k\}\), we denote the solution of \([2](https://arxiv.org/html/2607.13436#S2.E2)\) at time indexkkunder the inputuk=π​\(xk\)u\_\{k\}=\\pi\(x\_\{k\}\)starting fromξ\\xibyxk​\(ξ,π\)x\_\{k\}\(\\xi,\\pi\)\. We denote the expert policy byπ∗​\(xk\)\\pi\_\{\*\}\(x\_\{k\}\), and the discrepancy between any policyπ​\(xk\)\\pi\(x\_\{k\}\)and the expert policyπ∗​\(xk\)\\pi\_\{\*\}\(x\_\{k\}\)on a trajectory generated by the latter starting fromξ\\xibyΔkπ∗​\(ξ;π\)=π​\(xk​\(ξ,π∗\)\)−π∗​\(xk​\(ξ,π∗\)\)\\Delta\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi;\\pi\)=\\pi\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\-\\pi\_\{\*\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\.

The main objective of imitation learning \(IL\) is to learn a policy that induces a closed\-loop behavior similar to that of the expert policy, as quantified by the expected imitation gap,𝔼ξ​max1≤k≤Ld⁡‖xk​\(ξ,π\)−xk​\(ξ,π∗\)‖\\mathbb\{E\}\_\{\\xi\}\\operatorname\{max\}\_\{1\\leq k\\leq L\_\{d\}\}\\\|x\_\{k\}\(\\xi,\\pi\)\-x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\\\|\. The standard IL approach, known as BC, achieves the aforementioned objective by formulating a supervised learning problem that minimizes the empirical risk as follows:

π^∈argminπ∈Π​∑i=1nh​\(\{Δkπ∗​\(ξi;π\)\}k=0Ld\),\\hat\{\\pi\}\\in\\operatorname\{argmin\}\_\{\\pi\\in\\Pi\}\\sum\\nolimits\_\{i=1\}^\{n\}h\(\\\{\\Delta\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi\_\{i\};\\pi\)\\\}\_\{k=0\}^\{L\_\{d\}\}\),whereΠ\\Piis a preselected policy class, andh​\(⋅\)h\(\\cdot\)is a loss function designed to penalize the discrepanciesΔkπ∗​\(ξi;π\)\\Delta\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi\_\{i\};\\pi\)along the expert trajectories\.

### II\-BTaylor Series Imitation Learning

Despite the simplicity of BC, the learned policyπ^​\(xk\)\\hat\{\\pi\}\(x\_\{k\}\)can experience distribution shift due to compounding errors\. To this end, TaSIL replaces the standard BC loss function with

lpπ∗​\(ξ,π\)=∑j=0pmax0≤k≤Ld−1⁡‖∂xjΔkπ∗​\(ξ,π\)‖,l\_\{p\}^\{\\pi\_\{\*\}\}\(\\xi,\\pi\)=\\sum\\nolimits\_\{j=0\}^\{p\}\\operatorname\{max\}\_\{0\\leq k\\leq L\_\{d\}\-1\}\\\|\\partial\_\{x\}^\{j\}\\Delta\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi,\\pi\)\\\|,\(3\)where∂xj\\partial\_\{x\}^\{j\}denotes thejj\-th order partial derivative with respect toxxandpprefers to the highest order derivative considered in the loss function\. Note that any surrogate loss that upper bounds the supremum loss \([3](https://arxiv.org/html/2607.13436#S2.E3)\) can be used\. Thus, the empirical risk minimization problem becomes

π^p∈argminπ∈Π​∑i=1nlpπ∗​\(ξi,π\)\.\\hat\{\\pi\}\_\{p\}\\in\\operatorname\{argmin\}\_\{\\pi\\in\\Pi\}\\sum\\nolimits\_\{i=1\}^\{n\}l\_\{p\}^\{\\pi\_\{\*\}\}\(\\xi\_\{i\},\\pi\)\.The main idea behind TaSIL\[[14](https://arxiv.org/html/2607.13436#bib.bib13)\]is to minimize not only the discrepancy between the learned and expert policies, but also between their higher\-order derivatives up to orderpp, thereby mitigating policy\-induced distribution shifts as follows\.

Given a policyπ​\(xk\)\\pi\(x\_\{k\}\)and an additive input perturbationΔk\\Delta\_\{k\}, let the perturbed closed\-loop dynamics of the nominal determinstic system \([2](https://arxiv.org/html/2607.13436#S2.E2)\) be:

xk\+1=A​xk\+B​\(π​\(xk\)\+Δk\),x\_\{k\+1\}=Ax\_\{k\}\+B\(\\pi\(x\_\{k\}\)\+\\Delta\_\{k\}\),\(4\)and its solution at time indexkkbe denoted byxkπ​\(ξ,\{Δs\}s=0k−1\)x\_\{k\}^\{\\pi\}\(\\xi,\\\{\\Delta\_\{s\}\\\}\_\{s=0\}^\{k\-1\}\)\.

###### Assumption II\.1

The perturbed closed\-loop dynamics \([4](https://arxiv.org/html/2607.13436#S2.E4)\) under the expert policyπ∗​\(xk\)\\pi\_\{\*\}\(x\_\{k\}\)isη\\eta\-locallyδ\\delta\-input\-to\-state \(ISS\) stable, that is, for allΔk\\Delta\_\{k\}satisfyingsupk∈ℕ0⁡‖Δk‖≤η\\operatorname\{sup\}\_\{k\\in\\mathbb\{N\}\_\{0\}\}\\\|\\Delta\_\{k\}\\\|\\leq\\eta, there exists a classK​L\\pazocal\{KL\}functionβ​\(⋅,⋅\)\\beta\(\\cdot,\\cdot\)and a classK\\pazocal\{K\}functionγ​\(⋅\)\\gamma\(\\cdot\)such that for all initial conditionsξ1\\xi\_\{1\},ξ2∈X\\xi\_\{2\}\\in\\pazocal\{X\}, perturbation sequences\{Δk\}k≥0\\\{\\Delta\_\{k\}\\\}\_\{k\\geq 0\}andk∈ℕk\\in\\mathbb\{N\}, we have

‖xkπ∗​\(ξ1,\{Δs\}s=0k−1\)−xk​\(ξ2,π∗\)‖≤β​\(‖ξ1−ξ2‖,k\)\+γ​\(max0≤i≤k−1⁡‖Δi‖\)\.\\\|x\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi\_\{1\},\\\{\\Delta\_\{s\}\\\}\_\{s=0\}^\{k\-1\}\)\-x\_\{k\}\(\\xi\_\{2\},\\pi\_\{\*\}\)\\\|\\leq\\beta\(\\\|\\xi\_\{1\}\-\\xi\_\{2\}\\\|,k\)\\\\ \+\\gamma\(\\operatorname\{max\}\_\{0\\leq i\\leq k\-1\}\\\|\\Delta\_\{i\}\\\|\)\.\(5\)

