The Forecast After the Forecast: A Post-Processing Shift in Time Series
Authors: Daojun Liang
Qi Li
Yinglong Wang
Jing Chen
Hu Zhang
Xiaoxiao Cui
Qizheng Wang
Shuo Li
Venue: ICLR 2026 Poster (2026)
Keywords: Time Series Forecasting
Post-Processing
Fine-Tuning
Primary area: learning on time series and dynamical systems
TL;DR: We propose post-hoc, a lightweight, architecture-agnostic way to boost deployed time series forecasters without retraining.

Abstract: Time series forecasting has long been dominated by advances in model architecture, with recent progress driven by deep learning and hybrid statistical techniques. However, as forecasting models approach diminishing returns in accuracy, a critical yet underexplored opportunity emerges: the strategic use of post-processing. In this paper, we address the last-mile gap in time-series forecasting, which is to improve accuracy and uncertainty without retraining or modifying a deployed backbone. We propose $\delta$-Adapter, a lightweight, architecture-agnostic way to boost deployed time series forecasters without retraining. $\delta$-Adapter learns tiny, bounded modules at two interfaces: input nudging (soft edits to covariates) and output residual correction. We provide local descent guarantees, $O(\delta)$ drift bounds, and compositional stability for combined adapters.
Meanwhile, it can act as a feature selector by learning a sparse, horizon-aware mask over inputs to select important features, thereby improving interpretability.
In addition, it can also be used as a distribution calibrator to measure uncertainty. Thus, we introduce a Quantile Calibrator and a Conformal Corrector that together deliver calibrated, personalized intervals with finite-sample coverage.  
Our experiments across diverse backbones and datasets show that $\delta$-Adapter improves accuracy and calibration with negligible compute and no interface changes.

Published as a conference paper at ICLR 2026
THE FORECAST AFTER THE FORECAST: PROCESSING SHIFT IN TIME SERIES
A POST-
Daojun Liang
Qi 122
Yinglong Wang 122, Jing Chen
Hu Zhang
Xiaoxiao Cui
Qizheng Wang
1.2
Shuo Li 4.5
Key Laboratory of Computing Power Network and Information Security, Ministry of Education, Shandong Computer Science Center (National Supercomputing Center in Jinan),
Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250103, China
2 Shandong Provincial Key Laboratory of Computing Power Internet and Service Computing. Shandong Fundamental Research Center for Computer Science, Jinan, 250103, China
Joint SDU-NTU Centre for Artificial Intelligence Research (C-FAIR), Shandong University
4 Department of Computer and Data Sciences, Case Western Reserve University, Cleveland, USA
5 Department of Biomedical Engineering, Case Western Reserve University, Cleveland, USA
li i aanddjendeee {chenj j, zhanghu @ org, cuixiaoxiao @hotma com 574 5743706990ss 999qq.. com, shuc. www.lcase.eeeeeee.
org, {li.qi, wangyinglong) eel.llu.euuuu..eee..oo. edu. cn
ABSTRACT
Time series forecasting has long been dominated by advances model archi- tecture, with recent progress driven by deep learning and hybrid statistical tech- niques. However, as forecasting models approach diminishing returns in accu- racy, critical yet underexplored opportunity emerges: the strategic of post- processing. In this paper, we address the last-mile gap in time-series forecasting, which is improve accuracy and uncertainty without retraining modifying deployed backbone. We propose d-Adapter, lightweight, architecture-agnose way boost deployed time series forecasters without retraining. 8-Adapter learns tiny, bounded modules at two interfaces: input nudging (soft edits to covariates) and output residual correction. We provide local descent guarantees, (8) drift bounds, and compositional stability for combined adapters. Meanwhile, it can act as feature selector by learning sparse, horizon-aware mask over inputs to se- lect important features, thereby improving interpretabilityy In addition, it can also used as a distribution calibrator to measure uncertainty. Thus, we introduce Quantile Calibrator and Conformal Corrector that together deliver calibrated, personalized intervals with finite-sample coverage. Our experiments across di- backbones and datasets show that 8-Adapter improves accuracy and calibra- tion with negligible compute and no interface changes.
INTRODUCTION
Time Series Forecasting (TSF) powers decisions across energy Anderson (9766). finance Hyndman & Athanasopoulos (2018), retail Piccolo (1990), and the sciences Gardner Jr (1985). Despite im-
pressive gains from modern neural forecasters Ekambaram al. (2024); Hollmann al. (2025): 0 Liu et al. (2025). ranging from temporal convolutions Lea et al. (2016); [ Wu et al.
Liang 2025):
2019
2022 Li al. (2023) and Transformers Zhou et al. (2021): ): Nie al. (2023), k. Wang et al.
2024a): Liu et al. (2023): Ye et al. (2024); Wang et al. (2024a); Liang et al. (2024b) to hybrid
statistical neural models Liu et al. 2025);; Ekambaram et al. (2024), condition drift Baier et al. (2020): Liang et al. (2025) is still not alleviated. Conventional remedies, e.g., full fine-tuning, archi- tectural changes, or ensembling, either demand substantial compute, risk destabilizing a hardened system, complicate operations. To cope with this, testing-time adaptation (TTA) introduced into TSF. The testing- methods aim to mitigate test-time concept drift via linear adapter updates
Corresponning author.

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Published as a conference paper at ICLR 2026
Kim et al. (2025), dynamic gating Grover & Etemad (2025) parallel forecaster combines Lee al.
(2025). layer-wise wise adjustment Pham et al. (2023). and dynamic model selection Wen et al. (023)) ).
However, these methods rely, to varying degrees, on future labels for online model updates, thereby
introducing label leakage, where future ground-truth labels are unavailable when actually applied,
that causes model performance degradation Liang al. (2024a); yee Ava Lau et al. (2025). Further-
more, LoRA-style adapters Hu et al. (2022): ; Pfeiffer et al. (2020); Li & Liang (2021) 1) in NLP tend
lead to high performance variance, since the output range not fixed Biderman et (2024).
Thus, TSF in real deployments still faces the last-mile gap: 1) Conditions drift Baier et al. 20200)
which refers to gradual changes the data-gener process (e.g.. seasonal regime shifts, covariate shifts in demand patterns) that occur after the model has been deployed, making full retraining costly; 2) High performance variance. Existing post-processing techniques are prone have high performance variance due to unstable training: 3) Inefficient training/inference. Using complex modules or frequent updates to absorb low-complexity residuals Vovk al.) (2017) 2018) makes models suffer from inefficient training/inference.. Based on these, we ask different question: Can we really keep the strong forecaster intact and learn only a tiny, post-hoc module that makes small targeted corrections, accuracy and reliability improve without heavy retraining?
We answer "yes" with 6-Adapter, a lightweight, model-agnostic framework that augments a frozen forecaster F by learning tiny adapter A in two minimal placements: input-side nudging (softly editing covariates before inference) and output-side correction (residual refinement after inference). Concretely, we instantiate additive multiplicative forms for both placements, with small trust- region parameter 8 E (0, 1) that bounds edits for safety and stability. Since A is tiny network (c.g.. shallow MLP or low-rank head) trained while I' remains frozen, produces consistent gains with negligible training time and zero changes to F's inference interface.
Further, key instantiation of the input adapter sparse, nearly binary. horizon-aware aware mask M E [0. _| and applies it multiplicatively to the context X' X M. We train M end-to-end with sparsity, semporal-smotthnees regularizers so that the adapter preserves the base model's inductive biases while exposing the most consequential inputs for the frozen forecaster. This yields transparent selections, stable training. and strong empirical gains under tight compute budgets.
feature-selector (mask) adapter that learns
and budget
Beyond point accuracy, ô-Adapter also upgrades forecast uncertainty without modifying F. We present two distributional correctors: 1) Quantile Calibrator that learns horizon-wise quantile func- tions as bounded offsets from the point forecast, with a monotonic parameterization and pinball-loss training augmented by reliability regularization; and 2) Conformal Calibrator that learns scale function for normalized-residual conformal prediction, delivering finite-sample coverage with per- sonalized, heteroscedastic intervals. Empirically, both calibrators achieve -fate-of-the-art coverage intervals.
quality and produce tight,
Through 6-Adapter, this "last-mile" adjustment consistently improves forecasting accuracy in experiments across diverse backbones and datasets, with negligible training time and no change to inference interfaces. The main contributions are:
We formalize 8- A-Adapter and instantiate two placements (input nudging and output residual correction) in additivemmuliplicatiee forms, all drop-in and architecture-anoster
We introduce learnable, budgeted mask that identifies and preserves the most consequen- tial inputs, improving transparency and stability.
We propose quantile and conformal calibrators that deliver calibrated, heteroscedastic un- certainty with finite-sample coverage guarantees, all while keeping F frozen.
Across diverse backbones and benchmarks, 5-Adapter improves accuracy and calibration; ablations illuminate the roles of 8, capacity, horizon features, and residual structure.
2 METHODOLOGY
2.1 PROBLEM SETUP
Let D {(X(1), y()) denote training pairs of context windows X L.xd and future tar-
gets Y Hx (history length L,, horizon H, d covariates, target dimensions). A pre-trained
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Ouiput
Output
Output
Output
Pretrained
Post-Procees Processing
Pretrained
Model
Model
Pretrained
Model
Post Prot-Proeessing
Input
Input
Input
(a) Original Model
(b) Input-side Nudging
(c) Output- side Coneection
Figure d-Adapter performs input nudging and output correction on the frozen forecaster.
forecaster maps to predictions y = F(X) E #x We keep all parameters of fixed and introduce lightweight, learnable adapter Ag with parameters trained on D. The adapter composes with F via two families of edits:
Input-side nudging:
X x
(additive input)
(1.1)
X O (1+ 8 8 A
(multiplicative input) (1.2)
Output-side correction:
Y = =F(X) + AF()),
(additive output)
(1.3)
Y = =F(X) @ (1 + 8 A3" (F(X) x)),
(multiplicatiive output) (1.4)
The base risk of F under loss is
R(F) = E(,))~ ((8,))]
Here, we consider two adapters, trained by minimizing empirical risk over & with F' frozen, as shown in Eq. [ The key questions are: (i) when does small 5 provably help; (ii) why do lightweight adapters suffice; and (iii) How do we choose and what the stability of the adapter A? Now, let's answer these questions.
2.2 OUTPUT-SIDE ADAPTERS AS SHRINKAGE RESIDUAL LEARNING
Here, we consider the additive adapter, as shown in Eq. 1.3 = F(X) + (F(X),, With slight modifications, the relevant analyses and theories also apply to multiplicative adapters.
Let r(X) Y F(X) denote the residual process. For squared error {(Ý) Y)
= Yl the
population risk of the output adapter with a fixed cquals
Rout (8) 1
[l[(X) 6g(X)
9(X) == A (F(X).X).
(3)
Expanding,
Rowl (8) E [|r]l3]
E (rr 9)و
+ + [1g|
(4)
signal alignment
Proposition 2.1 (Small-step improvement). If Br, g)] > 0, then for all
2E (r.g)) g)]
0 <
(5)
E
we have (5) < Reu (0) 17. The quadratic in d has negative derivative at 0 and
unique minimizer 8*
(r(a))
Remark. Improvement hinges alignment between the learned correction g and the residual r. Even when is tiny, residuals have low-complexity structure (calendar offsets, horizon- bias, scale drift), small g can achieve positive alignment, and shrunken step 8 guarantees risk reduction. This is exactly the first step of boosting with shrinkage or a stacked residual learner with conservative learning rate.
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In practice, learn g from finite data with penalty (2(0) (e.g., l2, low rank, sparsity). The empirical objective
min 2 Σ 1lm 5go (Q(O)
(6)
yields a shrunken projection of residuals onto the function class of A. With small 5 and a low- capacity A, we target the dominant residual modes while avoiding variance blow-up.
2.3 INPUT-SIDE ADAPTERS VIA FIRST-ORDER LINEARIZATION
For the input-nudging adapter, as shown in Eq. of F around X:
1.1 X = x+5 (X), apply first-order expansion
F(x + 6u(X)) F(X) (X)(X))(XX,,
where u(X) A(X, h) and (x) E #xd the Jacobian of F w.r.t inputs. Under squared loss, replacing by JFu in the previous derivation yields
(i) 1E [[r]1"] DE [(r, JFu)] + J [[Jru
In general, for im 6X) F(X + ou( ox())), Ron = 3 || Via (X; (x: have
then there exists 6 > such that Rin < Rin (0) for all 8 E (0,6). And, if F is affine in the near of X, Prop. 2. is also hold.
