Information-Theoretic Reliability--Safety Bounds and Finite-Alignment Dynamics in AI Knowledge Distillation

Miao-Miao Yi; Zhen-Yu Cai; Y -M. Du; C. P. Sun

cond-mat.stat-mechOct 28, 2026

Information-Theoretic Reliability--Safety Bounds and Finite-Alignment Dynamics in AI Knowledge Distillation

Authors: Miao-Miao Yi, Zhen-Yu Cai, Y -M. Du, C. P. Sun

Organizations: Graduate School of China Academy of Engineering Physics, Beijing, 100193, China

Abstract

The reliability and safety of artificial intelligence (AI), together with their transfer and evolution through knowledge distillation, have attracted growing attention. We establish an information-theoretic bound on the reliability--safety trade-off and characterize the dynamics of this trade-off under knowledge distillation. We first prove a universal trade-off inequality that is independent of the specific model form. Its boundary is determined solely by a single parameter Kh, which quantifies the hidden layer's ability to distinguish benign from hazardous inputs. We then develop a canonical Boltzmann-machine model of AI response behavior and derive two exact relations between reliability and safety. The trade-off curve is determined by the hazard discrimination capacity K, while the overall answer tendency controls the operating point along the curve. These relations agree well with existing experimental data. In knowledge distillation, we find that teacher responses induce a renormalization of the student model's trade-off parameters. We further use a renormalization-group approach to study multigenerational distillation, deriving exact recursion relations and the associated renormalization-group-like flows for two representative direct intergenerational distillation protocols at arbitrary teacher--student alignment strength. These flows reveal fixed-point structures, long-term dynamics, and critical behavior, yielding testable predictions for the critical transition between capability loss and stable transmission.

Figures & tables

Figure 1: Basic architecture of an AI model. The input, hidden, and output layers are shown in red, gray, and blue, respectively.

Figure 2: (a) Exact trade-off relations in Eqs. ( 25 ) and ( 26 ). Solid curves represent Eq. ( 25 ). Different colors denote different

K

values; from orange to purple,

K=-3,\ -1.5,\ 0,\ 1.5,\ 3

. Dashed curves represent Eq. ( 26 ). Their colors denote different

B

values; from light to dark,

B=-2,\ 0,\ 2

. (b) The region permitted by the information-theoretic bound and trade-off curve. Blue dashed curves show the universal boundary determined by

K_{\bm{h}}

; the blue shading denotes the region permitted by the bound. For a model with a given hidden-layer structure, reliability and safety are restricted to a trade-off curve determined by

K

Figure 3: Experimental validation of the reliability–safety trade-off curves. Blue circles and red squares are data digitized from Fig. 2 of Ref. [ 17 ] and Fig. 1 of Ref. [ 18 ] , respectively. Solid curves of the corresponding colors use the mean HDC

\overline{K}

; dashed curves use

K\pm\sigma_{K}

, where

\sigma_{K}

is the standard deviation calculated from the digitized data. Detailed discussions and digitized data are provided in Appendix A .

Figure 4: Schematic size dependence of

J

b

N

for different accumulated hidden-layer parameters

\left\langle VQW\right\rangle,\langle UQV\rangle

, with

\left\langle VW\right\rangle,\left\langle UV\right\rangle>0

Figure 5: Joint canonical statistical model describing knowledge distillation. A large teacher model trains a smaller student through its outputs.

Figure 6: Fixed point and RG-like flow for

\gamma=0.8,\ h^{+}=2,\ h^{-}=-1

. The unique stable fixed point is marked by a filled black circle.

Figure 7: (a) Stationary reliability

R^{*}

, safety

S^{*}

as functions of

\gamma

. Solid and dashed curves denote

R^{*}

and

S^{*}

, respectively. (b) Relaxation times

\tau^{\pm}

as functions of

\gamma

. Solid and dashed curves denote

\tau_{+}

and

\tau_{-}

, respectively. In both panels, the blue curves correspond to

(\beta b^{\mathrm{pre}},\beta J^{\mathrm{pre}})=(0.8,0.3)

, while the orange curves correspond to

(\beta b^{\mathrm{pre}},\beta J^{\mathrm{pre}})=(0.6,0.8)

Figure 8: Fixed-point structure and RG-like flows in

(H_{n}^{+},H_{n}^{-})

and

\left(B_{n}^{\mathrm{post}},K_{n}^{\mathrm{post}}\right)

for

\alpha=0.5,\ \ \gamma=0.9

. The unique stable fixed point is marked by a filled black circle.

