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Positioning a Movable Antenna Without CSI: A Kernelized Bandit Under Costly Movement

Wonseok Choi, Yunseob Tae, Jeongjae Lee, Songnam Hong

arXiv:2609.27226v1eess.SP

TL;DR

Movable-antenna positioning is difficult when measuring a candidate requires slow relocation and the channel changes during transit. MoveUCB models the task as a non-stationary kernelized bandit with physical reachability and incorporates global target selection, persistence, and movement cost. It achieves sublinear dynamic regret under the standard variation-budget condition and, in simulations, the lowest regret with about half the travel of the closest moving competitor.

  • Problem

    Prior position optimization assumes candidate-position channels are known, but measuring a new position requires relocation and rewards may drift before measurement.

    Source: §I · p. 1

  • Method

    MoveUCB formulates positioning as a reachability-constrained, non-stationary kernelized bandit and combines global target search with target persistence and a transit-aware movement cost.

    Source: §I · p. 2, §VI · p. 11

  • Results

    About half the travel of GP-UCB; MoveUCB also attains the lowest regret in both simulated scenarios among the compared policies.

    Source: D. Results · p. 9

  • Takeaways & Limitations

    MoveUCB attains sublinear regret under the standard variation-budget condition alone, with physical parameters entering only through constants.

    Source: §I · p. 2, §VI · p. 11

Abstract

from arXiv · show

Movable antenna (MA) systems reconfigure the wireless channel by repositioning the antenna within a confined region. The position optimization behind this capability has been studied under the premise that the channel at every candidate position is known. However, acquiring that knowledge is costly: a position can be measured only after the antenna has moved there, which takes many slots at the speed of the actuator, and the channel drifts meanwhile. We formulate MA positioning as a non-stationary kernelized bandit with a reachability constraint, in which exploration, tracking, and the reward lost in transit are coupled through a single physical action. We propose MoveUCB, which addresses the reachability constraint through a global target search, a persistence rule that holds the target across slots, and a transit-aware movement cost term. We prove that MoveUCB attains sublinear dynamic regret under the standard variation-budget condition of non-stationary bandits alone, with the physical parameters entering only through constants. Simulations against a dynamic-programming oracle and movement-unaware baselines show that MoveUCB attains the lowest regret with about half the travel of the closest baseline that moves.

I. INTRODUCTION

Movable antennas offer spatial channel reconfiguration, but learning where to position one is difficult when measurements require slow movement and rewards drift in transit. The paper formulates this as a reachability-constrained, non-stationary kernelized bandit and proposes MoveUCB to address it.

I. INTRODUCTION

  • Existing position optimization generally assumes every candidate position’s channel and reward are known, although a new position can be measured only after costly relocation and rewards drift during transit.Prior CSI-free optimization instead treats position as a static optimization variable and uses zeroth-order gradient ascent.Source: §I · p. 1
  • The learning gap is the combination of bandit feedback, temporal drift, and physical reachability, which requires distant valuable positions to be approached over multiple slots.Prior kernelized and non-stationary bandits allow arbitrary arm choices each slot, while switching-cost models impose a distance penalty rather than a physical reachability constraint.Source: §I · pp. 1–2
  • The proposed MoveUCB combines a physics-informed kernel and movement cost with a persistent target and global target search followed by a reachable step.Its target is selected across the whole region rather than only among positions reachable in one slot.Source: §I · p. 2
  • MoveUCB attains sublinear regret under the standard variation-budget condition alone, and simulations report the lowest regret with about half the travel of the closest moving competitor.The formulation couples exploration, tracking, and reward lost in transit through a single physical action.Source: §I · p. 2

A. System Model

  • Each slot permits repositioning only within the actuator’s reachable region before transmission and one noisy reward observation at the resulting position.The actuator’s maximum displacement per slot is ρmax = vTa; when a selected destination is farther away, transmission occurs from intermediate positions during the journey.Source: A. System Model · p. 2, Figure 1 · p. 2

B. Problem Formulation

The formulation evaluates online positioning against an oracle under the same reachability constraint and movement cost. MoveUCB estimates drifting rewards from recent observations using a kernel designed to reflect the channel’s spatial structure.