Proposition 2.2 (General ô-step improvement). [r, JF]]
The proof given in Appendix B.2 This proposition states that for differentiable loss t, the loss gradient w.r.t. inputs satisfies v,((F(X), = J=(X)TV Vzl. Choosing u(X) = -BVx((((X).) v) for a small, learned preconditioner B recovers learned, damped gradient step in input space; training A on data finds such steps implicitly without computing Jp at test time.
3 THE STABILITY OF Ö-APATEER
3.1 PREDICTION STABILITY UNDER BOUNDED INPUT EDITS
Let X = X + (additive case). Then, we have
Proposition 3.1 (Drift bound). Assume the frozen forecuster F -pp-ipschitz, the change in pre- diction is bounded by
Y - 5 &8LFV Ld.
(9)
The proof is given in Appendix
B.3 Further, let X = X (x we have
Corollary (Multiplicativee input edits). | || - Bx, then
|
YI < de LF Bx A (X)
(10)
In particular, for 1, 1|- - 81 = O6)).
The proof is given in Appendix B.4
Corollary means that small 8 yields Lipschitz-stable predic-
tion changes for input adapters.
3.2 LOSS STABILITY AND GUARANTEED LOCAL IMPROVEMENT
Let 8 F(X) and consider an output edit γ Ý dd with d = Ag (X,X, we have
B 5391d11, g: Vul(u, -=x e(Y v) < ((8 8(g, d) 2
(11)
If d aligns with -. i.e. (g. d) < -a]lg|]|dd| we get
Theorem 2 (Descent for output adapters). If the per-sample prediction loss E(:, y) is B-smooth in its first argument (e.g., MSE, Huber), for any sample.
3 888 1 2
((Y,)) - {(L.v) v) < -8allglll|d|
(12)
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Hence, for any å (0. 2a [sll) Bl|d]
the loss strictly decreases. The optimal 8*
Bld]] yields
a²
x (() - E(8 5
2,3 199.
(13)
The proof is given in Appendix B.5
Remark. With MSE, Y y, so the improvement is proportional to the squared residual mag- nitude. Further, with bounded adapter family, the trained 4our (minimizing batch loss) produces that correlates with -g unless capacity is zero.
X du with = Assume F is Theorem 3 (Descent for input adapters). Let X
differentiable at X with Jacobian Jx(X). Define the effective prediction step Jy(X)u. Then for 5 small,
3 2101
((F(X), v) S E(Y .) + 5'g. s)
(14)
2
If s) = there exists 8 > O such that V8 E (0, 8) the loss strictly decreases. Moreover, optimizing the quadratic upper bound in 6 yields the same margin as Theorem 2 up to O(1) terms.
The proof is given in Appendix B.6 Theorems and 3 show that for sufficiently small ó and mild alignment, both adapter types reduce the loss locally, with explicit improvement margins.
3.3 COMPOSITIONAL STABILITY (INPUT + OUTPUT)
Let the full edit be X X ōu, 8 F()), then Ý - X''+od( X). Under the same conditions Prop. 3.I and Theorems and we have:
Proposition 3.2 (Composite drift and loss bound).
||8- Y1 1 - so the model drift is (6). Further, for the loss,
Y1l olld(Y"', X)I| < 8Lg &Ca,
(15)
( v) < <YY 9) + 0gg. d) + 6 + d| O
(16)
The proof is given Appendix B.7 the combined step s ++ aligns with -g by parameter-sharing learned gate, inherit the same descent guarantee Theorem 2
4
IMPLEMENTATION
4.1 5-ADAPTER
5- Adaapter targets structured residuals (bias, scale miscalibration, phase lag) while preserving F's inductive biases. We encode this through three principles: 1) Boundedness; Enforce small edits via i and penalties on A-)Il: 2) Low capacity: Use tiny architectures to avoid overfitting and respect production budgets. 3) Horizon awareness: Allow horizon-specife corrections without destabilizing temporal coherence. Concretely, we use tiny MLP as the backbone and impose:
(X)]l < 1, (-)l <
(17)
via tanh squashing and optional clipping, so that is direct bound the maximum per-entry change. For multiplicative edits we ensure positivity where required by applying exp( (a))))) as an alternative to 1 + Agg For compositional adapters (input+output), as stated in Prop. 3.2 their parameters can be optimized in parallel during the training process.
4.2 FEATURE SELECTOR
particularly transparent instantiation of our input adapter cast it as learnable mask (selector) that selects the parts of the input that are most consequential for the frozen forecaster F. Concretely, a context window X Rddd we parametrize an adapter Ao that outputs mask M
[0, 1] 1xd and apply it multiplicatively.
x'= = X M(
(18)
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The mask is trained end-to-end while keeping F fixed. Intuitively, MM plays the role of soft selector: values near keep information intact, values near 0 suppress it. To obtain discrete, human-readabee readable employ relaxed Bernoulli parameterizations, Let
selections without sacrificing differentiabilityy, ((x;0) 0) e RLxd be adapter logits. We form a Gumbel-Sigmoid (Concrete) relaxation
M(;;0,)) = ( log a (X; - G G).
(19)
where G is iid. Gumbel noise, (-.) is the logistic function, and > 0 is a temperature annealed from high value (smooth masks) to a low value (nearly binary). At inference, we may harden the mask via threshold Mard 1 1 0.5) keep it soft to avoid distributional brittleness. As a simpler alternative, we use a straight-through estimator: threshold in the forward pass, back- propagate through the corresponding sigmoid in the backward pass. Training the mask as selector requires explicit structure in the objective. Given predictions Y
F(X M), we minimize
min Cpred Y) +Y1 1 forecasting error
M + Aena
Σ (M
+Ai
TV(M)
+dbud (m K)
(20)
sparsity
entropy
temporal smoothness
budget
where m La Et Mr, is the average keep-rate and K E (0, 1] is a user-specified budget, which stabilizes selection under correlations by constraining the feasible keep set, e.g., use at most 10% of inputs. The é and entropy terms encourage sparse, nearly binary masks; the total-variation penalty TV(M) promotes temporal contiguity, reflecting the fact that relevant patterns often span short intervals rather than isolated instants. See the specific expressions of each part in Appendix C.3
4.3 DISTRIBUTION CALIBRATOR
Now, we introduce how the proposed adapter calibrator when the forecaster F is frozen and produces only fixed-point predictions.
4.3.1
QUANTILE CALIBRATOR
a distributional assumption is undesirable, the adapter can directly output horizon-wise quantiles bounded offsets from the point forecast:
1r,( = cas Xx x, o so(X,
(21)
where ag E [-1, 11,,11xxm xm and 80 Os learned scale. To ensure monotonicity in parameterize
Orj41.0 Iry.0 (xft ग. <
<T.;
(22)
where dz.0 is the adapter's raw increment for the gap between two adjacent quantile levels T, and T5+1. Eq. [22 anchored at central level (e.g., TJ/2) via the bounded offset around Ý. Then, for the training objective, we replace the point losses with pinball loss and add reliability regularization:
N
min 8
1
Iee C, (Y( (x,,0((()))) + Aca(Cre( + Ammgllagl13 jm
(23)
N
where & is the pinball loss; Cre can be the same soft-coverage penalty above, PIT-uniformity term computed by interpolating the predicted quantiles into a differentiable CDF and matching the PIT distribution to Uniform(O, 1).
4.3.2 CONFORMAL CALIBRATOR
When strict distribution- guarantees needed, we combine learned scale function with formal prediction, i.c., train wx(X, Ý) > (small adapter) predict residual magnitude while keeping the mean at Y:
N
y(i) - -100)001 /wo (x(1), ((0) + AlIwall
min
(24)
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subject to mild regularizer to keep we I on average. Then, use conformal scaling a held-out calibration set caal to compute normalized residuals as
| - 100 Y()) (x y()) Deal.
(25)
Then, the calibrated marginally valid prediction sets can be obtained by
(x) {y l1v y Na we (X, Y)},
(26)
where Ka is the empirical (1 - -uaanillee of {r This yields finite-sample coverage 1 1- aa under exchangeability. The adapter we personalizes interval width while F remains untouched.
EXPERIMENTS
We validate the 8-Adapter method variety of widely used datasets, see Appendix C.1 We test its gains when applied to pre-trained and state-of-the-art (SOTA) models (Section 5.1). its application feature selector (Section 5.2). and its effectiveness an interval calibrator (Section In this paper, we set 5 = 0.1 (0.01 for ETT datasets) and the learning rate of Adam to IE-4, and conduct an ablation study on them Section 5.4
5.1 EFFECTIVENESS OF 5-ADAPTER
Table 1: The improvement of 8-Adapter on Pre- Pre-Trained models.
Model
Sundial-S S (Univariate)
TTM R2 (Mullivariate)
Type
original
AdaiX
Ada-Y
original
Ada-X
Ada- Y
Dataset
MSE
MAE
MSE
MAE
IMP
MSE
MAE
IMP
MSE
MAE
MSE
MAE
IMP
MSE
MAE
IMP
ELC
0.427
0.463
0.334
0.410
17%
0.404
0.451
4%
0.180
0.272
0.167
0.262
6%
0.168
0.262
5%
Traffic
0.237
0.314
0.220
0.301
6%
0.224
0.302
5%
0.517
0.344
0.492
0.329
5%
0.492
0.325
5%
Exchange
0.312
2%
0.213
3%
0.210
Weather
0.427
0.463
0.025
0.005
96%
0.039
0.059
89%
0.150
0.196
0.148
0.193
2%
0.143
0.191
ETTml
0.121
0.217
0
0.190
24%
0.087
3.002
18%
0.338
0.357
0.329
0.357
1%
0.333
0,353
3%
ETTm2
0.348
0.420
0.201
0.325
32%
0.254
0.371
19%
0.259
0.174
4%
0.240
We first verify the performance gains of 6-Adapter on pre-trained models, including Sundial-S (Uni- variate) Liu et al. (2025) and TTM-R2 (Multivariate) Ekambaram et al, (2024). The experimental results in Table show that 5-Adapter consistently enhances forecasting performance across all datasets and backbone models, confirming its effectiveness and generality. Both the input adapter (Ada-X) and the output adapter (Ada-Y Y) have achieved significant performance gains. These results highlight that training lightweight adapters while keeping the backbone frozen is powerful and efficient way to boost predictive accuracy.
Table 2: Comparison of various adapter methods and online methods (averaged across all lengths).
Dataaci
Ofline
SOLID
TAFAS
LaRA
Ada-X-Y
Offlin:
SOLJD
TAFAS
LaRA
Ada-X+Y
Offlis:
SOLID
TAFAS
Aib-X4Y
OuzNal
FSNaT
0.182
0.182
0.515
0.510
ETThi
0.461
0.460
0.476
0.454
8.651
0.454
0.458
0.477
0.488
0.449
0.593
0.589
0.991
0.577
0.688
0,877
0.355
ETTml
0412
0.406
0411
0,407
0.396
0417
0.44
0.430
0414
0.483
0.64
0.661
0.688
0.997
0.56
0.851
ETTm2
0.255
0.235
0,288
0.211
0,293
0.39
1,171
Exchanng
0.300
0.347
0.363
0.366
6,297
0.383
0.376
0,302
0.376
0.316
0.599
0.491
0.385
B.465
0.647
0.778
0.43
0.433
Weither
0.266
0.255
0,256
0.251
6.342
0.259
0.257
0.259
0.255
0.244
0.324
0.316
0,325
0.299
0.360
0.541
DreNie and PSNc in implemented Ieozalon ho
the public lhoory ganrital in papar with lohel lesloge Fir morc denel, plaser nolor Tible
Ayperdidc::.m
Then, we compared the proposed &- A-aapter with other adapter methods and online learning methods by removing label leakage Liang et al. (2024a); yee Ava Lau al. (2025). It is worth noting that when removing label leakage, some methods have a certain degree of performance degradation. This may be because the design of these methods relies excessively that the 6-adapter achieves the lowest error on every dataset across all three backbones. The gains sizeable challenging sets, while remaining consistent the ETT variants. Moreover, when
future true values. Table shows

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Published as a conference paper at ICLR 2026
Table Gains of ô-Adapter on SOTA models (averaged across all lengths. See Table S for details).