In this setting, a system\-theoretic analysis can be sought where the learned policyπ^p​\(xk\)\\hat\{\\pi\}\_\{p\}\(x\_\{k\}\)is viewed as a perturbed version of the ISS expert policyπ∗​\(xk\)\\pi\_\{\*\}\(x\_\{k\}\), and the discrepancyΔkπ^p​\(ξ,π^p\)=π^p​\(xk​\(ξ,π^p\)\)−π∗​\(xk​\(ξ,π^p\)\)\\Delta\_\{k\}^\{\\hat\{\\pi\}\_\{p\}\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)=\\hat\{\\pi\}\_\{p\}\(x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)\)\-\\pi\_\{\*\}\(x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)\)acts as the perturbation\. In this case, the imitation gap defined asΓT​\(ξ,π^p\)=max1≤k≤Ld⁡‖xk​\(ξ,π^p\)−xk​\(ξ,π∗\)‖\\Gamma\_\{T\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)=\\operatorname\{max\}\_\{1\\leq k\\leq L\_\{d\}\}\\\|x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)\-x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\\\|is expressed asΓT​\(ξ,π^p\)=max1≤k≤Ld⁡‖xkπ∗​\(ξ,\{Δsπ^p​\(ξ,π^p\)\}s=0k−1\)−xk​\(ξ,π∗\)‖\\Gamma\_\{T\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)=\\operatorname\{max\}\_\{1\\leq k\\leq L\_\{d\}\}\\\|x\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi,\\\{\\Delta\_\{s\}^\{\\hat\{\\pi\}\_\{p\}\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)\\\}\_\{s=0\}^\{k\-1\}\)\-x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\\\|\. Under mild regularity assumptions \(see\[[14](https://arxiv.org/html/2607.13436#bib.bib13)\]for more details\), and for sufficiently large number of sampled trajectories, there exists a constantρT≥0\\rho\_\{T\}\\geq 0such that the imitation gap evaluated onξ∼D\\xi\\sim\\pazocal\{D\}satisfiesΓT​\(ξ,π^p\)≤ρT\\Gamma\_\{T\}\(\\xi,\\hat\{\\pi\}\_\{p\}\)\\leq\\rho\_\{T\}with high probability\.

Since TaSIL aims to reduce policy\-induced distribution shift, it performs well in mimicking the expert behavior on the nominal system used to collect the demonstrations\. However, it does not account for uncertainty\-induced distribution shifts arising from epistemic and aleatoric uncertainties in the true system\. Consequently, its performance is only optimized under nominal conditions\. Furthermore, it does not incorporate safety requirements during training\. In the sequel, we address these limitations by developing a distributionally robust and safe TaSIL framework\.

## IIIProposed Approach

In this section, we develop a distributionally robust and safe TaSIL framework that, \(I\) achieves robustness to uncertainty\-induced distribution shifts, \(II\) optimizes performance under distributional uncertainty, and \(III\) accounts for safety requirements during training\.

### III\-ARobustness

We employL1\\pazocal\{L\}\_\{1\}\-DRAC to achieve robustness of the true system \([1](https://arxiv.org/html/2607.13436#S2.E1)\) in the presence of uncertainties\. To this end, we define the nominal stochastic system:

d​Xt∗=\[Aμ​Xt∗\+Bc​Ut∗\]​d​t\+Aσ​d​Wt∗,dX^\{\*\}\_\{t\}=\\left\[A\_\{\\mu\}X^\{\*\}\_\{t\}\+B\_\{c\}U^\{\*\}\_\{t\}\\right\]dt\+A\_\{\\sigma\}dW^\{\*\}\_\{t\},\(6\)whereX∗∈ℝnX^\{\*\}\\in\\mathbb\{R\}^\{n\}andU∗∈ℝmU^\{\*\}\\in\\mathbb\{R\}^\{m\}represent the nominal state and input vectors, respectively, andWt∗∈ℝnwW^\{\*\}\_\{t\}\\in\\mathbb\{R\}^\{n\_\{w\}\}is another Brownian motion defined on\(Ω,F,ℙ\)\\left\(\\Omega,\\pazocal\{F\},\\mathbb\{P\}\\right\), and is independent ofWtW\_\{t\}\. Note that the nominal system \([6](https://arxiv.org/html/2607.13436#S3.E6)\) corresponds to the true system \([1](https://arxiv.org/html/2607.13436#S2.E1)\) withHμH\_\{\\mu\}andHσH\_\{\\sigma\}in \([1](https://arxiv.org/html/2607.13436#S2.E1)\) set to zero\. We also defineBu∈ℝn×\(n−m\)B\_\{u\}\\in\\mathbb\{R\}^\{n\\times\(n\-m\)\}such thatB¯=\[Bc​Bu\]\\bar\{B\}=\[B\_\{c\}\\ B\_\{u\}\]has full rank\.

By settingUt⋆U\_\{t\}^\{\\star\}as the control input obtained via a zero\-order hold of the proposed TaSIL policyπ^ps\\hat\{\\pi\}\_\{p\}^\{s\}\(derived in subsequent sections and inspired by the standard TaSIL policyπ^p\\hat\{\\pi\}\_\{p\}\), we consider the input to the true system \([1](https://arxiv.org/html/2607.13436#S2.E1)\) of the form:

U¯t=Ut∗\+UL1,t,\\bar\{U\}\_\{t\}=U^\{\*\}\_\{t\}\+U\_\{\\pazocal\{L\}\_\{1\},t\},\(7\)whereUL1,t∈ℝmU\_\{\\pazocal\{L\}\_\{1\},t\}\\in\\mathbb\{R\}^\{m\}is theL1\\pazocal\{L\}\_\{1\}\-DRAC input\[[7](https://arxiv.org/html/2607.13436#bib.bib20)\]\. This input is defined as the output of the low pass filter:

UL1,t=−ω​∫0te−ω​\(t−ν\)​Λ^m,ν​𝑑ν,\\displaystyle U\_\{\\pazocal\{L\}\_\{1\},t\}=\-\\omega\\int\_\{0\}^\{t\}e^\{\-\\omega\(t\-\\nu\)\}\\hat\{\\Lambda\}\_\{m,\\nu\}d\\nu,whereω∈ℝ\>0\\omega\\in\\mathbb\{R\}\_\{\>0\}is the filter bandwidth,0p×q0\_\{p\\times q\}denotes a zero matrix of sizep×qp\\times q, andΛ^m,t=\[Im0m×\(n−m\)\]​B¯−1​Λ^t\\hat\{\\Lambda\}\_\{m,t\}=\\begin\{bmatrix\}I\_\{m\}&0\_\{m\\times\(n\-m\)\}\\end\{bmatrix\}\\bar\{B\}^\{\-1\}\\hat\{\\Lambda\}\_\{t\}\. The variableΛ^t∈ℝm\\hat\{\\Lambda\}\_\{t\}\\in\\mathbb\{R\}^\{m\}is obtained via the adaptation law:

