Self-Calibrating Language Models via Test-Time Discriminative Distillation

Mohamed Rissal Hedna; Jan Strich; Martin Semmann; Chris Biemann

cs.CLMar 18, 2026

Self-Calibrating Language Models via Test-Time Discriminative Distillation

Authors: Mohamed Rissal Hedna, Jan Strich, Martin Semmann, Chris Biemann

Organizations: Hub of Computing and Data Science (HCDS), University of Hamburg, Germany

Abstract

Large language models (LLMs) are systematically overconfident: they routinely express high certainty on questions they often answer incorrectly. Existing calibration methods either require labeled validation data, degrade under distribution shifts, or incur substantial inference costs. Recent work has shown that LLMs already contain a better-calibrated signal than the one they verbalize: the token probability of "True" when the model is asked "Is this answer correct?" ( $P(\text{True})$ ) consistently outperforms their stated confidence, a gap that is theoretically grounded as generative error is lower-bounded by roughly twice the corresponding discriminative error. We introduce $\textbf{SECL}$ ( $\textbf{SE}$ lf- $\textbf{C}$ alibrating $\textbf{L}$ anguage Models), a test-time training (TTT) pipeline that exploits this gap as label-free self-supervision, requiring no labeled data or human supervision. SECL adapts only when the input distribution shifts, training on just 6--26% of the question stream at lower cost than the baseline it distills from. Across four small language models from three model families and four diverse domains, SECL reduces Expected Calibration Error (ECE) by 56--78%, outperforming its own supervision signal and matching or outperforming recent inference-time methods. SECL is the first method to apply TTT to calibration; seven ablations covering signal quality, gating strategy, weight accumulation, loss design, domain ordering, hyperparameter sensitivity, and layer selection confirm that each component is crucial and robust across configurations. Code: https://anonymous.4open.science/r/secl-emnlp26-submission-C890

Figures & tables

Figure 1: Calibration error vs. inference cost for Llama 3.2-3B (lower-left is better). Each point is a calibration method; the

x

-axis measures cost in forward-pass equivalents per question, and the

y

-axis measures Expected Calibration Error (ECE). SECL achieves the lowest calibration error at a fraction of the cost of existing methods.

Figure 2: Overview of SECL. (a) Test-Time Inference. An entropy-based change detector (Section 3.1 ) monitors the input stream. If no shift is detected, the adapted model $\theta^{\prime}_{t}$ is used directly; otherwise, a calibration burst updates it to $\theta^{\prime}_{t+1}$ . (b) Calibration Burst. For each of $B{=}50$ questions: the frozen model generates an answer with confidence $c_{t}$ and distractors, computes $\text{NormP}_{\text{True}}$ (Section 3.2 ), and applies a LoRA update when the two signals disagree by more than one bin (Section 3.3 ). Weights accumulate across questions without resetting.

		Llama 3.2-3B			Llama 3.1-8B			Gemma 2-2B			Phi 3.5-Mini
Method	Cost $\downarrow$	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$
Verbalized	1	.170	.292	.510	.225	.258	.684	.256	.314	.558	.251	.275	.600
P(True) Norm	6	.065	.223	.694	.120	.211	.718	.141	.259	.650	.154	.227	.675
DINCO †	$\sim$ 10	.101	.207	.762	.117	.210	.756	.408	.410	.566	.110	.212	.749
SECL (Ours)	1.8–4.6	.050	.241	.587	.083	.222	.643	.056	.254	.548	.110	.251	.521

Table 1: Overall results across the full 2,000-question stream. Best values per column in bold (lowest for ECE/Brier/Cost, highest for AUROC). Cost is amortized forward-pass equivalents per question; SECL’s range reflects model-dependent gating rates (Appendix M ). Accuracy is preserved within 1 percentage point for all models (Appendix D ). † Wang and Stengel-Eskin (2025) .

Model	Verbalized	SECL $\mathcal{O}$	SECL $\mathcal{R}$
Llama 3.2-3B	.170	.050 ( $-$ 71%)	.068 ( $-$ 60%)
Llama 3.1-8B	.225	.083 ( $-$ 63%)	.189 ( $-$ 16%)
Gemma 2-2B	.256	.056 ( $-$ 78%)	.149 ( $-$ 42%)
Phi 3.5-Mini	.251	.110 ( $-$ 56%)	.173 ( $-$ 31%)

Table 2: Domain order robustness (ECE).