Medel
DiafProd
iTransfemmrr
FoirerCNNN
FieTS
Antoforser
Dutnet
Original
Ada-X
Ads-
Original
Adax
Ads-Y
Original
AdrX
Ada-
OOiginal
Ada-X
Ads-Y
Original
Ada-X
AdorY
ELC
0.182
0.128
0.169
0.190
0.182
0.181
0.267
0.255
0.241
0.209
9.200
0.194
0.515
0.558
0.450
Exchasge
0.350
0.302
0.319
0:383
0.348
0.349
0.350
0.395
0.,779
0.416
0.412
0.422
01500)
0.481
0.462
Thaffic
0.453
0.448
0.442
0.45
0.420
0.461
0.777
0.760
0.740
0.596
0.500
0.572
0.972
0.950
0.918
Weather
0.256
0.251
0.245
0.259
0.200
0.345
0.255
0.251
0.244
0.255
0.359
0.243
0.325
0.306
0.290
CTThl
0.461
0.457
0.458
0.454
0.453
0.456
0.561
0,565
0.542
0.482
0.574
0.471
0.593
0.533
0.577
ETTR2
0.390
0.386
0.387
0.38%
0.385
0.390
0.545
0.499
1.506
0.537
0.492
0.4%%
0.438
0.430
0.423
ETTml
0.42
0.399
0.407
0.417
0.407
0,406
0.456
0.447
0.447
0:405
0.401
0.401
0.664
0.604
0.677
ETTm2
0.285
0.279
0.292
0.455
0.386
0.438
0.325
0.339
0.316
contrasted with OneNet and FSNet, 8-Adapter paired with standard backbones yields substantially lower errors on datasets, underscoring its plug-and-play effectiveness and robustness.
Next, we verify whether the 5-Adapter provides gains to the SOTA forecaster. Table 3 shows that 5-Adapter provides consistent and significant improvements across multiple SOTA models. For nearly all datasets, Ada-X and Ada-Y lead to lower prediction errors compared to the original mod- els, demonstrating that the proposed adapters generalize well to diverse forecasting architectures. Notably, Ada-X again delivers the largest gains, particularly change, Traffic, and ETT series, confirming that refining the input signals before model inference the most impactful strategy. Also, 5-Adapter yields clear benefits, highlighting its plug-and-play nature and ability to enhance high-performing models. These results further validate that 5-Adapter a broadly applicable, efficient, and effective enhancement method for modern forecaster.
challenging datasets such Ex-
0170
DisPred
Mansform
0.174 Lo221
0.17
0:2
0.165
0.168
1156
0:500
0.105
01155
4150
0162
0.302
0.150
0383
0.199
012
0.14
310X
0.156
Origral
ArboX
01140
Orign
Pinw (Radd)
Adoy
(Crdineé
(Orlew)
Figure 2: Performances of the forecaster F and 8-Adapter under batch or online training.
Finally, we test whether 8-Adapter is effective under different compositions and training methods. Implementation and training details of Ada-X+Y are in Appendix (C5).. Figure [ shows that 8- Adapter consistently reduces eπror under batch and online training. Each single adapter improves over the frozen forecaster and also outperforms conventional fine-tuning or continue- continuettraining. And training the adapters online yields further gains over batch. Importantly, A delivers the lowest MSE in all settings, indicating robust and statistically reliable improvements.
5.2
EFFECTIVENESS OF THE FEATURE SELECTOR
+ 10%
20%
30%
40%
50%
60%
-
70%
80%
900%
95%
0.9
0.79
0.85
0.7
030
0.69
0.6
Enoor
0.5
0.6
0.14
9.16
0.5
0.64
0.59
14
0.4
0.45
0.54
0.35
0.49
0.44
0.2
0.1
0.15
Rankon
Sponk
0.39
Manion
Randon
10
Soloot
Select
Learnable Select
Remove Learmable Select
Learnable Select
(a) Select the important features (ELCC
Remove important features (ELC)
(c) Select the important features (Weather)
Figure 3: Changes of forecaster's performance after selecting or removing valid features.
8

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Published as a conference paper at ICLR 2026
To verify the effectiivness of the mask adapter as a fea-
Table 4: Best performance of the mask adapter and its mask ratio.
Daraset
ELC
ETTh1
1:7h22
ETTml
ETTm2
Traffic
Weather
Exchange
ture selector, visualized its training process, as shown in Figure It demonstrates that a learnable mask adapter reliably identifies the most informative input features under varying spar- sity budgets. In subfigure (a), selected features yields markedly lower errors than random selec- tion across all retention rates (10-95%) and converges within a few epochs. Conversely, when the learned features are removed (b), the forecaster's error rises substantially, often worse than removing an equal number of randomly chosen features. This shows that these features are uniquely critical to performance rather than incidental. Table 4 shows mask ratio the mask adapter when the best performance is achieved (no budget added), and Figure 4 visualizes important features in dif- ferent proportions (most important features remain unchanged). These confirming that the learned selections consistently outperform random picks, and removing them degrades accuracy the most.
Original
0,163
0.390
0.296
0.345
0.182
0.444
0.173
0.099
Masked
0.334
0.436
0.171
Mask Rario
97%
96%
95%
97%
96%
98%
96%
92%
3.5
Buth Prediition
1.0
Frath
Trai Prediction
1.0
Brtti Prediction
Frnd ctlen Fraetures
3.5 Heatures
0.5
0.5
3.4
0.0
0.0
D.b
-3.5
-0.5
-0.5
-0.6
-3.0
-1.0
-1.0
-3.0
-1.5
50
100
350
100
150
(loue (15))
fentures Claus 0.5)
30% Gestlrets Colaua 0.33
Figure Visualization of different important features learned by the mask adapter.
5.3 PERFORMANCE OF CALIBRATOR
(a) ETT
(b) Traffi
(c) Eloctricity
(d) Weather
61,70 61.92
86.36
07:13
21,55
2488
25.43
00228
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56,
05,11
22:8
54-
63144
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CLB Pnhel SPLT
QC
0C
CCB 19b19
SPC1
OC
OC
COR PraM SPLI
QC
OC
OUR Embii SPCI
QC
CC
Figure 5: Comparisons among the Quantile (QC), Conformal (CC) calibrators and others.
Now, we verify the effect of 5-Adapter as the Quantile Calibrator (QC) and Conformal Calibrator (CC). As shown in Figure 5 calibrators consistently deliver the highest PICP, indicating better coverage reliability than strong baselines (CQR Romano et al. (2019), EnbPI & Xie (2021). SPCI Xu & Xie 2023 Further, in Figure 6 illustrate that both calibrators produce cell-calibrated intervals that expand near peaks and usually enclose the ground truth. QC tends to yield slightly wider, more conservative bands, while CC delivers comparably high coverage with tighter intervals.
Sample with Quontile bebtbatn PNCPL 8,225)
Sarppl with Canformall beadon SKCP 0.9991
Sample wihh Gunedt Cofirddo DeC 5991
Saciple Carte beaton pace 9.6711
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-0.
-30
-15
-3
[e] Resulls of Quant and Conformal Conector on test somple
(b) Results of Quantile and Conformal Comector on mnother teet semple
Figure 6: Visualization of the Quantile and Conformal calibrator predictions.

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Published as a conference paper at ICLR 2026
5.4 ABLATION STUDIES
The impact of is important. Figure 7 indi- cates that all adapter variants reduce error ver- sus the frozen model, but the combined adapter (Ada-X+Y) delivers the lowest median errors and the tightest variability. Across placements, a moderate adjustment size is the most reliable, e.g., pushing to d = 0.2 yields smaller or incon- sistent gains, suggesting overly aggressive cor- rections. It is confirmed that composing input and output adapters with a modest multiplica- tive trust-region produces the most accurate and stable forecasts.
Valodation
01772
0425
0118
0440
0.165
0.10
Adbx
Adol
A/bY MoY AdY MoXe MaX A
16.0.1
H6-02
seun.
1e402
1e-03
H4401
16-02
Figure 7: Performance of ò-Adapter's variants.
Then, we used PatchTST Nie et al. (2023) and Time.ixxr Wang al. (024a) backbones to compare the performance of additive and multiplicative composite 8- o-Adapter. As shown in Table [] after adding the 8-Adapter to PatchTST and TimeMixer, their performance has been significantly improved. the additive and multiplicative adapters reduce the MSE of PatchTST by 5.6% and 5. % respectively across various datasets. However, for TimeMixer, the MSE reductions from the additive and multiplicative adapters are 1.6% and 1.8% respectively. This indicates that both have their respective advantages, and the increase of the additive adapter is relatively more significant. In addition, for efficiency of the 8-Adapter, see Appendix C.8
Table 5: Comparison of additive and multiplicative composite 6-Adapter.
PatchTST
Ada-X+Y
Ada-Xx
TimeMixer
+ Ada-X+Y
+ Ada-Xx
MSE
MAE
MSE
MAF
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ELC
0,167
0.252
0.159
0.245
0.159
0.246
0.145
0.243
0.143
0.241
0.143
0.241
Weather
0.178
0.219
0.161
0.220
0.165
0.224
0.168
0,216
0.166
0.214
0.164
0.214
Traffic
0.463
0.297
0.451
0.292
0.448
0.290
0.475
0,317
0.465
0,307
0.467
0.310
Finally, we investigated the impact of two key factors: ō and its learning rate. As shown in Figure 8, despite the large variation ranges of 8 and the learning rate, the forecaster (iTransformer) can still maintain relatively stable prediction performance. However, other models, such as those that attempt to fine-tune pre-trained large models using Lora Hu et al. (2022), not only exhibit large variance but also lead to degradation in performance, which remains problem worthy of further exploration.
1R-2
56-2
1E.1
28-1
36-1
1R.2
16.3
1F.4
1R-5
1R-6
1B-5
IE-4
1B-2
0.185
0.156
0,172
0.164
MSE
0.180
0.154
MSE
0.170
1.3
Lomrst
0.102
0.175
01152
0.168
1.2.
0.100
0.166
1.1
BLC Waather
F.C
0.161
0.158
0.168
Weather
0.162
1.0
(e) Ablation Studly of 8
(b) Ablation Study of Leaming Rate
(c) Fine tuning Sundial
Figure 8: Variances of Ada-X with different 8(a) and learning rates (b) and fine-tuned Sundial (c).
6 CONCLUSION
We present 5-Adapter, lightweight and post-hoc framework that improves frozen forecasters via bounded input nudges and output residual corrections. we provide theory guaranteeing local descent and stable composition. To enhance interpretability and robustness, we introduce a feature-selector adapter that learns a sparse, horizon-aware mask under budget priors, exposing the most consequen- tial inputs while constraining edits. Beyond point forecasts, we deliver calibrated uncertainty via two distributional correctors: a Quantile Calibrator that learns quantile offsets trained with pinball loss, and Conformal Corrector that estimates heteroscedastic scales for normalized-residuss conformal prediction, yielding fiitt--ample coverage with personalized intervals. Across diverse backbones and datasets, 5-Adapter yields consistent accuracy and calibration gains.
10

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Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENTS
This work was supported by Project of Key R&D Program of Shandong Province, China (2025CXPT095, 2025CXGC010107), Taishan Scholars Program: NO.tspd22240814, the Pilot Project for the Integration of Science, Education and Industry of Qilu University of Technology (Shandong Academy of Sciences) (2025ZDZX01), the "20 Articles" Project for Universities in Ji- (202534093). the Pilot Project for Integrated Innovation of Science, Education, the Industry of Qilu University of Technology (Shandong Academy of Sciences) (2024ZDZX088, the Taishan Scholars Program of Shandong Province (No. 1sqnz20250747), and the National Natural Science Foundation of China (No. 62502250).
ETHICS STATEMENT
Biases in benchmark creation: The authors are aware of the potential for bias in the creation of our benchmark entries. The selection and definition of dark patterns, as well as the design of benchmark prompts, may inadvertently refect the authors' perspectives and biases. This includes assumptions about user interactions and model behaviors that may not universally accepted or relevant.
Misuse potential: While our intention with this benchmark is to identify and reduce the presence of dark design patterns in LLMs, we acknowledge the potential for misuse. There is risk that malicious actors could use this benchmark to fine-tune models in ways that intentionally enhance these dark patterns, thereby exacerbating their negative impact.
REPRODUCIBILITY STATEMENT
The code used in this paper can be found here. And we use notebooks write some simple examples so that readers can quickly implement the results of the paper. The steps to reproduce the paper are:
Download the code.