Λ^t=0n​1\[0,Ts\)\+λs​\(1−eλs​Ts\)−1​eλs​Ts​∑i=1⌊tTs⌋X~i​Ts​1\[i​Ts,\(i\+1\)​Ts\),\\hat\{\\Lambda\}\_\{t\}=0\_\{n\}\{1\}\_\{\[0,T\_\{s\}\)\}\+\\textstyle\\lambda\_\{s\}\(1\-e^\{\\lambda\_\{s\}T\_\{s\}\}\)^\{\-1\}e^\{\\lambda\_\{s\}T\_\{s\}\}\\sum\_\{i=1\}^\{\\lfloor\\frac\{t\}\{T\_\{s\}\}\\rfloor\}\\tilde\{X\}\_\{iT\_\{s\}\}\{1\}\_\{\[iT\_\{s\},\(i\+1\)T\_\{s\}\)\},withX~i​Ts=X^i​Ts−X¯i​Ts\\tilde\{X\}\_\{iT\_\{s\}\}=\\hat\{X\}\_\{iT\_\{s\}\}\-\\bar\{X\}\_\{iT\_\{s\}\},λs∈ℝ\>0\\lambda\_\{s\}\\in\\mathbb\{R\}\_\{\>0\},Ts∈ℝ\>0T\_\{s\}\\in\\mathbb\{R\}\_\{\>0\}, and1\[t1,t2\)1\_\{\[t\_\{1\},t\_\{2\}\)\}being11in the time interval\[t1,t2\)\[t\_\{1\},t\_\{2\}\)and 0 otherwise\. For an initial conditionX0X\_\{0\},X^t\\hat\{X\}\_\{t\}is the solution of the process predictor:

X^t=X0\+∫0t\(−λs​In​X~ν\+Aμ​X¯ν\+Bc​\(Uν∗\+UL1,ν\)\+Λ^ν\)​𝑑ν,\\hat\{X\}\_\{t\}=X\_\{0\}\\textstyle\+\\int\_\{0\}^\{t\}\\left\(\-\\lambda\_\{s\}\{I\}\_\{n\}\\tilde\{X\}\_\{\\nu\}\+A\_\{\\mu\}\\bar\{X\}\_\{\\nu\}\+B\_\{c\}\(U\_\{\\nu\}^\{\*\}\+U\_\{\\pazocal\{L\}\_\{1\},\\nu\}\)\+\\hat\{\\Lambda\}\_\{\\nu\}\\right\)d\\nu,whereX~t=X^t−X¯t\\tilde\{X\}\_\{t\}=\\hat\{X\}\_\{t\}\-\\bar\{X\}\_\{t\}\. We refer to\{ω,Ts,λs\}\\\{\\omega,T\_\{s\},\\lambda\_\{s\}\\\}as theL1\\pazocal\{L\}\_\{1\}\-DRAC parameters, and denote the proposedL1\\pazocal\{L\}\_\{1\}\-DRAC\-augmented TaSIL policy byπ^pl\\hat\{\\pi\}\_\{p\}^\{l\}\. Note that theL1\\pazocal\{L\}\_\{1\}\-DRAC augmentation is independent of the choice of the orderpp\.

Notice that, given a policyπ\\piand its correspondingL1\\pazocal\{L\}\_\{1\}\-DRAC\-augmented policyπl\\pi^\{l\}, it is shown in\[[7](https://arxiv.org/html/2607.13436#bib.bib20)\]that there exists a scalarρ≥0\\rho\\geq 0such that𝕎2​\(𝕏t​\(ξ,πl\),𝕏t∗​\(ξ,π\)\)≤ρ\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{t\}\(\\xi,\\pi^\{l\}\),\\mathbb\{X\}\_\{t\}^\{\*\}\(\\xi,\\pi\)\)\\leq\\rhofor allt∈\[0,Th\]t\\in\[0,T\_\{h\}\], under mild regularity assumptions on the nominal dynamics and the uncertain terms in the true system, and with an appropriate choice of theL1\\pazocal\{L\}\_\{1\}\-DRAC parameters\. Here,ThT\_\{h\}denotes the time horizon,𝕎2\\mathbb\{W\}\_\{2\}is the 2\-Wasserstein distance,𝕏t​\(ξ,πl\)\\mathbb\{X\}\_\{t\}\(\\xi,\\pi^\{l\}\)denotes the distribution of the true system \([1](https://arxiv.org/html/2607.13436#S2.E1)\) underπl\\pi^\{l\}starting fromξ\\xi, and𝕏t∗​\(ξ,π\)\\mathbb\{X\}\_\{t\}^\{\*\}\(\\xi,\\pi\)denotes the distribution of the nominal system \([6](https://arxiv.org/html/2607.13436#S3.E6)\) underπ\\pistarting fromξ\\xi\. The specific case in whichπ\\piis the standard TaSIL policy is addressed in\[[6](https://arxiv.org/html/2607.13436#bib.bib19)\]\. In the interest of space, we refer the reader to\[[6](https://arxiv.org/html/2607.13436#bib.bib19),[7](https://arxiv.org/html/2607.13436#bib.bib20)\]for further details on the considered assumptions and their implications\.

In the following subsections, we show howL1\\pazocal\{L\}\_\{1\}\-DRAC can inspire the offline training of TaSIL, by optimizing performance under distributional uncertainty while incorporating safety requirements\.

### III\-BPerformance

We propose a TaSIL formulation that optimizes performance under distributional uncertainty\. To this end, we use the Euler\-Marumaya scheme to reach a discretized version of the true system \([1](https://arxiv.org/html/2607.13436#S2.E1)\) given by

x¯k\+1=A​x¯k\+B​\(u¯k\+Hμ​\(x¯k\)\)\+\(Aσ\+Bc​Hσ​\(x¯k\)\)​Δ​wk,\\bar\{x\}\_\{k\+1\}=A\\bar\{x\}\_\{k\}\+B\(\\bar\{u\}\_\{k\}\+H\_\{\\mu\}\(\\bar\{x\}\_\{k\}\)\)\+\(A\_\{\\sigma\}\+B\_\{c\}H\_\{\\sigma\}\(\\bar\{x\}\_\{k\}\)\)\\Delta w\_\{k\},\(8\)whereΔ​wk​∼i\.i\.d\.​N​\(0nw,Δ​T​𝕀nw\)\\Delta w\_\{k\}\\overset\{\\text\{i\.i\.d\.\}\}\{\\sim\}\\pazocal\{N\}\(0\_\{n\_\{w\}\},\\Delta T\\mathbb\{I\}\_\{n\_\{w\}\}\), and a discretized version of the nominal system \([6](https://arxiv.org/html/2607.13436#S3.E6)\) given by

xk\+1∗=A​xk∗\+B​uk∗\+Aσ​Δ​wk∗,\\displaystyle x\_\{k\+1\}^\{\*\}=Ax\_\{k\}^\{\*\}\+Bu\_\{k\}^\{\*\}\+A\_\{\\sigma\}\\Delta w\_\{k\}^\{\*\},\(9\)whereΔ​wk∗​∼i\.i\.d\.​N​\(0nw,Δ​T​𝕀nw\)\\Delta w\_\{k\}^\{\*\}\\overset\{\\text\{i\.i\.d\.\}\}\{\\sim\}\\pazocal\{N\}\(0\_\{n\_\{w\}\},\\Delta T\\mathbb\{I\}\_\{n\_\{w\}\}\)\.

