Instance-Adaptive Scaling (IAS) Overview

• Instance-Adaptive Scaling (IAS) is a family of instance-conditioned mechanisms that adjust model features, scores, or compute allocations based on per-input signals.
• IAS is applied across various domains—including federated learning, robust classification, language-model inference, and adversarial training—to improve performance and resource efficiency.
• IAS leverages instance-specific metrics like feature similarity, gradient norms, or validation clusters to dynamically modulate model behavior without incurring extra inference overhead.

Instance-Adaptive Scaling (IAS) denotes a family of instance-conditioned mechanisms in which a model’s scale, modulation, decision rule, or inference budget is adjusted for each input rather than fixed globally. In recent arXiv literature, the term has been used for instance-wise feature modulation in federated learning, feature-cluster-conditioned class-score scaling for group robustness, per-query and per-step compute allocation in LLM reasoning, instancewise attack-step control in fast adversarial training, and adaptive prediction-grid refinement in online multicalibration (Feng et al., 2023, Seo et al., 2024, Park et al., 11 Jun 2025, Huang et al., 29 May 2026, Huang et al., 2022, Huang et al., 10 May 2026). The unifying theme is not a single architecture but a common design principle: instance-level signals are used to choose how strongly a system should modify features, scores, search width, optimization steps, or prediction resolution.

1. Terminology and conceptual scope

The phrase “Instance-Adaptive Scaling” is not used uniformly across subfields. In federated learning, IAS refers to instance-wise modulation of deep features through scale-and-shift parameters selected from a learned pool. In robust classification, it denotes post-hoc class-score scaling conditioned on feature-space clusters. In language-model inference, it denotes dynamic allocation of inference-time compute. In adversarial training, the term is used retrospectively to describe instancewise step-size adaptation. In online multicalibration, it describes adaptive refinement of the prediction grid (Feng et al., 2023, Seo et al., 2024, Huang et al., 29 May 2026, Huang et al., 2022, Huang et al., 10 May 2026).

Domain	Adapted quantity	Instance signal
Federated learning	SSF scale and shift parameters	query-key matching in a global pool
Group robustness	class-specific score vector	nearest-centroid feature cluster
LLM inference	sample budget or UIS action	calibrated success estimate or contextual features
Adversarial training	attack step size	input-gradient norm
Online multicalibration	active prediction intervals	interval play counts

A recurrent misconception is that IAS must be an inference-only technique. The literature is broader. Some methods are pure post-processing and require no retraining, such as class-specific scaling and cluster-conditioned scaling for group robustness. Others act during training, such as ATAS in adversarial robustness. Some are hybrid systems in which training builds the mechanism and inference performs instance-specific selection, as in FedIns and PRM-calibrated reasoning (Seo et al., 2024, Huang et al., 2022, Feng et al., 2023, Park et al., 11 Jun 2025).

2. Recurrent mathematical patterns

Despite the terminological dispersion, several mathematical templates recur. In feature-space IAS, a hidden representation is modulated by instance-specific affine parameters:

$$\tilde{h}_l^{(i)} = \gamma_l^{(i)} \odot h_l + \beta_l^{(i)}.$$

This is the formulation used by FedIns through scale-and-shift deep features (SSF), with instance-specific $\gamma_l^{(i)}$ and $\beta_l^{(i)}$ applied after operations such as MSA, MLP, and LN (Feng et al., 2023).

In logit-space IAS, the decision rule itself is altered by input-dependent rescaling of class scores. The robust-classification formulation first applies a class-specific vector $\mathbf{s}$ in CSS,

$\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$

and then extends it to cluster-conditioned IAS by assigning the input to a feature-space cluster $k^\*(x)$ and using $\mathbf{s}^{(k^\*(x))}$ instead (Seo et al., 2024).

In inference-budget IAS, the scaled quantity is compute. PRM-based reasoning methods infer the number of trajectories needed to meet a target success probability $C$:

$$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$

typically with a conservative lower confidence bound replacing $p$. UniScale generalizes this idea by defining a joint UIS action $\gamma_l^{(i)}$0, where model routing and test-time scaling are optimized together (Park et al., 11 Jun 2025, Huang et al., 29 May 2026).