2. Install the necessary environment.
3. Run "bash run.sh".
4. Or, run the provided notebook.
The code is given in this Repository
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Weiwei Ye, Songgaojun Deng, Qiaosha Zou, and Ning Gui. Frequency adaptive normalization for non-stationary time series forecasting. Advances in Neural Information Processing Systems, 37: 31350-31379 2024.
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Kun Yi, Jingru Fei, Qi Zhang, Hui He, Shufeng Hao, Defu Lian, and Wei Fan. Filternet: Harnessing frequency filters for time series forecasting. Advances in Neural Information Processing Systems, 17-55115-551444 2024a.
Kun Yi, Qi Zhang, Wei Fan, Shoujin Wang, Pengyang Wang, Hui He, Ning An, Defu Lian, Long- bing Cao, and Zhendong Niu. Frequency-domain mlps are more effective learners in time series forecasting, Advances in Neural Information Processing Systems, 36, 2024b.
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A RELATED WORK
A.1 CLASSICAL MODELS FOR TS FORECASTING
TS forecasting is a classic research field where numerous methods have been invented to utilize historical series predict future missing values. Early classical methods Piccolo (1990); Gardner Jr (185) are widely applied because of their well-defined theoretical guarantee and interpretability. For example, ARIMA Piccolo (1900) initially transforms non-stationary TS into stationary one via differencing, and subsequently approximates it using a linear model with several parameters. Exponential smoothing [Gardner Jr (1985) predicts outcomes future horizons by computing a weighted average across historical data. In addition, some regression-based methods, e.g., random forest regression (RFR) [Laaw et al. (2002) and support vector regression (SVR) Castro-Neto et al.) (009), etc., are also applied to TS forecasting. These methods are straightforward and have fewer parameters to tune, making them a reliable workhorse for TS forecasting. However, their short- coming is insufficient data fitting ability, especially for high-dimensional series, resulting in limited performance.
A.2 DEEP MODELS FOR TS FORECASTING
The advancement of deep learning has greatly boosted the progress of TS forecasting, Specifically, convolutional neural networks (CNNs) LeCun al. (1998) and recurrent neural networks (RNNs) Connor et al. (1994) have been adopted by many works to model nonlinear dependencies of TS, c.g., LST et al. (2018) improve CNNs by adding recursive skip connections capture long- and short-term temporal patterns; DeepAR Salinas et al. (2200) predicts the probability distribution by combining autoregressive methods and RNNs. Several works have improved the series aggregation forms of Attention mechanism, such as operations of exponential intervals adopted in Trans Li ct al. (019), ProbSparse activations in Informer Zhou et al. (2021), frequeney sampling in FED- former Zhou et al. (2022) and iterative refinement in Scaleformer Shabani et al. (2022). Besides, GNNs and Temporal convolutional networks (TCNs) Lea et al. (2016) have been utilized in some methods Wu et al. 2019); Li et al. (203);; ; Liu et al. (2022a); Wu et al. (2022) for TS forecasting on graph data. And some methods combine data-driven approaches with physical laws Hong et al.)
2023 2026 to achieve dynamic prediction of complex systems [Hong al. (2022). The afore- mentioned methods solely concentrate on the forms of aggregating input series, overlooking the
challenges posed by the concept drift problem.
A.3 TRANSFORMER-LIN -IKE MODELS
Since TS exhibit a variety of patterns, it meaningful and beneficial decompose them into several components, each representing
underlying category of patterns that evolving over time Anderson
(1976). Several methods, e.g.. STL Cleveland et al. (1990), Prophet Taylor & Letham (0018) and
N-BEATS Oreshkin et al.] (2019), commonly utilize decomposition historical series. There are also some methods, e.g., Autoformer Wu et al. (2021), FEDformer Zhou al. (2022), Non-stationary Transformers Liu al. (2022b) and DistPred Liang (2025), that harness decomposition into the Attention module. The aforementioned methods attempt to apply decomposition to input series enhance predictability, reduce computational complexity, or ameliorate the adverse effects of non-stationarity. Nevertheless, these prevalent methods are susceptible to significant concept drift when applied non-stationary TS.
preprocessing phase
Furthermore, there are four themes that use deep learning to predict time series: (1) smarter trans- formers Vaswani al. (2017). such as PatchTST Nie et al. (2022), iTransformer Liu et al. (223)) BasisFormer Ni et al. (2023). Minusformer Liang et al. (2024b). TimeXer |Wang et al. (2024b)
and Periodformer [iang et al. (202cc) which restructure attention or add learnable bases to extend context length, cut computation and boost accuracy; (2) competitive non-transformer backbones, in- cluding N-HiTS (hierarchical MLP) Challu et al. (2023), DLinear Zeng et al. (2023), PGN Jia et al. Challu
2024), PSL4).,SSDDEE LiLD et ( (2024d) and state-space models like TSMamba Ma et al. (2024). TimeMa- chine Ahamed & Cheng (2024) and FLDMamba Zhang et al. (2025). which deliver linear-time in- ference and rival or surpass transformers on long horizons Liang et al. (2023); 3); (3) foundation
initiatives, TimeGPT Garza et al. (2023), OneFitAll Zhou et al. (2023). TimeLLM Jin et al. (2024),
UniTime Liu al. (2024) and DAM Darlow et al. (2024) that pre-train on massive heterogeneous
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corpora and achieve impressive zero-shot few-shot performance across domains; and (4) training and interpretability advances, such as FreTS Yi al. (2024b), FilterNet Yi et al. (2024a), and decomposition-awann architectures Yu et al. (2025)), e.g. which tackle non-stationarity. quantify uncertainty and make forecasts more transpar- ent. ability to directly model the correlations among attributes within series by decoupling the dependency on input length.
normalization (FAN) Ye al. (2024). e.g.,
A.4 ONLINE LEARNING
Online learning strategies emhed concept drift adaptation Yu al. (2025a) within the forecasting models themselves. One example is FSNet which leverages complementary learning systems theory to pair slow-Iearning base forecaster with fast-adapting components. Another line of work is OneNet Wen et al. (2023), online ensembling approach that dynamically combines two neural models: one specializes in capturing temporal dependencies within each series, and the other focuses on cross-series (covariate) relationships. Each of these deep learning techniques illustrates how integrating drift-awareness (through dual-model architectures, ensembling, or proactive adjustment) improve TS forecasting performance in online Yu al. 2024)
A.5 POST-PROCESSIN METHODS IN TIME SERIES
Testing- Time adaption (TTA) is very important for Time Series Forecasting. The udapter-based methods include SOLID Chen et al. (2024), TAFAS Kim et al. (2025) and its follow-ups PETSA Medeiros et al. (205)) and Dyna Grover & Etemad (2025). ELF Lee et al. (2025), etc., and online approaches. c.g., FSNet Pham et al. (202)) and OneNet Wen et al. (203)) aim to mitigate test-time concept drift. Specifically, SOLID retrains selected predictor layers using the most recent similar samples; TAFAS updates linear adapters by online detection of temporal cycles; PETSA and DynaTTA extends TAFAS with additional losses and dynamic gating to further enhance adaptability. These methods are either based on linear adapters, parallel fusion, or overall fine-tuning and, they do not consider the impact of label leakage Liang et al. (024a)); [yee Ava Lau al.] (2025). On the contrary, 8-A 5-Adapter can perform non-linear adaptation on both input and output, with good theoretical guarantees. And it only relies on the most recent sample for fast updates. In addition, can be used as feature selector or a corrector.
A.6 POST-PROCESSING METHODS IN NLP
Our work conceptually related to the general parameter-efficient adaptation methods that have been developed primarily in NLP. Adapter modules for BERT and other Transformers add small task-specific bottleneck layers between pre-trained weights, keeping the backbone frozen while achieving fear-fine-tunin performance many downstream tasks (Houlsby al. (2019). This idea has been extended to multilingual and multi-task settings (e.g.. MAD-X), where language and task adapters are stacked enable cross-lingual transfer Pfeiifeer al. (2020). A complementary direction is low-rank adaptation (LoRA), which inserts trainable low-rank matrices into attention and feed-forward projections adapt large language models with only small number of additional parameters Hu et al. (2022). Another family of methods performs input-side adaptation via prompt and prefix tuning: instead of changing internal weights, they learn continuous prompts or prefixes at the embedding level that condition frozen language model for each task Li & Liang (2021).
Compared with the above methods, 5-Adapter adopts the same high-level principle of learning a small 8-module around frozen backbone, but it is tailored to TSF and operates strictly at the input/output interface of a possibly black-box forecaster. Specifically, we introduce horizon-aware input adapters, feature-masking modules, and output-side uncertainty adapters, and we analyze their behavior through Lipschitz-style stability and descent guarantees. To our knowledge, such an I/O level, theoretically characterized adapter framework for the existing NLP adapter or prompt-tuning literature, which primarily modifies internal layers or token embeddings of language models.
forecasting is not present in
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A.7 CONFORMAL PREDICTION
Conformal prediction presents an alternative framework for distribution prediction, diverging from traditional parametric approaches. their study, the authors in (Vovk et al., 2017: 2018) introduced random prediction system and proposed nonparametric prediction method grounded in confor- mal assumptions. By integrating conformal prediction with quantile regression in (Romano et al. 2019 Xu & Xie 2021 2023), they developed method for constructing prediction intervals for the response variable. However, the practical application of conformal prediction is not without limita- tions. Its effectiveness is often constrained by the assumption of exchangeability of residuals, which may not hold in all contexts, particularly in the presence of temporal dependencies. This limitation can lead less reliable prediction intervals when applied non-independent and identically dis- tributed (non-i.id.)) data, thereby challenging its robustness in real-world scenarios where data often exhibit complex dependencies.
A.8 TERMINOLOGG EXPLANATION
Conditions drift, which refers to gradual changes in the data-generating process (e.g., seasonal regime shifts, covariate shifts demand patterns) that occur after the model has been deployed, making full retraining costly; Low-complexity residual structure means that residual errors often exhibit simple patterns (e.g.. horizon-wise bias, scale miscalibration, calendar offsets) that can be captured by small function class (tiny MLPs/low-rank heads) rather than requiring new high- capacity backbone, but the base model fails to absorb them.
B THEORETICAL PROOF
B. PROOF OF PROPOSITION
Define risks (squared error):
Row(()) +( - (1(
(27)
and
(i) BBlll F(X 0u(x))111
(28)
Proof. Let A = [X) and B = E(R(X), (X)). Then
Romi (6) = E|R|1 - 8B + 62A
(29)
Hence Rou is a strictly convex quadratic in whenever A > O, with unique minimizer 6* B/A and minimal value
Ba
Reat (8*) = 1
R| R|
(30)
A
In particular, if B >> 0 and A > 0 then for all 0 < 8 2B/A, (8) Rum (0)
2|
Then, expand the square:
R fgl |R| 28 g) 11
(31)
Taking expectations and multiplying by 1 yields the displayed quadratic form. If A > 0, the deriva- tive d d5 -B + 5.A vanishes uniquely 6* B/A; strict convexity gives the minimal value above. If > 0, then near = 0 the derivative is negative, so every ō E (0, 2B AA) strictly improves the risk over 5 0. If A 0 then g a.s. and risk is constant; if B < 0 there is no positive 6 improving over 8 = 0.
Remark. (i) This is exactly the first (shrunk) step of residual boosting. (ii) The achievable drop at the optimal 6* is (B2 /A), which is positive iff B & 0 and A > 0.
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B.2 PROOF OF PROPOSITION 2.2
Let F I2d + R be differentiable, u X + Rd measurable nudging field, and define for d 2
Vin (X;)) = F( + (x()),, Rin (6) - = || - Uin (X;5)
(32)
Write - (x() for the Jacobian of F X, and A 18 (r((),, J=(X)u(X))]
If A > O, then there exists > such that (i) < Rmn for all 8 E (0, If. in addition, F is affine neighborhood of the support of X (Jp constant and the Hessian is zero), then
Ri = 2 E [|x(X) 8 JF u(x))
(33)
a quadratic function of whose unique minimizer
(x(),, JX
8 E [ 2
(34)
Based on the conditions, we know that: F' is C (continuously differentiable) on an open set con- taining {i + (u(
[0, do]} for some do And [|J=(X + du(X)) [X integrable uniformly for E [0, 6o]. and lyy F(X + du(X)) is integrable.
First, we have the following lemma.
Lemma 4 (Improvement via Jacobian Jacobian-aligned aligned nudging). Er, Jpu]] >> O, then sufficiently small > 0 reduces risk. As before, the optimal small-step size is 8 (r]je)) Jp
Proof. For each X, by the fundamental theorem of caleulus in Banach spaces,
F(X + 8x(x)) = F(X) + 1." Jr(X + tu(()) u(X) dt.
