Information Exponent: Asymptotic Rate Parameter

• Information exponent is a family of asymptotic rate parameters that quantifies how information-theoretic, statistical, or geometric quantities scale with problem size.
• It identifies the leading rate of distinguishability, recoverability, or singularity in diverse domains such as leaky PIR, SGD-based learning, hypothesis testing, and coding theory.
• This concept unifies various fields by providing a metaconcept that governs transitions in privacy leakage, gradient signal strength, error exponents, and Fisher-information curvature.

Searching arXiv for the primary and related papers on “information exponent” to ground the article. Searching for the primary L-PIR paper and several related uses of “information exponent.” Searching arXiv for ([2501.12310](/papers/2501.12310)) and related papers. to=arxiv_search 彩神争霸电脑版? Information exponent denotes a family of asymptotic rate parameters that quantify how sharply an information-theoretic, statistical, algorithmic, or geometric quantity scales with problem size. In the cited literature, the term appears in several distinct but structurally related senses: as the pure-differential-privacy leakage parameter in leaky private information retrieval, as the first nonzero Hermite degree governing SGD sample complexity in Gaussian single-index learning, and as the curvature-scaling exponent of the Fisher-information metric at criticality on microscopic coupling manifolds (Zhao et al., 21 Jan 2025, Tsiolis et al., 23 Oct 2025, Zhuravlev, 8 Mar 2026). In each case, the exponent isolates the leading asymptotic rate of distinguishability, recoverability, reliability, or singularity.

1. Terminological scope

The phrase information exponent is not attached to a single universal definition. Rather, it labels the dominant asymptotic rate in a given problem class. In some settings the exponent is literally exponential, such as an error exponent or a strong-converse exponent. In others it is an integer degree in an orthogonal expansion, or a power-law exponent in geometric criticality.

Domain	Exponent	Representative definition
Leaky PIR	Leakage ratio exponent $\epsilon$	Worst-case likelihood-ratio bound under pure differential privacy
Gaussian single-index learning	$\mathrm{IE}(g)$	Smallest Hermite index with nonzero coefficient
Fisher information geometry	$d_R$	Power in $|\mathcal R(J_c)| \sim n^{d_R}$
Weighted hypothesis testing	$I_w(p,q)$	Weighted Chernoff information
Quantum soft covering	$E_e(R), E_{sc}(R)$	Error and strong-converse exponents

This diversity is explicit in the literature. Quantum soft covering defines exponents for decay of trace distance and for exponential convergence to failure below threshold rate (Cheng et al., 2022). Context-sensitive hypothesis testing identifies the optimal weighted-loss exponent with a weighted Chernoff information (Kelbert et al., 9 Mar 2026). Channel and source-coding strong converses formulate exponent functions for correct-decoding or excess-distortion probabilities outside the achievable region (Oohama, 2017).

A plausible implication is that information exponent functions as a metaconcept: it denotes the leading asymptotic rate at which an informational task becomes possible, impossible, distinguishable, or singular, but the concrete mathematical object depends on the model.

2. Leakage ratio exponent in leaky private information retrieval

In leaky private information retrieval (L-PIR), the information exponent is the pure differential privacy parameter $\epsilon$, also called the leakage ratio exponent. Classical PIR requires perfect privacy in the sense that for every server $n$,

$$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$

L-PIR relaxes this to

$$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$

for every server $\mathrm{IE}(g)$0, every $\mathrm{IE}(g)$1, and every two demands $\mathrm{IE}(g)$2. Smaller $\mathrm{IE}(g)$3 means stronger privacy. The same work formulates the scheme through probabilities $\mathrm{IE}(g)$4 assigned to retrieval patterns $\mathrm{IE}(g)$5, under normalization, nonnegativity, download-cost, and DP constraints, and shows that joint optimization over all retrieval patterns materially improves the privacy–download tradeoff (Zhao et al., 21 Jan 2025).

The relevant schemes can be symmetrized so that the optimization reduces to probabilities indexed by the Hamming weight of a TSC-key payload vector $\mathrm{IE}(g)$6. Writing

$\mathrm{IE}(g)$7

the DP constraints reduce to adjacent-layer inequalities

$\mathrm{IE}(g)$8

and the optimization becomes a constrained minimization of the download cost. The resulting optimal distribution has a layered structure: keys of lower Hamming weight receive higher probability. With

$\mathrm{IE}(g)$9

the closed form is

$d_R$0

so that

$d_R$1

This establishes that lighter-weight keys are exponentially favored.

For fixed download cost $d_R$2 and fixed number of servers $d_R$3, the optimized distribution yields

$d_R$4

whereas the previous “UB” construction of Samy et al., which boosts only the clean retrieval pattern, gives

$d_R$5

hence $d_R$6. The significance is not merely numerical. The optimized scheme changes the scaling law itself: the privacy leakage needed to maintain fixed $d_R$7 and $d_R$8 grows only logarithmically in the number of messages $d_R$9, rather than linearly. The same exposition explains the terminology: $|\mathcal R(J_c)| \sim n^{d_R}$0 is an exponent because it controls worst-case likelihood-ratio distinguishability between two demands, and thus quantifies how rapidly a server can separate hypotheses as $|\mathcal R(J_c)| \sim n^{d_R}$1 grows.

