SAFE-T: Non-Stochastic Control Under Safety Constraints

• SAFE-T is a non-stochastic control framework for linear dynamical systems that robustly enforces time-varying convex safety constraints even under adversarial disturbances.
• It employs safety-certified gain sets and projection-based updates, enabling online gradient descent over dynamic, safe domains to guarantee zero per-step constraint violations.
• The approach balances safety and performance by achieving bounded dynamic regret and competitive computational efficiency, with demonstrated applications in tasks like quadrotor hovering.

SAFE-T is an overloaded acronym in the arXiv literature. Most prominently, it denotes “Safe control under Time-varying constraints with adversarial, non-stochastic disturbances,” introduced in “Safe Non-Stochastic Control of Linear Dynamical Systems,” where safe control is cast as an online sequential game over linear feedback gains with per-step safety and bounded dynamic regret guarantees (Zhou et al., 2023). The same acronym has also been used for “Safe Linear Thompson Sampling with Side Information” in stochastic linear bandits (Moradipari et al., 2019), for a future-aware safe active learning framework for time-varying systems using Gaussian processes (Lange-Hegermann et al., 2024), and for several unrelated frameworks in graph algorithms, experimentation, safe reinforcement learning, and holography. In current usage, however, SAFE-T most often refers to the non-stochastic control framework centered on robust safety under time-varying convex constraints and adversarial disturbances (Zhou et al., 2023).

1. SAFE-T as safe non-stochastic control

In its control-theoretic sense, SAFE-T studies linear dynamical systems corrupted with non-stochastic noise and seeks two simultaneous guarantees: zero constraint violation of convex time-varying constraints, and bounded dynamic regret against an optimal clairvoyant controller that knows the future disturbance sequence a priori (Zhou et al., 2023). The motivating setting is autonomy under real-world unpredictable disturbances such as wind and wake disturbances, with safety constraints that bound both the state and the control input.

The system model is linear and may be time-varying: $$x_{t+1} = A_t x_t + B_t u_t + w_t,\quad t=1,\ldots,T,$$ with state $x_t\in\mathbb{R}^{d_x}$, input $u_t\in\mathbb{R}^{d_u}$, known matrices $A_t,B_t$, and disturbance $w_t\in\mathbb{R}^{d_x}$. The time-invariant case is the special case $A_t\equiv A$, $B_t\equiv B$. SAFE-T assumes a known disturbance radius and imposes no probabilistic model: $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$

Safety is enforced through time-varying convex constraints on state and input: $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$ with polytopes as the leading special case. The per-round control objective is encoded through a convex loss $c_t(x_{t+1},u_t)$, with bounded values and gradients on bounded domains. A common example is the quadratic cost

$x_t\in\mathbb{R}^{d_x}$0

SAFE-T rewrites the problem as a sequential game. At each step $x_t\in\mathbb{R}^{d_x}$1, before observing $x_t\in\mathbb{R}^{d_x}$2, the controller selects a linear state-feedback gain $x_t\in\mathbb{R}^{d_x}$3 and applies

$x_t\in\mathbb{R}^{d_x}$4

After $x_t\in\mathbb{R}^{d_x}$5 is revealed through the next state, the controller incurs

$x_t\in\mathbb{R}^{d_x}$6

which is convex in $x_t\in\mathbb{R}^{d_x}$7 given $x_t\in\mathbb{R}^{d_x}$8. The controller therefore faces an online convex optimization problem with adversarial disturbances and time-varying safe domains.

2. Safety-certified gain sets and robust tightening

The central SAFE-T construction is the time-varying admissible set of feedback gains $x_t\in\mathbb{R}^{d_x}$9, built so that every chosen gain is robustly safe for all disturbances satisfying $u_t\in\mathbb{R}^{d_u}$0 (Zhou et al., 2023). The controller selects only gains in

$u_t\in\mathbb{R}^{d_u}$1

where $u_t\in\mathbb{R}^{d_u}$2 and $u_t\in\mathbb{R}^{d_u}$3 are design parameters.

