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

cs.LGJan 30, 2026

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

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

Organizations: School of Computing National University of Singapore · Université de Toulouse Fédération ENAC ISAE-SUPAERO ONERA · Institute for Infocomm Research A*STAR, Singapore

Abstract

Learning-to-defer (L2D) lets a predictor decide, at each round, whether to issue its own forecast or pay for an expert's. In non-stationary time series this decision must keep adapting, although deployment reveals only the consulted expert's forecast while a historical archive records every expert with the target. L2D-SLDS learns from this archive a switching state-space model of the target and all expert forecasts, whose shared and expert-specific states describe how experts move together and apart. Its predictive law supplies the internal forecast and the expected cost of every consultation, which a greedy router minimizes, and one consultation also updates the beliefs about unconsulted and unavailable experts. We prove sublinear regret against a changing conditional-risk oracle without exploration, when the candidate models are accurate and either the archive separates them or live feedback reveals cost differences. On three real datasets, L2D-SLDS has the lowest cost among eight bandit routers and adapts its consultation rate to the fee.

Figures & tables

Panel of Figure 1 — Figure 1: One generative step of L2D-SLDS. The regime selects dynamics and emission variances; the plate includes unavailable retained experts. After acting, $r_{t}$ is always observed and $d_{t,k}$ only if $I_{t}=k$ . The shared state $\boldsymbol{g}_{t}$ links queried and unqueried experts.

Figure 2: Consultation-cost sensitivity on Jena: (a) heldout combined cost and (b) percentage of rounds consulting an external expert, as the common fee

\beta

increases. Baselines are reselected on validation at each fee; mean

\pm

one sample SD over three seeds. The fee axis is linear near zero and logarithmic thereafter.

System	Delhi	Jena	Melbourne
L2D-SLDS	$\mathbf{0.954}\pm 0.000$	$\mathbf{2.502}\pm 0.000$	$\mathbf{2.321}\pm 0.000$
LinUCB	$0.966\pm 0.000$	$2.814\pm 0.000$	$2.394\pm 0.000$
Shared LinUCB	$0.966\pm 0.000$	$2.814\pm 0.000$	$2.372\pm 0.000$
Linear TS	$0.992\pm 0.019$	$2.930\pm 0.050$	$2.360\pm 0.023$
Ensemble	$1.003\pm 0.021$	$2.768\pm 0.021$	$2.428\pm 0.067$
Discounted LinUCB	$1.127\pm 0.000$	$2.899\pm 0.000$	$2.382\pm 0.000$

Table 1: Heldout combined cost (mean

\pm

sample SD, three seeds); best means in bold. All routers use full-covariance internal learners.

Variant	Delhi	Jena	Melbourne
Greedy (ours)	$\mathbf{0.954}\pm 0.000$	$\mathbf{2.502}\pm 0.000$	$\mathbf{2.321}\pm 0.000$
Inverse-gap	$1.095\pm 0.007$	$2.766\pm 0.022$	$2.361\pm 0.032$
Joint IX	$0.981\pm 0.003$	$3.071\pm 0.033$	$2.325\pm 0.008$
Structured	$0.954\pm 0.000$	$2.502\pm 0.000$	$2.325\pm 0.000$
Retain	$\mathbf{0.954}\pm 0.000$	$\mathbf{2.906}\pm 0.000$	$\mathbf{2.321}\pm 0.000$
Reset	$\mathbf{0.954}\pm 0.000$	$2.915\pm 0.000$	$2.327\pm 0.000$

Table 2: Ablations, heldout combined cost (mean

\pm

SD, three seeds). Top: routing rules on one learner. Middle: structured covariance. Bottom: withdrawn experts, retained or reset, and the best of LinUCB, discounted LinUCB, and linear TS.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$K,\ M,\ L$	Number of external experts, regimes, and predictive candidates
$\mathcal{E}_{t},\ \mathcal{A}_{t}$	Available external experts; $\{0\}\cup\mathcal{E}_{t}$
$I_{t},\ f_{t},\ h_{t,k},\ y_{t}$	Selected action, internal forecast, expert forecast, target
$B,\ \beta_{k},\ C_{*}$	Output bound, access fee, maximum action cost
$\mathcal{P}_{t}$	Observed pre-action information
$\boldsymbol{g}_{t},\ \boldsymbol{u}_{t,k},\ z_{t}$	Shared state, private state, regime

Appendix

Table 3: Principal notation. Bold symbols denote vectors or matrices; scalar targets, forecasts, scores, and costs are unbold.