B. Problem Formulation

  • The policy seeks to maximize cumulative reward net of actuation energy, but the reward sequence is revealed only causally through noisy observations.Dynamic regret is measured against an oracle that knows the reward sequence in advance while obeying the same reachability constraint.Source: B. Problem Formulation · pp. 2–3
  • Regret arises from insufficient exploration, insufficient tracking as rewards degrade, and excessive movement that loses transit reward and consumes actuation energy.The formulation penalizes travel when the learner moves more than the oracle.Source: §III · p. 3, B. Problem Formulation · p. 3

A. Reward Estimation with a Physics-Informed Kernel

  • MoveUCB estimates reward with kernel ridge regression over the w most recent observations, using spatial correlation while limiting the effect of outdated data.A Gaussian process prior yields a point estimate and uncertainty across the region; the window length balances estimation noise against temporal staleness.Source: A. Reward Estimation with a… · p. 3
  • The kernel represents nonoscillatory aggregate power and residual content alongside damped sinusoids for spatial fringes in the channel reward.The fringe frequencies span [0, 2/λ], and their coherence length decreases as angular spread increases.Source: A. Reward Estimation with a… · pp. 3–4
  • The kernel’s oscillatory terms capture fringes that the default squared-exponential kernel lacks, while its nonoscillatory terms represent reward energy near zero frequency.Setting Q = 0 and σ²_c = 0 recovers the squared-exponential kernel described in the passage.Source: A. Reward Estimation with a… · p. 4
  • Kernel hyperparameters are tied to stable angular-sector and angular-spread quantities, with fringe periods set from directional-cosine differences.The passage states that no kernel hyperparameter requires knowledge of the instantaneous channel.Source: A. Reward Estimation with a… · p. 4
  • The kernel is twice differentiable at the origin and has a spectrum confined to a point mass and Gaussian lobes, properties used to bound discretization error and information gain.Its output scale is based on warm-up rewards net of observation-noise variance.Source: A. Reward Estimation with a… · p. 4

B. Movement-Aware Position Optimization

MoveUCB uses an upper confidence bound to value uncertain positions, then modifies position selection to account for slow physical movement. Relocation costs include energy expenditure, lost reward in transit, and estimate staleness.

B. Movement-Aware Position Optimization

  • The standard upper confidence bound combines estimated reward and uncertainty, but by itself it omits transit losses and the delay before reaching a distant target.A larger confidence parameter weights uncertainty more heavily and promotes exploration.Source: B. Movement-Aware Position Optimization · p. 4
  • A displacement incurs actuation energy, lost reward while transmitting from intermediate positions, and staleness because the reward can drift before arrival.The passage describes each cost as proportional to distance; the latter two arise from the actuator’s limited speed.Source: 1) Accounting for a relocation cost · p. 4

1) Accounting for a relocation cost:

MoveUCB accounts for distance-dependent movement costs and maintains a target during travel, while still allowing a sufficiently better target to replace it. Its two-stage selection searches globally, then advances only as far as the actuator permits.

1) Accounting for a relocation cost:

  • Transit loss and estimate staleness scale as σf/ρmax per unit distance, so MoveUCB combines them under a movement-cost weight alongside actuation energy.The window length w divides both terms, and no online estimate of the drift rate is required.Source: 1) Accounting for a relocation cost · p. 4

2) Target persistence:

  • The persistence rule replaces a target only when another position exceeds it by margin ∆, suppressing mid-journey reversals while bounding the per-slot acquisition-value loss by ∆.The rule limits target changes rather than freezing the antenna; the passage contrasts it with travel budgets and distance penalties.Source: 2) Target persistence · p. 5

3) Two-stage selection:

  • MoveUCB maximizes its acquisition function over the entire region to select a target, then projects that target onto the reachable interval for the next position.The global search can select a distant peak that a search restricted to one-slot-reachable positions would miss; the one-dimensional search uses a uniform grid.Source: 3) Two-stage selection · p. 5

4) Complexity:

MoveUCB’s analysis gives its per-slot computational cost and establishes the assumptions and information-gain tools used to bound regret. The guarantees apply to a fixed but unknown reward sequence whose accumulated variation is sublinear in the horizon.

4) Complexity:

  • Per slot, MoveUCB costs O(w^3 + Gw^2) for a grid of G points, with rank-one updates and downdates avoiding repeated cubic factorization.The movement-aware components add no computational cost beyond the posterior computation.Source: 4) Complexity · p. 5

IV. REGRET ANALYSIS

  • The regret analysis handles estimation and staleness as in non-stationary kernelized bandits, while bounding travel through target switches and reverse information-gain decomposition.The number of target switches is tied to persistence, and the reverse decomposition is used to bound it.Source: §IV · p. 5, A. Preliminaries · p. 6

A. Preliminaries

  • Rewards are fixed but unknown per sequence, and sublinear regret requires only that the variation budget V_T=o(T), without a statistical drift model.Reward regularity bounds the RKHS norm and reward magnitude; observation noise is conditionally R-sub-Gaussian.Source: A. Preliminaries · pp. 5–6
  • The chain decomposition bounds posterior uncertainty in both forward and reverse sample order, and its determinant-based argument covers adaptively chosen positions.For the specified kernel, the maximum information gain is O((log n)^2), evaluated at the window size w.Source: A. Preliminaries · p. 6
  • A dwell slot ends at the target, while a transit slot leaves the antenna en route; T_trans counts transit slots.Source: A. Preliminaries · p. 6

B. Bounding the Travel

MoveUCB bounds travel by making movement depend on target switches rather than the trajectory alone. Persistence limits switches, whose number is controlled by how quickly the acquisition function changes.

