Self-Divergence Steps in Learning

• Self-divergence steps are mechanisms that use divergence measures to compare a model to its updated states, providing regularization and controlled evolution.
• They employ integral and differential operator techniques to create monotonic sequences that offer stability in optimization and prevent issues like collapse.
• Applications span semi-supervised, federated learning, variational inference, and even quantum field theory, highlighting their broad impact on model robustness.

Self-divergence steps encompass a wide class of mechanisms, operators, and algorithmic updates that explicitly utilize divergence measures to manage, regularize, or analyze the evolution of statistical models, distributions, and learning algorithms—either with respect to themselves at different states or in relation to related iterative approximations. In information geometry, optimization, semi-supervised learning, federated learning, statistical mechanics, and large-scale neural training, self-divergence steps formalize and control the effect of self-comparison, incremental upgrades, and regularized evolution by leveraging convexity, decomposition, and operator-induced transformations of divergence functions.

1. Divergence Foundations and Self-Divergence Formalism

Divergences—functions such as $f$-divergence, $\alpha$-Rényi divergence, and Bregman divergence—quantify the discrepancy between two probability distributions $P = \{p_1, ..., p_k\}$ and $Q = \{q_1, ..., q_k\}$.

Key definitions:

• $f$-divergence: $D_f(P \,\|\, Q) = \sum_{i=1}^k q_i\, f(p_i / q_i)$, with $f:(0,\infty) \rightarrow \mathbb{R}$ convex, $f(1)=0$.
• $\alpha$-Rényi divergence: $$D_\alpha(P \,\|\, Q) = \frac{1}{\alpha-1} \log \left( \sum_{i=1}^k p_i^\alpha q_i^{1-\alpha}\right)$$.

A "self-divergence step" (an Editor's term) refers to the evaluation or manipulation of a divergence where, through a sequence of updates or operator applications, one compares a distribution/model or its representation to itself after some transformation, perturbation, or learning event (Felice et al., 2018, Nishiyama, 2018, Nishiyama, 2019).

Formally, such steps appear as integrals along mixture paths, as gradient or operator-induced increments, or as terms that regularize self-updating model parameters:

• Path mixing: $R(t) = (Q-P)t + P$
• Operator transformation: $$Y[D](P \,\|\, R(t)) = \frac{1}{t} \int_0^t [-D(P \,\|\, R(s))] ds$$
• Update-based divergences: VCD, spread-based, or iterative preference steps.

These constructions endow learning and optimization processes with structured, often monotone, control over model progress, mitigating issues like confirmation bias, collapse, or instability.

2. Operator-Induced Self-Divergence and Monotonicity

The introduction of integral and differential operators acting on divergences yields systematic self-divergence steps along path-connected distributions (Nishiyama, 2019). For a convex divergence $D(\cdot\|\cdot)$ and $R(t)$ as a convex interpolation between two distributions, define:

• Integral operator: $Y[D](P\|R(t)) = (1/t)\int_0^t [-D(P\|R(s))]ds$
• Sequence construction: $Y^{k+1}[D] = Y[Y^k[D]]$, $Y^0[D]=D$

Monotonicity result:

$$D(P\|R(t)) \geq Y[D](P\|R(t)) \geq Y^2[D](P\|R(t)) \geq \dots \geq Y^k[D](P\|R(t)) \geq 0,\ \text{for }t\in[0,1]$$

Such self-divergence steps allow one to define a monotonically decreasing sequence of divergences, providing lower bounds to more complex divergences (e.g., KL or reverse KL), and facilitate the analysis and regularization of unsupervised or semi-supervised learning dynamics (Nishiyama, 2019).

This framework also encompasses polylogarithmic sequences of divergences, explicitly embedding measures like $\chi^2$-divergence (PL$_0$), KL-divergence (PL$_1$), Jeffreys divergence (SL$_0$), and reverse KL (SL$_1$).

