Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning to Defer in Non-Stationary Time Series via Switching State-Space Models

Yannis Montreuil, Letian Yu, Axel Carlier, Lai Xing Ng, Wei Tsang Ooi

TL;DR

This work addresses online routing for non-stationary time series with partial feedback and a changing expert pool by introducing L2D-SLDS, a factorized switching linear-Gaussian state-space model over signed expert residuals. The model couples a global factor $\mathbf{g}_t$ with per-expert idiosyncratic states $\mathbf{u}_{t,k}$ and a context-driven regime $z_t$, enabling cross-expert information transfer when only one expert is observed. An IDS-inspired routing rule leverages one-step predictive costs and information gain about the latent state to balance exploitation and exploration, while a dynamic registry manages expert entry/exit without destabilizing retained marginals. Empirical results on synthetic regime-transfer data and Melbourne temperatures show improved routing performance over contextual-bandit baselines and ablations lacking the shared factor. The approach offers a principled framework for robust, data-efficient decision-making in non-stationary, partially observed, and resource-constrained settings.

Abstract

We study Learning to Defer for non-stationary time series with partial feedback and time-varying expert availability. At each time step, the router selects an available expert, observes the target, and sees only the queried expert's prediction. We model signed expert residuals using L2D-SLDS, a factorized switching linear-Gaussian state-space model with context-dependent regime transitions, a shared global factor enabling cross-expert information transfer, and per-expert idiosyncratic states. The model supports expert entry and pruning via a dynamic registry. Using one-step-ahead predictive beliefs, we propose an IDS-inspired routing rule that trades off predicted cost against information gained about the latent regime and shared factor. Experiments show improvements over contextual-bandit baselines and a no-shared-factor ablation.

Learning to Defer in Non-Stationary Time Series via Switching State-Space Models

TL;DR

This work addresses online routing for non-stationary time series with partial feedback and a changing expert pool by introducing L2D-SLDS, a factorized switching linear-Gaussian state-space model over signed expert residuals. The model couples a global factor with per-expert idiosyncratic states and a context-driven regime , enabling cross-expert information transfer when only one expert is observed. An IDS-inspired routing rule leverages one-step predictive costs and information gain about the latent state to balance exploitation and exploration, while a dynamic registry manages expert entry/exit without destabilizing retained marginals. Empirical results on synthetic regime-transfer data and Melbourne temperatures show improved routing performance over contextual-bandit baselines and ablations lacking the shared factor. The approach offers a principled framework for robust, data-efficient decision-making in non-stationary, partially observed, and resource-constrained settings.

Abstract

We study Learning to Defer for non-stationary time series with partial feedback and time-varying expert availability. At each time step, the router selects an available expert, observes the target, and sees only the queried expert's prediction. We model signed expert residuals using L2D-SLDS, a factorized switching linear-Gaussian state-space model with context-dependent regime transitions, a shared global factor enabling cross-expert information transfer, and per-expert idiosyncratic states. The model supports expert entry and pruning via a dynamic registry. Using one-step-ahead predictive beliefs, we propose an IDS-inspired routing rule that trades off predicted cost against information gained about the latent regime and shared factor. Experiments show improvements over contextual-bandit baselines and a no-shared-factor ablation.
Paper Structure (69 sections, 6 theorems, 77 equations, 7 figures, 7 tables, 4 algorithms)

This paper contains 69 sections, 6 theorems, 77 equations, 7 figures, 7 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Offline Learning-to-Defer
  5. Non-Stationary Time Series and SSMs
  6. Context-Aware Routing in Non-Stationary Environments
  7. Problem Formulation
  8. Generative Model: Factorized SLDS
  9. Latent state hierarchy
  10. Implications of the Hierarchy
  11. Exploration via Information-Directed Sampling
  12. Dynamic Registry Management
  13. Experiments
  14. Synthetic: Regime-Dependent Correlation and Information Transfer
  15. Melbourne Daily Temperatures
  16. ...and 54 more sections

Key Result

Proposition 1

Fix $t$ and $z_t=m$, and let $\mathcal{G}_t\coloneqq \sigma(\mathcal{F}_t,I_t,z_t=m)$. Let $j\neq I_t$ and let $(e_{t,j}^{\mathrm{pred}},e_{t,I_t}^{\mathrm{pred}})$ denote the one-step-ahead predictive residuals under $p(e_{t,\cdot}\mid \mathcal{F}_t,z_t=m)$. Assume that this predictive pair is join In particular, when the predictive cross-covariance is non-zero, observing $e_t=e_{t,I_t}$ shifts t

Figures (7)

  • Figure 1: Regime-0 expert dependence in the synthetic transfer experiment. Each heatmap shows the pairwise Pearson correlation (color: $[-1,1]$) between experts' per-round losses (experts indexed $0$--$3$). Top row: partial feedback (only queried losses observed). Columns (left-to-right) show the ground-truth correlation implied by the simulator and the correlations estimated by each method. L2D-SLDS best recovers the block-structured correlations (experts $\{0,1\}$ vs. $\{2,3\}$).
  • Figure 2: L2D-SLDS with partial feedback and context-dependent regime switching: $p(z_t\mid z_{t-1},\mathbf{x}_t)$. The plate $j\in\mathcal{K}_t$ indexes experts whose idiosyncratic states are stored. Each $e_{t,j}$ is a potential residual, but only $e_{t,I_t}$ is revealed at round $t$.
  • Figure 3: Regime-0 expert dependence in the synthetic transfer experiment. Each heatmap shows the pairwise Pearson correlation (color: $[-1,1]$) between experts' per-round losses (experts indexed $0$--$3$). Top row: partial feedback (only queried losses observed). Columns (left-to-right) show the ground-truth correlation implied by the simulator and the correlations estimated by each method. L2D-SLDS best recovers the block-structured correlations (experts $\{0,1\}$ vs. $\{2,3\}$), highlighting the benefit of modeling shared latent factors for cross-expert information transfer under censoring.
  • Figure 4: We report the selection frequency of each expert over time as a function of the underlying regime for the synthetic experiment \ref{['sec:exp_synthetic_transfer_appendix']}. The top figure corresponds to the oracle, while the bottom figure shows our approach evaluated against the baselines. By construction, experts 0 and 1 perform better in regime 1, whereas experts 2 and 3 perform better in regime 2. Accordingly, a well-adapted router should select experts 0 and 1 more frequently in regime 1 and experts 2 and 3 more frequently in regime 2. L2D-SLDS (with and without $g_t$) is the only method that captures this structure, closely matching the oracle’s selection behavior. In contrast, LinUCB and NeuralUCB fail to adapt their selection frequencies to the regimes.
  • Figure 5: Synthetic: Regime-Dependent Correlation and Information Transfer: (a) Expert selection frequency over time. (b) Time series of the synthetic target (Section \ref{['sec:exp_synthetic_transfer_appendix']}) along with expert prediction errors. Figure (a) illustrates how methods adapt their routing under regime switches and temporary expert unavailability (Expert 1 is unavailable on a contiguous interval). L2D-SLDS shifts selection toward the currently well-calibrated experts, while LinUCB/NeuralUCB exhibit more diffuse exploration.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1: L2D-SLDS reliability and residual emission
  • Proposition 1: Information transfer under a shared factor
  • Proposition 1: Pruning does not affect retained experts
  • Proposition 1: Coupling at birth through the shared factor
  • Remark 2: $(z_t,\mathbf{g}_t)$-Information Gain for Non-Stationary Routing
  • Proposition 2: Information transfer under a shared factor
  • Proposition 2: Pruning does not affect retained experts
  • Proposition 2: Coupling at birth through the shared factor