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Distributional Uncertainty and Adaptive Decision-Making in System

This paper extends the monotone co-design framework to handle distributional uncertainty by modeling design outcomes as probability distributions and enabling adaptive decision-making through Markov-kernel re-parameterizations, thereby overcoming the limitations of traditional interval-based models in capturing probabilistic risks and multi-stage design choices.

Original authors: Yujun Huang, Gioele Zardini

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yujun Huang, Gioele Zardini

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the architect of a complex machine, like a delivery drone. You need to choose the best combination of parts: a motor, a battery, a camera, and a flight computer. But here's the catch: nothing is perfect or certain.

The battery might last 10% longer or 10% shorter than the label says.
The wind might be calm or a hurricane.
You might need to carry a light package today and a heavy one tomorrow.

This paper presents a new, smarter way to design these machines. It moves away from "guessing the worst case" to understanding the full range of possibilities and making decisions that adapt as you learn more.

Here is the breakdown using simple analogies:

1. The Old Way: The "Worst-Case" Blanket

The Problem:
Traditionally, engineers design for the "worst-case scenario." Imagine you are packing for a trip. You don't know if it will rain or shine, so you bring a heavy, waterproof tent, a raincoat, and an umbrella, just in case.

The Flaw: This is safe, but it's heavy and expensive. You might end up carrying a tent when it's a sunny day.
The Limitation: This method treats uncertainty like a simple box (e.g., "The battery is between 90% and 110% of capacity"). It can't tell you how likely it is to be 90% vs. 110%, nor can it help you change your mind halfway through the trip if you see the sun coming out.

2. The New Way: The "Probabilistic Weather Forecast"

The Solution:
The authors propose a framework that treats uncertainty like a weather forecast. Instead of just saying "It might rain," they say, "There is a 70% chance of light rain, a 20% chance of a storm, and a 10% chance of sunshine."

Distributional Uncertainty: Instead of a simple box, they use a probability cloud. They know exactly how likely every possible outcome is.
The Magic Ingredient (Quasi-Measurable Spaces): To make the math work without getting stuck in impossible loops (like trying to measure the size of a shape that changes while you measure it), they use a special mathematical tool called "Quasi-Measurable Spaces." Think of this as a universal translator that allows different types of uncertainty to talk to each other without breaking the rules of logic.

3. The Superpower: Adaptive Decision-Making

The "Wait-and-See" Strategy:
The most powerful part of this new framework is Adaptivity.

The Old Way: You must buy your tent, umbrella, and sunglasses before you leave the house. You are stuck with them.
The New Way: You leave the house with a plan, but you make decisions as you go.
- Stage 1: You check the sky (observe the data).
- Stage 2: If it looks cloudy, you grab the umbrella. If it's sunny, you leave it behind.
- Stage 3: If the wind picks up, you switch to a more stable flight mode.

In the paper, this is modeled using Markov Kernels. Imagine a flowchart that updates itself. As you get new information (like a sensor reading or a task change), the flowchart reroutes you to the best decision for that specific moment, rather than forcing you to stick to a rigid plan made in the dark.

4. The Case Study: The Delivery Drone

To prove this works, the authors tested it on a delivery drone.

The Scenario: The drone needs to deliver packages of different weights over different distances. The battery performance and motor efficiency vary randomly.
The Result:
- Old Method: The drone was designed to carry the heaviest possible load in the worst possible weather. It was heavy, expensive, and inefficient for normal days.
- New Method: The drone's design adapts.
  - If the battery is performing well, it chooses a lighter, cheaper motor.
  - If the battery is weak, it switches to a more efficient (but heavier) motor.
  - The Outcome: The drone saves a massive amount of money and energy on average, while still being safe enough for the rare bad days. It understands the risk rather than just fearing it.

5. Why This Matters (The "So What?")

This paper gives engineers a language for risk.

It allows them to ask: "What is the probability that this design will fail?" instead of just "Will it fail?"
It allows them to ask: "If I wait to see the weather before buying the battery, how much money can I save?"
It turns system design from a rigid, static puzzle into a dynamic, living strategy that learns and adapts.

Summary Analogy

Think of designing a system like cooking a meal for a party:

Old Way (Interval): You assume everyone will be hungry and eat a huge steak. You buy 50 steaks. If only 10 people show up, you waste money. If 60 show up, you run out.
New Way (Distributional & Adaptive): You know the guest list is a bit fuzzy. You buy ingredients for a flexible menu.
- You start by buying the basics.
- As guests arrive (observations), you check how hungry they are.
- If they are light eaters, you serve a salad. If they are hungry, you grill the steaks.
- You use a "probability map" of how hungry people usually get to decide how much to buy initially.

This paper provides the mathematical recipe to make that flexible, adaptive cooking possible for complex machines like robots, self-driving cars, and power grids.

1. Problem Statement

Complex engineered systems (e.g., Cyber-Physical Systems, robotics) require coordinated design choices across heterogeneous components (hardware and software) under multiple conflicting objectives.

Limitations of Existing Models: Current "monotone co-design" frameworks model subsystems as Design Problems (DPs)—relations between functionalities and resources in partially ordered sets (posets). However, existing methods handle uncertainty primarily via interval bounds (worst-case and best-case scenarios).
The Gap: Interval models are too coarse. They cannot:
1. Represent probabilistic risk (e.g., chance constraints, confidence levels, quantiles).
2. Support multi-stage adaptive decisions where later choices depend on intermediate observations (e.g., test outcomes or realized environmental conditions).
3. Distinguish between designs that are "almost always safe" versus those that are uniformly conservative.