The first inequality is a robust tightening of the next-state constraint. Starting from

$u_t\in\mathbb{R}^{d_u}$4

and requiring $u_t\in\mathbb{R}^{d_u}$5 for all $u_t\in\mathbb{R}^{d_u}$6, one obtains

$u_t\in\mathbb{R}^{d_u}$7

Using the disturbance bound and triangle inequality yields the worst-case term $u_t\in\mathbb{R}^{d_u}$8, hence

$u_t\in\mathbb{R}^{d_u}$9

Similarly, the input constraint becomes

$A_t,B_t$0

The resulting guarantee is per-step and robust: if $A_t,B_t$1, then for any disturbance $A_t,B_t$2,

$A_t,B_t$3

Thus SAFE-T enforces zero per-step violations of the time-varying convex constraints. The framework requires one-step-ahead state-constraint information $A_t,B_t$4 to construct $A_t,B_t$5, and it uses the technical assumption of recursive feasibility, $A_t,B_t$6 for all $A_t,B_t$7. A plausible implication is that feasibility management is part of the safety architecture rather than a secondary implementation detail.

3. Safe-OGD and the online update mechanism

SAFE-T solves the sequential game with “Safe Online Gradient Descent” (Safe-OGD), which generalizes Online Gradient Descent to time-varying safe domains by projecting onto the next admissible set (Zhou et al., 2023). The per-round update is:

1. Pick $A_t,B_t$8 and apply $A_t,B_t$9.
2. Observe $w_t\in\mathbb{R}^{d_x}$0 and compute $w_t\in\mathbb{R}^{d_x}$1.
3. Suffer loss $w_t\in\mathbb{R}^{d_x}$2 and form $w_t\in\mathbb{R}^{d_x}$3.
4. Compute $w_t\in\mathbb{R}^{d_x}$4.
5. Obtain $w_t\in\mathbb{R}^{d_x}$5.
6. Perform a gradient step and projection: $w_t\in\mathbb{R}^{d_x}$6 where $w_t\in\mathbb{R}^{d_x}$7 denotes projection onto $w_t\in\mathbb{R}^{d_x}$8.

The boundedness and stability constraints in $w_t\in\mathbb{R}^{d_x}$9 are not merely regularizers; they are part of the regret analysis. The conditions $A_t\equiv A$0 and $A_t\equiv A$1 imply

$A_t\equiv A$2

This, in turn, yields a Frobenius-norm gradient bound

$A_t\equiv A$3

and an admissible-set diameter bound

$A_t\equiv A$4

This structure distinguishes SAFE-T from standard online convex optimization with static domains. The projection set $A_t\equiv A$5 depends on the next-step safety description and also on the current state $A_t\equiv A$6. As a result, even when $A_t\equiv A$7 and $A_t\equiv A$8 are time-invariant, the feasible gain domain can still vary over time.

4. Dynamic regret, comparator structure, and theoretical guarantees

SAFE-T evaluates performance through dynamic policy regret against a safe clairvoyant sequence of linear feedback gains $A_t\equiv A$9 that knows the disturbances in advance (Zhou et al., 2023). The regret is

$B_t\equiv B$0

where $B_t\equiv B$1, $B_t\equiv B$2, and both trajectories are evaluated under the same disturbance sequence.

The regret bound depends on two variation quantities. The first is the comparator path length,

$B_t\equiv B$3

which measures how rapidly the clairvoyant optimal controller changes. The second is the domain-variation term,

$B_t\equiv B$4

which quantifies the effect of changing the projection set from $B_t\equiv B$5 to $B_t\equiv B$6.

The main SAFE-T bound is

$B_t\equiv B$7

With $B_t\equiv B$8, this yields

$B_t\equiv B$9

This bound extends the familiar dynamic-regret scaling from online convex optimization to time-varying safe domains. When the domain is time-invariant, $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$0, and the bound reduces to $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$1. This suggests that the extra term $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$2 is the precise analytical price of safety-domain variation, rather than a generic looseness of the proof.

