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Conservative Continuous-Time Treatment Optimization

Nora Schneider, Georg Manten, Niki Kilbertus

Abstract

We develop a conservative continuous-time stochastic control framework for treatment optimization from irregularly sampled patient trajectories. The unknown patient dynamics are modeled as a controlled stochastic differential equation with treatment as a continuous-time control. Naive model-based optimization can exploit model errors and propose out-of-support controls, so optimizing the estimated dynamics may not optimize the true dynamics. To limit extrapolation, we add a consistent signature-based MMD regularizer on path space that penalizes treatment plans whose induced trajectory distribution deviates from observed trajectories. The resulting objective minimizes a computable upper bound on the true cost. Experiments on benchmark datasets show improved robustness and performance compared to non-conservative baselines.

Conservative Continuous-Time Treatment Optimization

Abstract

We develop a conservative continuous-time stochastic control framework for treatment optimization from irregularly sampled patient trajectories. The unknown patient dynamics are modeled as a controlled stochastic differential equation with treatment as a continuous-time control. Naive model-based optimization can exploit model errors and propose out-of-support controls, so optimizing the estimated dynamics may not optimize the true dynamics. To limit extrapolation, we add a consistent signature-based MMD regularizer on path space that penalizes treatment plans whose induced trajectory distribution deviates from observed trajectories. The resulting objective minimizes a computable upper bound on the true cost. Experiments on benchmark datasets show improved robustness and performance compared to non-conservative baselines.
Paper Structure (44 sections, 2 theorems, 54 equations, 3 figures, 2 tables)

This paper contains 44 sections, 2 theorems, 54 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setting
  4. Methodology
  5. Structural Dynamics Model for Identifiability
  6. Conservative Optimization of Treatment Plans
  7. Experimental Setup
  8. Experimental Results
  9. Conclusion
  10. Summary of Notation
  11. On Identifiability for Continuous-Time Treatment Effects
  12. Structural Dynamics Model.
  13. The Class of Admissible Treatments.
  14. Details on Our Proposed Approach
  15. Proof of the telescope lemma
  16. ...and 29 more sections

Key Result

Proposition 1

Under assumptions assum:fcr_formalassum:positivity_formalassum:sde_estimation_assumption and the structural dynamics assum:mechanistic_model such that $X = X^{(U, \mathrm{pot})}$ a.s., is identifiable from the observational law. In particular, for the value $\mathcal{J}(t_0, X^{(u,\mathrm{pot})},u)$ is identified.

Figures (3)

  • Figure 1: Effect of increasing regularization strength $\lambda$. (Left) Improvement of the ground truth costs for $\lambda = \{1, 10, 100\}$ relative to the ground truth costs with no regularization, i.e., $\lambda=0$ on three optimization problems across $15$ different initial conditions (Cancer (Explicit), Cancer (Relative), Covid-19 (Target Trajectory). Boxplots show medians and interquartile ranges (IQR). Whiskers extend to the farthest point within $1.5 \times \text{IQR}$ from boxes. (Right) Example of predicted optimal treatment plans for $\lambda \in \{0, 1, 10, 100 \}$ together with treatment plans seen during training and ground truth optimal treatment plan.
  • Figure 2: Example patient trajectories under treatment plans optimized with different conservatism levels ($\lambda$). Tumor volume over time for two optimized treatment plans. Shaded regions indicate variability across simulated trajectories, the bold curve denotes the mean trajectory, and the dot marks the mean final tumor volume.
  • Figure 3: Effects of regularization strength parameter $\lambda$.

Theorems & Definitions (11)

  • Remark 1: Neural SDEs
  • Proposition 1: Identifiability
  • Lemma 1: Value Error Telescope
  • Remark 2
  • proof
  • Definition 1: Integral probability metric (IPM)
  • Remark 3
  • Example 1
  • Definition 2: Maximum Mean Discrepancy
  • Remark 4: Expression of MMD, definiteness
  • ...and 1 more