Empirical PAC-Bayes Bound for Markov Chains

• The paper introduces a fully empirical PAC-Bayes bound that replaces the unknown dependency constant with an estimator computed from observed Markov chain transitions.
• The method leverages concentration inequalities to estimate the pseudo-spectral gap, providing robust generalization guarantees even under temporal dependence.
• The approach adapts to finite and certain infinite state spaces, yielding nearly tight bounds similar to those for i.i.d. data when the chain exhibits rapid mixing.

Empirical PAC-Bayes Bound for Markov Chains refers to a data-driven generalization inequality that extends the classical PAC-Bayes learning theory to temporally dependent data generated by a Markov chain, with an explicit, empirically-estimated dependency constant. The critical advance is the ability to replace unknown process properties, such as the @@@@1@@@@ that governs the chain's mixing and concentration behavior, by an estimator computed entirely from observed data—yielding fully empirical bounds even in the presence of temporal dependence (Karagulyan et al., 25 Sep 2025).

1. Background: Generalization in Dependent Data and Pseudo-Spectral Gap

Classical PAC-Bayes bounds quantify the generalization error of randomized predictors under the assumption of independent observations (i.i.d. data). For dependent sequences, such as those generated by Markov chains, the standard proofs break down; the empirical average cannot be viewed as a sum of independent variables, and concentration inequalities inherit constants determined by the degree of dependence. In Markov chains, this dependence is captured by the spectral gap in reversible cases, or the pseudo-spectral gap in the non-reversible case—a spectral quantity associated with the transition operator (and its time-reversal). For a Markov kernel $P$ with time-reversal $P^*$, the pseudo-spectral gap is defined as

$$\gamma_{\text{ps}} = \max_{k \ge 1} \frac{\gamma((P^*)^k P^k)}{k},$$

where $\gamma(\cdot)$ denotes the spectral gap (i.e., one minus the largest non-unit eigenvalue). For reversible chains, $\gamma_\text{ps}$ coincides with the usual spectral gap; for non-reversible chains, $\gamma_\text{ps}$ is strictly more general and measures the effective rate of decorrelation in the chain (Paulin, 2012).

2. Non-Empirical PAC-Bayes Bound for Markov Chains

Assume $(X_1, ..., X_n)$ forms a stationary Markov chain with pseudo-spectral gap $\gamma_\text{ps} > 0$. For any prior $\mu$ over the hypothesis space $\Theta$, any posterior $\rho$, loss function $\ell$ bounded by $c$, and $0 < \lambda < n/10$, with probability at least $1-\delta$: $$\mathbb{E}_{\theta \sim \rho}[R(\theta)] \leq \mathbb{E}_{\theta \sim \rho}[r(\theta)] + \frac{2 \lambda c^2 (1 + 1/(n\gamma_\text{ps}))}{n-10\lambda} + \frac{\mathrm{KL}(\rho\|\mu) + \log(1/\delta)}{\lambda \gamma_\text{ps}}$$ where $r(\theta)$ is the empirical risk, $R(\theta)$ is the population risk, and $\mathrm{KL}(\rho\|\mu)$ is the Kullback-Leibler divergence. The dependence on $\gamma_\text{ps}$ quantifies the cost of temporal correlation: when the chain mixes slowly (small $\gamma_\text{ps}$), the effective sample size is reduced, and the bound weakens accordingly. For i.i.d. data, $\gamma_\text{ps} = 1$, recovering the classical PAC-Bayes rate (Paulin, 2012).

3. Fully Empirical Estimation of the Pseudo-Spectral Gap

A central innovation is the empirical estimation of $\gamma_\text{ps}$ in finite state Markov chains. Let $P$ be the empirically estimated transition matrix (from observed transitions) and let $K$ be a bounded integer parameter. Define

$$\widehat{\gamma}_{\text{ps}}[K] = \max_{k \in [K]} \frac{\gamma((\widehat{P}^T)^k \widehat{P}^k)}{k}$$

where $\widehat{P}^T$ is the transpose of the transition matrix (the empirical time-reversal). Under concentration results established in related work, for any $\epsilon > 0$,

$$\mathbb{P} \left( \left| \frac{\widehat{\gamma}_{\text{ps}}}{\gamma_{\text{ps}}} - 1 \right| \leq \epsilon \right) \geq 1 - \alpha(n, \gamma_{\text{ps}}, \epsilon)$$

where $\alpha$ is a (small) failure probability depending on $n$, $\gamma_\text{ps}$, and $\epsilon$. This enables the substitution of $\gamma_\text{ps}$ by its estimator in the PAC-Bayes bound, yielding a data-driven generalization guarantee (Karagulyan et al., 25 Sep 2025).

