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Dynamic programming principle in cost-efficient sequential design: optimal update scheduling under cost constraints

This paper applies the dynamic programming principle to formulate an optimal, cost-efficient sequential experimental design for D-optimality in binary response models, specifically addressing the constraints of switching measurements on superconducting Josephson junctions where fixed time costs for covariate updates are significant.

Original authors: Jeongmin Han, Juha Karvanen, Mikko Parviainen

Published 2026-09-11

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CC BY 4.0

Original authors: Jeongmin Han, Juha Karvanen, Mikko Parviainen

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

In the world of quantum computing, the building blocks are not made of silicon and wire, but of superconducting circuits that operate at temperatures near absolute zero. One critical component in these circuits is the Josephson junction, a tiny device that acts as a switch for electrical current. Before these junctions can be used in a computer, scientists must measure their physical properties with extreme precision. This measurement process involves sending a series of electrical pulses through the junction and watching to see if it switches to a voltage state. The likelihood of this switch happening depends on the strength of the pulse. Because the equipment must be kept at such a frigid temperature, the window of time available to run these experiments is incredibly short; the cooling system can only maintain stability for a limited duration. This creates a race against time where every second counts, and the way researchers choose to gather their data can mean the difference between a successful measurement and a wasted experiment.

The challenge lies in how to schedule these measurements. To get the most accurate picture of the junction's behavior, scientists need to adjust the strength of the electrical pulses based on what they have learned so far. This is a sequential process: measure, learn, adjust, and measure again. However, changing the settings on the pulse generator takes a significant amount of time—much longer than the split second it takes to actually run a single measurement. If a researcher adjusts the settings after every single measurement, the time spent adjusting would eat up almost all the available experimental window, leaving little time for gathering data. The goal, therefore, is to find the perfect balance: how often should the settings be changed to get the best possible results without wasting precious time on adjustments?

A team of researchers led by Jeongmin Han, Juha Karvanen, and Mikko Parviainen tackled this problem by applying a mathematical strategy known as dynamic programming. This approach, often used in fields like economics and engineering to solve complex decision-making problems, works by breaking a long, complicated journey into a series of smaller, manageable steps. Instead of trying to plan the entire experiment from start to finish at once, the method looks at the immediate next step and asks what the best move is right now, given the information currently available and the time remaining. The researchers used this logic to determine the optimal moments to update the pulse settings. They built a model that predicts how much new information is gained from a set of measurements and how that information grows over time, while also accounting for the heavy time penalty of making an adjustment.

Their simulations, which tested their new method against older techniques used in previous studies, revealed a clear advantage. In the past, researchers often used a simple rule of thumb, such as increasing the number of measurements by a fixed percentage at each stage. While this worked reasonably well, it was not perfectly efficient. The new dynamic programming approach, by contrast, calculated the exact timing for updates based on the specific costs and the current state of knowledge. In their tests, the researchers set up a scenario where the initial estimates of the junction's properties were quite poor, a common situation in real-world experiments. They found that their method required significantly less time to reach the same level of accuracy as the older methods. Specifically, when aiming for a high level of precision, the new approach reduced the total time cost by approximately 13.58 percent compared to the previous best approximation.

The study highlights that the efficiency of an experiment is not just about how fast you can measure, but how wisely you schedule your adjustments. By treating the experiment as a series of calculated steps rather than a rigid schedule, the researchers showed that it is possible to extract more value from the limited time available in superconducting electronics. The method is flexible enough to handle different goals, such as maximizing the accuracy of the data within a fixed time limit, or reaching a specific target of accuracy in the shortest time possible. While the specific numbers and models used in the study were tailored to the unique physics of Josephson junctions, the underlying logic offers a powerful tool for any scientific field where data is collected in batches and where changing the experimental conditions carries a high cost. The work demonstrates that with the right mathematical framework, scientists can squeeze more insight out of every fleeting moment of stability in the lab.

Technical Summary: Dynamic Programming Principle in Cost-Efficient Sequential Design

Problem Statement
This study addresses the challenge of optimal sequential experimental design in scenarios where updating covariates incurs a significant fixed time cost compared to the duration of a single measurement. The specific motivation arises from parameter estimation in switching measurements of superconducting Josephson junctions (JJs), components critical for quantum computing and superconducting electronics. In these experiments, a component is cooled to near absolute zero to achieve superconductivity, a state that remains stable only for a limited duration ( $T$ ). A sequence of current pulses is applied, and a binary voltage response is observed. The relationship between the current pulse height (covariate $x$ ) and the probability of a voltage response is modeled using a binary response with a complementary log-log link.