3. Information exponent in gradient-based learning

In Gaussian single-index models, the information exponent is defined through the Hermite expansion. If

$|\mathcal R(J_c)| \sim n^{d_R}$2

then

$|\mathcal R(J_c)| \sim n^{d_R}$3

Equivalently, if $|\mathcal R(J_c)| \sim n^{d_R}$4 is smooth near the origin and $|\mathcal R(J_c)| \sim n^{d_R}$5 as $|\mathcal R(J_c)| \sim n^{d_R}$6, then $|\mathcal R(J_c)| \sim n^{d_R}$7. The same framework defines the generative exponent

$|\mathcal R(J_c)| \sim n^{d_R}$8

with $|\mathcal R(J_c)| \sim n^{d_R}$9 (Tsiolis et al., 23 Oct 2025).

For vanilla one-pass SGD on a two-layer network with one hidden neuron, the update

$I_w(p,q)$0

followed by normalization, has sample complexity governed by $I_w(p,q)$1. If $I_w(p,q)$2 and $I_w(p,q)$3, then

$I_w(p,q)$4

iterations suffice, and are necessary, for weak recovery. Thus the information exponent captures the flatness of the early-learning signal: higher first nonzero Hermite degree implies a weaker alignment signal and higher sample complexity.

The same paper shows that non-correlational updates can break this barrier. Reusing a sample or using a two-timescale update produces a polynomial oracle $I_w(p,q)$5 with Hermite–Hermite coefficients

$I_w(p,q)$6

An informal master expression gives

$I_w(p,q)$7

For batch-reuse SGD, if $I_w(p,q)$8 is the first power whose information exponent equals $I_w(p,q)$9, then whenever

$E_e(R), E_{sc}(R)$0

the algorithm enters the generative exponent regime,

$E_e(R), E_{sc}(R)$1

whereas for $E_e(R), E_{sc}(R)$2 below

$E_e(R), E_{sc}(R)$3

it reverts to the information-exponent regime. The same phase-transition structure appears in alternating layer-wise SGD, where the update

$E_e(R), E_{sc}(R)$4

leads to

$E_e(R), E_{sc}(R)$5

and if $E_e(R), E_{sc}(R)$6, then

$E_e(R), E_{sc}(R)$7

A complementary refinement appears for orthogonal multi-index models. There, using only the lowest active Hermite degree can be misleading: when only degree $E_e(R), E_{sc}(R)$8 is active, SGD recovers only the relevant subspace because of rotational invariance; when the lowest active degree is $E_e(R), E_{sc}(R)$9, classical information-exponent theory predicts $\epsilon$0 samples. For targets of the form

$\epsilon$1

a two-stage procedure first uses the second-order term for subspace recovery and then the higher-order term for direction recovery, yielding

$\epsilon$2

with strong recovery (Ren et al., 2024). This suggests that, in multi-index settings, the full active-degree pattern can matter more than the single lowest degree.

4. Distinguishability exponents in hypothesis testing and soft covering

In context-sensitive binary hypothesis testing, the relevant exponent is the weighted Chernoff information. Given a nonnegative multiplicative weight $\epsilon$3 and simple hypotheses $\epsilon$4 and $\epsilon$5, the weighted Bhattacharyya affinity is

$\epsilon$6

with

$\epsilon$7

The weighted Chernoff information is

$\epsilon$8

If

$\epsilon$9

then the optimal total weighted loss satisfies

$n$0

equivalently

$n$1

The derivation embeds weighted geometric mixtures into an exponential family, with

$n$2

where $n$3 is the log-normalizer (Kelbert et al., 9 Mar 2026).

Quantum soft covering uses a different but closely related exponent formalism. For a classical–quantum state $n$4 and an i.i.d. random codebook of size $n$5, the average trace distance

$n$6

obeys, for $n$7,

$n$8

Using additivity yields the achievability exponent

$n$9

which is strictly positive if and only if $$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$0. In the opposite regime, the strong-converse exponent is

$$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$1

positive if and only if $$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$2 (Cheng et al., 2022).

These constructions share a common role for the exponent: it is the sharp asymptotic rate at which distinguishability or covering error decays, or at which failure becomes overwhelming below threshold.

5. Exponent functions in channel coding and source coding

For stationary memoryless channels with an input-cost constraint, the strong-converse exponent is formulated through the correct-decoding probability. If

$$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$3

over codes of rate at least $$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$4 satisfying the block average-cost constraint, then

$$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$5

In the finite-alphabet case this exponent equals the Dueck–Körner minimax form

$$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$6

and also equals the dual Arimoto–Oohama representation. Thus the exponent exactly determines how fast the correct-decoding probability must decay when $$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$7 (Oohama, 2017).