Assume that there exists a scalarρ≥0\\rho\\geq 0such that𝕎2​\(𝕏t​\(ξ,π^pl\),𝕏t∗​\(ξ,π^ps\)\)≤ρ​for all​t∈\[0,Th\]\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{t\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{l\}\),\\mathbb\{X\}\_\{t\}^\{\*\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)\)\\leq\\rho\\text\{ for all \}t\\in\[0,T\_\{h\}\]\. It should be noted that this bound may not hold exactly for the corresponding discrete\-time systems\. As a result, it is necessary to consider the errors introduced by temporal discretization\. As mentioned in\[[8](https://arxiv.org/html/2607.13436#bib.bib21)\], one can inflate the constantρ\\rhoto account for the discretization error\. For simplicity, we maintain the constantρ\\rhoand assume that the Wasserstein distance between𝕏k​\(ξ,π^pl\)\\mathbb\{X\}\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{l\}\)and𝕏k∗​\(ξ,π^ps\)\\mathbb\{X\}\_\{k\}^\{\*\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)satisfies𝕎2​\(𝕏k​\(ξ,π^pl\),𝕏k∗​\(ξ,π^ps\)\)≤ρ\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{l\}\),\\mathbb\{X\}\_\{k\}^\{\*\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)\)\\leq\\rhowhere, for an initial conditionξ\\xi,𝕏k​\(ξ,π^pl\)\\mathbb\{X\}\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{l\}\)and𝕏k∗​\(ξ,π^ps\)\\mathbb\{X\}\_\{k\}^\{\*\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)are the distributions ofx¯k\\bar\{x\}\_\{k\}underπ^pl\\hat\{\\pi\}\_\{p\}^\{l\}andxk∗x\_\{k\}^\{\*\}underπ^ps\\hat\{\\pi\}\_\{p\}^\{s\}, respectively\.

###### Proposition III\.1

Letfc​l​\(x\)=A​x\+B​π^ps​\(x\)f\_\{cl\}\(x\)=Ax\+B\\hat\{\\pi\}\_\{p\}^\{s\}\(x\)be Lipschitz continuous with constantLL\. Consider the systems in \([2](https://arxiv.org/html/2607.13436#S2.E2)\) and \([9](https://arxiv.org/html/2607.13436#S3.E9)\), both initialized atξ\\xiand evolving underπ^ps\\hat\{\\pi\}\_\{p\}^\{s\}\. Letδxk​\(ξ,π^ps\)\\delta\_\{x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)\}be the Dirac distribution of the nominal deterministic state in \([2](https://arxiv.org/html/2607.13436#S2.E2)\) at timekk\. Then, there exists a constantρa\\rho\_\{a\}such that𝕎2​\(𝕏k∗​\(ξ,π^ps\),δxk​\(ξ,π^ps\)\)≤ρa\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{k\}^\{\*\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\),\\delta\_\{x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)\}\)\\leq\\rho\_\{a\}for allk∈\{0,…,⌊Th/Δ​T⌋\}k\\in\\\{0,\\ldots,\\lfloor T\_\{h\}/\\Delta T\\rfloor\\\}\.

###### Proof:

Define the errorek=xk∗−xke\_\{k\}=x\_\{k\}^\{\*\}\-x\_\{k\}at timekk\. The error dynamics becomeek\+1=fc​l​\(xk∗\)−fc​l​\(xk\)\+Aσ​Δ​wk∗e\_\{k\+1\}=f\_\{cl\}\(x\_\{k\}^\{\*\}\)\-f\_\{cl\}\(x\_\{k\}\)\+A\_\{\\sigma\}\\Delta w\_\{k\}^\{\*\}\. Sincefc​lf\_\{cl\}is Lipschitz with constantLL, we have‖ek\+1‖≤L​‖ek‖\+‖Aσ​Δ​wk∗‖\\\|e\_\{k\+1\}\\\|\\leq L\\\|e\_\{k\}\\\|\+\\\|A\_\{\\sigma\}\\Delta w\_\{k\}^\{\*\}\\\|\. SinceΔ​wk∗\\Delta w\_\{k\}^\{\*\}is independent ofeke\_\{k\}and has zero mean, we reach𝔼​\[‖ek\+1‖2\]≤L2​𝔼​\[‖ek‖2\]\+tr⁡\(Aσ​Aσ⊤\)​Δ​T\\mathbb\{E\}\[\\\|e\_\{k\+1\}\\\|^\{2\}\]\\leq L^\{2\}\\mathbb\{E\}\[\\\|e\_\{k\}\\\|^\{2\}\]\+\\operatorname\{tr\}\(A\_\{\\sigma\}A\_\{\\sigma\}^\{\\top\}\)\\Delta T\. Withe0=0e\_\{0\}=0, this recursion implies thateke\_\{k\}is finite for any finitekk\. Due to the Dirac distributionδxk​\(ξ,π^ps\)\\delta\_\{x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)\}, the 2 Wasserstein distance𝕎2​\(𝕏k∗​\(ξ,π^ps\),δxk​\(ξ,π^ps\)\)2=𝔼​\[‖ek‖2\]≤ρa\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{k\}^\{\*\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\),\\delta\_\{x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)\}\)^\{2\}=\\mathbb\{E\}\[\\\|e\_\{k\}\\\|^\{2\}\]\\leq\\rho\_\{a\}whereρa=maxk∈\{0,…,⌊Th/Δ​T⌋\}⁡𝔼​\[‖ek‖2\]\\rho\_\{a\}=\\max\_\{k\\in\\\{0,\\ldots,\\lfloor T\_\{h\}/\\Delta T\\rfloor\\\}\}\\mathbb\{E\}\[\\\|e\_\{k\}\\\|^\{2\}\]\. ∎

The Lipschitz condition in Proposition[III\.1](https://arxiv.org/html/2607.13436#S3.Thmprop1)is mild, as many neural networks are Lipschitz under standard weight and activation assumptions\. By definingρL=ρ\+ρa\\rho\_\{L\}=\\rho\+\\rho\_\{a\}and using the triangle inequality, it is deduced that𝕎2​\(𝕏k​\(ξ,π^pl\),δxk​\(ξ,π^ps\)\)≤ρL\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{l\}\),\\delta\_\{x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)\}\)\\leq\\rho\_\{L\}\. To optimize performance in the presence of distributional uncertainty, we propose the risk minimization problem:

π^pr∈argminπ∈Π⁡𝔼ξ∼D​\[lp,L1π∗,r​\(ξ,π\)\],\\hat\{\\pi\}\_\{p\}^\{r\}\\in\\operatorname\{argmin\}\_\{\\pi\\in\\Pi\}\\mathbb\{E\}\_\{\\xi\\sim\\pazocal\{D\}\}\\left\[l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},r\}\(\\xi,\\pi\)\\right\],where the loss functionlp,L1π∗,r​\(ξ,π\)l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},r\}\(\\xi,\\pi\)is given by