In training-time IAS for adversarial robustness, the scaled quantity is the attack step size:

$\gamma_l^{(i)}$1

Large-gradient instances receive smaller steps, and low-gradient instances receive larger steps (Huang et al., 2022).

In online multicalibration, the adapted object is the active prediction partition. An interval $\gamma_l^{(i)}$2 is split when its expected play count reaches $\gamma_l^{(i)}$3, balancing discretization error against online-learning regret (Huang et al., 10 May 2026).

Taken together, these formulations suggest that IAS is best understood as an instance-conditioned control mechanism whose target variable depends on the application: features, logits, compute, steps, or discretization scale.

3. Feature-space IAS in federated learning

The paper "Towards Instance-adaptive Inference for Federated Learning" introduces FedIns, which targets both inter-client and intra-client heterogeneity by combining a frozen pre-trained backbone with federated pools of SSF parameters (Feng et al., 2023). The backbone is denoted $\gamma_l^{(i)}$4, with $\gamma_l^{(i)}$5 frozen during federated fine-tuning. SSF modules are inserted after core operations such as MSA, MLP, and LN, and perform feature-wise linear modulation through

$\gamma_l^{(i)}$6

Each client learns an SSF pool together with a small classifier head while keeping $\gamma_l^{(i)}$7 fixed. The client-side learnable set is

$\gamma_l^{(i)}$8

where $\gamma_l^{(i)}$9 contains SSF items and $\beta_l^{(i)}$0 contains learnable keys. After local training, the server aggregates client pools by data-size weighting:

$\beta_l^{(i)}$1

At inference time, a query vector is computed from the frozen backbone, cosine similarity is evaluated against all keys in the global pool, and the top $\beta_l^{(i)}$2 SSF items are uniformly averaged:

$\beta_l^{(i)}$3

The resulting instance-wise SSF is then reparameterized into the backbone, so the method introduces no extra inference-time parameters or FLOPs after reparameterization.

The reported implementation uses ViT-B/16 pre-trained on ImageNet-21K, SGD, batch size $\beta_l^{(i)}$4, learning rate $\beta_l^{(i)}$5, $\beta_l^{(i)}$6 communication rounds, $\beta_l^{(i)}$7 local epochs per round, default pool size $\beta_l^{(i)}$8, and best-matched count $\beta_l^{(i)}$9. Communication remains small because only SSF parameters, keys, and a small head are exchanged: in the ViT-B/16 setup, FedIns communicates about $\mathbf{s}$0M parameters per round versus about $\mathbf{s}$1M for full-model FedAvg, i.e., less than $\mathbf{s}$2 of full-model communication (Feng et al., 2023).

Empirically, the method is evaluated on Label Shift benchmarks (CIFAR-100, Tiny-ImageNet) and Feature Shift (DomainNet). Reported accuracies include DomainNet $\mathbf{s}$3 for full FedAvg versus $\mathbf{s}$4 for FedIns, CIFAR-100 $\mathbf{s}$5 versus $\mathbf{s}$6, and Tiny-ImageNet $\mathbf{s}$7 versus $\mathbf{s}$8. The abstract states “a $\mathbf{s}$9 improvement against the top-performing method with $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$0 communication cost on Tiny-ImageNet” (Feng et al., 2023). Ablations further indicate that SSF outperforms prompt-based alternatives, that pool size improves performance up to about $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$1, and that $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$2 is generally optimal.

Conceptually, FedIns sits close to FiLM, adapters, LoRA, FedBN, and lightweight MoE. Its distinguishing property is that the adaptation is both federated and instance-adaptive, yet can still be merged back into the model without inference-time overhead.

4. Logit-space IAS for robustness and bias control

The paper "Re-evaluating Group Robustness via Adaptive Class-Specific Scaling" formulates IAS as an extension of class-specific scaling (CSS) for group robustness under spurious correlations and dataset bias (Seo et al., 2024). A trained classifier $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$3 produces a score vector $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$4. CSS applies a nonnegative class-specific coefficient vector $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$5 and predicts via

$\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$6

IAS makes the scaling vector depend on cluster identity in feature space. Using penultimate-layer embeddings $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$7, K-means is run on validation features to obtain centroids $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$8. For each cluster $\arg\max_c (\mathbf{s}\odot\hat{\mathbf{y}}(x))_c,$9, a cluster-specific scaling vector $k^\*(x)$0 is learned on that cluster’s validation examples. At test time,

$k^\*(x)$1

and prediction uses $k^\*(x)$2.

