⚡ electrical engineering

Conformal Predictive Programming for Chance Constrained Optimization

This paper introduces Conformal Predictive Programming (CPP), a framework that transforms chance-constrained optimization into a deterministic problem using conformal prediction's quantile lemma and an independent calibration step to provide robust a posteriori guarantees, even in settings where standard assumptions fail or distribution shifts occur.

Original authors: Yiqi Zhao, Xinyi Yu, Matteo Sesia, Jyotirmoy V. Deshmukh, Lars Lindemann

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yiqi Zhao, Xinyi Yu, Matteo Sesia, Jyotirmoy V. Deshmukh, Lars Lindemann

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 trying to program a robot to navigate a maze. You want it to take the shortest path possible, but there's a catch: the robot's sensors are a bit fuzzy. Sometimes it sees a wall where there isn't one, and sometimes it misses a wall that is actually there. You need a plan that guarantees the robot won't crash, even with those fuzzy sensors, but you don't know exactly how "fuzzy" the sensors are.

This is the problem of Chance Constrained Optimization. It's like trying to find the best route while saying, "I need to be 90% sure I won't hit a wall," without knowing the exact rules of the maze.

The Old Way: Guessing with Scenarios

Traditionally, engineers have used a method called the "Scenario Approach." Imagine you ask a group of friends to guess where the walls might be. You then build a path that avoids every single wall your friends guessed. If you ask enough friends, you might get lucky and find a safe path.

But here's the problem: if your friends are wrong about the type of walls (maybe they think walls are always straight, but in reality, they are curved), your path might fail. Also, if the maze is huge and complex, asking enough friends to be safe becomes a math nightmare that takes forever to solve. The paper argues that these old methods often rely on strict rules about the shape of the maze (like it must be perfectly round or smooth) that don't always exist in the real world.

The New Idea: Conformal Predictive Programming (CPP)

The authors of this paper propose a new tool called Conformal Predictive Programming (CPP). Think of CPP as a two-step "test and certify" process that works even when you don't know the exact rules of the game.

Step 1: The Practice Run (Optimization)
First, the robot runs a simulation using a set of practice data (let's say 200 practice runs). It tries to find the best path that avoids the walls in these specific practice runs. It's like the robot saying, "Okay, based on these 200 tries, this looks like the best route."

Step 2: The Safety Check (Calibration)
Here is where CPP gets clever. Instead of just trusting the practice run, the robot grabs a second, completely separate set of data (another 200 runs) to act as a referee. It checks the path it found in Step 1 against this new referee data.

Using a statistical trick called the "Quantile Lemma," CPP calculates a safety margin. It asks: "How often did this path fail in the referee data?" If it failed too often, the path is rejected. If it passed enough times, the system gives it a "certificate" saying, "We are 90% sure this path is safe."

Why This is a Big Deal

The paper highlights three main superpowers of CPP:

It Doesn't Need to Know the Rules: Unlike the old methods, CPP doesn't need to assume the maze is a perfect circle or that the walls are smooth. It works even if the data is weird, messy, or non-convex (twisted and turned).
It Has a "Safety Net" for the Future: The paper shows that CPP can give you a guarantee after you've done the math (called "a posteriori"). It's like building a bridge, testing it with a heavy truck, and then saying, "Okay, now we know for sure this bridge can hold 10 tons." The paper explicitly states that for many complex problems, the old methods couldn't give you this kind of "after-the-fact" guarantee without doing impossible math.
It Adapts to Changes: The authors show that CPP can handle "distribution shifts." Imagine the robot is trained in a sunny maze but has to drive in a rainy one. The "Robust CPP" version of the tool can adjust the safety margin to account for the rain, ensuring the robot stays safe even when the conditions change.

What the Numbers Say

The authors didn't just theorize; they ran simulations to prove it works.

In a convex (smooth) test case, they ran 300 experiments. The new method successfully kept the robot safe about 91% of the time (matching their target of 90% safety), while the old "discard" method only hit 87%.
In a non-convex (twisted) test case, which is much harder, the new method still hit 90% and 89% safety rates.
They also tested a "Stochastic Optimal Control" scenario (a robot moving in 2D space). When they introduced a "distribution shift" (simulating a change from a variance of 0.012 to 0.013), the standard method failed to cover the safety target (only 80%), but the new "Robust CPP" method hit 96%.

What They Don't Claim

It's important to note what this paper doesn't say. It doesn't claim to have solved every possible optimization problem in the universe. It specifically notes that if you don't have a second set of data for the "referee" step, you can't get these specific guarantees. Also, while the method is efficient, it still requires two datasets (one for planning, one for checking), which is a trade-off compared to methods that try to do it all in one go.

The Bottom Line

The paper suggests that by splitting the problem into a "planning" phase and a "calibration" phase, we can build robots and systems that are safer and more reliable, even when the world is messy and unpredictable. It's a way of saying, "We don't need to know everything about the future to make a safe plan today; we just need a good way to check our work."