\mathcal{O}

: GSM8K

\to

MMLU

\to

ARC

\to

TruthfulQA (default).

\mathcal{R}

: reversed. SECL improves over the verbalized baseline under both orderings for all models; per-domain breakdowns in Appendix L .

Strategy	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Trained
Always-on MSE	0.047	0.240	0.593	100%
Bin-gate ( $\leq\!1$ )	0.052	0.242	0.585	55.8%
Bin-gate ( $\leq\!2$ )	0.044	0.242	0.578	32.6%
Entropy-gated ( $B\!=\!50$ )	0.050	0.241	0.587	25.6%

Table 3: Gating strategy ablation (Llama 3.2-3B). Entropy-gated bursts achieve comparable ECE to always-on training while requiring only 25.6% of questions.

Appendix figures & tables30 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Total params	LoRA layers	LoRA params	% of total
Llama 3.2-3B	3.21B	4	327,680	0.010%
Llama 3.1-8B	8.03B	8	1,048,576	0.013%
Gemma 2-2B	2.61B	8	491,520	0.019%
Phi 3.5-Mini	3.82B	8	786,432	0.021%

Appendix

Table 4: LoRA trainable parameters per model (rank

r{=}8

). Llama 3.2-3B: last 4 layers (24–27 of 28); Llama 3.1-8B: last 8 layers; Gemma and Phi: last 8 layers (Gemma: 18–25 of 26; Phi: 24–31 of 32).

Component	Parameter	Value
LoRA	Rank $r$	8
	Scaling factor $\alpha$	16
	Target layers (Llama 3.2-3B)	Last 4
	Target layers (Llama 3.1-8B, Gemma, Phi)	Last 8
	Target modules (Llama, Gemma)	q_proj , v_proj
	Target modules (Phi)	qkv_proj

Appendix

Table 5: Full hyperparameter settings used across all experiments. Model-specific values are noted where applicable.

Model	Method	ECE $\downarrow$	AdaECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Llama 3.2-3B	Verbalized	.170	.167	.292	.510	.576
	Self-Consistency †	.093	.096	.211	.728	.624
	+ Temp Scaling ∗	.047	.075	.250	.504	.576
	P(True) Norm	.065	.067	.223	.694	.576
	+ Temp Scaling ∗	.029	.030	.218	.692	.576
	SECL (Ours)	.050	.060	.241	.587	.577

Appendix

Table 6: Overall results across all models and methods (2,000 questions each). Bold : best overall per model for ECE, AdaECE, Brier, and AUROC. Underline : best among label-free methods.

Metric	Seed 42	Seed 43	Seed 44	Mean $\pm$ Std
Verbalized ECE	.1695	.1757	.1757	.1736 $\pm$ .0029
SECL ECE	.0496	.0502	.0392	.0464 $\pm$ .0051
Verbalized AUROC	.5095	.5119	.5119	.5111 $\pm$ .0011
SECL AUROC	.5870	.6005	.5920	.5932 $\pm$ .0056

Appendix

Table 7: Llama 3.2-3B multi-seed robustness (seeds 42, 43, 44). SECL remains strongly better calibrated than the verbalized baseline across seeds and also improves AUROC on this model.

Model	Method	ECE $\downarrow$	AdaECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Conf. range
Llama 3.2-3B	Verbalized + Temp ( $T{=}17.6$ )	.047	.075	.250	.504	[.44, .56]
	Verbalized + Platt	.021	.057	.244	.481	[.56, .61]
	P(True) Norm + Temp ( $T{=}1.8$ )	.029	.030	.218	.692	[.14, .86]
	SECL (no labels)	.050	.060	.241	.587	[.05, .85]
Gemma 2-2B	Verbalized + Temp ( $T{=}11.2$ )	.047	.037	.249	.550	[.41, .59]
	Verbalized + Platt	.035	.050	.250	.550	[.42, .55]

Appendix

Table 8: Post-hoc calibration baselines (5-fold CV). Temperature scaling compresses the confidence range substantially, achieving low ECE by predicting near-constant values for models with weak verbalized discrimination (fitted

T

shown).