Dataset	Prefix	Fit	Warmup	Validation	Heldout	Heldout target dates
Delhi	627	313	78	78	472	2016-01-02–2017-04-24
Jena	2400	1200	300	300	1800	2010-12-02–2011-09-27
Melbourne	1458	729	182	182	1096	1987-12-31–1990-12-31

Appendix

Table 4: Chronological sample counts.

Method	Tuned settings and fixed settings
LinUCB / Shared LinUCB	Bonus multiplier $\{0,0.05,0.2,1\}$ ; ridge $\{0.01,0.1,1,10\}$ .
Linear TS	Posterior scale $\{0.01,0.05,0.2,1\}$ ; ridge $\{0.01,0.1,1,10\}$ .
Ensemble	Perturbation standard deviation $\{0.01,0.05,0.2,1\}$ ; ridge $\{0.01,0.1,1,10\}$ ; 20 members.
Discounted LinUCB	Discount $\{0.90,0.97,0.99,0.999\}$ ; bonus multiplier $\{0.01,0.05,0.2,1\}$ ; ridge 1, noise scale $0.5$ , parameter-norm bound 1, confidence parameter $0.05$ .
CUSUM-LinUCB	Bonus $\{0.05,0.5\}$ ; threshold $\{0.1,0.3,1,3\}$ ; slack $\{0.005,0.02\}$ ; warmup and reference length 25, ridge 1.
GLR-LinUCB	Bonus $\{0.05,0.5\}$ ; detector level $\{0.001,0.01,0.05,0.2\}$ ; minimum window $\{10,25\}$ ; reference length 25, ridge 1.

Appendix

Table 5: Declared baseline search grids. Cartesian products give exactly 16 configurations per method. Each dataset selects its configuration on chronological validation with seed 101, before heldout evaluation.

Figure 3: Heldout cumulative combined cost of each method minus that of L2D-SLDS, averaged over seeds 11, 13, and 17; positive values favor L2D-SLDS. Error bars show one sample SD of the paired differences. Each method follows its own deployed feedback history, and panel scales differ.

Fee	L2D-SLDS cost	Best contextual baseline	Baseline cost	L2D-SLDS queries (%)
0	$\mathbf{2.502}\pm 0.000$	Ensemble	$2.768\pm 0.021$	$100.0\pm 0.0$
0.05	$\mathbf{2.552}\pm 0.000$	Ensemble	$2.789\pm 0.029$	$95.4\pm 0.0$
0.10	$\mathbf{2.585}\pm 0.000$	LinUCB	$2.779\pm 0.000$	$91.6\pm 0.0$
0.20	$\mathbf{2.669}\pm 0.000$	LinUCB	$2.874\pm 0.000$	$85.8\pm 0.0$
0.40	$\mathbf{2.843}\pm 0.000$	Shared LinUCB	$3.112\pm 0.000$	$75.3\pm 0.0$
0.80	$\mathbf{3.078}\pm 0.000$	LinUCB	$3.290\pm 0.000$	$60.6\pm 0.0$

Appendix

Table 6: Jena cost sensitivity. Mean

\pm

sample SD over three seeds. The best contextual baseline is identified separately at each fee; this column is a descriptive minimum over the eight baselines, not an additional deployed policy. Bold marks the lowest displayed combined cost, including ties. Query percentages are descriptive and are not ranked.