The core problem is to determine the optimal times to update the covariate values (current pulse heights) to maximize the efficiency of parameter estimation (specifically, D-optimality) within the constrained stable time window. While sequential designs allow for updating estimates after each observation, the high cost of adjusting the pulse generator necessitates a "batch-sequential" approach. The key challenge is determining the optimal update intervals that balance the information gain against the time cost of the update ( $C_s$ ). Previous approaches, such as those by Karvanen et al. (2007) and Karvanen (2009), utilized ad hoc methods or approximate asymptotic models to determine update times, which could lead to inefficiencies, particularly when initial parameter estimates are poor.

Methodology
The authors formulate the problem using the Dynamic Programming Principle (DPP) from the mathematical theory of optimal control. The approach is grounded in D-optimal design theory for the complementary log-log model, where the goal is to maximize the square root of the determinant of the Fisher information matrix ( $D$ ).

Information Accumulation Model: The study utilizes an approximation for the expected change in the D-criterion ( $h(D)$ ) derived from Karvanen (2009). This function models the expected increase in information when two additional measurements are made, assuming the parameters follow a normal distribution centered on the current estimates. The model assumes that if covariates are not updated, the information gain rate remains constant based on the current design.
Dynamic Programming Formulation: The authors define two distinct optimization problems solvable via backward iteration:
- Maximizing D given a fixed time $T$ : A value function $u(D, t)$ represents the maximum additional D-criterion achievable starting from information state $D$ at time $t$ . The DPP equation is:
  $u(D, t) = \sup_{t_{next} \in (t+C_s, T]} \{ u(D_{next}, t_{next}) + (D_{next} - D) \}$
  where $D_{next}$ is the projected information after an update at time $t_{next}$ .
- Minimizing time given a target $D_{final}$ : A value function $v(D)$ represents the minimum time required to reach a target accuracy $D_{final}$ from current state $D$ . The DPP equation is:
  $v(D) = \inf_{\Delta t \in (C_s, \infty)} \{ v(D + (\Delta t - C_s)h(D)) + \Delta t \}$
Numerical Solution: The equations are solved numerically using R. The state space (time and information level) is discretized. The algorithm iterates backward from the terminal condition (e.g., $t = T - C_s$ ) to determine the optimal update schedule. The method accounts for the initialization state ( $D_0$ ), the update cost ( $C_s$ ), and the specific target (time or accuracy).

Key Contributions

First DPP Framework for Binary Response: While Huan and Marzouk (2016) applied DPP to optimal sequential design with continuous responses, this paper presents the first formulation of a DPP-based experimental design framework specifically for binary response models.
Precise Optimization of Update Times: Unlike previous ad hoc methods (e.g., increasing sample size by a fixed percentage at each stage), this approach precisely solves for optimal update times using optimal control theory. It explicitly handles the trade-off between the cost of updating and the information gained.
Handling Variable Stages: The framework accommodates situations where the number of stages is not predetermined, unlike some previous dynamic programming applications in design that assumed a fixed number of stages.
Robustness to Initial Estimates: The method automatically incorporates the initial information level ( $D_0$ ) and the update cost ( $C_s$ ) into the optimization, making it adaptable to scenarios where initial parameter estimates are highly inaccurate.

Results
The authors conducted simulations to benchmark the proposed DPP method against the ad hoc method from Karvanen et al. (2007) and the approximate asymptotic method from Karvanen (2009).

Simulation Setup: The experiment simulated switching measurements with a true parameter set of $(a, b) = (0.24, -61)$ . The initial estimates were intentionally poor to yield a low initial $D_0$ (0.1408), a scenario where cost-efficiency is most critical. The update cost $C_s$ was set to approximately 228.4 units relative to a single measurement cost of 1.
Performance: The DPP-based design demonstrated superior cost-efficiency. In the simulation with a fixed time horizon ( $T=3500$ ), the DPP method achieved the same level of accumulated information ( $D_{accum}$ ) with a total time cost of 14.77 seconds, whereas the method from Karvanen (2009) required more time.
Quantitative Gain: Compared to the approximate cost-efficient method in Karvanen (2009), the DPP method reduced the time cost by 13.58%. The results indicate that the DPP method avoids the excessive early-stage updates often seen in previous methods when initial estimates are poor, thereby conserving the limited stable temperature time.

Significance and Claims
The paper claims that the application of optimal control theory, specifically the Dynamic Programming Principle, offers a rigorous and more cost-efficient alternative to existing heuristic methods for sequential design in high-cost update environments. The authors emphasize that their method is not limited to the specific example of Josephson junctions but is applicable to any sequential design where data is collected in batches and updates carry a significant cost.

The authors remain modest regarding the theoretical underpinnings, acknowledging that the method relies on approximations from prior literature (specifically the asymptotic normality of Maximum Likelihood Estimators and the specific form of the Fisher information accumulation). However, they assert that these approximations allow the problem to be formulated in a way that is solvable by efficient computational tools. The study concludes that the proposed approach finds wide applicability in cost-efficient sequential designs across different statistical models, provided the information accumulation function can be characterized.

Technical Summary: Dynamic Programming Principle in Cost-Efficient Sequential Design

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