Model	Method	ECE $\downarrow$	AdaECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$
Llama 3.2-3B	P(True) Norm + Temp ∗	.029	.030	.218	.692
	SECL	.050	.060	.241	.587
	SECL + Temp ∗	.049	.053	.241	.584
Gemma 2-2B	P(True) Norm + Temp ∗	.037	.033	.233	.647
	SECL	.056	.060	.254	.548
	SECL + Temp ∗	.011	.037	.249	.542

Appendix

Table 9: SECL combined with temperature scaling (5-fold CV). SECL+Temp achieves the best or near-best ECE overall while preserving SECL’s discrimination improvements.

Method	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Verbalized	.170	.292	.510	.576
Raw P(True)	.161	.248	.673	.576
P(True) Norm $\tau{=}0.3$	.118	.238	.685	.576
P(True) Norm $\tau{=}0.7$	.065	.223	.694	.576
P(True) Norm $\tau{=}1.5$	.169	.254	.658	.579

Appendix

Table 10: Effect of P(True) normalization (Llama 3.2-3B,

N{=}2{,}000

). Raw P(True) improves over the verbalized baseline. Normalization with temperature

\tau

yields further gains. The optimal

\tau

varies by model (see Table 11 ).

$\boldsymbol{\tau}$	Llama 3.2-3B	Gemma 2-2B	Phi 3.5-Mini	Llama 3.1-8B
0.3	.118	.255	.172	.270
0.7	.065	.215	.168	.234
1.0	.083	.180	.161	.195
1.5	.169	.141	.154	.153
3.0	.247	.143	.245	.120

Appendix

Table 11: P(True) Norm ECE as a function of normalization temperature

\tau

. Bold: selected value per model.

Parameter	Value	ECE $\downarrow$	AdaECE $\downarrow$
$\alpha_{\text{step}}$ †	0.2	.067	.083
	0.3	.066	.069
	0.5	.052	.073
$\delta$ †	0.15	.052	.073
$\delta$ †	0.20	.064	.067
Loss †	Plain MSE (no directional)	.085	.088

Appendix

Table 12: Hyperparameter sensitivity (Llama 3.2-3B). Bold indicates the selected configuration. † Without entropy gating. ‡ With entropy gating.

Model	Method	Domain	ECE $\downarrow$	AdaECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Llama 3.2-3B	Verbalized	GSM8K	.218	.215	.294	.565	.472
		MMLU	.106	.109	.253	.601	.562
		ARC	.095	.112	.207	.568	.726
		TruthfulQA	.372	.371	.414	.301	.544
	P(True) Norm	GSM8K	.133	.138	.264	.594	.476
		MMLU	.091	.090	.239	.652	.566

Appendix

Table 13: Per-domain results for Llama 3.2-3B, Gemma 2-2B, and Phi 3.5-Mini. Best ECE and AdaECE per domain are in bold .

Setting	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
SECL (accumulate)	.050	.241	.587	.577
Reset per question	.237	.327	.484	.576

Appendix

Table 14: Weight accumulation ablation (Llama 3.2-3B).

Setting	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Trained %
Always-train	.047	.240	.593	100%
Entropy-gated (SECL)	.050	.241	.587	25.6%

Appendix

Table 15: Entropy gating vs. always-train (Llama 3.2-3B). Entropy gating matches always-train ECE at a fraction of the cost.

Model	$\boldsymbol{\beta}$	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Llama 3.2-3B	0 (default)	.050	.241	.587	.577
	0.01	.044	.242	.591	.573
	0.1	.149	.279	.527	.569
Gemma 2-2B	0 (default)	.056	.254	.548	.515
	0.01	.127	.271	.486	.514
	0.1	.238	.305	.551	.520

Appendix

Table 16: KL regularization ablation across models (SECL + entropy gating).

Model	Layers	Range	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Llama 3.2-3B	Early	0–3	.046	.235	.619	.591
	Mid	12–15	.039	.233	.628	.573
	Late (default)	24–27	.039	.241	.592	.570
	Last 8	20–27	.044	.240	.609	.558
	All	0–27	.058	.232	.640	.588
Gemma 2-2B	Last 4	22–25	.116	.277	.464	.513

Appendix

Table 17: LoRA layer ablation. Top: layer position on Llama 3.2-3B (4 layers each except "All"). Bottom: layer count on Gemma 2-2B and Phi 3.5-Mini (3.8B).