lp,L1π∗,r​\(ξ,π\)\\displaystyle l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},r\}\(\\xi,\\pi\)=∑j=0pmax0≤k≤Ld−1​supνk∈𝔹ρL​\(δxk​\(ξ,π∗\)\)Ykj​\(ξ\),\\displaystyle=\\sum\\nolimits\_\{j=0\}^\{p\}\\max\_\{0\\leq k\\leq L\_\{d\}\-1\}\\sup\_\{\\nu\_\{k\}\\in\\mathbb\{B\}\_\{\\rho\_\{L\}\}\(\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}\)\}Y\_\{k\}^\{j\}\(\\xi\),\(10\)Ykj​\(ξ\)\\displaystyle Y\_\{k\}^\{j\}\(\\xi\)=𝔼yk∼νk​‖∂ykjπ​\(yk\)−∂ykjπ∗​\(yk\)‖,\\displaystyle=\\mathbb\{E\}\_\{y\_\{k\}\\sim\\nu\_\{k\}\}\\\|\\partial\_\{y\_\{k\}\}^\{j\}\\pi\(y\_\{k\}\)\-\\partial\_\{y\_\{k\}\}^\{j\}\\pi\_\{\*\}\(y\_\{k\}\)\\\|,and𝔹ρL​\(δxk​\(ξ,π∗\)\):=\{ℚ\|𝕎2​\(ℚ,δxk​\(ξ,π∗\)\)≤ρL\}\\mathbb\{B\}\_\{\\rho\_\{L\}\}\\left\(\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}\\right\):=\\left\\\{\\mathbb\{Q\}\\ \|\\ \\mathbb\{W\}\_\{2\}\\left\(\\mathbb\{Q\},\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}\\right\)\\leq\\rho\_\{L\}\\right\\\}\. Note that we construct the ball𝔹ρL\\mathbb\{B\}\_\{\\rho\_\{L\}\}around the expert trajectoryxk​\(ξ,π∗\)x\_\{k\}\(\\xi,\\pi\_\{\*\}\), since it is known a priori, whereasxk​\(ξ,π^ps\)x\_\{k\}\(\\xi,\\hat\{\\pi\}\_\{p\}^\{s\}\)is not\. The accuracy of this approximation depends on how closely the learned policy matches the expert; in general,ρL\\rho\_\{L\}can be enlarged to account for the approximation error\.

The intuition behind \([10](https://arxiv.org/html/2607.13436#S3.E10)\) is as follows\. Instead of aligning the learned policy \(and its derivatives\) with the expert policy \(and its derivatives\) only along the expert trajectory, we consider a ball of distributions centered at the Dirac distribution of the expert trajectory\. Trajectories are then sampled from the worst\-case distribution within this ball, and the learned policy \(and its derivatives\) are aligned along them\. This approach ensures that the TaSIL policy aligns as closely as possible with the expert policy under uncertainty\.

The main limitation of this loss function is that the expert policy and its derivatives \(up to orderpp\) are not available along the sampled trajectories\. However, they are available along the expert trajectories, in whose neighborhood the sampled trajectories lie\. Thus, we approximate the expert policy locally using a Taylor expansion around the expert trajectory\. In particular, for a stateyky\_\{k\}in the vicinity ofxk​\(ξ,π∗\)x\_\{k\}\(\\xi,\\pi\_\{\*\}\), we consider the surrogate,π∗​\(yk\)≈π∗​\(xk​\(ξ,π∗\)\)\+Jπ∗​\(xk​\(ξ,π∗\)\)​\(yk−xk​\(ξ,π∗\)\)\+⋯,\\pi\_\{\*\}\(y\_\{k\}\)\\approx\\pi\_\{\*\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\+J\_\{\\pi\_\{\*\}\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\(y\_\{k\}\-x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\+\\cdots,up to orderpp, whereJπ∗​\(xkπ∗​\(ξ\)\)J\_\{\\pi\_\{\*\}\}\(x\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi\)\)denote the Jacobian of the expert policy evaluated atxkπ∗​\(ξ\)x\_\{k\}^\{\\pi\_\{\*\}\}\(\\xi\), and is available by assumption\.

###### Theorem III\.1

If no epistemic or aleatoric uncertainty is present, then the distributionally robust TaSIL formulation in \([10](https://arxiv.org/html/2607.13436#S3.E10)\) coincides with the standard TaSIL formulation in \([3](https://arxiv.org/html/2607.13436#S2.E3)\)\.

###### Proof:

In the absence of epistemic and aleatoric uncertainty, the system in \([8](https://arxiv.org/html/2607.13436#S3.E8)\) boils down to the system in \([2](https://arxiv.org/html/2607.13436#S2.E2)\)\. In this case, the distribution𝕏k​\(ξ,π∗\)\\mathbb\{X\}\_\{k\}\(\\xi,\\pi\_\{\*\}\)of the true system matches the Dirac distributionδxk​\(ξ,π∗\)\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}of the nominal deterministic system\. Thus,𝕎2​\(𝕏k​\(ξ,π∗\),δxk​\(ξ,π∗\)\)=0\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{k\}\(\\xi,\\pi\_\{\*\}\),\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}\)=0, and hence,𝔹ρL​\(δxk​\(ξ,π∗\)\)=\{δxk​\(ξ,π∗\)\}\\mathbb\{B\}\_\{\\rho\_\{L\}\}\\left\(\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}\\right\)=\\\{\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}\\\}\. To this end,νk=δxk​\(ξ,π∗\)\\nu\_\{k\}=\\delta\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}in \([10](https://arxiv.org/html/2607.13436#S3.E10)\), the inner supremum problem reduces to‖∂xk​\(ξ,π∗\)jπ​\(xk​\(ξ,π∗\)\)−∂xk​\(ξ,π∗\)jπ∗​\(xk​\(ξ,π∗\)\)‖\\\|\\partial\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}^\{j\}\\pi\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\-\\partial\_\{x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\}^\{j\}\\pi\_\{\*\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\\\|, and the formulations in \([3](https://arxiv.org/html/2607.13436#S2.E3)\) and \([10](https://arxiv.org/html/2607.13436#S3.E10)\) are equivalent\. ∎

Following\[[17](https://arxiv.org/html/2607.13436#bib.bib22)\],one can replace the inner supremum problem by its relaxed Lagrangian in \([10](https://arxiv.org/html/2607.13436#S3.E10)\) and use a robust surrogate of the resulting problem for training\. In this case, gradient ascent can be used to solve the supremum problem\.