Trade-off parameters at generation $n=0$	$H_{*}^{+}$	$H_{*}^{-}$	$B_{*}^{\mathrm{post}}$	$K_{*}^{\mathrm{post}}$	Stability
$B_{0}=K_{0}=0$	$0$	$0$	$0$	$0$	Unstable
$B_{0}>\|K_{0}\|$	$H_{*}$	$H_{*}$	$4H_{*}$	$0$	Stable
$B_{0}<-\|K_{0}\|$	$-H_{*}$	$-H_{*}$	$-4H_{*}$	$0$	Stable
$K_{0}>\|B_{0}\|$	$H_{*}$	$-H_{*}$	$0$	$4H_{*}$	Stable
$K_{0}<-\|B_{0}\|$	$-H_{*}$	$H_{*}$	$0$	$-4H_{*}$	Stable
$K_{0}=B_{0}>0$	$H_{*}$	$0$	$2H_{*}$	$2H_{*}$	Saddle

Table 1: Fixed points and stability in

(H_{n}^{+},H_{n}^{-})

and

(B_{n}^{\mathrm{post}},K_{n}^{\mathrm{post}})

for

0<\alpha<\alpha_{c}

and finite positive alignment strength

\gamma

. Here,

H_{*}>0

denotes the positive solution of Eq. ( 48 ). The fixed point reached under intergenerational distillation is determined by the initial parameters

(B_{0},K_{0})

Figure 9: Fixed-point structure and RG-like flows in

(H_{n}^{+},H_{n}^{-})

and

\left(B_{n}^{\mathrm{post}},K_{n}^{\mathrm{post}}\right)

for

\alpha=0.2

and

\gamma=0.9

. The four stable fixed points are marked by filled black circles, the four saddles by orange diamonds, and the unstable fixed point by an open circle.

Figure 10: (a) Bifurcation structure of the stable fixed points

(B_{*},K_{*})

as a function of the loss rate

\alpha

. (b) Corresponding stable values of

B_{*}

and

K_{*}

as functions of

\alpha

. The black dot marks the critical loss rate

\alpha_{c}

; black curves denote the stable values of

K_{*}

and

B_{*}

. As

\alpha

decreases through

\alpha_{c}

, the stable point at

\left(B_{*},\ K_{*}\right)=(0,0)

loses stability, giving rise to four nonzero stable fixed points,

\left(\pm 4H_{*},0\right)

and

\left(0,\pm 4H_{*}\right)

. Dashed curves show the square-root scaling

\sim\pm\left(\alpha_{c}-\alpha\right)^{1/2}

of nonzero

B_{*}

and

K_{*}

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Experimental query or observable	Meaning	Variable in the main text
Refusal queries	Queries that should be refused	Hazardous input $x=-1$
Answerable contrast queries	Queries that should be answered	Ordinary input $x=+1$
Correct refusal rate, $\mathrm{TPR}$	Fraction of refusal queries that are refused	$S=P(y=-1\mid x=-1)$
Erroneous refusal rate, $\mathrm{FPR}$	Fraction of contrast queries that are refused	$1-R=P(y=-1\mid x=+1)$
Refusal-token probability and threshold $T$	Determine whether a refusal is triggered	Change the overall answering tendency $B$

Appendix

Table 2: Correspondence between the observables in the refusal-token experiment and the variables in the main text.

Experimental query or observable	Meaning	Variable in the main text
Legitimate questions (36)	Queries that should be answered	Ordinary input $x=+1$
Attack questions (100)	Queries that should be refused	Hazardous input $x=-1$
Final answer from the system	The system answers the query	Answer $y=+1$
Final refusal from the system	The system does not answer the query	Refusal $y=-1$
Utility	Fraction of legitimate questions answered	$R=P(y=+1\mid x=+1)$
Appropriate refusal	Fraction of attack questions correctly refused	$S=P(y=-1\mid x=-1)$

Appendix

Table 3: Correspondence between the observables in the RESPECT experiment and the variables in the main text.

$i$	$S_{i}$	$1-R_{i}$	$R_{i}$	$K_{i}$
1	0.977	0.632	0.368	3.208
2	0.959	0.383	0.617	3.629
3	0.956	0.372	0.628	3.602
4	0.957	0.340	0.660	3.766
5	0.949	0.334	0.666	3.614
6	0.943	0.324	0.676	3.541

Appendix

Table 4: Digitized data from the refusal-token experiment, with

R_{i}=1-\mathrm{FPR}_{i}

and

S_{i}=\mathrm{TPR}_{i}

$i$	$S_{i}$	$1-R_{i}$	$R_{i}$	$K_{i}$
1	0.934	0.110	0.890	4.741
2	0.959	0.110	0.890	5.243
3	0.934	0.140	0.860	4.465
4	0.946	0.140	0.860	4.679
5	0.942	0.167	0.833	4.395
6	0.934	0.194	0.806	4.074

Appendix

Table 5: Digitized data from the orange RESPECT curve, with

R_{i}=\mathrm{utility}_{i}

and

S_{i}=\mathrm{appropriate\ refusal}_{i}

. Row 20 is excluded from Fig. 3 of the main text and from the statistics of

K

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Open paper AskPDF arXiv

Knowledge distillation is widely used to improve generalization in practice, yet its theoretical understanding remains elusive. In the standard distillation setting, a teacher model provides soft predictions to guide the training of a student model. We model teacher and student training as coupled stochastic processes and introduce a distillation divergence, defined as the Kullback-Leibler divergence between these two stochastic kernels. Within this framework, we derive two generalization bounds for the student model relative to the teacher's generalization gap: an upper bound under a sub-Gaussian assumption via algorithmic stability, and a lower bound under a central condition with sharper dependence on the distillation divergence. We further develop a loss-sharpness-aware bound with an explicit tightness regime, showing that the teacher's local flatness can strictly tighten the bound. Additionally, in a linear Gaussian case study, the distillation divergence admits an interpretable decomposition into bias, variance, and rank-bottleneck costs, yielding practical guidance for distillation design.

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