B. Bounding the Travel

  • Actuation energy and transit reward loss both scale with distance, so bounding travel bounds their combined contribution to regret.The transit reward loss is bounded by 2g_max per transit slot; a smaller ∆ raises the travel contribution while reducing per-slot loss.Source: B. Bounding the Travel · pp. 6–7
  • Without persistence, expiring observations can repeatedly draw the learner back to old positions, producing Θ(ρ_max T) travel even for a static reward function.The persistence rule ties travel to the number of target switches.Source: B. Bounding the Travel · p. 6
  • With a fixed target, each run moves monotonically toward it and covers at most D, so total travel is bounded by D times the number of target runs.Each transit slot displaces the antenna by exactly ρ_max, linking transit count to total travel.Source: B. Bounding the Travel · p. 6
  • A switch requires the current target to fall more than ∆ behind the best position, and the switch count is bounded by the accumulated movement S_T of the acquisition function.The acquisition function’s movement accounts for posterior updates, reward drift, and the moving reference point of the movement-cost term.Source: B. Bounding the Travel · pp. 6–7

C. Main Result

Under the stated assumptions and theorem settings, MoveUCB achieves sublinear regret whenever the variation budget is o(T). The movement-related exponent is 7/8, while physical actuator parameters affect constants rather than the exponent.

C. Main Result

  • A longer window trades noise averaging for staler observations, while a smaller margin reduces per-slot acquisition loss at the expense of more switching and travel.The movement-cost coefficient does not appear in the guarantee; it governs switch distance rather than switch frequency.Source: C. Main Result · p. 7
  • Regret is sublinear whenever V_T=o(T), with probability at least 1−δ under the theorem’s assumptions and parameter settings.The bound is established against the per-slot maximizer over D, counting slots when the antenna is en route in full.Source: C. Main Result · p. 7
  • The regret exponent is 7/8 from movement, versus 3/4 for estimation and staleness; D, ρ_max, and c_e affect only the constant.The movement-related exponent reflects the cost of a decision variable that must travel to where it is evaluated.Source: C. Main Result · p. 7

V. SIMULATION RESULTS

The section validates the theoretical analysis and evaluates MoveUCB against a movement-aware oracle and several baselines.

V. SIMULATION RESULTS

  • The simulations validate the theoretical analysis and evaluate MoveUCB against a movement-aware oracle and several baselines.Source: §V · p. 7

A. Benchmarks

The evaluation compares MoveUCB with a movement-aware oracle and baselines under shared movement costs and reachability constraints. It specifies the simulation setup and varies scenario and drift model to test global-search opportunities and variation-budget regimes.

A. Benchmarks

  • The oracle uses non-causal knowledge and exact backward dynamic programming to define regret; GP-UCB, Sweep-and-hold, ϵ-greedy, and Fixed serve as baselines.GP-UCB shares MoveUCB’s estimator, kernel, and window but searches only reachable positions; the other baselines use sweeping, random exploration, or no repositioning.Source: A. Benchmarks · pp. 7–8

B. Setup

  • The evaluation horizon is 15,300 slots (76.5 s); results average 48 independent realizations of the channel, blockage position, and noise, with error bars showing ± one standard error.Every policy except the fixed antenna uses 100 spatial samples from a single frozen channel snapshot at t=0; Sweep-and-hold uses them to select its first holding position, not to calibrate an estimator. These pre-evaluation samples have no reachability constraint; evaluated GP-based trajectories obey reachability constraint (1).Source: B. Setup · p. 8

C. Scenarios and Drift Models

  • Dense multipath and partial blockage vary the opportunity for global search, while continuous drift has V_T = Θ(T) and piecewise-stationary drift has V_T = Θ(S).In the partial-blockage scenario, a search begun in shadow has no reachable direction of improvement; letting S grow more slowly than T yields V_T = o(T).Source: C. Scenarios and Drift Models · pp. 8–9

D. Results

MoveUCB leads the reported comparisons in regret while using less travel, and the results examine how that performance depends on movement cost, variation, kernel choice, confidence, and design components. The conclusion states a sublinear dynamic-regret guarantee under sublinear channel variation and identifies two open extensions.