A recurring theme is the safety–performance trade-off. Robust tightening subtracts $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$3 from the state constraint, and the stability condition $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$4 restricts the feasible set further. These features guarantee zero violations, but may shrink $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$5 and degrade achievable performance. Conversely, in high-rate control with slowly varying constraints, $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$6 can be small, improving the regret bound.

5. Computation, empirical validation, and limitations

The per-round computation in SAFE-T has two main parts: gradient evaluation for $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$7, and projection onto $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$8 (Zhou et al., 2023). The projection is a convex optimization problem with linear inequalities and convex norm constraints $$\|A_t\|\le\kappa_A,\quad \|B_t\|\le\kappa_B,\quad w_t\in\mathcal{W}\triangleq\{w:\|w\|\le W\}.$$9 and $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$0. The latter is convex in $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$1 as an operator-norm ball, and the projection can be solved via second-order cone or semidefinite relaxation depending on norm choice. Its complexity scales polynomially in the problem size and number of constraints.

The step size is set to $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$2 for the regret guarantee. Larger $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$3 accelerates adaptation but increases the $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$4 term; smaller $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$5 decreases that term but enlarges the $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$6-scaled terms. The stability margin $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$7 and the gain bound $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$8 also affect the bound through $$\begin{aligned} x_t &\in \mathcal{X}_t \triangleq \{x : L_{x,t} x \le l_{x,t}\},\ u_t &\in \mathcal{U}_t \triangleq \{u : L_{u,t} u \le l_{u,t}\}, \end{aligned}$$9 and $c_t(x_{t+1},u_t)$0.

The reported simulation domain is a quadrotor hovering task with LTI dynamics

$c_t(x_{t+1},u_t)$1

a 6D state, 3D inputs, disturbance bound $c_t(x_{t+1},u_t)$2, horizon $c_t(x_{t+1},u_t)$3, state constraints

$c_t(x_{t+1},u_t)$4

input constraints

$c_t(x_{t+1},u_t)$5

and cost

$c_t(x_{t+1},u_t)$6

The reported outcomes are threefold. First, all methods considered, including SAFE-OGD, ensure zero violations of state and input constraints. Second, SAFE-OGD achieves competitive or superior cumulative loss relative to safe $c_t(x_{t+1},u_t)$7 and $c_t(x_{t+1},u_t)$8 controllers with shorter horizons such as $c_t(x_{t+1},u_t)$9, while $x_t\in\mathbb{R}^{d_x}$00 baselines can achieve lower loss at substantially higher computational cost. Third, SAFE-OGD is significantly faster per round, reported as up to $x_t\in\mathbb{R}^{d_x}$01 faster than $x_t\in\mathbb{R}^{d_x}$02 and $x_t\in\mathbb{R}^{d_x}$03 faster than $x_t\in\mathbb{R}^{d_x}$04 at $x_t\in\mathbb{R}^{d_x}$05.

The framework nonetheless rests on restrictive assumptions: known disturbance bound $x_t\in\mathbb{R}^{d_x}$06, known $x_t\in\mathbb{R}^{d_x}$07, convex safety sets, one-step-ahead knowledge of safety parameters, and recursive feasibility. The paper notes that robust MPC or tube-MPC ideas with a terminal invariant set and a baseline safe stabilizing controller can be used to enforce recursive feasibility when a lookahead horizon is available. Extensions proposed in the source include nonlinear systems, model uncertainty, partial observations, and stochastic disturbances treated through a high-probability or moment bound playing the role of $x_t\in\mathbb{R}^{d_x}$08. These are described as directions rather than established guarantees.

6. Other arXiv uses of the acronym

The acronym SAFE-T has been reused in several technically unrelated literatures. The following usage patterns appear in the supplied arXiv corpus.