Dataset	Policy	Prediction loss	Fee	Query fraction	Internal MSE
Delhi	L2D-SLDS	$\mathbf{0.950}\pm 0.000$	$0.004\pm 0.000$	$0.017\pm 0.000$	$\mathbf{0.968}\pm 0.000$
	Inverse-gap	$0.956\pm 0.011$	$0.140\pm 0.005$	$0.636\pm 0.022$	$0.970\pm 0.001$
	Joint IX	$0.966\pm 0.002$	$0.015\pm 0.001$	$0.067\pm 0.007$	$0.981\pm 0.000$
Jena	L2D-SLDS	$\mathbf{2.502}\pm 0.000$	$0.000\pm 0.000$	$1.000\pm 0.000$	$\mathbf{4.068}\pm 0.000$
	Inverse-gap	$2.766\pm 0.022$	$0.000\pm 0.000$	$0.919\pm 0.011$	$4.099\pm 0.010$
	Joint IX	$3.071\pm 0.033$	$0.000\pm 0.000$	$0.986\pm 0.002$	$4.140\pm 0.003$

Appendix

Table 7: Routing-rule ablation with a common full-covariance learner. Mean

\pm

sample SD over three seeds. Prediction loss plus fee gives the combined cost; internal MSE follows each system’s feedback history. Bold marks the lowest prediction loss and internal MSE within each dataset.

Data	Full	One regime	$10^{3}\Delta C$
Switching	$\mathbf{0.101}\pm 0.004$	$0.178\pm 0.015$	$76.861\pm 17.047$
Stationary	$0.116\pm 0.005$	$\mathbf{0.114}\pm 0.007$	$-2.183\pm 1.993$

Appendix

Table 8: Synthetic regime study. Combined cost, mean

\pm

sample SD over three trajectories;

\Delta C

is one regime minus full, so positive values favor the full dictionary.

Dataset	L2D-SLDS	LinUCB	Disc. LinUCB	Linear TS	Reset	$10^{3}\Delta C$
Delhi	$\mathbf{0.954}\pm 0.000$	$0.966\pm 0.000$	$1.119\pm 0.000$	$0.992\pm 0.020$	$\mathbf{0.954}\pm 0.000$	$0.000\pm 0.000$
Jena	$\mathbf{2.906}\pm 0.000$	$3.089\pm 0.000$	$3.221\pm 0.000$	$3.166\pm 0.074$	$2.915\pm 0.000$	$8.693\pm 0.000$
Melbourne	$\mathbf{2.321}\pm 0.000$	$2.394\pm 0.000$	$2.377\pm 0.000$	$2.355\pm 0.023$	$2.327\pm 0.000$	$6.192\pm 0.000$

Appendix

Table 9: Expert withdrawal and reentry on the real datasets. Combined cost, mean

\pm

sample SD over three seeds. Right:

\Delta C

is reset minus retain, so positive values favor retention.

Policy	Delhi	Jena	Melbourne
L2D-SLDS (structured)	$\mathbf{0.954}\pm 0.000$	$\mathbf{2.502}\pm 0.000$	$\mathbf{2.325}\pm 0.000$
Inverse-gap	$1.095\pm 0.007$	$2.766\pm 0.022$	$2.360\pm 0.033$
Joint IX	$0.981\pm 0.003$	$3.073\pm 0.027$	$\mathbf{2.325}\pm 0.008$

Appendix

Table 10: Routing rules with the structured-covariance learner. Mean combined cost

\pm

sample SD over three seeds.

Figure 4: Jena expert-pool scaling. Mean and one sample SD across three fitting seeds on the same heldout series: (a) prediction cost at zero fee, (b) live selection and feedback time, excluding fitting and historical replay, and (c) retained posterior-array storage. Pool sizes are spaced logarithmically.