Technical Summary: Conformal Predictive Programming for Chance Constrained Optimization

Problem Statement
The paper addresses Chance Constrained Optimization (CCO) problems, which involve minimizing a cost function $J(x)$ subject to constraints that must hold with a high probability ( $1-\delta$ ) over a random vector $Y$ . Formally, the problem is:
$\min_{x \in X} J(x) \quad \text{s.t.} \quad P(f(x, Y) \leq 0) \geq 1 - \delta$
where $f$ is a Borel measurable function and $Y$ follows an unknown distribution $P_Y$ . The primary challenge in solving CCO is that the distribution $P_Y$ is typically unknown, and even if known, evaluating the probability constraint requires solving complex, often intractable integrals. Existing sampling-based approaches, such as the Scenario Approach (SA) and Sample Average Approximation (SAA), often rely on strong structural assumptions (e.g., convexity of $J$ , $f$ , and $X$ ) to provide a priori guarantees. When these assumptions do not hold, or when a priori guarantees are too conservative or computationally expensive to compute (e.g., requiring the identification of support constraints), existing methods struggle to provide reliable a posteriori feasibility guarantees.

Methodology: Conformal Predictive Programming (CPP)
The authors propose Conformal Predictive Programming (CPP), a framework that transforms the stochastic CCO problem into a deterministic optimization problem using Conformal Prediction (CP). CPP operates in two distinct phases:

Optimization Step (Quantile Reformulation):
Instead of using the raw chance constraint, CPP replaces it with a quantile constraint defined over an optimization dataset $\{Y^{(1)}, \dots, Y^{(K)}\}$ . The problem is reformulated as:
$\min_{x \in X} J(x) \quad \text{s.t.} \quad \hat{Q}_{\alpha(K)}(f(x, Y^{(1)}), \dots, f(x, Y^{(K)})) \leq 0$
Here, $\hat{Q}_{\alpha(K)}$ is the empirical quantile at a specific confidence level $\alpha(K)$ . The authors demonstrate that this reformulation is an instantiation of SAA, allowing CPP to inherit a priori guarantees from SAA literature for specific problem classes (e.g., separable constraints).
Calibration Step (A Posteriori Guarantees):
To provide rigorous a posteriori feasibility guarantees without relying on structural assumptions like convexity, CPP utilizes a second, independent calibration dataset $\{Y^{(K+1)}, \dots, Y^{(K+L)}\}$ .
- Marginal Guarantees: Using the Quantile Lemma from CP, CPP computes a quantile $C_m$ over the calibration data. Theorem 3.4 establishes that the solution $x^*(K)$ satisfies $P(f(x^*(K), Y) \leq C_m) \geq 1-\delta$ with respect to the product measure of the data.
- Conditional Guarantees: By employing a tightened confidence level during quantile computation (Conditional Quantile Lemma), CPP provides guarantees that hold with high confidence ( $1-\beta$ ) conditioned on the calibration dataset.
- Quantile Shift: To address cases where the computed bound $C$ is positive, the authors propose a "quantile shift" method (Theorem 3.6). This calculates an adjusted failure probability $\delta^*$ based on the number of satisfied constraints in the calibration set, ensuring $P(f(x^*(K), Y) \leq 0) \geq 1-\delta^*$ with high confidence.

Key Contributions

New Framework: The introduction of CPP as a sampling-based approach that requires limited structural assumptions on the CCO problem.
Statistical Guarantees: The provision of both conditional and marginal a posteriori feasibility guarantees. Unlike standard SA, these guarantees do not depend on the dimension of the decision variable $n$ , reducing conservatism for high-dimensional problems.
Modularity and Versatility: The framework is designed to easily incorporate various Conformal Prediction variants. The authors demonstrate this by extending CPP to:
- Robust CPP (RCPP): Handles distribution shifts (e.g., sim-to-real gaps) using robust conformal prediction and ambiguity sets.
- Mondrian CPP (MCPP): Addresses class-conditional chance constraints (MCCO), ensuring safety guarantees hold for specific subgroups of the random variable distribution.
Connection to Existing Theory: The paper establishes that CPP inherits a priori guarantees from SAA and that its conditional a posteriori guarantees asymptotically reduce to Chernoff bounds.

Results and Case Studies
The authors validate CPP through numerical case studies involving both convex and nonconvex problems, as well as a stochastic optimal control task.

Convex and Nonconvex Optimization: In experiments comparing CPP against the Scenario Approach (SA) for nonconvex problems, CPP demonstrated competitive performance. Notably, while nonconvex SA methods (e.g., [18, 26]) require computationally expensive steps to find irreducible support subsets for a posteriori guarantees (taking ~10 seconds in the study), CPP's quantile shift calculation is significantly faster (linear time).
Empirical Coverage: The experiments confirmed that the empirical coverage of the solutions converges to the theoretical bounds ( $1-\delta$ or $1-\delta^*$ ). The histograms of the computed bounds centered near zero, indicating that the quantile reformulation effectively approximates the original chance constraint.
Robust and Mondrian Extensions: In a stochastic optimal control problem (robot navigation), RCPP successfully handled a distribution shift (simulating a change in noise variance), achieving higher empirical coverage (0.96) compared to the standard CPP baseline (0.80). Similarly, Mondrian CPP successfully maintained coverage guarantees for specific disturbance magnitude groups where standard CPP failed to do so.

Significance and Claims
The paper claims that CPP offers a computationally efficient and extensible framework for CCO problems where standard assumptions (convexity, known distributions) are unavailable or difficult to verify. Its primary significance lies in providing dimension-independent certificates of feasibility. Unlike the Scenario Approach, where guarantees become looser as the number of decision variables increases, CPP's guarantees remain robust regardless of dimensionality, provided a second calibration dataset is available. The authors emphasize that this approach bridges the gap between the theoretical rigor of conformal prediction and the practical needs of control and planning under uncertainty, offering a practical alternative to methods that rely on complex structural analysis or yield overly conservative solutions.