Figure 5: Raw

P(\text{True})

( left ) vs. distractor-normalized

\text{NormP}_{\text{True}}

( right ) on Llama 3.2-3B. Normalization suppresses suggestibility bias, producing a better-calibrated supervision signal (ECE: 0.161

\to

0.065).

Figure 6: Raw

P(\text{True})

( left ) vs.

\text{NormP}_{\text{True}}

( right ) for the three non-8B models. Distractor normalization consistently reduces calibration error across model families.

Figure 7: Reliability diagrams: Verbalized baseline ( left ) vs. SECL ( right ) for Llama 3.2-3B (ECE: 0.170

\to

0.050), Gemma 2-2B (ECE: 0.256

\to

0.056), and Phi 3.5-Mini (3.8B) (ECE: 0.251

\to

0.110). SECL consistently shifts predictions toward the diagonal, eliminating the systematic overconfidence visible in the verbalized baselines.

Model	Order	ECE $\downarrow$	AdaECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Llama 3.2-3B	Verbalized	.170	.167	.292	.510	.576
	SECL $\mathcal{O}$	.050	.060	.241	.587	.577
	SECL $\mathcal{R}$	.068	.082	.248	.578	.575
Gemma 2-2B	Verbalized	.256	.252	.314	.558	.516
	SECL $\mathcal{O}$	.056	.060	.254	.548	.515
	SECL $\mathcal{R}$	.149	.133	.258	.625	.520

Appendix

Table 18: Domain order sensitivity (overall metrics).

\mathcal{O}

: GSM8K

\to

MMLU

\to

ARC

\to

TruthfulQA.

\mathcal{R}

: TruthfulQA

\to

ARC

\to

MMLU

\to

GSM8K.

Model	Order	GSM8K	MMLU	ARC	TruthfulQA
Llama 3.2-3B	$\mathcal{O}$	.070	.067	.112	.068
Llama 3.2-3B	$\mathcal{R}$	.117	.032	.081	.134
Gemma 2-2B	$\mathcal{O}$	.356	.054	.144	.210
Gemma 2-2B	$\mathcal{R}$	.396	.096	.046	.189
Phi 3.5-Mini (3.8B)	$\mathcal{O}$	.283	.054	.129	.229
Phi 3.5-Mini (3.8B)	$\mathcal{R}$	.068	.213	.099	.343

Appendix

Table 19: Per-domain ECE for original (

\mathcal{O}

) vs. reversed (

\mathcal{R}

) domain order.

Model	Trained	Skipped	SECL	P(True)
Llama 3.2-3B	512 (25.6%)	1,488	9,168	12,000
Llama 3.1-8B	251 (12.6%)	1,749	5,514	12,000
Gemma 2-2B	119 (5.9%)	1,881	3,666	12,000
Phi 3.5-Mini (3.8B)	160 (8.0%)	1,840	4,240	12,000

Appendix

Table 20: Computational cost comparison (in FWD-equivalents). Trained = questions receiving TTT; Skipped = entropy-gated questions (generation only). Per trained question:

{\approx}15

FWD-eq (1 gen

+

5 P(True)

+

3 epochs

\times

3 FWD-eq/step). Per skipped question:

1

FWD-eq. P(True) Norm baseline: 6 FWD-eq

\times

2,000

=12{,}000

Method	ECE $\downarrow$
Verbalized	.247
P(True) Norm $\tau{=}0.3$	.290
P(True) Norm $\tau{=}0.7$	.263
P(True) Norm $\tau{=}1.0$	.257
P(True) Norm $\tau{=}1.5$	.265
P(True) Norm $\tau{=}2.0$	.267

Appendix

Table 21: Qwen 2.5-3B: no generation–discrimination gap. All P(True) Norm temperatures yield worse ECE than the verbalized baseline.

Model	Total	Trained (%)	Skipped (%)
Llama 3.2-3B	2,000	512 (25.6%)	1,488 (74.4%)
Gemma 2-2B	2,000	119 (5.9%)	1,881 (94.1%)
Phi 3.5-Mini (3.8B)	2,000	160 (8.0%)	1,840 (92.0%)
Llama 3.1-8B	2,000	251 (12.6%)	1,749 (87.4%)

Appendix

Table 22: Entropy gating trigger statistics per model. Trained% is the fraction of questions receiving TTT adaptation.