$K$	Full	Structured	Full	Structured	Reduction
	Prediction cost		Posterior (KiB)
4	$\mathbf{2.5013}\pm 0.0029$	$\mathbf{2.5013}\pm 0.0029$	0.727	0.352	51.6%
8	$\mathbf{2.4081}\pm 0.0003$	$\mathbf{2.4081}\pm 0.0003$	2.133	0.633	70.3%
16	$\mathbf{2.4242}\pm 0.0124$	$2.4259\pm 0.0099$	7.195	1.195	83.4%
32	$\mathbf{2.4241}\pm 0.0012$	$2.4275\pm 0.0076$	26.320	2.320	91.2%
64	$\mathbf{2.5084}\pm 0.0126$	$2.5107\pm 0.0277$	100.570	4.570	95.5%

Appendix

Table 11: Heldout prediction cost and posterior storage on Jena. Costs are mean

\pm

sample SD over three fitting seeds; posterior-array sizes have zero SD. Bold marks the lower displayed cost or storage within each pair.

Method	$K=4$	$K=8$	$K=16$	$K=32$	$K=64$
L2D-SLDS (full)	$0.383\pm 0.010$	$0.762\pm 0.018$	$1.544\pm 0.057$	$3.059\pm 0.043$	$\mathbf{6.118}\pm 0.091$
L2D-SLDS (structured)	$\mathbf{0.377}\pm 0.007$	$\mathbf{0.756}\pm 0.013$	$1.542\pm 0.034$	$\mathbf{3.027}\pm 0.049$	$\mathbf{6.118}\pm 0.115$
LinUCB	$\mathbf{0.377}\pm 0.002$	$0.765\pm 0.014$	$1.546\pm 0.036$	$3.162\pm 0.223$	$6.120\pm 0.119$
Shared LinUCB	$0.391\pm 0.019$	$0.769\pm 0.012$	$\mathbf{1.518}\pm 0.004$	$3.064\pm 0.044$	$6.203\pm 0.097$
Linear TS	$0.379\pm 0.002$	$0.769\pm 0.012$	$1.533\pm 0.041$	$3.088\pm 0.052$	$6.144\pm 0.142$
Ensemble	$\mathbf{0.377}\pm 0.004$	$0.774\pm 0.008$	$1.519\pm 0.008$	$3.060\pm 0.074$	$6.177\pm 0.091$

Appendix

Table 12: End-to-end seconds per round on Jena, mean

\pm

sample SD of three per-seed timing averages. Each average uses 60 measured rounds after ten warmup rounds. All methods include their internal learner’s computation. Bold marks the lowest displayed mean in each column, including ties.

Method	Posterior (KiB)	Policy buffers (KiB)	Policy time (ms)	Peak RSS (MiB)
L2D-SLDS (full)	100.570	0.000	$0.0210\pm 0.0014$	$118.73\pm 0.19$
L2D-SLDS (structured)	4.570	0.000	$\mathbf{0.0203}\pm 0.0019$	$118.53\pm 0.38$
LinUCB	100.570	6.602	$1.73\pm 0.29$	$118.53\pm 0.20$
Shared LinUCB	100.570	542.945	$69.95\pm 0.37$	$118.41\pm 0.12$
Linear TS	100.570	542.945	$5.68\pm 0.20$	$118.35\pm 0.19$
Ensemble	100.570	666.227	$5.29\pm 0.23$	$118.35\pm 0.38$

Appendix

Table 13: Storage and policy overhead at the largest tested pool (

K=64

). Posterior and policy-buffer sizes are in KiB and have zero SD across seeds; policy time and peak process memory are mean

\pm

sample SD. Bold marks the lowest displayed value in each column, including ties.

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Learning-Augmented Algorithms Experts Bandits

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Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. While existing work studies this problem in batch settings, real-world deployments require handling streaming data, changing expert availability, and shifting expert distribution. We introduce the first online L2D algorithm for multiclass classification with bandit feedback and a dynamically varying pool of experts. Our method achieves regret guarantees of

O((n+n_e)T^{2/3})

in general and

O((n+n_e)\sqrt{T})

under a low-noise condition, where

T

is the time horizon,

n

is the number of labels, and

n_e

is the number of distinct experts observed across rounds. The analysis builds on novel

\mathcal{H}

-consistency bounds for the online framework, combined with first-order methods for online convex optimization. Experiments on synthetic and real-world datasets demonstrate that our approach effectively extends standard Learning-to-Defer to settings with varying expert availability and reliability.

Abstract

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