### III\-CSafety

![Refer to caption](https://arxiv.org/html/2607.13436v1/x1.png)Figure 1:Monte Carlo simulations of the task under different controllers and operating conditions\.\(Top\-Left\)Standard TaSIL policy in the absence of uncertainties\.\(Top\-Right\)Standard TaSIL policy in the presence of uncertainties\.\(Bottom\-Left\)Standard TaSIL policy augmented withL1\\pazocal\{L\}\_\{1\}\-DRAC in the presence of uncertainties\.\(Bottom\-Right\)Distributionally robust and safe TaSIL policy augmented withL1\\pazocal\{L\}\_\{1\}\-DRAC in the presence of uncertainties\.While the expert can account for safety requirements when collecting demonstrations under nominal conditions, TaSIL does not explicitly take these requirements into account\. Here, we consider a safe region whose admissible set isX=\{x∈ℝn:cj⊤​x≤dj,∀j∈\{1,…,nc\}\},\\pazocal\{X\}=\\\{x\\in\\mathbb\{R\}^\{n\}:c\_\{j\}^\{\\top\}x\\leq d\_\{j\},\\ \\forall j\\in\\\{1,\\ldots,n\_\{c\}\\\}\\\},wherencn\_\{c\}is the number of constraints withcj∈ℝnc\_\{j\}\\in\\mathbb\{R\}^\{n\}anddj∈ℝd\_\{j\}\\in\\mathbb\{R\}for allj∈\{1,…,nc\}j\\in\\\{1,\\ldots,n\_\{c\}\\\}\. Starting from an expert data pointxk​\(ξ,π∗\)x\_\{k\}\(\\xi,\\pi\_\{\*\}\), we define the next nominal and true states under a policyπ\\pi, respectively, aszk​\(ξ\)=A​xk​\(ξ,π∗\)\+B​π​\(xk​\(ξ,π∗\)\)z\_\{k\}\(\\xi\)=Ax\_\{k\}\(\\xi,\\pi\_\{\*\}\)\+B\\pi\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)andz¯k​\(ξ\)=A​xk​\(ξ,π∗\)\+B​\(π​\(xk​\(ξ,π∗\)\)\+Hμ​\(xk​\(ξ,π∗\)\)\)\+\(Aσ\+Bc​Hσ​\(xk​\(ξ,π∗\)\)\)​Δ​wk\\bar\{z\}\_\{k\}\(\\xi\)=Ax\_\{k\}\(\\xi,\\pi\_\{\*\}\)\+B\(\\pi\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\+H\_\{\\mu\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\)\+\(A\_\{\\sigma\}\+B\_\{c\}H\_\{\\sigma\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\)\)\)\\Delta w\_\{k\}following \([2](https://arxiv.org/html/2607.13436#S2.E2)\) and \([8](https://arxiv.org/html/2607.13436#S3.E8)\)\. Note that thez¯k​\(ξ\)\\bar\{z\}\_\{k\}\(\\xi\)has a distribution𝕏1​\(xk​\(ξ,π∗\),π\)\\mathbb\{X\}\_\{1\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\),\\pi\)\.

Due to the stochastic nature ofz¯k​\(ξ\)\\bar\{z\}\_\{k\}\(\\xi\), we consider the chance constraints,ℙ​\(z¯k​\(ξ\)∈X\)≥1−δs\\mathbb\{P\}\(\\bar\{z\}\_\{k\}\(\\xi\)\\in\\pazocal\{X\}\)\\geq 1\-\\delta\_\{s\}, whereδs∈\(0,1\)\\delta\_\{s\}\\in\(0,1\)is a user specified risk parameter\. Owing to their convexity and ability to distinguish tail events, we approximate the chance constraints using the Conditional\-Value\-at\-Risk \(CVaR\) constraints as follows:

CVaR1−δsz¯k​\(ξ\)∼𝕏1​\(xk​\(ξ,π∗\),π\)⁡\(maxj∈\{1,…,nc\}⁡cj⊤​z¯k​\(ξ\)−dj\)≤0\.\\operatorname\{CVaR\}\_\{1\-\\delta\_\{s\}\}^\{\\bar\{z\}\_\{k\}\(\\xi\)\\sim\\mathbb\{X\}\_\{1\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\),\\pi\)\}\\left\(\\max\_\{j\\in\\\{1,\\ldots,n\_\{c\}\\\}\}c\_\{j\}^\{\\top\}\\bar\{z\}\_\{k\}\(\\xi\)\-d\_\{j\}\\right\)\\leq 0\.\(11\)The main challenge in \([11](https://arxiv.org/html/2607.13436#S3.E11)\) is that𝕏1​\(xk​\(ξ,π∗\),π\)\\mathbb\{X\}\_\{1\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\),\\pi\)is unknown due to the existing uncertainties\. Note, however, that𝕎2​\(𝕏1​\(xk​\(ξ,π∗\),π\),δzk​\(ξ\)\)≤ρL\\mathbb\{W\}\_\{2\}\(\\mathbb\{X\}\_\{1\}\(x\_\{k\}\(\\xi,\\pi\_\{\*\}\),\\pi\),\\delta\_\{z\_\{k\}\(\\xi\)\}\)\\leq\\rho\_\{L\}from Proposition[III\.1](https://arxiv.org/html/2607.13436#S3.Thmprop1)\. Thus, we can replace \([11](https://arxiv.org/html/2607.13436#S3.E11)\) withZk​\(ξ\)≤0Z\_\{k\}\(\\xi\)\\leq 0where

Zk​\(ξ\)=supνk∈𝔹ρL​\(δzk​\(ξ\)\)CVaR1−δsz¯k​\(ξ\)∼νk⁡\(maxj∈\{1,…,nc\}⁡cj⊤​z¯k​\(ξ\)−dj\)\.Z\_\{k\}\(\\xi\)=\\sup\_\{\\nu\_\{k\}\\in\\mathbb\{B\}\_\{\\rho\_\{L\}\}\(\\delta\_\{z\_\{k\}\(\\xi\)\}\)\}\\operatorname\{CVaR\}\_\{1\-\\delta\_\{s\}\}^\{\\bar\{z\}\_\{k\}\(\\xi\)\\sim\\nu\_\{k\}\}\\left\(\\max\_\{j\\in\\\{1,\\ldots,n\_\{c\}\\\}\}c\_\{j\}^\{\\top\}\\bar\{z\}\_\{k\}\(\\xi\)\-d\_\{j\}\\right\)\.\(12\)Inspired by \([12](https://arxiv.org/html/2607.13436#S3.E12)\), we aim to add a safety component to the TaSIL loss function to promote safety\. To this end, we propose the risk minimization problem:

π^ps∈argminπ∈Π⁡𝔼ξ∼D​\[lp,L1π∗,s​\(ξ,π\)\],\\hat\{\\pi\}\_\{p\}^\{s\}\\in\\operatorname\{argmin\}\_\{\\pi\\in\\Pi\}\\mathbb\{E\}\_\{\\xi\\sim\\pazocal\{D\}\}\\left\[l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},s\}\(\\xi,\\pi\)\\right\],where the loss functionlp,L1π∗,s​\(ξ,π\)l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},s\}\(\\xi,\\pi\)combines the distributionally robust term \([10](https://arxiv.org/html/2607.13436#S3.E10)\) with an additional physics\-informed component, and is defined as

lp,L1π∗,s​\(ξ,π\)=lp,L1π∗,r​\(ξ,π\)\+λ​∑k=0Ld−1ReLU⁡\(Zk​\(ξ\)\),\\displaystyle l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},s\}\(\\xi,\\pi\)=l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},r\}\(\\xi,\\pi\)\+\\lambda\\sum\\nolimits\_\{k=0\}^\{L\_\{d\}\-1\}\\operatorname\{ReLU\}\\left\(Z\_\{k\}\(\\xi\)\\right\),\(13\)whereλ∈ℝ\+\\lambda\\in\\mathbb\{R\}\_\{\+\}andReLU\\operatorname\{ReLU\}denotes the rectified linear unit\.

The intuition behind this loss function is as follows\. When the CVaR constraint is satisfied \(i\.e\., its value is negative\), the ReLU term evaluates to zero and the loss reduces to the distributionally robust TaSIL objective, which encourages the learned policy to mimic the expert policy under uncertainty\. However, if the CVaR constraint is violated, typically because the expert demonstrations were collected under conditions that differ from those at deployment, the ReLU becomes positive and adds a penalty\. This discourages the policy from blindly imitating the expert and instead guides it to remain as close as possible while respecting the constraint\. Note that ReLU is only one possible choice for penalization; smooth alternatives include ELU and GeLU\.