The paper aims to extend monotone co-design to handle distributional uncertainty and adaptive decision-making while preserving the compositional structure of the framework.

2. Methodology

The authors develop a mathematical framework that lifts standard co-design operations from deterministic sets to probability distributions over design problems.

A. Mathematical Foundations: Quasi-Measurable Spaces

To avoid classical measurability pathologies (e.g., the impossibility of defining a sigma-algebra on function spaces that makes evaluation measurable), the authors utilize Quasi-Measurable Spaces (QMS) and Quasi-Universal Spaces (QUS).

QMS: A set $A$ equipped with a distinguished set of "random variables" (maps from a sample space $\Omega$ to $A$ ).
QUS: A QMS where the sample space $\Omega$ is a Polish space (e.g., $\mathbb{R}^n$ ).
Key Property: This framework ensures that the space of Design Problems ($DP$) and the operations on them (series, parallel, feedback) remain quasi-measurable, allowing for the composition of probability distributions over DPs.

B. Distributional Design Problems

Random DPs: Instead of a single DP, the system is modeled as a random variable taking values in the space of DPs.
Compositionality: The standard co-design operations (Series, Parallel, Feedback, Union, Intersection) are proven to be quasi-measurable. This means they can be "lifted" to operate on distributions of DPs.
- Example: The "Union" of two random DPs represents a design choice where the designer can select the best outcome after observing the random realization.

C. Adaptive Decision-Making via Markov Kernels

The framework introduces Markov kernels to model multi-stage adaptive processes.

Re-parameterization: Design processes are modeled as a sequence of stages where specifications and implementations are updated based on observations.
Structure: A policy is defined as a Markov kernel that maps previous observations and decisions to the next set of design choices. This allows the framework to distinguish between "here-and-now" decisions (made before uncertainty is resolved) and "wait-and-see" decisions (made after observing outcomes).

D. Queries and Observations

The authors define interfaces to extract engineering-relevant metrics from the distributional models:

Queries: Maps from distributions of DPs to scalar or probabilistic metrics (e.g., probability of feasibility, expected resource cost).
Observations: Maps from a specific DP to a value (e.g., the minimal resource required for a fixed functionality). These can be lifted to queries on distributions.
Confidence Bounds: The framework generalizes interval bounds to inner and outer confidence bounds, providing probabilistic guarantees (e.g., "with 95% probability, the design lies within this performance envelope").

3. Key Contributions

Distributional Extension of Co-design: The first framework to simultaneously handle general distributional uncertainty over compositional design models and multi-stage adaptivity, while remaining closed under standard co-design compositions.
Mathematical Rigor: The use of Quasi-Measurable Spaces to rigorously define probability measures over Design Problems, overcoming measurability issues that plague higher-order probability on function spaces.
Adaptive Policy Formalization: A formal language for multi-stage design policies using Markov kernels, allowing designers to explicitly model when decisions are made relative to information acquisition.
Probabilistic Query Interface: A set of operators to compute risk-aware metrics (feasibility probabilities, quantiles, confidence bounds) directly from the compositional model.

4. Results: UAV Case Study

The framework was validated using a task-driven Unmanned Aerial Vehicle (UAV) co-design problem involving actuation and battery subsystems.

Comparison of Models:
- Deterministic: Shows the optimal trade-off curve.
- Interval Uncertainty: Produces an optimistic/pessimistic envelope. However, the paper demonstrates that this envelope is misleading for "here-and-now" decisions because it assumes the designer can switch components after uncertainty is realized (wait-and-see), which is often not feasible.
- Distributional & Adaptive: Uses Monte Carlo sampling to generate distributions of minimal lifetime costs for various payloads.
Adaptivity Levels: The study compared three levels of adaptivity:
1. Non-adaptive: Component choices (actuator/battery type) made before observing parameters.
2. Partly adaptive: Actuator chosen first; battery chosen after observing actuator specs.
3. Fully adaptive: All component choices made after observing all parameters.
Findings:
- Increased adaptivity significantly reduced the expected lifetime cost, particularly near feasibility boundaries.
- However, adaptivity did not always improve the worst-case (upper tail) performance, highlighting the trade-off between average performance and risk.
- Interval models failed to capture these nuanced trade-offs, whereas the distributional framework explicitly quantified the probability of different outcomes.

5. Significance

This paper bridges the gap between compositional system design and stochastic optimization.

Risk-Aware Design: It moves beyond worst-case robustness, enabling designers to quantify and optimize for specific risk levels (e.g., 99% feasibility).
Information-Dependent Design: It provides a rigorous mathematical tool to evaluate the value of information. Designers can now formally ask: "How much does our performance improve if we delay the selection of the battery until after we test the actuator?"
Scalability: By preserving the compositional nature of co-design, the framework allows complex systems to be built from uncertain subsystems without losing the ability to reason about the whole system's probabilistic behavior.

In summary, the authors provide a foundational theory and practical tools for designing complex systems where uncertainty is probabilistic and decisions are made sequentially as information becomes available.