This IAS is explicitly training-free. It is a pure post-processing on prediction scores and does not change losses, gradients, sample weights, or training. Scaling factors are chosen on an attribute-annotated validation set by greedy per-class search over

$k^\*(x)$3

with the objective chosen from worst-group accuracy (WA), unbiased accuracy (UA), or average accuracy (AA). The paper uses standard K-means with $k^\*(x)$4, and reports that results are stable and superior to CSS when $k^\*(x)$5 (Seo et al., 2024).

A central contribution is the explicit treatment of the robustness–average-accuracy trade-off. Robustness metrics are defined from group accuracies $k^\*(x)$6, with

$k^\*(x)$7

To summarize the entire trade-off curve, the paper introduces robust coverage, the area under the Pareto frontier of robust accuracy versus average accuracy.

The empirical results are notable because naïve ERM plus post-hoc scaling often matches or exceeds more elaborate debiasing pipelines. On CelebA, the ERM baseline reports WA $k^\*(x)$8, UA $k^\*(x)$9, and AA $\mathbf{s}^{(k^\*(x))}$0, while ERM+IRS reports WA $\mathbf{s}^{(k^\*(x))}$1, UA $\mathbf{s}^{(k^\*(x))}$2, and AA $\mathbf{s}^{(k^\*(x))}$3. On Waterbirds, ERM goes from WA $\mathbf{s}^{(k^\*(x))}$4, UA $\mathbf{s}^{(k^\*(x))}$5, AA $\mathbf{s}^{(k^\*(x))}$6 to ERM+IRS WA $\mathbf{s}^{(k^\*(x))}$7, UA $\mathbf{s}^{(k^\*(x))}$8, AA $\mathbf{s}^{(k^\*(x))}$9. On CivilComments-WILDS, ERM goes from WA $C$0, UA $C$1, AA $C$2 to ERM+IRS WA $C$3, UA $C$4, AA $C$5. On FMoW-WILDS, ERM goes from WA $C$6, UA $C$7, AA $C$8 to ERM+IRS WA $C$9, UA $$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$0, AA $$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$1 (Seo et al., 2024).

The paper’s broader implication is methodological rather than architectural. It shows that some reported robustness gains can be reinterpreted as movement along a score-scaling frontier rather than as fundamentally different learned representations. IAS, in this sense, is a decision-rule adjustment mechanism rather than a feature-learning mechanism.

5. Compute-allocation IAS for language-model inference

A distinct line of work uses IAS to allocate inference-time compute in reasoning systems. One formulation begins with calibrated process reward models (PRMs). In "Know What You Don't Know: Uncertainty Calibration of Process Reward Models," a PRM evaluates a partial reasoning trajectory $$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$2 for a query $$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$3 and is calibrated via quantile regression to estimate the probability that continuing from that prefix will eventually yield a correct final answer (Park et al., 11 Jun 2025). For a target overall success probability $$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$4, the instance-adaptive sample budget is

$$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$5

and the deployed rule uses a lower confidence bound:

$$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$6

The same logic extends to beam search through IAS-of-$$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$7 and IAS-of-$$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$8, which adapt either the number of continuations per prefix or the number of retained prefixes.

Calibration is crucial in this formulation. On AIME24–25 with Llama-3.2-1B, ECE drops from $$N_{\mathrm{IAS}}(p,C) = \frac{\log(1-C)}{\log(1-p)},$$9 to $p$0 and Brier score from $p$1 to $p$2 after calibration; on MATH500 with the same model, ECE drops from $p$3 to $p$4 and Brier from $p$5 to $p$6 (Park et al., 11 Jun 2025). For MATH500 with Qwen2.5-7B, fixed BoN at $p$7 yields $p$8, while BoN+IAS with calibrated PRMs achieves about $p$9 at budget $\gamma_l^{(i)}$00, i.e., about $\gamma_l^{(i)}$01 fewer samples. The paper also reports that IAS spends about $\gamma_l^{(i)}$02 fewer samples on level-1 MATH500 problems than on level-5 problems. A central caveat is explicit: uncalibrated PRMs overestimate success probabilities and therefore cause severe under-allocation.

