Signal-Dependent Strategies

• Signal-dependent strategies are decision protocols that adjust actions and information acquisition based on the statistical properties of observed signals.
• They are applied in adaptive Bayesian learning, image restoration, radar design, and optimal trading to manage noise and dynamic feedback.
• By internalizing the dynamic value and structure of signals, these strategies optimize learning efficiency, risk management, and overall system performance.

Signal-dependent strategies are decision protocols in which the agent's actions, information acquisition, or response are explicitly adapted to the realized or anticipated statistical properties of observed signals—particularly when the signal's informativeness, cost, or structure is endogenously modulated by the agent’s own choices or by latent states of the environment. Such frameworks permeate adaptive Bayesian learning, stochastic control, information economics, sensing, radar/communication system design, and decentralized multi-agent coordination, as well as chemotactic modeling and social learning. Core to these strategies is the recognition that the value, reliability, or externality of a signal is itself action- or state-contingent, mandating policies that internalize not just direct payoffs but the dynamic, often nonlinear, feedback between decisions and future information structure.

1. Bayesian Frameworks with Action- and State-Dependent Signal Variance

In adaptive Bayesian learning models, signal-dependent strategies arise prominently when the signal noise variance is a function of both the agent's chosen action and the underlying (possibly unknown) state. At each period $t$, the agent selects action $a_t \in \mathbb{R}$, then observes a noisy signal $y_t = \theta + \epsilon_t$, with $\epsilon_t \sim N(0, \sigma^2(a_t, s_t))$, where $s_t$ may be $\theta$ or a known transformation thereof. The Bayesian update for the posterior $\theta \mid F_t$ is Gaussian if the prior and noise models are, with posterior variance evolving as

$$\sigma_t^{-2} = \sigma_{t-1}^{-2} + f(a_t, s_t)^{-1}$$

where $f$ is the action-state-dependent variance function. This framework allows for optimal decision rules such as utility maximization

$$a_t^* = \argmax_{a \in \mathcal{A}} \mathbb{E}\big[ U(a, \theta) \mid F_{t-1} \big]$$

or variance minimization (Bayesian A-optimal design):

$$a_t^* = \argmin_{a \in \mathcal{A}} \mathbb{E}[\sigma_t^2 \mid F_{t-1}, a_t=a]$$

Derivatives with respect to action determine the information-maximizing policy when $f$ is smooth. Signal-dependent strategies thus endogenize experiment design, collective learning (including social learning/externalities), and create economic feedbacks such as “uncertainty traps” or action-dependent observability in engineering (Hou, 2023).

2. Canonical Models and Applications

Signal-dependent strategies are widely instantiated in models across disciplines:

• Experiment Design: In sequential data collection, the agent’s choice of sampling effort or allocation (e.g. number of i.i.d. subsignals) sets the signal-to-noise ratio, directly controlling learning efficiency (Hou, 2023).
• Social Learning: When many agents choose whether to generate signals, the aggregate precision becomes endogenous to participation levels. Coordination failures can create equilibria where no agent finds it worthwhile to produce information due to others’ inaction (“uncertainty trap”) (Hou, 2023).
• Volatility/Non-Gaussian Uncertainty: With state-dependent signal variance (e.g. $f(s) = s^2$), posterior updating is non-conjugate and necessitates numerical approaches, underscoring regimes where higher volatility begets reduced informativeness, entangling learning with endogenous uncertainty (Hou, 2023).
• Forecast/Tracking Error Models: If $f(a, s) = k|a-s|^2$, signal precision is maximized by accurate forecasting, making information acquisition itself dependent on action quality; this appears in iterated forecasting, robust controls, and mechanism design (Hou, 2023).

3. Information Acquisition, Dynamic Learning, and Time-Risk Preferences

Signal-dependent learning strategies intersect with preferences over information acquisition under resource constraints. In belief updating with entropy-rate constraints, the agent can select among processes subject to a fixed rate of uncertainty reduction. Strategies maximizing time-risk (“Greedy Exploitation”) target thresholds via state-dependent jump processes biased to the closest target in Bregman divergence, producing highly dispersed stopping-time distributions. Conversely, pure accumulation strategies minimize time risk via jumps that preserve entropy, resulting in deterministic hitting times. The specific realized beliefs and subsequent information-acquisition schedule are thus signal- and state-dependent, with the optimal strategy parameterized by the agent’s convexity in time-risk preferences (Chen et al., 2018).

4. Signal-Dependent Strategies in Engineering: Sensing, Image Restoration, and Radar

In signal processing and imaging, signal-dependent noise arises fundamentally from physics:

• Image Restoration: Camera observations exhibit signal-dependent variance, typically as mixed Poisson-Gaussian noise. MAP estimation frameworks explicitly incorporate this structure, and update rules or noise-adaptive proximal iterations are derived for efficient denoising and deblurring. Quantization and other digitization artifacts, whose statistics may also be signal-level dependent, are incorporated through pixelwise cost constructions and integrated into variable-splitting and augmented Lagrangian optimization loops (Chakrabarti et al., 2012).
• Radar and Communications: Radar STAP (space-time adaptive processing) and FDA (frequency-diverse array) radar problems feature signal-dependent interference, e.g. when ground clutter or mainlobe jamming power depends on the current transmit waveform. Optimal waveform and filter design then jointly adapt to the induced interference structure, with constraints and objectives quartic in the design variables. State-of-the-art methods use biquadratic SDP relaxations, majorization-minimization, or coordinate descent algorithms to navigate the nonconvexity introduced by the signal-dependent terms (O'Rourke et al., 2017, Jia et al., 2022, Yang et al., 2021).
• Photon-Counting Communications: Under Poisson shot noise, variance intrinsically scales with (action-dependent) optical intensity. Frame synchronization and multi-user detection in optical wireless communications then require iterative, statistically principled estimators that directly account for the signal-dependent shot-noise model, outperforming standard AWGN/linear approximations (Li et al., 18 Dec 2025).