SAFE-T usage	Area	Core idea
“Safe Linear Thompson Sampling with Side Information” (Moradipari et al., 2019)	Linear stochastic bandits	Stage-wise linear safety constraints with side measurements $x_t\in\mathbb{R}^{d_x}$09
Future-aware safe active learning (Lange-Hegermann et al., 2024)	Gaussian-process active learning	T-IMSPE minimizes posterior variance over current and future states under safety filtering
“Safety in $x_t\in\mathbb{R}^{d_x}$10-$x_t\in\mathbb{R}^{d_x}$11 Paths, Trails and Walks” (Cairo et al., 2020)	Graph algorithms	Safe subwalks common to all $x_t\in\mathbb{R}^{d_x}$12-$x_t\in\mathbb{R}^{d_x}$13 paths, trails, or walks
“Safe Testing” (Beasley, 2023)	Sequential experimentation	Anytime-valid inference via e-values and e-processes
Safe online RL in 1D LQR (Schiffer et al., 25 Apr 2025)	Reinforcement learning	High-probability safety and $x_t\in\mathbb{R}^{d_x}$14-regret with truncated linear controllers
“Safe Gauge-String Correspondence” (Rey et al., 2019)	High-energy theory	Safe gauge theories dual to safe noncritical strings on asymptotically AdS
Safe sets in weighted trees (Ehard et al., 2017)	Combinatorics	PTAS for the connected safe number of a weighted tree

In linear bandits, SAFE-T denotes a frequentist Thompson-sampling algorithm for linear reward maximization under an unknown linear safety constraint $x_t\in\mathbb{R}^{d_x}$15, with side information $x_t\in\mathbb{R}^{d_x}$16 observed at each played action (Moradipari et al., 2019). The safe action set is the robust inner approximation

$x_t\in\mathbb{R}^{d_x}$17

and the reported regret order matches that of linear Thompson sampling without safety constraints up to logarithmic factors.

In Gaussian-process active learning, SAFE-T refers to a future-aware safe acquisition rule for time-varying systems. Its central objective is T-IMSPE,

$x_t\in\mathbb{R}^{d_x}$18

optimized subject to safety filtering $x_t\in\mathbb{R}^{d_x}$19, with $x_t\in\mathbb{R}^{d_x}$20 used in the experiments (Lange-Hegermann et al., 2024).

In graph theory, SAFE-T is the study of safe subwalks with respect to all $x_t\in\mathbb{R}^{d_x}$21-$x_t\in\mathbb{R}^{d_x}$22 paths, trails, or walks of a directed graph (Cairo et al., 2020). The paper shows linear-time characterizations and algorithms in several cases, an $x_t\in\mathbb{R}^{d_x}$23 compact representation for maximal safe walks, and an NP-hardness dichotomy for visible-subset variants of the path and trail problems.

In large-scale experimentation, Safe Testing uses e-values and e-processes for anytime-valid inference (Beasley, 2023). The defining condition is

$x_t\in\mathbb{R}^{d_x}$24

with stopping rule $x_t\in\mathbb{R}^{d_x}$25. This enables continuous monitoring without inflating Type I error under optional stopping.

In safe online reinforcement learning for one-dimensional LQR, SAFE-T denotes a high-probability safety-constrained learning framework with regret

$x_t\in\mathbb{R}^{d_x}$26

measured relative to a baseline of truncated linear controllers that clip controls at the safety boundary (Schiffer et al., 25 Apr 2025). The safety requirement is formulated as

$x_t\in\mathbb{R}^{d_x}$27

with high probability for all times.

The remaining uses are unrelated to control or learning. “Safe Gauge-String Correspondence” studies holographic duality for safe gauge theories with non-Gaussian ultraviolet fixed points (Rey et al., 2019). “Approximating Connected Safe Sets in Weighted Trees” studies $x_t\in\mathbb{R}^{d_x}$28-safe sets and proves a PTAS for the connected safe number of a weighted tree (Ehard et al., 2017).

Across these usages, the shared lexical theme is invariance of a “safe” set under uncertainty, but the mathematical object varies widely: convex gain domains, ellipsoidal confidence-safe action sets, posterior safe regions, safe subwalks, e-processes, truncated controllers, or graph cuts. A plausible implication is that SAFE-T is better understood as a family of domain-specific safety formalisms than as a single unified framework.