Method	ECE $\downarrow$	AdaECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Verbalized	.225	.225	.258	.684	.644
+ Temp Scaling ∗	.080	.089	.213	.681	.644
P(True) Norm	.120	.117	.211	.718	.646
+ Temp Scaling ∗	.108	.103	.208	.716	.646
SECL (Ours)	.083	.080	.222	.643	.646

Appendix

Table 23: Llama 3.1-8B results. SECL reduces ECE by 63% relative to the verbalized baseline while adapting only 12.6% of questions. ∗ Requires ground-truth labels (5-fold CV).

Method	Domain	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Verbalized	GSM8K	.298	.305	.714	.602
	MMLU	.208	.251	.644	.670
	ARC	.108	.164	.633	.798
	TruthfulQA	.287	.310	.669	.506
SECL	GSM8K	.066	.238	.602	.572
	MMLU	.120	.223	.591	.686

Appendix

Table 24: Per-domain results for Llama 3.1-8B.

Training Target	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
None (Verbalized baseline)	.170	.292	.510	.576
Self-Consistency ( $N{=}10$ )	.432	.470	.443	.564
P(True) Norm (ours)	.050	.241	.587	.577

Appendix

Table 25: Effect of training target on Llama 3.2-3B. All rows use the same SECL pipeline; only the pseudo-label differs. Self-Consistency as a target degrades calibration below the untrained baseline.

Figure 8: Reliability diagrams comparing candidate training targets on Llama 3.2-3B. Self-Consistency ( left , ECE

=

0.093) produces a reasonable calibration curve but with systematic bias. Raw P(True) ( center , ECE

=

0.161) exhibits upward bias from suggestibility. P(True) Norm ( right , ECE

=

0.065) tracks the diagonal most closely, confirming it as the best-calibrated supervision signal for SECL training.

Figure 9: Confidence score distributions for correct (green) and incorrect (red) predictions on Llama 3.2-3B. Self-Consistency and Raw P(True) both cluster near 1.0 with poor class separation; P(True) Norm spreads scores across

[0,1]

with better separation between correct and incorrect predictions.

Model	Method	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Llama 3.2-3B	Verbalized (4,000q)	.190	.313	.508	.577
Llama 3.2-3B	SECL (4,000q)	.051	.239	.607	.581
Gemma 2-2B	Verbalized (4,000q)	.276	.328	.548	.508
Gemma 2-2B	SECL (4,000q)	.086	.263	.500	.512
Phi 3.5-Mini (3.8B)	Verbalized (4,000q)	.192	.245	.593	.687
Phi 3.5-Mini (3.8B)	SECL (4,000q)	.259	.260	.712	.696

Appendix

Table 26: 4,000-question comparison against same-stream verbalized baselines. SECL substantially improves calibration on Llama and Gemma. For Phi, stronger long-stream calibration requires sufficient distillation; with the 4,000q run, calibration remains worse than verbalized while discrimination and accuracy improve.

Model	Method	FWD-eq	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Llama 3.2-3B	Verbalized	1	.170	.292	.510	.576
	DINCO	$\sim$ 10	.101	.207	.762	.466
	SECL (ours)	4.6	.050	.241	.587	.577
Phi 3.5-Mini (3.8B)	Verbalized	1	.251	.275	.600	.667
	DINCO	$\sim$ 10	.110	.212	.749	.560
	SECL (ours)	2.1	.110	.251	.521	.665

Appendix

Table 27: SECL vs. DINCO across all models. SECL achieves better calibration (ECE) at lower amortized cost, while DINCO provides superior discrimination (AUROC) at higher cost.

Method	Order	Stream	ECE $\downarrow$	Brier $\downarrow$	AUROC $\uparrow$	Acc
Verbalized	Forward	2k	.195	.287	.586	.506
P(True) Norm	Forward	2k	.173	.273	.625	.506
SECL (Ours)	Forward	2k	.047	.239	.618	.500
SECL (Ours)	Reversed	2k	.047	.241	.625	.497
Verbalized	Forward	4k	.179	.278	.598	.525
SECL (Ours)	Forward	4k	.061	.237	.625	.522

Appendix

Table 28: TruthfulQA generation-domain evaluation (Llama 3.2-3B). SECL reduces ECE by 76% on the 2k forward stream (0.195

\to

0.047), remains stable under reversed ordering, and at 4k still improves over the same-stream verbalized baseline by 66% (0.179

\to

0.061).

Inference-Time Adaptation of Language Models

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