Controllable Information Acquisition

• Controllable information acquisition is a framework that optimizes the selection, timing, and precision of data collection by balancing benefits against costs.
• The approach integrates formal models from stochastic control, online learning, and LQG to derive optimal probing and stopping policies.
• Algorithms use threshold-based decisions to achieve near-optimal performance while managing resource, time, and risk constraints across diverse applications.

Controllable information acquisition refers to decision-theoretic and algorithmic frameworks in which the selection, frequency, precision, or timing of information acquisition actions are themselves subject to optimization, rather than being passively determined by system dynamics or exogenous schedules. In such models, the information-gathering process is shaped by explicit costs, constraints, or trade-offs, and the optimal information–acquisition policy is derived jointly with the task-specific decision or control strategy. This concept is central in areas ranging from stochastic control, online learning, contract theory, economics, to networked systems, where the choice of how much, when, and what type of information to acquire must be balanced against resource, time, or risk constraints.

1. Formal Models and Problem Structure

In formal treatments, controllable information acquisition is embedded within sequential decision problems in which the value and cost of information are incorporated explicitly. A paradigmatic example appears in multichannel wireless networks, where a transmitter seeks to maximize its transmission rate by probing a subset of $n$ channels, each with $K$ possible states, incurring channel-specific costs $c_j$ (0804.1724):

• System Configuration: Each channel $j$ is in state $i \in \{0,\ldots, K-1\}$ with reward $r_i$ and occurrence probability $p_{ij}$. The decision process is a sequential tree wherein, at each node, the sender decides whether to probe another channel and, if so, which one.
• Objective: Maximize a utility (gain) function given by

$$G = \sum_{i=0}^{K-1} q_i r_i - \sum_{j \in \mathcal{P}} c_j$$

where $q_i$ is the probability the transmission uses a channel in state $i$ and $\mathcal{P}$ indexes probed channels.

• Policy Actions: The decision-maker controls which probes to perform, their order, and when to stop and transmit using either a probed or a (potentially unprobed) “backup” channel.

Analogous structures appear in online decision-making where the action space includes both information–acquisition and task–execution actions (Atan et al., 2016), as well as in continuous-time control, where costly experiments can be performed to directly shape the conditional distribution of hidden state parameters (Knochenhauer et al., 19 Aug 2024, Liang et al., 17 Aug 2025).

2. Utility Functions and Trade-off Formulation

Controllability is achieved by quantifying both the value and the cost of information within a utility framework. The canonical formulation balances expected benefit from improved decisions against the cumulative acquisition cost:

• Wireless diagnosis (0804.1724):

$$G = \mathbb{E}[\text{Transmission Reward}] - \sum \text{Probing Costs}$$

In unsaturated regimes (i.e., with input queues), additional constraints, such as transmission rate stability, are introduced:

$$\max G \quad \text{subject to average rate} \geq \lambda$$

• Online learning (Atan et al., 2016):

$$\text{Gain} = \frac 1 T \sum_{t=1}^T \left[ \beta r_t - \sum_{i \in \mathcal{I}_t} c_i \right]$$

where the cost $c_i$ is for observation $i$, and $\beta$ scales the benefit.

• Dynamic control (Knochenhauer et al., 19 Aug 2024): The expected cost in an LQG problem includes a quadratic penalty on information acquisition rate $h_t$:

$\int_0^\infty c(h_t) dt$

with $c(h)$ convex and increasing, e.g., $c(h) = \zeta h^2$ or $c(h) = \zeta h^{1+\varepsilon}$, enforcing a diminishing returns property.

• Portfolio selection (Liang et al., 17 Aug 2025): Acquisition cost is coupled directly to the chosen instantaneous signal precision $\theta_t$, penalizing $k(\theta_t^2)$ per unit time in the wealth evolution SDE.

The trade-off is therefore modeled by explicit marginal analysis: information is acquired only up to the point where its expected marginal benefit matches or exceeds its marginal cost, leading to the appearance of sharp thresholds or optimal stopping rules.