Since the safe region is represented using linear constraints,Zk​\(ξ\)≤0Z\_\{k\}\(\\xi\)\\leq 0can be expressed, following\[[8](https://arxiv.org/html/2607.13436#bib.bib21)\], as

Zkj​\(ξ\)=cj⊤​zk​\(ξ\)−dj\+‖cj‖2δs/nc​ρL≤0,∀j∈\{1,…,nc\}\.Z\_\{k\}^\{j\}\(\\xi\)=c\_\{j\}^\{\\top\}z\_\{k\}\(\\xi\)\-d\_\{j\}\+\\frac\{\\\|c\_\{j\}\\\|\_\{2\}\}\{\\sqrt\{\\delta\_\{s\}/n\_\{c\}\}\}\\rho\_\{L\}\\leq 0,\\forall j\\in\\\{1,\\ldots,n\_\{c\}\\\}\.\(14\)In this case, the loss functionlp,L1π∗,s​\(ξ,π\)l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},s\}\(\\xi,\\pi\)is given by

lp,L1π∗,s​\(ξ,π\)=lp,L1π∗,r​\(ξ,π\)\+λ​∑j=1nc∑k=0Ld−1ReLU⁡\(Zkj​\(ξ\)\)\.l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},s\}\(\\xi,\\pi\)=l\_\{p,\\pazocal\{L\}\_\{1\}\}^\{\\pi\_\{\*\},r\}\(\\xi,\\pi\)\+\\lambda\\sum\_\{j=1\}^\{n\_\{c\}\}\\sum\_\{k=0\}^\{L\_\{d\}\-1\}\\operatorname\{ReLU\}\\left\(Z\_\{k\}^\{j\}\(\\xi\)\\right\)\.\(15\)While the safety term considers one\-step\-ahead states initialized from expert data points, it naturally extends to multi\-step\-ahead predictions\. Moreover, instead of relying on expert trajectories, one may use the trajectories sampled in \([10](https://arxiv.org/html/2607.13436#S3.E10)\) via gradient ascent\. Finally, the proposed approach employs the boundρL=ρ\+ρa\\rho\_\{L\}=\\rho\+\\rho\_\{a\}for learning the TaSIL policy, as in \([15](https://arxiv.org/html/2607.13436#S3.E15)\)\. The computation of theL1\\pazocal\{L\}\_\{1\}\-DRAC bound, however, depend on the nominal dynamics under the learned policy, leading to a coupled co\-design problem\. This challenge is further compounded by the requirement of a contractive nominal policy for the computation of theL1\\pazocal\{L\}\_\{1\}\-DRAC bound\. While this requirement is discussed in\[[6](https://arxiv.org/html/2607.13436#bib.bib19)\], this aspect would benefit from a more thorough analysis\. Tools from the co\-design theory\[[3](https://arxiv.org/html/2607.13436#bib.bib23)\]provide an interesting framework to jointly design TaSIL andL1\\pazocal\{L\}\_\{1\}\-DRAC; this direction is left for future work\. In the present work,ρL\\rho\_\{L\}is chosen empirically\.

## IVSimulation Results

In this section, we demonstrate the effectiveness of the proposed framework on a UAV case study, in which the goal is to emulate an expert planner in an uncertain environment while accounting for safety requirements\.

We consider a UAV model where the state vector is given byx=\[p⊤,v⊤,η⊤,ω⊤\]⊤x=\[p^\{\\top\},v^\{\\top\},\\eta^\{\\top\},\\omega^\{\\top\}\]^\{\\top\}, comprising the positionp=\[px,py,pz\]⊤p=\[p\_\{x\},p\_\{y\},p\_\{z\}\]^\{\\top\}, linear velocityv=\[vx,vy,vz\]⊤v=\[v\_\{x\},v\_\{y\},v\_\{z\}\]^\{\\top\}, Euler anglesη=\[ϕ,θ,ψ\]⊤\\eta=\[\\phi,\\theta,\\psi\]^\{\\top\}, and body angular ratesω=\[p,q,r\]⊤\\omega=\[p,q,r\]^\{\\top\}of the vehicle\. The input vector is given byu=\[u1,u2,u3,u4\]⊤u=\[u\_\{1\},u\_\{2\},u\_\{3\},u\_\{4\}\]^\{\\top\}, whereu1u\_\{1\}denotes the total thrust andu2,u3,u4u\_\{2\},u\_\{3\},u\_\{4\}denote the roll, pitch, and yaw moments, respectively\. We assume a low\-level controller, in the form of an LQR law given byu=−K​x\+L​du=\-Kx\+Ld, is used to stabilize the UAV, whered=\[xd,yd,zd,ψd\]⊤d=\[x\_\{d\},y\_\{d\},z\_\{d\},\\psi\_\{d\}\]^\{\\top\}denotes the desired trajectory for the UAV positionppand headingψ\\psi\. The matricesKKandLLare control gains of appropriate dimensions\. At time indexkk, the desired trajectory is updated asdk=dk−1\+Δ​dk−1d\_\{k\}=d\_\{k\-1\}\+\\Delta d\_\{k\-1\}\. Our objective is to learn a policy that imitates the expert planner generating the trajectory incrementsΔ​dk−1\\Delta d\_\{k\-1\}\.

Letmm,JxJ\_\{x\},JyJ\_\{y\},JzJ\_\{z\}, andggdenote the UAV’s mass, moments of inertia, and gravitational acceleration, respectively\. The UAV model can then be written in the form \([1](https://arxiv.org/html/2607.13436#S2.E1)\) withXt∈ℝ12X\_\{t\}\\in\\mathbb\{R\}^\{12\}representing the UAV state vector,Ut∈ℝ4U\_\{t\}\\in\\mathbb\{R\}^\{4\}denoting the UAV desired trajectory,Aμ=Aq−Bq​KA\_\{\\mu\}=A\_\{q\}\-B\_\{q\}K,Bc=Bq​LB\_\{c\}=B\_\{q\}L, whereBq=\[1m​e6​1Jx​e10​1Jy​e11​1Jz​e12\]B\_\{q\}=\\left\[\\frac\{1\}\{m\}e\_\{6\}\\ \\frac\{1\}\{J\_\{x\}\}e\_\{10\}\\ \\frac\{1\}\{J\_\{y\}\}e\_\{11\}\\ \\frac\{1\}\{J\_\{z\}\}e\_\{12\}\\right\],Aσ=\[σv​e4​σv​e5​σv​e6​ση​e10​ση​e11​ση​e12\]A\_\{\\sigma\}=\\left\[\\sigma\_\{v\}e\_\{4\}\\ \\sigma\_\{v\}e\_\{5\}\\ \\sigma\_\{v\}e\_\{6\}\\ \\sigma\_\{\\eta\}e\_\{10\}\\ \\sigma\_\{\\eta\}e\_\{11\}\\ \\sigma\_\{\\eta\}e\_\{12\}\\right\], and