3. Self-Divergence Steps in Optimization and Learning Algorithms

Self-divergence measures fundamentally structure or regularize iterative updates in learning algorithms, often leading to improved stability, diversity, or robustness.

• Iterative Self-Improvement (ISI):

In LLM training, ISI frameworks (e.g., DIVE) alternate between model generations and selection steps, but suffer diversity collapse—where self-training on generated data leads to narrowed solution space (Qin et al., 1 Jan 2025). DIVE integrates Sample Pool Expansion (accumulating responses across all past iterations) and Data Selection (greedy selection for maximal diversity, after outlier filtering) as explicit self-divergence steps, leading to substantial gains (10–45% relative in diversity metrics) without lowering solution quality.

Data selection algorithms compute diversity increase over sets in embedding or output space, and preference learning (with DPO loss) integrates these diverse examples, directly guiding model self-evolution toward more robust behavior.

• Variational Inference & Contrastive Divergence:

Variational Contrastive Divergence (VCD) (Ruiz et al., 2019) computes the difference between the initial variational distribution $q_\theta(z)$ and its improved version after $t$ MCMC steps, $q_\theta^{(t)}(z)$:

$$L_{VCD}(\theta) = KL(q_\theta(z)\|p(z|x)) - KL(q_\theta^{(t)}(z)\|p(z|x)) + KL(q_\theta^{(t)}(z)\|q_\theta(z))$$

Rearrangement yields $$L_{VCD}(\theta) = -\mathbb{E}_{q_\theta}[f_\theta(z)] + \mathbb{E}_{q_\theta^{(t)}}[f_\theta(z)]$$, with $f_\theta(z) = \log p(x,z) - \log q_\theta(z)$.

The VCD is a canonical self-divergence step, integrating "feedback" from updated samples to tune the variational family away from confirmation traps. As $t\to\infty$, $L_{VCD}$ converges to the symmetrized KL divergence.

• Spread Divergence:

When model and data distributions have mismatched supports, the spread divergence smooths both using a convolution kernel prior to divergence calculation (Zhang et al., 2018). Self-divergence steps are realized by "self-normalizing" both outputs (via spreading), ensuring the divergence remains a faithful discrepancy measure and enabling learning in implicit generative models.

• ADAM and Non-convergence under Self-Divergence:

Self-divergence steps can also exhibit pathological behavior: for ADAM with fixed stepsize, explicit constructions show perpetual nonzero divergence steps yielding runaway iterates, independent of algorithmic parameters, for certain smooth univariate functions (Toint, 2023). This highlights limitations in update mechanisms not designed to decay or regulate their increments.

4. Canonical Decomposition and Information Geometry

Information geometry provides deeper structural insights into self-divergence via geometric and canonical decompositions (Felice et al., 2018, Nishiyama, 2018):

• Pseudo-squared distance:

Defined via dual exponential maps:

$$r(p,q) = \langle \exp_p^{-1}(q), (\ast\exp_p)^{-1}(q) \rangle_p$$

From this, tangent fields along geodesics $\Pi_t(p)$ and $\Pi_t^*(p)$ are defined, leading to the potential property:

$\nabla_q r(p,q) = \Pi_q(p) + \Pi_q^*(p)$

Integration along these fields reconstructs the canonical divergence and its dual.

• Self-divergence as Geodesic Step:

Decomposition yields

$$\Pi_q(p) = \nabla_q D(p,\cdot) + X_q, \quad \langle X_q, \dot{\sigma}(1)\rangle = 0$$

The gradient $\nabla_q D(p,\cdot)$ traces the self-divergence step along the geodesic, while $X_q$ is orthogonal. In dually flat or self-dual cases, this reduces to classical Bregman or Riemannian divergence measures. This unifies information-geometric, potential-theoretic, and algorithmic perspectives on self-divergence steps.