5. Adaptive Sensing, Sampling, and Multi-Agent Signal Dependence

Optimal sampling in adaptive estimation and graph signal processing exhibits signal-dependent design:

• Adaptive LMS/RLS over graphs utilizes node sampling probabilities $\mathbf{p}$ that solve for coverage/convergence/accuracy tradeoffs. The optimal $\mathbf{p}^*$ depends on the spectral projections of the underlying signal and the noise variance, such that nodes “rich” in spectral content or with lower noise receive higher sampling probability. These distributions are explicitly determined via optimization programs constrained by desired mean-square deviation and learning rate, often requiring fractional or successive convex approximations (Lorenzo et al., 2017).
• In multi-agent coordination and reinforcement learning, signal-mediated strategies employ exogenous signals as coordination devices, enabling decentralized agents to share latent intent and synchronize action without in-game communication. Policy structures are conditioned on random signals whose distribution is optimized during centralized training, then used for decentralized equilibrium play in zero-sum games and adversarial environments (Cacciamani et al., 2021).
• Cooperative Greedy Pursuit in Sparse Approximation: Block-wise pursuit algorithms dynamically adjust which signal blocks to upgrade or prune based on instantaneous residuals, yielding a global greedy allocation that is dependent on local signal properties and cross-block residual structure (Rebollo-Neira, 2015).

6. Signal-Dependent Strategies in Economics, Social Science, and Control

Economic and trading models critically embed signal-dependent strategies:

• Optimal Trading: When asset prices are mean-reverting and predictive signals are themselves Markovian (e.g. order book imbalance), optimal liquidation/execution policies are derived from variational calculus over cost functionals with signal-indexed drift. The trading schedule depends not only directly on the current signal value but anticipates the evolution of signal-driven expected price moves and transient market impact. Explicit solutions (singular or continuous depending on impact decay) are provided for Ornstein-Uhlenbeck signal processes and transient/instantaneous impact kernels (Lehalle et al., 2017, Bellani et al., 2018).
• Social Learning and Uncertainty Traps: When agents' data generation increases precision for themselves and others, and when observation costs or incentives for information production are endogenous, nontrivial equilibria can arise. Endogenous “uncertainty traps” occur if agents wait for others to produce information, and everyone’s incentive to act is tied to the anticipated informational environment (Hou, 2023).
• Communication Games and Multi-Agent Signaling: In theoretical games such as the Signaler-Responder, equilibrium play can yield signal-dependent communication (e.g., send a signal only when in need), learned via distributed Bayesian updating and Thompson sampling. The emergent communication regime depends delicately on the reward-cost-penalty structure and endogenous beliefs over need and response (Bhuckory et al., 2024).

7. Theoretical and Practical Implications

Signal-dependent strategies unify a mathematical insight: the value, cost, and structure of information acquisition, action choice, and policy optimization are typically intertwined via endogenous or feedback mechanisms that depend on the realized or chosen signals. This duality between actions shaping observations and observations shaping actions is fundamental in:

• Active learning/experiment design, where information gains can be efficiently budgeted by targeting actions that maximize future information precision.
• Adaptive filtering and sensing, where resource-constrained signal acquisition is allocated optimally in accordance with prior signal structure and desired performance.
• Strategic and economic environments, where agents rationally internalize signal-dependent utilities, risks, or social externalities, leading to richer equilibrium predictions and informing mechanism design in environments with data production feedbacks.
• Engineering systems, where robust system performance is best achieved by action policies that anticipate and shape the statistics of the future information environment.

Mathematically, such strategies require sophisticated coupling of Bayesian inference, stochastic control, convex/nonconvex optimization, game-theoretic equilibrium analysis, and information-theoretic bounds.

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References

• Adaptive Bayesian Learning with Action and State-Dependent Signal Variance (Hou, 2023)
• Information Acquisition and Time-Risk Preference (Chen et al., 2018)
• Image Restoration with Signal-dependent Camera Noise (Chakrabarti et al., 2012)
• Multi-Spectrally Constrained Transceiver Design against Signal-Dependent Interference (Yang et al., 2021)
• Cooperative Greedy Pursuit Strategies for Sparse Signal Representation by Partitioning (Rebollo-Neira, 2015)
• Static vs Adaptive Strategies for Optimal Execution with Signals (Bellani et al., 2018)
• Incorporating Signals into Optimal Trading (Lehalle et al., 2017)
• Adaptive Graph Signal Processing: Algorithms and Optimal Sampling Strategies (Lorenzo et al., 2017)
• Multi-Agent Coordination in Adversarial Environments through Signal Mediated Strategies (Cacciamani et al., 2021)
• Synchronization, Identification, and Signal Detection for Underwater Photon-Counting Communications With Input-Dependent Shot Noise (Li et al., 18 Dec 2025)
• Unsupervised Denoising for Signal-Dependent and Row-Correlated Imaging Noise (Salmon et al., 2023)
• Signal Synchronization Strategies and Time Domain SETI with Gaia DR3 (Nilipour et al., 2023)
• The Signaler-Responder Game: Learning to Communicate using Thompson Sampling (Bhuckory et al., 2024)