Aq=\[09×3diag⁡\(I3,A23,0,I3\)0303×9\],A23=\[sψ0​gcψ0​g−cψ0​gsψ0​g\]\.A\_\{q\}=\\begin\{bmatrix\}0\_\{9\\times 3\}&\\operatorname\{diag\}\(I\_\{3\},A\_\{23\},0,I\_\{3\}\)\\\\ 0\_\{3\}&0\_\{3\\times 9\}\\end\{bmatrix\}\\hskip\-2\.84544pt,\\ A\_\{23\}=\\begin\{bmatrix\}s\_\{\\psi\_\{0\}\}g&c\_\{\\psi\_\{0\}\}g\\\\ \-c\_\{\\psi\_\{0\}\}g&s\_\{\\psi\_\{0\}\}g\\end\{bmatrix\}\\hskip\-2\.84544pt\.The parameterψ0\\psi\_\{0\}denotes the heading around which the UAV operates,ei∈ℝ12e\_\{i\}\\in\\mathbb\{R\}^\{12\}denote unit vectors whoseii\-th element is 1 for alli∈\{1,…,12\}i\\in\\\{1,\\ldots,12\\\},σv∈ℝ\+\\sigma\_\{v\}\\in\\mathbb\{R\}\_\{\+\},ση∈ℝ\+\\sigma\_\{\\eta\}\\in\\mathbb\{R\}\_\{\+\},sθs\_\{\\theta\}andcθc\_\{\\theta\}denotesin⁡\(θ\)\\sin\(\\theta\)andcos⁡\(θ\)\\cos\(\\theta\), respectively\.

In numerical simulations, the model parameters are set asm=1​k​gm=1\\mathrm\{kg\},g=9\.81​m/s2g=9\.81\\mathrm\{m/s^\{2\}\},Jx=Jy=0\.01​kg,m2J\_\{x\}=J\_\{y\}=0\.01\\mathrm\{kg,m^\{2\}\},Jz=0\.02​kg,m2J\_\{z\}=0\.02\\mathrm\{kg,m^\{2\}\}, andψ0=0\\psi\_\{0\}=0, whileσv=0\.1\\sigma\_\{v\}=0\.1andση=0\.1\\sigma\_\{\\eta\}=0\.1\. The weighting matricesQ=diag⁡\(10,10,100,0,0,0,0,0,10,0,0,0\)Q=\\operatorname\{diag\}\(10,10,100,0,0,0,0,0,10,0,0,0\)andR=diag⁡\(1,0\.1,0\.1,0\.1\)R=\\operatorname\{diag\}\(1,0\.1,0\.1,0\.1\)are used to compute the LQR gainsKKandLL\. The expert planner generates trajectories that drive the UAV to follow a circular horizontal trajectory of radius0\.80\.8in a counterclockwise direction, from any initial condition in the vicinity of the trajectory\. The safe region is defined as\{\(x,y\)∈ℝ2∣x≤0\.8,y≤0\.8\}\\\{\(x,y\)\\in\\mathbb\{R\}^\{2\}\\mid x\\leq 0\.8,\\;y\\leq 0\.8\\\}\.

Following the notation in \([1](https://arxiv.org/html/2607.13436#S2.E1)\), we consider the epistemic uncertaintyHμ​\(x\)H\_\{\\mu\}\(x\)that affects the UAV through the same channels asUtU\_\{t\}, i\.e\., the desired trajectory\. This disturbance can equivalently be viewed as disturbing forces and momentsH​\(x\)=L​Hμ​\(x\)H\(x\)=LH\_\{\\mu\}\(x\)acting through the same channels as the control input vectoruu\. The epistemic uncertaintyHμ​\(x\)H\_\{\\mu\}\(x\)is modeled as a sum of sinusoids with randomly generated amplitudes and frequencies\. The frequencies are uniformly sampled from\[3,4\]​rad/s\[3,4\]\\mathrm\{rad/s\}\. The amplitudes in thedxd\_\{x\},dyd\_\{y\},dzd\_\{z\}anddψd\_\{\\psi\}channels are uniformly sampled from\[0\.02,0\.025\]\[0\.02,0\.025\],\[0\.02,0\.025\]\[0\.02,0\.025\],\[0\.04,0\.05\]\[0\.04,0\.05\], and\[0\.04,0\.05\]\[0\.04,0\.05\], respectively\. In this case study, we setHσ​\(x\)H\_\{\\sigma\}\(x\)to zero\.

We consider learned policies comprising two hidden layers, each of which has 32 neurons and uses GELU as activation functions\. In all simulations, we use TaSIL of orderp=1p=1\. The policy is trained over 1000 iterations using1010expert trajectories, each of which has150150data points and starts from an initial condition sampled from a uniform distribution in the vicinity of the desired trajectory\. TheL1\\pazocal\{L\}\_\{1\}\-DRAC parameters are given byω=5\\omega=5,Ts=0\.01T\_\{s\}=0\.01, andλ=5\\lambda=5\.

We analyze the system under the following policies and operating conditions: standard TaSILπ^p\\hat\{\\pi\}\_\{p\}in the absence of uncertainties, standard TaSILπ^p\\hat\{\\pi\}\_\{p\}in the presence of uncertainties, standard TaSILπ^p\\hat\{\\pi\}\_\{p\}augmented withL1\\pazocal\{L\}\_\{1\}\-DRAC in the presence of uncertainties, and distributionally robust and safe TaSILπ^ps\\hat\{\\pi\}\_\{p\}^\{s\}augmented withL1\\pazocal\{L\}\_\{1\}\-DRAC in the presence of uncertainties\. We simulate \([1](https://arxiv.org/html/2607.13436#S2.E1)\) using the Euler\-Maruyama scheme with a sampling timeΔ​T=0\.01\\Delta T=0\.01\. The Brownian motion increments are modeled as i\.i\.d\. Gaussian random variables with zero mean and varianceΔ​T\\Delta T\. The planner operates at a slower time scale, with a0\.10\.1sampling time\.

Figure[1](https://arxiv.org/html/2607.13436#S3.F1)shows Monte Carlo simulations of the system behavior under varying initial conditions and epistemic uncertainties, while performing the desired tasks across the four aforementioned cases\. The initial conditions are sampled near the left quarter of the trajectory, whereas the epistemic uncertainties are constructed as sums of sinusoids, as described earlier\. As shown in the top\-left panel, the standard TaSIL policy successfully mimics the expert behavior, following a circular trajectory of radius0\.80\.8, in uncertainty\-free environments\. However, in the presence of uncertainty \(top\-right panel\), the policy deviates from the expert\. With theL1\\pazocal\{L\}\_\{1\}\-DRAC augmentation \(bottom\-left panel\), the policy remains closer to the desired trajectory under uncertainty\. Notably, in all of the above cases, the UAV fails to avoid the unsafe region\. In contrast, the distributionally robust and safe TaSIL \(bottom\-right panel\) closely follows the expert \(e\.g\., along the bottom\-left portion of the trajectory\) while intentionally deviating near unsafe regions \(e\.g\., along the top\-right portion\) to ensure safety\.

## VCONCLUSIONS

We propose a novel distributionally robust and safe imitation learning framework\. The developed framework is designed to be robust to both policy\-induced and uncertainty\-induced distribution shifts, optimize performance under distributional uncertainty, and account for safety requirements during training\. We validate the proposed framework through extensive simulations of a UAV flying in uncertain environments while avoiding unsafe regions\.

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