4. Empirical PAC-Bayes Bound: Finite-State Case

Plugging the estimator $\widehat{\gamma}_{\text{ps}}$ (up to a $(1+\epsilon)$ slack) into the non-empirical PAC-Bayes framework, one obtains the main empirical result: with high probability (up to $\delta+\alpha(n, \gamma_{\text{ps}}, \epsilon)$),

$$\mathbb{E}_{\theta \sim \rho}[R(\theta)] \leq \mathbb{E}_{\theta \sim \rho}[r(\theta)] + \frac{2 \lambda c^2 \left( 1 + 1/n^{1-a} \right)}{n-10\lambda} + \frac{\mathrm{KL}(\rho \| \mu) + \log(1/\delta)}{\lambda \, \widehat{\gamma}_{\text{ps}} (1+\epsilon)}$$

for any parameter $a \in (0,1)$. All terms are observable from the data except for the loss bound $c$, which is assumed known. This is the first PAC-Bayes generalization guarantee for Markov chains where the dependency constant is estimable from the observed trajectory without knowledge of the true transition kernel (Karagulyan et al., 25 Sep 2025).

5. Generalization, Applicability, and Relation to Classical Bounds

This empirical PAC-Bayes bound maintains uniform generalization control over the choice of posterior and is sharply sensitive to the observed dependence in the data. When the chain mixes rapidly, $\widehat{\gamma}_\text{ps}$ is close to 1 and the bound is nearly as tight as in the i.i.d. case. In comparison, earlier approaches for dependent data introduced explicit or implicit constants (mixing time, spectral gap, mixing coefficients) that must be provided as assumptions or estimated with strong prior knowledge (Paulin, 2012, Cuong et al., 2014, Rivasplata et al., 2020). The present result removes this limitation: dependency is automatically reflected through $\widehat{\gamma}_\text{ps}$, bypassing the need for unverifiable assumptions.

6. Extensions to Infinite State Spaces

While the empirical estimator is immediately computable in finite-state spaces, the Markov chain PAC-Bayes framework can be extended to certain infinite or continuous state spaces given sufficient structure. For example, in the case of an AR(1) process

$U_t = a U_{t-1} + \zeta_t, \quad |a| < 1,$

the pseudo-spectral gap is $$\gamma_\text{ps} = 1 - a^2 = \operatorname{Var}(U_1)^{-1}$$. In this case, the estimator

$$\widehat{\gamma}_\text{ps} = \min \left\{ 1,\ \frac{1}{\frac{1}{n} \sum_{t=1}^n U_t^2} \right\}$$

is used, and tail bounds are proven for its concentration. However, in such settings, additional knowledge of the noise distribution or mixing conditions may be required to ensure the validity of the empirical concentration, and thus more care is needed to justify fully empirical bounds (Karagulyan et al., 25 Sep 2025).

7. Experimental Behavior and Practical Tightness

Simulated studies, including binary classification with finite state spaces $d = 4, 10, 20, 50, 100$, confirm that both the non-empirical (true $\gamma_\text{ps}$) and empirical (estimated $\widehat{\gamma}_\text{ps}$) bounds vary in parallel across sample sizes. For moderate to large $n$, the empirical bound is essentially as tight as the non-empirical one, confirming that the estimator does not meaningfully degrade the quality of the PAC-Bayes guarantee in practice. For very small $n$, both bounds remain vacuous, reflecting the inherent statistical difficulty. The estimator's accuracy—and hence the tightness of the bound—increases with the mixing rate of the chain: as $\gamma_\text{ps}$ becomes small, both the bound and its empirical version become loose, reflecting reduced effective sample size (Karagulyan et al., 25 Sep 2025).

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In summary, the empirical PAC-Bayes bound for Markov chains is a generalization guarantee for randomized predictors trained on temporally dependent data, in which the dependency constant (pseudo-spectral gap) is empirically estimated from the observed sequence. This approach bridges the gap between theoretical and practical generalization bounds in the temporally dependent setting, providing guarantees that adapt to the observed degree of dependence and are directly applicable in data-rich, real-world Markovian settings.