(35)
Hence,
Rs Ro / J= +tu(X)) u u(X)dt.
(36)
Let F(8) = Rin = Rs 12 Using 25 || 2 = 2(v,
F' (6) E (Rs, -F (x )u(x)) u(X)).
(37)
Under the domination assumption, dominated convergence allows 5 - O inside the expectation, giving
F0 E(Ro, ((X))(X))
(38)
If C O. then F'(0) -C By continuity of F' 0 (again from dominated convergence and continuity of Jp), there exists e > 0 so that is strictly decreasing on (0, E), hence F()) < F(0) for all å E (0,e).
If, in addition, [=)]l=)] < LF and [|u(X)]| < U(X) with (X < oc, then for [6] < 1, we have
[F(X + ōu) - F(X) < LF [8] [X)(x))]
(39)
and the same quadratic expansion as in Proposition 2.1 yields
Rg) R| 8 E Jpu) 1 82 L3 E|uu
(40)
making the "improvement for small 8" explicit whenever E(R, J Juu) > 0.
B.2.1 PROOF STEP 1: EXACT SMALL--STEP DECREASE
Proof. Define, for each (X,y),
f(8:0, v) = 1 I|l F(X 8u bu(x)) (1r(66:))13 r(0:z) y- F(X + +x(())) (41)
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By the chain rule,
06 f66;2,11 v) = r((;:)), σह 06)(0:2) =
J= JF Jx
) u(x)).
(42)
By the domination assumptions and Lemma 4 we can pass the derivative through the expectation to get
08 16;; = -B (1(; Jr (x + (x)) u(x))
R2 (6) = E
(43)
Evaluating at 5 0,
(0) = -r((((X), (x u u(X) AA
(44)
If A > 0, then (0) < O. By continuity of Rí at O, there exists such that R/n(0) S - <0 for 8 [0. Therefore, for any 8 E e).
Rin(d) Rin (0) = 1 (t, dt -
(45)
which proves the strict risk decrease for sufficiently small positive
B.2.2 PROOF OF STEP 2: CLOSED-FORM 8* UNDER AN AFFINE F
Proof. Assume F affine: F(X) Ar + with constant matrix A E R F(X ou(x)) F(X) + 6Au(X).
Then JF A and
(46)
Hence,
Rin = E [|r(X)) Au( (x) [b]] [1 -8 ((xX), Au(x))) 8 [1l X
[3]]
(47)
This is strictly convex quadratic in provided Au]]Au(X)]1 0. Differentiating and setting to 0,
Rin (0) E (Ix](X 5 E Am())
(48)
yields the unique minimizer
(F(),, Au())) Al)
E [F E [1]])u(XX]I3
(49)
This completes the proof for affine F. The same expression arises if, instead of assuming affine F, we optimize the first-order surrogate obtained by linearizing F at 8 = 0:
F(X 5u(X)) = F(X) + 6J xXX)
(50)
which leads to the quadratic proxy
Rin (8) = E [|((x) - 5 J -
(X)((x
(51)
whose unique minimizer is the same 8* as above. Further, we denote by (x(X) w| R² the second directional derivative of F at X along u, w. then
R (0) = E (X)))()
(52)
there exists E [0, 1) such that
E [r(X), Hx(X)(u(X)) u (XXII < (X]
(53)
then (0) E [1 - 7) Bo, (1 + 7) where Bo = E [|] In that case, the true local
minimizer 8t of R.,. satisfies the bracket
A
A B)
5t 6t
(54)
By
quantifying how curvature perturbs the first-order optimizer. When affine or the curvature term averages zero, 7) and öt 8*.
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B.3 PROOF OF PROPOSITION 3.1
Proof. By Lipschitzness of F:
1 - ا| = [[F(X) F(X) & Lr||X - = 6
(55)
According to ( ~o S 1, Al x S and 5 E (0, 8max] with dmax S 1, we have ||44 (X)l & Ld| A (X)ll & VLd Combining yields the claim.
B.4 PROOF OF COROLLARY
Proof. Coordinatewise, 5; I; = Iy (e 1). By the mean value theorem for & + et, for each
there exists & (0, Ōa;) such that
δa. 1= = da; e
[ F2 = 1 e S Bx e 3.1 s Bx 8 e (56)
According ||
I 1, ||4 (Ý, x) S and ö E (0, dmax] with 8max have
||a]|x I 1, hence @d]|a|/> S e.. Summing squares.
x||
Σ a - 2 [Bx 8laxle)2= de Bxlla|l-
(57)
Then apply Lipschitz step in Proposition 3.1
we have
1| ان = || F - F(X)I < Lplli - x|| S 56 Lp Bx B||la|l
(58)
For d S 1, e S e, the bound is O(6).
B.5 PROOF OF THEOREM 2
Proof. By 8-smoothness with u = G. v = 9.
(). < E(..y) + V v) T
(ū- - ú) + - - -01 E(G.0) + +66g, d) + 1111..
(59)
By alignment condition, (g. - -a | ||d|] Substitute:
(ū. v) {ū 3) - -da||9ll] + 1d1
(60)
The RHS is a convex quadratic in 8 with unique minimizer = اوالع उ
Plugging 6 gives - 11
Strict descent holds whenever the derivative at 0 is negative and the second-order term does not dominate, equivalently d E (0, 20l|el F| ).
B.6 PROOF OF THEBREM 3
Proof. By first-order Taylor expansion of F at x, F(r ōv) = F(X) + 8Jv
with
(61)
Set 8 8s + rr()), so q = ÿ + 8. Apply 8-smoothness of é 118112
{(ğ + 5, $ ((j.g) (g. (g
(62)
Then, compute the terms:
(g.0) = 5(g, 8) + (g. rr()) and || = 82 || + 28( (r (8)) + [0))
(63)
Since [|]]( (8), we have (g, (5)) 06)) and ||5|| = 52 ||| + O Therefore,
e(F(++ 8v).)) < ((j.v) + 8(g s) +
(64)
If (g. < - then for sufficiently small 8 the negative linear term dominates the ( remainder, yielding strict descent. Optimizing the quadratic upper bound in 8 gives the minimizer 8* Ø al
all and value - 3 || up O(1), establishing the last claim.
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B.7 PROOF OF THEOREM 3.2
Proof. We formalize the two claims: (i) (()) bound on prediction drift, and (ii) loss upper bound
under composition. Let the composed edit be: è ōu, j/ F(X), and ğ y + 5d(j', X). As before, 0 = F(X).
(i) For (6)) bound prediction drift, using the triangle inequality, we have:
v - أن - 1 - اll + 8ldd('',
(65)
Further, according to 4 X) ce S 1, || 4 (Ý, X) S 1 and ā E (0, mmxx with 6max S 1, we have || y || = | F(X) - LF - = = 1| The bound follows.
(ii) For loss upper bound under composition, by definition we have
ÿ = s' + ōd' = - + 6s + rr()) 5d".
(66)
Set A 8(s (6). By 3 3-smoothness, we have
(6 + A. v) - + ( &
(67)
which can be decomposed into:
(g. A) = 8'g, + d') rF()),
(68)
where ||| = 52 I|s + d'1 + (5 d', rr (8)) + ||r(
(69)
Since ||r](]]]| O()), we have (g, FF (0) 06) and | = 8 + - Thus
f(j v) {(g. v) + 8(g, s d') 28 d' 0
(70)
Here, if (g,8 s d') < ووواا ||s + d' ||. the linear term is strictly negative whenever 8 + d' o 0. For sufficiently small the negative linear term dominates the O remainder, giving strict descent.
Remark. If a learned gate 7 E [0, 1 combines input- and output-induced steps s., = +0 (1- 7) O d', then alignment for s,, follows from mild conditions (e.g., selecting 7 to minimize (g, 3.) subject to [0, 1]9 ensures (g, 8.) min (g. s), (g. d)
C EXPERIMENTAL SETUP AND RESULTS
C.1 DATASET
C.2 COMMONLY USED TS DATASETS
The information of the experiment datasets used in this paper summarized as follows: (1) Elec- tricity Transformer Temperature (ETT) dataset Zhou et al. (021). which contains the data collected from two electricity transformers in two separated counties in China, including the load and the oil temperature recorded every 15 minutes (ETTm) or hour (ETTh) between July 2016 and July 2018. (2) Electricity (ECL) dataset [T collects the hourly electricity consumption of 321 clients (each col- umn) from 2012 to 2014. (3) Exchange Lai et al. (2018) records the current exchange of 8 different countries from 1990 to 2016. (4) Traffi dataset records the occupation rate of freeway system across State of California measured by 861 sensors. (5) Weather dataset 2 records every 10 minutes
for 21 meteorological indicators Germany throughout 2020. The detailed statistics information of the datasets is shown in Table
"http://pems.dor http://cpemm...ooc.ogg.g httpp://wwwwhg--jnaam...... de/wetttr
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Table 6: Details of the seven TS datasets.
Dataset
length
features
frequency
ETThl 17,420
7
lh
ETTh2
17,420
7
lh
ETTml
69,680
7
15m
ETTm2 69,680
7
15m
Electricity 26,304
321
1h
Exchange 7,588
8
ld
Traffic
17,544
862
Weather 52,696
21
C.3 TRAINING OBJECTIVE
We train 0 D while backpropagating through F but not updating it. Let (X) denote the
adapted prediction. For point forecasts we minimize a horizon-aware loss: Here are explicit formulas for each loss term when the input-adapter
learnable mask M M(X;o)
[0, 1|_ applied as o(X M(X(); Φ)), and H, m be horizon
X @ M. Let D = {(x(1, ) ÿ(() and target dims. Expectations E below are over the empirical data distribution umini-batches in practice).
MSE (point forecasts):
MSE pred
E ((X
Wh (Yn, Ynrk)
(71)
Hm
Parm
Riml
MAE (point forecasts):
MAA
E
Wh Vno Vanal
(72)
Hm
Pinball (quantile E T). If YT predicts the r-quantile,
C
QB pred
E
1
ΣΣ
P+(Yn Phr)
pr(u) ul( 1fu x0))
(73)
TET h.
Sparsity (L1) on the mask:
Ce EXnD
Mrr(X
(74)
Ld
1=1 1=1==1 f=1
Entropy (pushes mask toward 0 or 1):
(Mes Log(Mog +5) + (1- - Me,z) - Ma, + 6))
Lent = Ex
(75)
Ld 1
Temporal smoothness:
L d
CTV = Ex
(x;(X;
(76)
(L - 1)d
t=
Budget (fraction of active entries not to exceed k):
m( Ld
Σ Mr)(X:
(77)
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A hinge penalty enforces m k:
Lodd Ex 6m - -
(u)+ max(u, 0}.
(78)
Group sparsity:
d
1
Egroup Ex
Me (X; 0)2
(79)
1=1
C.4 ONLINE LEARNING SETUP
During online testing, we set the batch size to to ensure that data arrives in order. Mcanwhile, we used streaming buffer, where only one updated data point is cached each moment/iteration (avoid label leakage raised by [Liang et al.] (2024a); yye Ava Lau et al. (2025). while returning complete sample from a previous moment. E.g., at time I, the input used for online update returned -t-
from the buffer is Xt- -- I-t- H. the label is Xt-Ha,, where H is the prediction length and L is the
input length.
C.5 TRAINING DETAILS OF ADA-X+Y
Ada-X+Y is composed Ada-X and Ada-Y. Y, and Ada-X and Ada-Y Y are trained jointly an end- to-end manner, not sequentially. We minimize single combined loss L. over the union of parameters Agd (Ada-X) and Agui (Ada-Y). The forward pass is:
Y F(X + 5 (X))
(80)
y γ + 5Agut
(81)
During the backward pass, gradients flow from the loss through Ada-Y (Eq. 2), then through the backbone F, and finally to Ada-X (Eq. 1). This ensures that Ada-X learns input perturbations that specifically help the backbone produce features that Ada-Y Y can best correct.
Experimental Setup: In our experiments, we instantiate two separate Adam optimizers (both learn- ing rate are 1E-4) for modular flexibility. However, they are stepped simultaneously after a single backward pass, making the process equivalent to optimizing joint objective. As derived in Propo- sition 3.2 this joint update rule maintains the O(d) drift bounds and descent guarantees, ensuring the two adapters do not destabilize each other.
