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In-Context Operator Learning for Linear Propagator Models

Tingwei Meng, Moritz Voß, Nils Detering, Giulio Farolfi, Stanley Osher, Georg Menz

TL;DR

This paper addresses optimal liquidation under linear propagator price-impact models by introducing ICON-OCnet, a two-stage framework that first learns the unknown price-impact operator from a few in-context examples and then solves the resulting stochastic control problem using a surrogate operator in a neural policy (OCnet). The core method combines offline transformer-based pre-training of the price-impact operator with online few-shot prompting to adapt to unseen propagator kernels, enabling data-efficient inference of the non-Markovian dynamics. The results show accurate operator inference with as few as five context examples and successful retrieval of ground-truth optimal execution strategies when the ICON surrogate is used in the OCnet, demonstrating the practical potential of few-shot operator learning for complex control problems. This approach offers a flexible, data-efficient pathway to handle unknown market impact regimes in real-time trading, with extensions to nonlinear kernels and real-data settings as promising future work.

Abstract

We study operator learning in the context of linear propagator models for optimal order execution problems with transient price impact à la Bouchaud et al. (2004) and Gatheral (2010). Transient price impact persists and decays over time according to some propagator kernel. Specifically, we propose to use In-Context Operator Networks (ICON), a novel transformer-based neural network architecture introduced by Yang et al. (2023), which facilitates data-driven learning of operators by merging offline pre-training with an online few-shot prompting inference. First, we train ICON to learn the operator from various propagator models that maps the trading rate to the induced transient price impact. The inference step is then based on in-context prediction, where ICON is presented only with a few examples. We illustrate that ICON is capable of accurately inferring the underlying price impact model from the data prompts, even with propagator kernels not seen in the training data. In a second step, we employ the pre-trained ICON model provided with context as a surrogate operator in solving an optimal order execution problem via a neural network control policy, and demonstrate that the exact optimal execution strategies from Abi Jaber and Neuman (2022) for the models generating the context are correctly retrieved. Our introduced methodology is very general, offering a new approach to solving optimal stochastic control problems with unknown state dynamics, inferred data-efficiently from a limited number of examples by leveraging the few-shot and transfer learning capabilities of transformer networks.

In-Context Operator Learning for Linear Propagator Models

TL;DR

This paper addresses optimal liquidation under linear propagator price-impact models by introducing ICON-OCnet, a two-stage framework that first learns the unknown price-impact operator from a few in-context examples and then solves the resulting stochastic control problem using a surrogate operator in a neural policy (OCnet). The core method combines offline transformer-based pre-training of the price-impact operator with online few-shot prompting to adapt to unseen propagator kernels, enabling data-efficient inference of the non-Markovian dynamics. The results show accurate operator inference with as few as five context examples and successful retrieval of ground-truth optimal execution strategies when the ICON surrogate is used in the OCnet, demonstrating the practical potential of few-shot operator learning for complex control problems. This approach offers a flexible, data-efficient pathway to handle unknown market impact regimes in real-time trading, with extensions to nonlinear kernels and real-data settings as promising future work.

Abstract

We study operator learning in the context of linear propagator models for optimal order execution problems with transient price impact à la Bouchaud et al. (2004) and Gatheral (2010). Transient price impact persists and decays over time according to some propagator kernel. Specifically, we propose to use In-Context Operator Networks (ICON), a novel transformer-based neural network architecture introduced by Yang et al. (2023), which facilitates data-driven learning of operators by merging offline pre-training with an online few-shot prompting inference. First, we train ICON to learn the operator from various propagator models that maps the trading rate to the induced transient price impact. The inference step is then based on in-context prediction, where ICON is presented only with a few examples. We illustrate that ICON is capable of accurately inferring the underlying price impact model from the data prompts, even with propagator kernels not seen in the training data. In a second step, we employ the pre-trained ICON model provided with context as a surrogate operator in solving an optimal order execution problem via a neural network control policy, and demonstrate that the exact optimal execution strategies from Abi Jaber and Neuman (2022) for the models generating the context are correctly retrieved. Our introduced methodology is very general, offering a new approach to solving optimal stochastic control problems with unknown state dynamics, inferred data-efficiently from a limited number of examples by leveraging the few-shot and transfer learning capabilities of transformer networks.
Paper Structure (14 sections, 1 theorem, 24 equations, 10 figures, 2 tables)

This paper contains 14 sections, 1 theorem, 24 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Optimal liquidation with linear propagator models
  3. Model setup
  4. Optimization problem and optimal strategy
  5. Model parameters
  6. Price impact operator learning and neural network solver
  7. ICON training and few-shot learning
  8. Neural network solver with ICON surrogate
  9. Numerical results
  10. Data generation
  11. ICON performance
  12. ICON-OCnet performance
  13. Conclusion and outlook
  14. Proof of Proposition \ref{['prop:solution']}

Key Result

Proposition 2.1

Assume that $\varepsilon > 0$ and $\varrho, \phi \geq 0$. Then, the unique optimal strategy $u^\star \in \mathcal{U}$ in def:optproblem with propagator kernels of type (I) and (II) is given by where the stochastic process $(a_t)_{0 \leq t \leq T}$ and the kernel $B: [0,T]^2 \rightarrow \mathbb{R}$ are defined as with $K(t,s) := \tilde{C}(t-s) + \lambda G(t-s) 1_{\{ s \leq t\}}$ for all $0 \leq s

Figures (10)

  • Figure 1: Illustration of the ICON-OCnet structure. In both steps, the red rounded rectangle represents the training of a neural network, while the blue rounded rectangle indicates a pre-trained neural network with frozen parameters.
  • Figure 2: Illustration of 10 training trajectories (selling rates $u$ and corresponding price impact $Y$) for the three different kernels with parameters $\lambda = 0.2$, $\beta = 0.5$, $\gamma = 0.45$.
  • Figure 3: ICON error (on the training set and a separate test set) versus training iterations with 100,000 training steps in total.
  • Figure 4: Heatmaps for ICON in-distribution errors for different types of in-context examples (first label) and ICON models trained on a specific dataset (second label). The value of each box represents the mean error over 16 random samples of sets of hyperparameters $\theta$ from the corresponding ranges, generating the five in-context examples ($x$-axis represents values for $\lambda$, $y$-axis represents values for $\beta$ and $\gamma$, respectively).
  • Figure 5: Heatmaps similar to Figure \ref{['fig:icon_heatmap_id']} for ICON errors for different types of in-distribution in-context examples (first label) and ICON models trained on a specific dataset (second label). Here, the question condition for the ICON prediction is the out-of-distribution optimal execution strategy $u^\star$ of the corresponding propagator model associated with hyperparameter $\theta$. The value of each box represents the mean error over 16 random samples of sets of hyperparameters $\theta$ from the corresponding ranges ($x$-axis represents values for $\lambda$, $y$-axis represents values for $\beta$ and $\gamma$, respectively).
  • ...and 5 more figures

Theorems & Definitions (3)

  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3