For constant-composition codes on discrete memoryless channels, the random-coding error exponent under maximum mutual information decoding coincides with that under maximum likelihood decoding. With Gallager’s function

$$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$8

dual-domain analysis shows

$$\Pr\bigl(Q_n,A_n\mid k_1\bigr)=\Pr\bigl(Q_n,A_n\mid k_2\bigr)\quad \forall k_1,k_2.$$9

The same method extends to joint source–channel coding, where the generalized MMI decoder achieves the same random-coding exponent as the MAP decoder (Moeini et al., 23 Jan 2025).

In the binary symmetric channel with noisy feedback, the zero-rate reliability function also has an exponent interpretation. For the forward channel $$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$0,

$$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$1

while the noiseless-feedback exponent is

$$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$2

If the feedback link is $$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$3, then

$$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$4

with $$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$5 and $$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$6. The main theorem states that if $$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$7, then

$$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$8

so sufficiently reliable noisy feedback strictly improves the zero-rate exponent (0808.2092).

Source-coding strong converses lead to analogous exponent functions. In the one-helper problem, the correct-decoding probability exponent

$$\frac{\Pr\bigl(Q_n^{[k_1]}=q,\,A_n^{[k_1]}=a\bigr)} {\Pr\bigl(Q_n^{[k_2]}=q,\,A_n^{[k_2]}=a\bigr)} \le e^\epsilon,$$9

admits an explicit single-letter lower bound

$\mathrm{IE}(g)$00

with

$\mathrm{IE}(g)$01

thereby strengthening the strong converse to an exponential statement outside the Ahlswede–Körner–Wyner rate region (Oohama, 2015). For Wyner–Ziv coding, the excess-distortion exponent

$\mathrm{IE}(g)$02

satisfies

$\mathrm{IE}(g)$03

and every $\mathrm{IE}(g)$04 with $\mathrm{IE}(g)$05 obeys

$\mathrm{IE}(g)$06

yielding an exponential strong converse (Oohama, 2016).

Across these problems, the exponent function is the quantitative form of impossibility outside the operational region: it states not merely that performance fails, but how fast it fails.

6. Information-geometric exponent at criticality

In microscopic Fisher-information geometry, the information exponent is a power-law exponent for scalar curvature divergence. For a $\mathrm{IE}(g)$07-dimensional lattice with periodic boundary conditions and $\mathrm{IE}(g)$08 sites, the microscopic coupling manifold has dimension $\mathrm{IE}(g)$09, one parameter per bond. The Fisher information metric is

$\mathrm{IE}(g)$10

and defines a Riemannian metric $\mathrm{IE}(g)$11. From $\mathrm{IE}(g)$12 one constructs the Christoffel symbols, Riemann tensor, and scalar curvature $\mathrm{IE}(g)$13 in the usual way (Zhuravlev, 8 Mar 2026).

At a second-order critical point under periodic boundary conditions, Fourier diagonalization separates soft and hard momentum sectors. The bond-operator connected two-point function decays as

$\mathrm{IE}(g)$14

and the Fisher eigenvalues satisfy

$\mathrm{IE}(g)$15

near $\mathrm{IE}(g)$16. The resulting curvature scaling is

$\mathrm{IE}(g)$17

This is the information-geometric exponent.

The paper gives explicit universality-class predictions and numerical checks. For 2D Ising, with $\mathrm{IE}(g)$18 and $\mathrm{IE}(g)$19, the prediction is

$\mathrm{IE}(g)$20

confirmed by exact transfer-matrix computations for $\mathrm{IE}(g)$21--$\mathrm{IE}(g)$22 with $\mathrm{IE}(g)$23 and by multi-seed MCMC through $\mathrm{IE}(g)$24. For 3D Ising, with $\mathrm{IE}(g)$25 and $\mathrm{IE}(g)$26, the prediction is $\mathrm{IE}(g)$27, consistent with MCMC on $\mathrm{IE}(g)$28 tori up to $\mathrm{IE}(g)$29 and a power-law fit $\mathrm{IE}(g)$30. For 2D Potts $\mathrm{IE}(g)$31, the predicted value is $\mathrm{IE}(g)$32, while the observed effective exponent oscillates non-monotonically around $\mathrm{IE}(g)$33, consistent with $\mathrm{IE}(g)$34 logarithmic corrections; for $\mathrm{IE}(g)$35, the predicted value is $\mathrm{IE}(g)$36, again with strong logarithmic corrections.

A further structural feature is the Ricci decomposition identity

$\mathrm{IE}(g)$37

verified to $\mathrm{IE}(g)$38--$\mathrm{IE}(g)$39 significant figures for all models and sizes considered. The paper emphasizes that this exponent is distinct from Ruppeiner thermodynamic curvature. Its operational meaning is geometric rather than inferential: it measures how the scalar curvature of the full microscopic Fisher manifold diverges as the manifold dimension itself grows with the system size.

Taken together, these literatures show that information exponents organize asymptotic theory at several levels: privacy leakage in L-PIR, gradient-signal strength in nonlinear learning, distinguishability in testing and soft covering, reliability decay beyond coding thresholds, and curvature divergence at criticality. This suggests that the unifying role of an information exponent is not its formal expression, but its status as the leading asymptotic rate parameter governing an informational transition.