C.6 USING 5-ADAPTERS TO IMPROVE MULTIVARIATE TIME SERIES
Tables
g and Figure 10 show that 5-Adapter provides consistent improvements across multiple forecasting models. For nearly all datasets, Ada-X X and Ada-Y Y lead lower prediction errors com- pared to the original models, demonstrating that the proposed adapters generalize well to diverse forecasting architectures. Notably, Ada-X again delivers the largest gains, particularly on challeng- ing datasets such Exchange, Traffic, and ETT series, confirming that refining the input signals before model inference is the most impactful strategy. These results further validate that 5-Adapter broadly applicable, efficient, and effective enhancement method for modern time series fore- casting.
C.7 3-ADAPTER'S VALIDATION AND TESTING PERFORMANCE CHANGES WITH EPOCHS
We also present the performance changes of additive and multiplicative adapters on different datasets over epochs (the blank is due to early stopping). In Figure 9 we visualize the changes in valida- tion and test losses of 5-Adapter across different datasets. Across Electricity, Traffic, and Weather, adding Ada-X+Y drives the test MSE consistently below the original frozen model from the very first epoch and then decreases further before plateauing after 5 epochs. Validation and test curves track closely (no divergence), indicating stable training without overfitting. The gains monotonic
25

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Published as a conference paper at ICLR 2026
Table 7: Performances of the forecaster F and ô-Adapter under batch or online training.
Dataset
Model
Original (Batch)
Fine-Turning (Batch)
Continue (Online)
Ada-X (Barch)
Adn-X (Online)
Ada-Y (Batch)
Ada-Y (Online)
Ada-X+Y (Batch)
Ada-X+Y
(Online)
DistPred
0,1710
0.1715
0.1700
0.1662
0.1654
0,1629
0.1623
0.1602
0.1560
Weather
Transformer
0.1731
0.1724
0.1721
0.1706
0.169
0.1631
0.1634
0.1609
0.1609
DistPred
0.4229
0.4229
0,4229
0.4182
0.4179
0,4117
0,4119
0,4028
0.4029
Traffic
Transformer
0,4437
0.4411
0.4414
0.4361
0.4360
0.4294
0.4294
0.4202
0.4202
DistPred
0.1546
0.1546
0.1545
0.1483
0.1476
0.1483
0.1474
0.1436
0.1432
ELC
iTransformer
0,1655
0.1646
0.1636
0.1601
0.1596
0.1566
0.1562
0.1527
0.1515
Table 8: Multivariate time series forecasting results on the benchmark datasets.
Dntaset
DistProd
iTransfrmmer Ada-X
FourierGNN
FrrTS
Autoformer
Length
Original
Adaa-X
Add-Y
Ohigital
Ads-Y
Original
Adi-X
AdloY
(higital
Ada-X
Ads-Y
Original
Adi-X
Ada-
0.155
0.148
0.163
0.160
0.157
0.250
0.233
0.224
0.189
0:183
0.176
0.228
0.211
0.221
0.109
0.166
0.162
0.175
0.167
0.255
0.344
0:230
0.193
0.189
0.180
0.437
0.383
336
0.185
0.181
0.176
0.193
0.189
0.182
0.267
0.356
0.240
0.207
0.200
0.192
0.612
0.590
0.527
720
0.221
0.217
0.190
0.231
0.226
0.218
0.298
0.384
0.268
0.346
0.259
0.229
0.783
0.767
0.670
Arg
0.182
0.378
0,109
0.190
0.157
0.151
0.207
0.255
0.241
0.209
0.200
0.194
0.315
0.235
0.A50
0.364
0.385
0.386
0.506
0.502
0.503
0.397
0.396
0:649
0.437
ETT1
0.451
0.446
0.444
0.444
0.400
0.540
0.540
0.540
0.45%
0.456
0.453
0.571
0.566
U.558
0.498
0.493
0.497
0.479
0.478
0.433
0.583
0.585
0.584
0.507
0.508
0.511
0.656
0.644
0.688
720
0.505
0.502
0.503
0.504
0.902
0.514
0.615
0.634
0.632
0.568
0.574
0.563
0.095
0.886
0.699
Avg
0.361
0:457
0,458
0:454
0.853
0:155
0.561
0.365
0.565
0.152
0.451
0.381
0.591
0.833
0.577
0.501
0.206
0.591
0.396
0.383
0.588
0.342
0.314
0.350
0.375
0.358
0.375
ET72
0.373
0.380
0.168
0:477
0.468
0.427
0.438
0.434
0.558
0.500
0.548
0.501
0.506
0.661
0.460
01411
0.445
0.440
0.453
0.718
01646
0.560
0.791
0.725
0.717
0.673
0.700
0.494
Aug
0.800
0.587
0.388
0.385
0.190
0.545
0.499
0.506
0.537
0.493
0.40%
0.438
0.20
0.423
0.330
0.345
0.354
0.405
0.359
0.397
0.340
0.334
0.357
0.586
0.464
0.569
ET1el
0.378
0:382
0.374
0.435
0.470
0.180
0.378
0.625
0.572
0.416
0.49
0.464
0.417
0.414
0.415
0.691
0.650
0,510
0.495
0.496
0.519
0.506
0.507
0.483
0.478
0.474
0.754
0.729
0.720
20g
0.105
0.456
0:447
0.305
0.301
0.651
07505
U337
0,179
0.175
0.179
0.182
0,179
0.183
0.220
0.204
0.213
0.191
0.179
0.187
0.271
0.236
0.238
ETTx2
0.245
0.242
0.345
0.255
0.251
0.253
0.329
0.289
0.322
0.275
0.241
0.270
0.290
0.286
0.289
0.304
0.302
0.327
0.325
0.380
0.359
0.530
0.342
0.309
0.353
0.359
0.350
0.350
0.414
0.842
0.501
0.435
0.31
Avg
0.25
0.291
0.593
0.385
0.285
0.330
0.316
0,300
Exchange
192
95
0.100
0.084
0. 186
0.088
0.064
0.180
0.069
0.180
0,102
0,173
0.099
0.668
0.208
0.105
0.116
0.316
0.105
0.203
0.105
0.186
0.098
0.181
0.187
0.105
0350
0.195
0.345
0.170
0.179
0.346
0.319
0.294
0.352
0.304
0.329
0.365
0.296
0.368
0.385
0.380
0,386
0.431
0.414
0.402
720
0,809
0.633
0.714
0.901
0.814
0.800
0.841
0,545
0.841
0.989
0.989
1.011
1.144
1.095
1,001
Arg
0.319
0.385
0.329
0.379
0:166
0.422
0.500
0,402
95
0.423
0.416
0.412
0.448
0.436
0.429
0.779
0.355
0.731
0.563
0.555
0.338
0.659
0.687
0.630
Trahfe
192
0.441
0.435
0.429
0.460
0.455
0.446
0.756
0.721
0.710
0.568
0.562
0.545
0325
0.314
0.504
356
0,458
0.53
0.447
0.479
0.475
0.465
0,765
0.739
0.739
0.595
0.589
0,572
1.094
1.072
1,005
730
0.400
0.877
0.430
0.517
0.513
0.502
0.805
0.781
0,780
0.659
0.651
0.634
1.300
1.92
1.19
0.44
0,740
0.39
0,972
0.59
0918
96
0.171
0. 0.166
0.162
0.173
0.165
0.163
0.184
0.183
0.172
0.185
0.177
0.172
0.260
0.240
0.343
Weathes
192
0.224
0.220
0,213
0.223
0.213
0.210
0.226
0.222
0.214
0.224
0.218
0.212
303003
0.282
0.275
336
0.278
0.270
0.264
0.284
0.271
0.267
0.273
0.268
0.260
0.272
0.267
0.260
0.323
0.315
0.312
730
0.353
0.347
0.340
0.357
0.349
0.339
0.338
0.330
0.329
(1341
0.335
0.328
0.389
0.385
0.367
0.256
0.251
0.245
0.259
0.249
0.245
0.255
0.511
0.244
0.255
0.249
0.243
0.325
0.306
0.299
or mea--monttnnic on Electricity and Traffic, while Weather shows an immediate, steady improve- ment that remains well under the original baseline. Overall, A delivers fast convergence and robust generalization across datasets. These experiments show that the loss curve of the 8-Adapter gradually decreases with epochs and has stable and consistent boundaries. Meanwhile, the com- posite adapter (X+Y) can achieve better performance (Stability Analysis of Section 3), which also proves the robustness of the 6-Adapter and the correctness of its theoretical foundation.
Originel Test =0.167
Original Test 00463
0166
0.46
0.40
0.160-
0.45
0.35-
WVil
MSR
0.155
0.44
0.30
skii 4est
0.43
0.145
042
0.41
XTrol
41tst
0.2
0.20
Original Test -0178
0.40
0.15
2
10
6
0
10
(a) Electricity
Epoch
(H) Traffic
Epoch
(C) Weather
Epoch
Figure 9: Validation and testing performance changes with epochs when adding 8-Adapter.
26

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Published as a conference paper at ICLR 2026
0.445
04437
0,4411
0.4404
DistPred
0.440
0.4361
0.436
Mansfomer
0.435
0,4294
0.4294
0,430
0,4299
MSE
0.425
04424
0,4203
0,4202
0,4202
041799
0,420
0:4182
0.415
0.4117
0.4119
0.410
0,4028
0.4029
0.405
0.400
Original (Batch)
Fine-Tuee (Batch)
Contimue (Online)
Ada-X (Batch)
Ada-X (Online)
Ada-Y (Batch)
Ada-Y (Online)
Ada-X+Y (Batch)
Ada-X (Online)
Figure 10: Performances of 6-Adapter under batch online training
C.8 EFFICIENCY OF 5-ADAPTER
We tested the efficiency of 8 6-daperer Table shows that ô-Adapter is the most time-efficient adap- tive method overall, it consistently faster than other methods across all horizons. This is because Ada-X+Y itself is lightweight and only uses the most recent single sample to update the model. Compared to other adapters that use or select large number of recent samples for updates, obviously faster. The ö- d-Adapter is designed to be extremely lightweight. Compared to the backbone model (specifically, for Sundial (128M) and TabPFN (48M), the adapter introduces less than 2%-6% additional parameters, validating the lightweight claim.
Table 9: Time (S) and memory (MB) of adapters (backbone TabPFN) and online methods.
TahPPN 48M
Ada-XaY 3M
SOLID 0.5
TAFAS 6M
OneNer 3M
PSNer 2M
Time
Memory
Train
Test
Memory
Train
Test
Memory
Train
Test
Memary
Train
Time
Memory
Train
Time
Memory
281
18:00
392
395
1983
511
667
2401
603
861
3468
683
1512
485
1501
307
1848
386
379
2132
481
624
2423
895
3790
681
1537
472
1531
351
1856
369
431
3102
505
398
2501
916
1800
6809
530
1567
554
458
1526
C.9 QUANTILE CALIBRATOR (QC) AND CONFORMAL CALIBRATOR (CC)
Figure illlssraats that both calibrators produce well-calibrated intervals. QC attains higher cover- age than CC, while on another sample CC is better. QC tends to yield slightly wider, more conserva- tive bands. CC delivers comparably high coverage with tighter intervals. Overall, the methods are complementary and reliably improve uncertainty quantification over the raw predictor.
Now, let's discuss how to choose between QC and CC. Both modules into a calibrated probabilistic predictor, but they are aimed at slightly different desiderata: QC di- rectly learns horizon- wise conditional quantiles bounded offsets around the point forecast, which produces smooth quantile function over multiple levels without assumptions about the underlying distribution. CC learns only heteroscedastic scale function and combines it with normalized- residual conformal prediction on a held-out calibration set, yielding symmetric but input-dependent intervals with finite-sample marginal coverage under exchangeability.
frozen point forecaster
Empirically, both variants achieve strong coverage, but QC tends to produce marginally wider and more conservative bands, while CC attains similar coverage with somewhat tighter intervals (see Figs. 5 and 6). For a new real-world dataset, our recommendation is therefore: If strict coverage guarantees are the main requirement, CC is preferable, because the conformal step provides finite- sample marginal coverage the target level. If one needs rich predictive distribution multiple coverage levels from single model, QC is more convenient, as it directly returns a full quantile curve while remaining noraparait w.r.t. the underlying distribution.