3. Algorithmic and Policy Characterizations

Algorithms for controllable information acquisition exploit the structure of the value–cost trade-off to produce succinct, efficiently computable policies:

• Structural theorems (0804.1724): There exists an optimal probing policy with a unique backup channel, yielding “threshold” and “order” properties—channels are probed in decreasing order of value-to-cost ratio (e.g., $p_{ij}/c_j$).
• Greedy and threshold policies: Policies probe sequentially, with each decision conditioned on the observed state and the current estimate of marginal value. If the expected incremental benefit of probing another channel falls below its cost, probing stops.
• Lagrangean relaxation: Constraints (e.g., minimum transmission rate) are incorporated as penalties, transforming constrained dynamic programs into tractable unconstrained forms (0804.1724).
• Dynamic programming reduction: In continuous-time LQG problems, the partially observed problem, after filtering, reduces to a deterministic optimal control problem for the conditional variance $\gamma_t$ (Knochenhauer et al., 19 Aug 2024). The HJB equation for the value function admits semi-explicit solution in terms of $\gamma$, with the optimal acquisition rate $h^*(\gamma)$ determined via feedback.
• Verification via method of characteristics: For Bayesian adaptive investment with costly information (Liang et al., 17 Aug 2025), the HJB equation for the value function is solved explicitly using an ansatz and the method of characteristics. This yields deterministic feedback forms for both signal precision and trading policy.

Illustrative table:

Setting	Acquisition Control	Policy Structure
Wireless multi-channel (0804.1724)	Channel probes	Threshold/ordered, backup
Online bandits (Atan et al., 2016)	Observation subset	Optimistic, confidence-driven
Continuous LQG (Knochenhauer et al., 19 Aug 2024)	Test rate $h_t$	Feedback in $\gamma_t$, threshold
Portfolio (Liang et al., 17 Aug 2025)	Signal precision $\theta_t$	Deterministic, decoupled, threshold

4. Characteristic Phenomena and Threshold Effects

Analysis of optimal policies in several frameworks reveals threshold-type behavior:

• Critical acquisition threshold: There exists a value $\gamma_D$ such that no information is gathered if the filtered uncertainty $\gamma$ is below $\gamma_D$ (Knochenhauer et al., 19 Aug 2024). Above this level, optimal acquisition is continuous and increasing in $\gamma$.
• Separation or decoupling: In models with certain utility structures (e.g., CARA utility and Gaussian priors), the optimal information acquisition rule and the trading or state control rule are deterministically decoupled (Liang et al., 17 Aug 2025). The information acquisition policy is a function only of the current posterior variance (filter), while the control strategy is of certainty-equivalent type.
• Asymptotic regimes: For quadratic cost, the optimal acquisition rate $h^*(\gamma)$ obeys

$$\lim_{\gamma\to\infty} \frac{h^*(\gamma)}{\gamma^{1/2}} = \text{constant}$$

reflecting more aggressive information collection under high uncertainty, and

$h^*(\gamma) \to 0 \quad \text{as} \ \gamma \to 0$

with polynomial vanishing depending on cost convexity (Knochenhauer et al., 19 Aug 2024).

5. Empirical and Practical Implications

Controllable information acquisition frameworks yield significant insights for real-world and engineered systems:

• Resource-constrained network protocols: Selective probing based on value-to-cost ordering drastically reduces energy and time consumption in wireless systems while retaining near-optimal throughput (0804.1724).
• Learning and experimentation policies: Online learning strategies that integrate the selection of costly observations with actions (as in healthcare diagnostics or finance) outperform naive full-information or myopic baselines by adaptively adjusting the degree of information acquisition as data accrues (Atan et al., 2016).
• Robustness in control: In partially observed control, explicit management of information cost yields more stable and predictable control under uncertainty, and clarifies the importance of only gathering information when uncertainty is economically meaningful (Knochenhauer et al., 19 Aug 2024, Liang et al., 17 Aug 2025).
• Algorithm complexity: The use of structure theorems, orderings, and relaxation ensures that practical algorithms for large-scale systems have polynomial complexity and admit performance guarantees (e.g., constant-factor optimality in gain or loss).

6. Applications and Extensions

Controllable information acquisition models have found applications in network scheduling, sequential experiment design, active sensing and robotics, online personalized recommendation, and contract and mechanism design. Notably:

• Wireless MAC and channel access: Foundation for scheduling and probing in multi-channel MAC protocols.
• Healthcare: Adaptive sequential testing strategies for diagnostics, with sublinear regret achieved by optimizing over joint spaces of test actions and treatments (Atan et al., 2016).
• Robotics: Active sensor management where the marginal benefit of additional information is continuously quantified and regulated (Jeong et al., 2019, Wakulicz et al., 2021).
• Finance: Portfolio optimization where the degree of costly signal acquisition is directly controlled and optimized over time (Liang et al., 17 Aug 2025).

7. Theoretical Significance

The rigorous modeling of controllable information acquisition challenges the classical separation principle in stochastic filtering and decision theory. In many frameworks, optimal learning is tightly coupled with the acquisition cost, inducing endogenous experimentation schedules, threshold effects, and rich feedback structures. The formalism underscores the necessity of treating information acquisition as a decision variable and drives the development of computationally tractable, provably good control, learning, and scheduling policies in information-rich and resource-limited modern systems.