• Divergence Sum Decomposition:

Symmetric Bregman divergence admits the decomposition:

$$\sum_v B_{F,\mathrm{sym}}(p_v,c) = J_{F,a}(p) + J_{F^*,a}(p^*) + B_F(c,c)$$

separating distinct sources of divergence—Jensen, conjugate Jensen, and residual Bregman—suggestive for controlling self-divergence in iterative or self-referential estimation procedures (Nishiyama, 2018).

5. Applications to Semi-Supervised, Robust, and Federated Learning

Self-divergence steps have broad applications and regularization roles in both classical statistical learning and distributed systems:

• Semi-Supervised Learning via Divergence-Based Empirical Risk:

Reformulating the empirical risk in self-training (including pseudo-labeling and entropy minimization) as divergence minimization between the (possibly noise-corrupted) pseudo-label distribution and the model predictions, self-divergence steps enable boundedness and robustness to label noise (Aminian et al., 1 May 2024). Using f-divergences and $\alpha$-Rényi divergences with properties (e.g., metric or bounded divergences), one derives risks less susceptible to confirmation bias and more robust to pseudo-label noise than conventional cross-entropy, especially for Jensen–Shannon or other bounded divergences.

The divergence-based risk is:

$$\hat{R}_D(\theta,Z) = D(\hat{P}(Y|X)\,\|\,P_\theta(Y|X))$$

(with $\hat{P}$ a convex combination of empirical labels and pseudo-labels). Regularization terms include $D$-entropy and divergence to the uniform distribution over predictions, discouraging degenerate output distributions.

• Federated and Self-Supervised Learning:
  • Layer-wise Divergence-Aware Weight Aggregation (L-DAWA):
  • Aggregates layer weights by angular divergence with the global model to mitigate client bias, acting as a decentralized self-divergence regulator (Rehman et al., 2023).
  • Federated EMA with Divergence-Adaptivity (FedEMA):
  • Dynamically adapts exponential moving average rates via $\|W_g-W_k\|$-based divergence, preserving local knowledge under high data heterogeneity (Zhuang et al., 2022).

Both examples use explicit model-to-self or model-to-global divergence assessments to guide update steps and prevent model collapse.

6. Self-Divergence in Physical Models and Field Theory

In quantum field and many-body statistical physics, self-divergence elimination is critical for mathematically consistent thermodynamics:

• Self-Consistent Hartree-Fock-Bogoliubov Theory (SSED):

When modeling interacting Bose gases, the naive self-consistency equation for the anomalous average (gap parameter) includes divergent terms for contact interactions. The divergence is "self-eliminated" by recognizing that in the limit of large cutoff, the gap parameter must vanish:

$$\Delta^{(\text{SSED})} = \lim_{I\to\infty} \frac{n_0 U}{1 + UI} = 0$$

The theory thus replaces ad hoc regularization by a self-divergence step leading to a unique, finite solution—affecting the chemical potential, excitation spectrum, thermodynamic stability, and phase transition phenomenology (Bulakhov et al., 16 Jun 2025).

The resulting physical implications include a first-order phase transition, nontrivial dependence of condensate density on temperature, and a demonstration that a pure pair condensate is thermodynamically unstable (negative compressibility), in contrast with the Popov approximation.

7. Impact and Theoretical Significance

Self-divergence steps, algorithms, and operator-induced sequences are a unifying theme in modern statistical learning, information theory, optimization, and physics. They provide:

• Rigorous mechanisms for structuring iterative improvements and regularizing model evolution.
• Monotonicity and boundedness properties for risk control and robustness.
• Decomposition frameworks to disentangle sources of divergence and error.
• Paths for bridging geometry, probabilistic inference, and statistical physics via canonical transformations and operator calculus.

Critical open problems include characterizing the trade-offs between diversity and performance under explicit self-divergence constraints in large models (as in ISI/DIVE), establishing conditions under which monotonic divergence sequences yield optimal statistical or optimization guarantees, and extending divergence-aware regularizations to new architectures and scientific models.