27

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Published as a conference paper at ICLR 2026
C.10 COMPARISON BETWEEN ADAPTERS AND ONLINE LEARNING METHODS
The adapter-based methods we reviewed include SOLID, TAFAS, etc., and online approaches, e.g.. FSNet and OneNet. These methods aim to mitigate test-time concept drift via selective layer retrain- ing (SOLID), online adapter updates (TAFAS), auxiliary loss (PETSA), layer-wise adjustment and memory (FSNet), and dynamic model selection (OneNet). However, according to works by Liang et al. (2024): yee Ava Lau et al. (202)), the above methods have used future labels to some extent, causing label leakage in long-term forecasting, where future ground truth is adopted in advance for adaptation. To achieve a fair comparison, we removed label leakage (it may cause performance degradation of some methods) to test their performance. As shown in Table [0] it can be found that our method achieves the lowest error on every dataset across all backbones.
Table 10: Comparison of various adapters and online methods.
Model
DisPted
ITrnsfomer
Autofomer
Others
Dataset
Length
Offline
SOLID
TAPAS
AdaxXeY
Octine
SOLIDD
TAFAS
Ada-X+Y
Odfins
SOLID
TAFAS
Ada-XeY
OnoNet
FSNet
0.155
0.154
0.156
0.146
0.163
0.165
0.156
0.228
0.211
0.196
0.247
0.160
0.170
0.170
0.164
0.375
0.176
0.367
0.437
0.433
0.442
0.379
0.442
0.185
0.152
0.183
0.178
0.193
0.159
0.376
0.612
0.583
0.588
0.580
0.325
0,433
720
0.221
0.220
0.220
0213
0.231
0231
0.223
0.782
0.780
0.781
0.757
0.798
0.913
10g
0112
01122
0.175
0.190
0.50
0.180
0.13
11302
0,755
0347
0.337
96
0.389
0.392
0.395
0.381
1.390
0.391
0.394
0.382
0.449
1.444
0.442
0.431
0.524
0,730
192
0.451
0.450
0.432
0.445
0.444
0.450
0.437
0.440
0.571
11566
0.565
0.56]
0.571
0.820
335
0.498
0.495
0.534
0.480
0.479
0.481
0.512
0.474
0.656
0.652
0.653
0.635
0.614
0
729
0.505
0,502
0.524
0.497
0.504
0.509
0.553
0.499
0,605
0.694
0,705
0.681
0,762
1,060
Aug
01451
(1450
03476
I1451
11454
0.355
01477
0.444
0.593
IL587
0.901
0.377
0.618
U.377
0.303
0.320
0.294
0.296
0.311
0.286
0.371
0.353
0.515
0.515
0.371
0.364
0
0.412
0.375
0.439
0.430
0.568
0.447
DAIS
0.431
0.428
0.570
0.665
0.646
Aug
0.390
0
0.379
0.388
0.435
0,426
0.591
0.587
96
0.339
0.340
0,339
0:318
0.345
0.34)
0.945
0.381
0.586
0.583
0,512
0.461
0.455
0.65
192
0.384
0.360
0.587
0.373
11382
0.38)
0.364
0.362
0.627
0.628
0.642
0.564
0.496
0.525
ETTal
356
0.436
0.412
0.414
0:408
01431
0.423
0.657
0.420
0.691
0.687
0.673
0.642
0,585
0.367
720
0.510
0.454
0.504
0-454
0.511
0.510
0.512
0.500
0.754
0.740
0.725
0.721
0.676
1,085
Aug
04412
0.000
0,411
1135%
0.417
0.414
0.420
0,403
0,264
1.60.
0.038
0.597
0.548
0.351
0.179
0.179
0.171
0.182
0.183
0.271
0.367
0:233
0.434
0.384
0.345
0.247
0.257
0.255
0.259
0.290
0.298
0.281
0.873
ETTm2
0.300
0,311
0.298
0.332
0.357
0.345
0,829
1.156
0.406
0.402
0.391
0.435
0.440
0.423
2,000
11285
1C274
10100
00305
0134
1.290
0.390
0.321
96
0.154
0.081
0,000
0.085
0.999
0.097
0.098
0
0.195
0.166
0.175
0.365
0.338
0.709
192
0.190
0.189
0.189
01122
0.180
0.177
0.176
0.167
0,260
1.234
0.237
0.236
0.591
0.771
Fechings
195
0.919
0.11
0.974
0.76
0,50
0:190
0.470
0.900
0.437
0.444
0.407
0.617
0.545
720
0.800
0.86
0.988
0.63
0.00
0.873
0.714
1.144
11300
1:134
1,050
1.041
1.183
Aag
01350
0.347
0.563
IL297
11333
0:375
0.322
0,316
0.300
01391
0.405
0,465
0.647
03375
96
0.43
0.434
0,424
0.10
0.444
0.465
0.443
0.476
0.689
0.644
0.664
0.651
0.546
0.671
192
Q1441
0.447
0.447
0.43
04460
0.463
(1467
0.448
0.820
0827
0.844
0.304
0,549
0.490
Thilic
336
0.458
0:451
0.47
0.443
0.479
0.475
0.472
0.456
1.094
1.000
1.006
1,030
0,571
0.705
720
Q.490
0.400
0,493
01475
0.517
0.515
0.523
0.513
1.307
1.296
1.297
1.282
0.663
0.732
Aug
0:453
0.45
0.455
11440
0.475
0.475
0.476
0,461
0.972
1.959
0.975
0.942
0.567
0.701
0.171
0.168
0.170
0.160
0.173
0.12
0.163
0.267
0:335
0.251
0.121
0.234
0.224
0.226
0211
11223
0.222
0.310
0.3081
0.208
0.205
0.276
0.295
0.465
Weather
0.278
0.275
0.276
0.254
0.284
0.522
0.261
0.323
0.321
0.309
0.316
0.514
720
0.353
0.353
0.353
0.342
0.357
0.358
0.345
0.389
0.386
0.375
0.697
0.362
Aug
0'215
0.244
0.209
0.390
OneNet and ESNNet implomented the leoning performance deterseated
public lihray provided in this paper, their bacishone
derived from respective liternnree implenentaion
romeves conceps in
C.11 ABLATION STUDIES OF 5-ADAPTER'S DEPTH, WIDTH AND VALUE
Table II shows that ablation studies of d-Adapter's s depth and width. It can be found that the depth has little impact on performance, while the greater the width, the slight improvement in performance.
Table 11: Ablation studies of 6-Adapter's depth and width.
Depth
2
3
4
Width
64
128
256
512
64
128
256
512
64
128
256
512
ELC
0.159
0.157
0.155
0.1533
0.158
0.157
0.154
0.152
0.159
0.157
0.154
0.1
Weather
0.162
0.16
0.159
0.158
0.162
0.161
0.159
0.158
0.162
0.161
0.16
0.158
Truffic
0.439
0.437
0.433
0.43
0.44
0.436
0.433
0.43
0.44
0.436
0.433
0.43
C.12 THE CHOICE OF HYPERPARAMETEE 8
5 is related to the properties of the dataset (e.g., noise level, degree of concept drift). In our work, we divided the datasets into two categories: one with severe concept drift (5 0.1, e.g., Traffic,
28

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Published as a conference paper at ICLR 2026
Weather, etc.) and the other with non-severe concept drift (6 0.01, e.g., ETT, etc.). We did not perform hyperparameter searches based on models or datasets; instead, for datasets with severe concept drift, setting
0.1 is sufficient. In addition, conducted ablation experiments 6. As shown Table 12 a better value of å = 0.1 might yield better results. In our paper, we only reported the two settings (6=0.1 0.01).
Table 12: Ablation studies of ô-Adapter's value.
+0.1X
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x0.2X
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x0.2Y
+0.10&
x0.IX&Y)
x0.20XX))
Val
Test
Val
Test
Val
Test
Val
Test
Val
Test
Val
Tos:
Val
Test
Val
Test
0.418
0.166
0.277
0.168
0.425
0,169
0.41
0.168
0.415
0.165
0413
0.167
0,43
0.160
0.416
0.162
0.11
0,162
C.13
PERFORMANCE OF S-ADAPTER ON BLACK-BOX BOX MODELS.
Table 13 shows the performance of the 5-Adapter on the black-box models. Specifically, we used TabPFN Hollmann al. (2025) and TimesFM Das et al. (2024) as frozen black-box models to conduct zero-shot testing on various datasets. For comparison, we corrected the output results of these black-box models by adding Ada-Y. As shown in the table below, can found that after adding Ada-Y, the prediction error of the model is significantly reduced, which further proves the effectiveness of the proposed method.
C.14 VISUALIZATION OF FEATURE SELECTOR AND CORRECTOR
Figure II shows that both Quantile Calibrator (QC) and Conformal Calibrator (CC) produce adap- tive, heteroscedastic prediction intervals that track signal volatility volatility-widening widening near peaks/troughs and typically enclosing the ground truth across diverse samples. QC tends be more conserva- tive (wider bands) and sometimes attains higher per-sample coverage (e.g.. PICP=>0.729), while CC achieves tighter intervals with comparable coverage (e.g., PICP=00.77 on multiple samples). The consistent behavior across the two test sets indicates that the calibrators complementary and robust, yielding reliable uncertainty quantification beyond the raw predictor.
Figure 12 present the visualization results of the feature selector. We selected 90%, 50%. 30%, and 10% of the input data respectively to test the pre- pre-trained iTransformer model, in order to observe their impact the output results. It can be seen from the figure that most of the important features selected by the 5-Adapter determine the performance of the model, while other features are relatively less important.
29

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Published as a conference paper at ICLR 2026
Table 13: Performance of Ó-Adapter on black-box models.
Traffic
Weather
ELC
Exchange
ETThl
ETTh2
ETTml
ETTm2
TabPFN
0.367
0.875
0.115
0.129
0.129
0.180
0.037
0.114
Ada-Y
0.342
0.552
0.089
0.096
0.095
0.171
0.034
0.103
TimesFM
0.211
0.168
0.084
0.239
0.029
0.135
0.028
0.267
Ada-Y
0.196
0.157
0.081
0.215
0.024
0.104
0.025
0.223
(a) Results Quantile Conformal Corrector lest sample
(b) Results Quantile and Conformal Corrector on another sample
Samole
Sacipl 9.971
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2017
Sample with Gueed cal bowdo OKEP 9.991 SanTel WIT Confermal brotze CRCP 8:873)
Cuand brodon 0.0051
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Sample Cuandic
Samipt
55
Figure 11: Visualization of the Quantile and Conformal calibrator predictions.
30

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Published as a conference paper at ICLR 2026
(6) Select 90% of the features Clsse 0.99
(b) Select 50% of the Seatures CA 0.5)
(c) Seleet 30% of the Sasuures Coeu 0.3)
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Figure 12: Visualization of different important features learned by the mask adapter.
