Optimization of Bregman Variational Learning Dynamics

TL;DR

This work introduces Bregman--Variational Learning Dynamics (BVLD), a time-varying operator framework that unifies Bayesian updates, mirror descent, and proximal methods under nonstationary environments via $T_t(p)=\arg\min_q\{f_t(q)+D_\psi(q\|p)\}$. It establishes that BVLD operators are $\kappa$-averaged with $\kappa=\mu/(\mu+L)$ in the $D_\psi$ geometry, yielding contraction, Fejér monotonicity, and exponential convergence, and it derives a continuous-time evolution variational inequality (EVI) showing exponential decay of a Bregman energy up to a drift term. The framework is extended to distributionally robust optimization and Pareto/multiobjective variants, with preserved stability and well-posedness, and is complemented by exact, inexact, and inner-quasi-Newton implementations. Empirical analyses on synthetic and real data demonstrate the predicted contraction and drift-resilience, highlighting BVLD's potential for robust adaptive learning, digital twins, and control in nonstationary settings. Overall, BVLD provides a rigorous operator-level foundation for stable, geometry-aware learning under drift, bridging convex analysis, dynamical systems, and robust optimization.

Abstract

We develop a general optimization-theoretic framework for Bregman-Variational Learning Dynamics (BVLD), a new class of operator-based updates that unify Bayesian inference, mirror descent, and proximal learning under time-varying environments. Each update is formulated as a variational optimization problem combining a smooth convex loss f_t with a Bregman divergence D_psi. We prove that the induced operator is averaged, contractive, and exponentially stable in the Bregman geometry. Further, we establish Fejer monotonicity, drift-aware convergence, and continuous-time equivalence via an evolution variational inequality (EVI). Together, these results provide a rigorous analytical foundation for well-posed and stability-guaranteed operator dynamics in nonstationary optimization.

Figures (4)

• Figure 1: Empirical geometry of $\kappa$–stability under synthetic BVLD. This figure is generated purely from synthetic simulations to illustrate the theoretical stability geometry of BVLD. Left: calibration reliability showing near-ideal confidence–accuracy alignment. Right: stability map illustrating normalized regret regimes $R_T/T$ as a function of environmental noise $\sigma_{\mathrm{env}}$ and observational noise $\sigma_{\mathrm{obs}}$. Stable dynamics ($R_T/T<0.05$) correspond to strongly averaged mappings (green); the transition zone ($0.05\le R_T/T\le0.15$) denotes weakly contractive but monotone behavior (lavender); beyond this threshold, instability emerges (coral).
• Figure 2: Empirical BVLD convergence and hybrid stability geometry under real and synthetic data. (a) Convergence trajectories of BVLD--Exact, BVLD--Inexact, and BVLD--QN illustrate progressively faster convergence and lower steady--state error, confirming theoretical $\kappa$--stability. (b) Intel field ($\sigma_{\mathrm{env}}=0.755$) visualizes long--horizon environmental stability with low--frequency envelope patterns. (c) Hybrid mirror field integrates both environmental and process variations, showing oscillatory adaptation consistent with mirror--space dynamics. (d) SECOM field ($\sigma_{\mathrm{obs}}=0.335$) captures periodic contraction bands reflecting process noise and equipment fluctuations. Together, these panels demonstrate that BVLD maintains its curvature--aligned stability structure even under heterogeneous stochastic conditions.
• Figure 3: Visualization of robust envelope formation, Pareto frontier evolution, and bilevel coupling dynamics under the BVLD framework. Panel (a) illustrates the robust envelope $E_\rho(p)$ under increasing robustness parameter $\rho$, showing smooth inflation of level sets. Panel (b) shows the emergent Pareto front with random perturbations and the ideal frontier (black dashed), confirming convex dominance behavior. Panel (c)–(f) depict the contraction relation, cumulative drift bound, and Lyapunov orbit corresponding to Theorems \ref{['thm:drift']}--\ref{['thm:evi']}. These results verify the theoretical guarantees of contraction, drift resilience, and exponential stability in the BVLD system.
• Figure 4: Empirical real–hybrid extensions of the BVLD framework. (a) Robust envelope $E_\rho(p)$ derived from real Intel sensor data, illustrating curvature regularization under increasing ambiguity $\rho$. (b) Pareto front between environmental variability (Intel) and observation reliability (SECOM), with color-coded normalized time and ideal frontier (black dashed). (c) Bilevel coupling dynamics between $\sigma_{\mathrm{env}}(t)$ and $\sigma_{\mathrm{obs}}(t)$, showing convergence toward an empirical equilibrium (red marker). These hybrid experiments confirm that BVLD preserves its theoretical stability guarantees under nonstationary and real-world noise conditions, as verified over 500-step hybrid simulations.

Theorems & Definitions (41)

• Lemma 2.1: Co-coercivity (Baillon--Haddad) Baillon1977Bauschke2011
• proof
• Proposition 3.1: KKT $\Longleftrightarrow$ Fixed Point of $T_t$
• proof
• Theorem 3.2: Averagedness and Firm Nonexpansiveness
• proof
• Proposition 3.3: Smoothness and contraction of the Bregman--Moreau envelope
• proof
• Definition 3.4: PL and Quadratic Growth (QG)
• Proposition 3.5: PL $\Rightarrow$ QG and Linear Rate