A second formulation generalizes compute allocation beyond trajectory count. "UniScale: Adaptive Unified Inference Scaling via Online Joint Optimization of Model Routing and Test-Time Scaling" defines Unified Inference Scaling (UIS) as a joint action space

$\gamma_l^{(i)}$03

where $\gamma_l^{(i)}$04 is the chosen base model, $\gamma_l^{(i)}$05 is Question Parallelism, $\gamma_l^{(i)}$06 is Candidate Parallelism, and $\gamma_l^{(i)}$07 is Beam Size (Huang et al., 29 May 2026). The method models adaptive UIS as a contextual multi-armed bandit learned via LinUCB. Reward combines correctness, verifier score, and normalized cost:

$\gamma_l^{(i)}$08

The cost model is hardware-aware, expressed in eFLOPs, and decomposes total UIS cost into prefill, incremental decoding, and verification.

UniScale supplements the bandit formulation with path-aware early exiting and dense verification feedback. Reported verifier diagnostics are AUROC $\gamma_l^{(i)}$09 and point-biserial correlation $\gamma_l^{(i)}$10 between PRM score and correctness. In the full UIS setting, UniScale reports Cost-Sensitive reward $\gamma_l^{(i)}$11, accuracy $\gamma_l^{(i)}$12, and cost $\gamma_l^{(i)}$13 TeFLOPs, and in Quality-Priority mode reward $\gamma_l^{(i)}$14, accuracy $\gamma_l^{(i)}$15, and cost $\gamma_l^{(i)}$16 (Huang et al., 29 May 2026). Early exit reduces total cost by $\gamma_l^{(i)}$17 with an accuracy change of $\gamma_l^{(i)}$18 percentage points, and dense feedback improves Quality-Priority accuracy by $\gamma_l^{(i)}$19 percentage points while reducing cost by $\gamma_l^{(i)}$20.

Within this literature, IAS no longer means “scale a tensor” or “scale a logit.” It means allocating search width, model capacity, or budget on a per-request basis, often under explicit cost-aware objectives.

6. Training-time instance-adaptive updates

In fast adversarial training, IAS is instantiated as instancewise adaptation of the inner-maximization step size. "Fast Adversarial Training with Adaptive Step Size" shows that catastrophic overfitting is instance-dependent and that instances with larger input-gradient norms are more likely to trigger it (Huang et al., 2022). The method defines an average input-gradient norm per instance,

$\gamma_l^{(i)}$21

and uses a moving average of squared norms to set

$\gamma_l^{(i)}$22

The adversarial example update is

$\gamma_l^{(i)}$23

The implementation uses $\gamma_l^{(i)}$24, $\gamma_l^{(i)}$25, $\gamma_l^{(i)}$26 for CIFAR-10/100, and $\gamma_l^{(i)}$27 for ImageNet. Previous-epoch initialization is adopted as in ATTA so that the entire $\gamma_l^{(i)}$28 ball remains reachable even when $\gamma_l^{(i)}$29 is small.

The empirical pattern is consistent across datasets. On CIFAR-10 with $\gamma_l^{(i)}$30 and ResNet-18, ATAS reports Clean $\gamma_l^{(i)}$31, PGD-10 $\gamma_l^{(i)}$32, PGD-50 $\gamma_l^{(i)}$33, AA $\gamma_l^{(i)}$34, with training time about $\gamma_l^{(i)}$35 h, while ATTA reports PGD-50 $\gamma_l^{(i)}$36 and AA $\gamma_l^{(i)}$37. On ImageNet with $\gamma_l^{(i)}$38 and ResNet-50, ATAS reports Clean $\gamma_l^{(i)}$39, PGD-50 $\gamma_l^{(i)}$40, AA $\gamma_l^{(i)}$41, compared with ATTA PGD-50 $\gamma_l^{(i)}$42 and AA $\gamma_l^{(i)}$43 (Huang et al., 2022). The theoretical analysis, under convex-concave and smooth assumptions, gives a tighter regret term for the adaptive scheme when per-instance gradient norms are long-tailed.