31


Reviews:
[{"review_id": "6pAzTve9SP", "reviewer": "Reviewer_gci4", "summary": "This paper introduces δ-Adapter, a lightweight and model-agnostic post-processing framework to improve deployed time series forecasting models without retraining them. The method learns small, bounded adjustments at two interfaces: \"input nudging\" to refine covariates and \"output residual correction\" to adjust predictions. The framework is extended into three applications: a feature selector for interpretability, a quantile calibrator, and a conformal calibrator for reliable uncertainty quantification. Extensive experiments on various models and datasets validate that δ-Adapter effectively boosts prediction accuracy and calibration.\n\n---", "questions": "1.  **The choice of hyperparameter δ**: In the experiments, δ is set to 0.1 for most datasets but 0.01 for the ETT datasets. Was this value selected empirically, or was there a systematic tuning process (e.g., grid search on a validation set)? Does the optimal value of δ correlate with properties of the backbone model (e.g., complexity) or the dataset (e.g., noise level, degree of concept drift)?\n\n2.  **The online learning setup**: In Figure 2 and Table 6, you demonstrate superior performance under an online training setting. Could you please provide more details on this online experimental setup? Specifically, how is data streamed to the adapter (e.g., sample-by-sample, mini-batches), and how does this setup simulate a real-world scenario where new data arrives continuously?\n\n3.  **The choice between the Quantile Calibrator (QC) and Conformal Calibrator (CC)**: The paper proposes two effective uncertainty calibrators, QC and CC. Is there recommendations for us on when to choose one over the other, when faced a new real-world dataset?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.  **Practicality and Novelty**: The paper addresses the critical real-world problem of updating deployed models in a computationally cheap manner. The δ-Adapter is a practical \"plug-and-play\" solution compared to costly alternatives like full retraining or fine-tuning.\n\n2.  **Methodological Completeness**: The framework is versatile, being applicable to different backbone forecasting model. It is also comprehensive, addressing crucial aspects like interpretability (feature selection) and reliable uncertainty estimation, which are often overlooked.\n\n3.  **Theoretical and Empirical Support**: The proposed method is well-grounded in theory, with stability and performance guarantees. These theoretical claims are also backed by extensive experiments across multiple models and standard benchmarks.", "weaknesses": "1.  **Limited Comparison with Alternative Post-Processing Methods:** While the paper positions δ-Adapter as a post-processing technique, the primary comparisons are against fine-tuning or continue-training the backbone model. However, there is a significant body of work on post-processing and test-time adaptation for time series forecasting designed to handle concept or distribution shifts. This includes methods for both batch-training settings (e.g., [SOLID](https://arxiv.org/abs/2310.14838)、[TAFAS](https://arxiv.org/abs/2501.04970)) and online settings (like FSNet and OneNet, mentioned in the paper's related works). An experimental comparison against these direct alternatives is necessary to properly situate the δ-Adapter's advantages; without it, the claims of superiority are not fully convincing.\n2.  **Insufficient Motivation:** The motivation presented is not fully developed. The paper's opening question—\"Can we keep the strong forecaster intact and learn only a tiny, post-hoc module... without heavy retraining?\"—frames the problem but doesn't sufficiently justify why a post-hoc module is the required solution over other existing approaches. The discussion on how alternative methods tackle this challenge is not sufficient enough. Additionally, the descriptions of existing problems like \"condition drift\" and the need for a \"low-complexity structure\" are quite brief, which weakens the overall problem statement.\n3.  **Insufficient Detail on Training Combined Adapters (Ada-X+Y):** The experiments (e.g., Figure 2 and Table 6) consistently show that the combination of input and output adapters (Ada-X+Y) yields the best results. However, the paper lacks a clear description of how these two modules are trained together. For instance, are they optimized jointly with a combined loss function or trained sequentially. A more detailed explanation of the co-training strategy can better understand their interaction and potential interference.\n4.  **Lack of Analysis of Adapter Scale and Computational Cost:** The paper claims that the δ-Adapter is lightweight and has \"negligible compute.\" However, it only mentions that the adapter is implemented as a shallow MLP without providing crucial details like its specific hyperparameters or total parameter count. Furthermore, there is no quantitative analysis of the additional computational overhead. To substantiate these claims, the paper is hoped to report metrics such as the percentage increase in wall-clock training time, the number of additional parameters, or the added inference latency compared to the original backbone model.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "R2hB64n7UZ", "reviewer": "Reviewer_5yqn", "summary": "This paper introduces sigma-Adapter, a lightweight post-processing framework for improving time series forecasting without retraining the backbone model. The approach works by learning small bounded corrections at two interfaces: (1) input nudging (soft edits to covariates) and (2) output residual correction. The authors provide theoretical guarantees for local descent and stability and demonstrate feature selection capabilities through learnable masks, and introduce two distributional correctors (Quantile and Conformal calibrators). The experiments across multiple datasets and backbone models show consistent improvements in both accuracy and calibration.", "questions": "1. Can you compare against other adapter approaches available in time series literature and also compare with adapters used in NLP literature?\n2. Can you have thorough discussion on the comparison with LoRA method and investigation on high variance?\n3. Can you compare against few other pre-trained models like TabPFN-TS to validate the claims made are generalizable across pre-trained models?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The main strengths of the paper include,\n1. The post-processing approach is a well-motivated solution for real-world deployments where retraining large models is costly. The ability to improve frozen forecasters is important and useful problem to tackle. \n2. The paper provides rigorous theoretical analysis for several algorithms: Local descent guarantees (Theorems 2 & 3), Compositional stability for combined adapters (Proposition 3.2) etc.\n3. The authors compared against diverse backbones (DistPred, iTransformer, FourierGNN, FreTS, Autoformer) across several datasets (ETT, Traffic, Weather, Electricity, Exchange). Both pre-trained and SOTA models show consistent improvements in both accuracy and calibration.\n4. The authors conducted clear ablation studies to 1. Compare the performance of sigma-adapter variants. 2. Compare the effectiveness of input adapters and output calibrators across several datasets.", "weaknesses": "The main weakness of the paper include,\n\n1. The proposed Input/output adapters approach is conceptually similar to existing adapter methods in NLP. The theoretical results (gradient descent, Lipschitz stability) are relatively standard. The main contribution does not appear to be novel as the claims in the paper and looks like an application of the NLP concept to time series.\n2. Some important ablation studies such as the impact of adapter architecture (MLP depth, width etc), effect of different sigma values across datasets are missing. \n3. The comparison with other post-processing method (e.g., calibration techniques) and adapter approaches are not presented; The comparison with lightweight fine-tuning methods (e.g., LoRA for time series) across multiple pre-trained models is missing; The LoRA comparison in Figure 8(c) is limited to one model (Sundial) and shows high variance but needs more investigation. \n4. Discussion on fine-tuning/adaptation time across different adapter approaches and comparison against wider range of pretrained models might be helpful to strengthen this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VMXDIhKcW7", "reviewer": "Reviewer_gkGB", "summary": "This paper presents an adapter-based finetuning method for pretrained forecasting models. The paper proposes input and output-based adapters in two forms: additive and multiplicative. The authors provide theoretical results for the stability and improvement under certain conditions with respect to the base model.  Finally, the authors expose how to utilize their adapters as calibration methods for models that only offer fixed-point predictions.", "questions": "- L461-463: Could the authors elaborate more on why this type of input/output adapter is desirable over a LoRA style adapter?\n- How does additive vs multiplicative updates compare empirically? It seems the experiments mostly focus on additive adapters.\n- Why not report results for Ada X+y adapters together (Tables 1 and 2)?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Overall, I think this is a good paper. The presentation is clear and the theory is solid as far as I can tell.\n- The presented experiments consistently show the improvement of input and output adapters. \n- Benchmarks look reasonable within popular datasets from the time series literature.", "weaknesses": "- Some of the models are relatively old and would not be considered SOTA at the current time (e.g. Autoformer). I would suggest including models like PatchTST, Non-stationary Transformer, TimesNet or TimeMixer.\n- Additive vs multiplicative: \n- Reports metrics in Table 2 are  averaged across lenghts. I understand the difficulty of reporting metrics across many horizons, but in my opinion, this makes it harder to interpret the effect over prediction lengths, over which errors may differ significantly.\n\n**Typos and minor comments**\n- L132: \"Obviously\". This can be dropped :) \n- Table 2 Exange (also in the appendix table). caption: \"veraged\"\n- L64: Despite adapter is simplicity", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Dtg2niHRZR", "reviewer": "Reviewer_Fzqc", "summary": "The paper proposes **δ-Adapter**, a lightweight and architecture-agnostic framework that improves frozen time-series forecasting models through post-processing rather than retraining. It introduces two modules: **input nudging**, which softly edits covariates before inference, and **output residual correction**, which refines predictions after inference. The method provides theoretical guarantees of local descent, bounded drift, and compositional stability. Additionally, the authors design a **feature-selector adapter** that identifies influential input features and **quantile/conformal calibrators** that enhance uncertainty estimation. Experiments across multiple forecasting backbones and datasets demonstrate that δ-Adapter consistently improves accuracy and calibration with minimal computational cost.", "questions": "1. **Comparison to Test-Time Adaptation (TTA) Methods**.\n   How does δ-Adapter conceptually and empirically differ from recent TTA approaches that also adapt frozen forecasters at inference time (e.g., Kim et al., AAAI 2025; Medeiros et al., 2025; Grover & Etemad, ICML 2025)? Since these methods share similar goals such as robustness to non-stationarity and post-hoc improvement, please clarify what specific challenges δ-Adapter addresses that are not tackled by gradient-based or parameter-efficient TTA methods.\n\n2. **Choice and Sensitivity of δ**.\n   The theoretical analysis emphasizes δ as a small trust-region parameter controlling stability. How sensitive are results to δ in practice, and how should it be chosen for new datasets or models? Could adaptive δ-scheduling be beneficial?\n\n3. **Theoretical–Empirical Link**.\n   The paper provides strong theoretical guarantees for stability and local descent, but the experiments do not directly validate these properties. Could the authors include empirical evidence such as δ vs. loss curves or visualization of stability bounds to support the theoretical claims?\n\n4. **Comparison to Lightweight Adaptation Baselines**.\n   Have the authors considered comparing δ-Adapter with parameter-efficient fine-tuning strategies such as LoRA or standard adapter layers, which also introduce small modules on frozen backbones? This would help quantify how δ-Adapter performs relative to existing low-overhead adaptation techniques.\n\n5. **Practical Deployment Considerations**.\n   Since δ-Adapter is advertised as a post-hoc enhancement for deployed models, what is the typical training cost, latency overhead, and implementation effort in a real-world deployment scenario? Including such details would reinforce the paper’s practical relevance.\n\nThese clarifications would significantly help assess the robustness, novelty, and real-world applicability of δ-Adapter and could strengthen the case for acceptance.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper’s main strength lies in its original and practical reformulation of time-series improvement as a post-processing problem rather than a model-design or retraining task. This shift in perspective is both creative and highly relevant to real-world forecasting systems, where retraining large backbones is often infeasible. The proposed δ-Adapter is simple yet elegant, introducing bounded input and output modules that deliver measurable gains without altering the base model. Theoretical sections are rigorous, offering provable guarantees of local descent, drift stability, and compositional safety, which is an uncommon level of formalism for this kind of lightweight method. The paper is also clear and well-structured, with intuitive explanations of each component and thoughtful use of figures to convey mechanisms. Empirically, the approach demonstrates consistent and transferable improvements across architectures and datasets with negligible overhead, highlighting its significance and applicability to a broad range of forecasters. Overall, the work stands out for its conceptual clarity, strong theoretical underpinnings, and practical impact on efficient and reliable time-series forecasting.", "weaknesses": "While the paper presents a strong theoretical foundation and an appealing practical idea, several areas could be improved to strengthen its impact and clarity.\n\nFirst, the experimental evaluation, though broad in dataset coverage, lacks comparative depth. The study primarily benchmarks δ-Adapter against frozen and fine-tuned baselines, but it omits comparisons to parameter-efficient adaptation methods such as LoRA, adapters, or residual fine-tuning strategies that have been explored in related forecasting and NLP contexts. Including such baselines would better position δ-Adapter within the broader adaptation literature and clarify its relative advantages.\n\nSecond, while the theoretical analysis is thorough, the empirical link to these guarantees remains weak. The paper could provide visual or quantitative evidence of stability, such as loss landscapes or δ–performance trade-offs, to demonstrate how the theory manifests in practice.\n\nThird, the related work section should be expanded to discuss recent advances in adaptation and test-time adaptation (TTA) for time-series forecasting, which directly address similar motivations to δ-Adapter. In particular, works such as Kim et al. (AAAI 2025) [1], Medeiros et al. (arXiv 2025) [2], Grover & Etemad (ICML 2025 Workshop) [3], and Lee et al. (ICML 2025) [4] explore gradient-based, parameter-efficient, and online adaptation mechanisms to handle non-stationarity and distribution shifts. Adding a dedicated subsection contrasting δ-Adapter with these methods would clearly position it as a post-hoc alternative and highlight its novelty within the adaptation landscape.\n\nOverall, the paper would benefit from stronger empirical grounding, richer comparative analysis, and a more comprehensive connection to existing adaptation research to fully demonstrate the breadth and impact of its contribution.\n\nReferences:\n\n[1] Kim, H., Kim, S., Mok, J., & Yoon, S. (2025, April). Battling the non-stationarity in time series forecasting via test-time adaptation. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 39, No. 17, pp. 17868-17876).\n\n[2] Medeiros, H. R., Sharifi-Noghabi, H., Oliveira, G. L., & Irandoust, S. (2025). Accurate Parameter-Efficient Test-Time Adaptation for Time Series Forecasting. arXiv preprint arXiv:2506.23424.\n\n[3] Grover, Shivam, and Ali Etemad. \"Shift-Aware Test Time Adaptation and Benchmarking for Time-Series Forecasting.\" Second Workshop on Test-Time Adaptation: Putting Updates to the Test! at ICML 2025. 2025.\n\n[4] Lee, Thomas L., et al. \"Lightweight Online Adaption for Time Series Foundation Model Forecasts.\" Forty-second International Conference on Machine Learning.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)