D. Results

  • MoveUCB attains the lowest regret in both scenarios and uses about half the travel of GP-UCB, the closest moving baseline.The conclusion also reports the lowest regret among the baselines while covering about half the travel of the closest one.Source: D. Results · p. 9, §VI · p. 11
  • As movement cost rises to ce = 66, MoveUCB travel falls from 99λ to 58λ and its margin over the closest competitor widens from 0.20 to 0.51.The closest baseline changes from GP-UCB to Sweep-and-hold between ce = 10 and 30.Source: D. Results · p. 9
  • MoveUCB leads at every tested variation budget, but its advantage is largest when channel reconfiguration is rare relative to the time needed to relocate.The reported traversal takes 100 slots; the advantage shrinks as the variation budget grows.Source: D. Results · p. 9
  • The Physics kernel attains the lowest RMSE when evaluated alongside three alternatives on the same MoveUCB observation windows.Those windows contain 300 observations at only nine distinct positions, so estimates elsewhere are extrapolations.Source: D. Results · pp. 9–10
  • MoveUCB has its lowest regret at β = 1.5; beyond that, its travel increases by an order of magnitude and transit approaches half the horizon.At their respective optima, GP-UCB remains 55% above MoveUCB in regret and travels twice as far.Source: D. Results · p. 10
  • In the ablation, global search nearly halves regret from the local rule; adding movement cost and persistence then sharply reduces travel and switches with little regret change.The switch count falls from Nsw = 4778 to 2770 with movement cost, then to 298 with persistence; at ce = 30, movement cost also reduces regret relative to global search by 0.417.Source: D. Results · pp. 10–11, Figure 7 · p. 11

VI. Conclusion

  • The conclusion gives two open directions: multiple movable antennas with coupled rewards, and displacement-dependent actuation intervals requiring a revised optimism argument.It also states that sublinear dynamic regret holds whenever channel variation is sublinear, with physical parameters entering only through constants.Source: §VI · p. 11

APPENDIX A PROOF OF THEOREM 1

The proof controls estimation error under drift, bounds travel through a reverse information-gain argument, and combines those bounds with slot-wise regret accounting. Its time-varying confidence bound separates observation noise from staleness in the window of retained samples.

APPENDIX A PROOF OF THEOREM 1

  • The proof aligns the reward under gt with the position selected under the preceding function, charging at most VT and an endpoint constant for the index shift.It then bounds dwell-slot regret using confidence widths, staleness, movement cost, and the persistence margin, while transit slots are bounded by 2gmax.Source: Appendix A · p. 11
  • Reverse chain decomposition bounds travel, and combining its switch-count bound with movement expenditure and estimation error yields the regret bound.The grid approximation contributes O(LU∆gT), which is O(1) under ∆g = O(D/T).Source: Appendix A · pp. 11–12

A. Confidence Bound Under Drift

  • The time-varying confidence bound separates static kernel-regression error from staleness caused by earlier functions retained in the window.A union bound gives the simultaneous confidence event, while staleness is controlled by the window's accumulated variation and contributes the w^3/2 VT scaling.Source: A. Confidence Bound Under Drift · p. 12

B. Uncertainty Sums

The uncertainty-sum argument bounds both estimation uncertainty and travel-related uncertainty by using forward and reverse conditioning over blocks. The surface-drift analysis controls how the target objective changes as the observation window advances.

B. Uncertainty Sums

  • Forward conditioning on past samples governs estimation, while reverse conditioning on future samples governs travel; blockwise bounds yield uncertainty sums of order O(Tγw/w).The lemma applies to any adaptively chosen or fixed position sequence.Source: B. Uncertainty Sums · p. 12

C. Surface Drift

  • Surface drift is bounded by tracking changes in the windowed mean and variance as one sample is exchanged, with movement-cost variation absorbed by the leading bound.The movement-cost contribution is O(T/w), and the surface-drift lemma holds on the noise and confidence events.Source: C. Surface Drift · pp. 12–13

D. Dwell Regret

The persistence rule keeps the selected target within a margin of the current acquisition-function maximum. This invariant supports the dwell-regret bound without requiring the target to be exactly optimal.

D. Dwell Regret

  • The retained target is never more than ∆ below the maximum of Ft, whether it remains from the prior slot or is reset to the maximizer.On dwell slots, this invariant supports a regret bound combining uncertainty, staleness, movement cost, and ∆.Source: D. Dwell Regret · p. 13
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