A different training-time use appears in generalized zero-shot learning. "Instance Adaptive Prototypical Contrastive Embedding for Generalized Zero Shot Learning" defines IAS as an instance adaptive re-scaling mechanism inside a margin-based prototypical contrastive loss (Paul et al., 2023). For prototype $\gamma_l^{(i)}$44, positive sample $\gamma_l^{(i)}$45, and negative sample $\gamma_l^{(i)}$46, the adaptive weights are

$\gamma_l^{(i)}$47

These weights modulate the logits in the loss so that hard pairs receive larger scale and easy pairs receive smaller scale. The paper reports that $\gamma_l^{(i)}$48 and $\gamma_l^{(i)}$49 work best across benchmarks. In ablations on AWA2, constant scaling with margin gives $\gamma_l^{(i)}$50, $\gamma_l^{(i)}$51, $\gamma_l^{(i)}$52, while IAS with margin gives $\gamma_l^{(i)}$53, $\gamma_l^{(i)}$54, $\gamma_l^{(i)}$55; on CUB, the corresponding numbers are $\gamma_l^{(i)}$56 versus $\gamma_l^{(i)}$57 (Paul et al., 2023).

These works share a hardness-aware interpretation of IAS. The scale is not merely instance-dependent; it is often derived from a surrogate of instance difficulty, such as gradient norm or distance from a target similarity margin.

7. Online calibration, neighboring notions, and limitations

In online learning, IAS has been used to denote automatic adaptation to feature scales without manual tuning. "Adaptive scale-invariant online algorithms for learning linear models" proposes ScInOL1 and ScInOL2, which maintain per-coordinate statistics

$\gamma_l^{(i)}$58

and use the invariant quantity

$\gamma_l^{(i)}$59

Under diagonal feature scaling $\gamma_l^{(i)}$60, $\gamma_l^{(i)}$61 is unchanged and the weights transform so that predictions remain invariant. The regret bounds match oracle-tuned per-coordinate OGD up to logarithmic factors, with $\gamma_l^{(i)}$62 time and memory per round (Kempka et al., 2019). Here IAS means scale adaptation by invariance rather than by explicit per-instance routing.

In online multicalibration, "Instance-Adaptive Online Multicalibration" uses dynamic dyadic partitions and splits an interval when its accumulated expected play reaches $\gamma_l^{(i)}$63. The main theorem recovers the worst-case-optimal $\gamma_l^{(i)}$64 rate while adapting to easier sequences: in the marginal stochastic setting the rate becomes $\gamma_l^{(i)}$65, and for piecewise-stationary means with $\gamma_l^{(i)}$66 segments it becomes $\gamma_l^{(i)}$67 (Huang et al., 10 May 2026). The error is controlled by the number of leaves in the refinement tree, so IAS is realized as adaptive refinement of prediction resolution rather than scaling of activations or logits.

Adjacent terminology appears in other literatures. "Adaptive Scaling for Sparse Detection in Information Extraction" uses the term Adaptive Scaling rather than IAS, and computes a batch-level scaling factor

$\gamma_l^{(i)}$68

to weight negative losses under an expected $\gamma_l^{(i)}$69 objective (Lin et al., 2018). "Dynamic Low-Rank Instance Adaptation for Universal Neural Image Compression" does not explicitly describe or cite IAS, but compares low-rank decoder updates against bias-only and multiplicative per-feature baselines that are conceptually close to additive and multiplicative scaling (Lv et al., 2023). This terminological spread matters: not every “adaptive scaling” method is an IAS method in the same technical sense.

Across applications, reported limitations follow the mechanism being adapted. FedIns notes pool-size scaling, selection stability, prototype or key quality, and edge-memory costs (Feng et al., 2023). PRM-based IAS depends on calibration quality, independence assumptions, and robustness under distribution shift (Park et al., 11 Jun 2025). UniScale depends on verifier quality, cost-model fidelity, and the tractability of very large action spaces (Huang et al., 29 May 2026). ATAS relies on hyperparameters that keep adaptive steps within stable ranges and is analyzed theoretically only under convex-concave assumptions (Huang et al., 2022). These limitations reinforce a general point: IAS is powerful when the instance descriptor is reliable, but fragile when the conditioning signal is noisy, miscalibrated, or poorly matched to the deployment shift.