Two-Phase Learning Dynamic in Agents

• Two-phase learning dynamic is defined as a phenomenon where adaptive agents split into ordered and disordered regimes driven by majority and performance update rules.
• The model employs local update rules with tunable memory, creating parallel, disordered, and anti-parallel phases with clearly defined critical transition points.
• It aligns with voter-model universality, demonstrating that memory and update scheduling critically shape the phase transitions and coarsening dynamics.

A two-phase learning dynamic refers to the empirically and theoretically observed phenomenon that the training of certain learning systems—including competitive learning models, deep neural networks, and game-theoretic agent-based models—naturally divides into two distinct regimes or "phases" distinguished by characteristic structural and dynamical properties. In the context of competitive learning modeled as coupled agent systems, the two phases often manifest as ordered and disordered regions in parameter space and are associated with transitions governed by update rules, memory effects, and statistical mechanics universality classes (Bhat et al., 2011).

1. Model Description and Local Update Rules

The model studied consists of agents on a lattice, each choosing between two strategies denoted "+" and "–". Agents iteratively update their strategy based on two rules:

• Majority Rule: Adaptation according to the strategy adopted by neighbors—a local consensus mechanism.
• Performance Rule: Switching based on the historical success rates (pay-offs) of the strategies among neighbors, with the influence of past outcomes modulated by memory parameters (ε₊, ε₋).

At each update, the local state of an agent is determined by stochastically applying the two rules, with individual update probabilities controlled by parameters p₊, p₋ for strategy success rates, and memory "sharpness" via ε.

2. Phase Diagram and Two-Phase Regimes

The phase diagram in the $(p, \varepsilon)$ plane displays three regions:

Phase Type	Description	Order Parameter/Signature
Parallel Frozen Phase (PFP)	Ordered: All agents aligned (parallel order).	Low energy $E$ (aligned spins)
Disordered Phase	Paramagnetic: No long-range order, high entropy.	Intermediate energy $E$, mixed configuration
Anti-parallel Frozen Phase (AFP)	Ordered: Alternating arrangement of strategies (anti-parallel spins).	High energy $E$ (alternating spins)

• For sequential updates (ss) and memoryless agents ($\varepsilon = 1$), there exists a disordered phase between two critical points $p_{c_1}\simeq 0.56$ and $p_{c_2}\simeq 0.70$.
• For parallel or mixed update schemes (pp, ps), a symmetric phase diagram (about $p=0.5$) emerges, with critical points shifted and ordered phases—PFP (parallel order, low $p$) and AFP (anti-parallel, high $p$)—bracketing the disordered phase.

The boundary between the phases is determined by the system's energy, and the presence of two ordered phases separated by a disordered (“paramagnetic”) phase constitutes the two-phase regime.

3. Memory Effects and Compensatory Dynamics

Memory, parameterized by $\varepsilon_+$ and $\varepsilon_-$, affects the agent's responsiveness to outcome-based updates:

• Short memory ($\varepsilon\to 1$): Agents react quickly to immediate outcomes, promoting rapid adaptation but supporting a broad disordered phase.
• Long memory ($\varepsilon\ll 1$): Agents accumulate longer histories, causing sluggish switching and effectively suppressing the disordered phase.

A critical insight is that an inferior strategy can sometimes outperform a superior one if it is paired with longer memory. That is, high outcome volatility associated with shorter memory may reduce relative performance, while enhanced persistence via long memory stabilizes an otherwise low-performing strategy.

4. Update Paradigms and Their Dynamical Impact

The update sequence has both quantitative and qualitative effects:

Update Type	Majority Rule	Performance Rule	Disordered Phase?	Critical Exponent (log-slope of $1/E$)
ss	Sequential	Sequential	Yes	$2/\pi$ (voter model universality)
pp	Parallel	Parallel	Yes	$\sim 1/(2\pi)$ or $-1/(5\pi)$
ps / sp	Parallel/S.	Sequential/P.	Yes / No	$4/(3\pi)$ or none (sp: rapid consensus)

• In the ss case, energy relaxation obeys $E(t)\approx (\pi/2)/\ln t$, the exact scaling of the voter model.
• In pp/ps, the energy decay remains logarithmic but with different prefactors, still within the generalized voter universality class.
• For sp (sequential for majority, parallel for performance), the majority rule dominates, consensus is driven rapidly, and the disordered phase disappears.

Update-induced differences in coarsening dynamics highlight that the phase transitions and critical behavior depend on the interplay between environmental (majority) and performance-based (outcome) pressures.

5. Linear Response in Asymmetric Regimes

When the two strategies have different (but not drastically distinct) success probabilities ($p_+ = p \pm H/2$), the system exhibits a linear response:

$M = \tanh(bH)$

where $M$ is global magnetization and $b$ is a constant dependent on the phase-center $p_\mathrm{central}$ (the mean of the critical points), $b\propto [p_\mathrm{central}+p]^2$.

This proportionality between strategy bias $H = p_+ - p_-$ and the population-level ordering $M$ demonstrates that near coexistence the system's response to small asymmetries is of linear type, characteristic of classical statistical models.

6. Universality Classes and Critical Dynamics

The observed phase transitions align with the generalized universality class of the voter model for almost all update paradigms (except sp). This universality is evidenced by the logarithmic scaling of the energy coarsening:

$E(t) \sim \frac{\text{const}}{\ln t}$

with the slope determined by the type of update. The underlying mechanism is interfacial noise and the absence of surface tension—core features of the voter model's ordering kinetics—rather than energy minimization via deterministic gradient descent.

In cases with bias or memory-driven asymmetry, the system retains the voter scaling, signifying robustness of the universality class against detailed structural changes.

7. Implications and Broader Significance

The two-phase learning dynamic elucidated in the model has several broader implications:

• The emergence of two ordered regimes separated by a disordered intermediate phase reflects the competition between social conformity (majority rule) and individualistic adaptation (outcome rule).
• Memory parameters afford agents the ability to modulate the collective dynamics; tuning memory transforms the qualitative phase structure and can even render disadvantageous strategies viable.
• The explicit characterization of phase diagrams, critical points, order parameters, and scaling laws can inform the design of distributed, adaptive systems, socioeconomic agent models, and statistical inference networks that operate under competing update pressures.
• The correspondence with the voter model universality highlights that rich, nonequilibrium phase structures can emerge even in these stylized, minimal settings.

In summary, a two-phase learning dynamic in competitive learning models manifests as two ordered phases—parallel and anti-parallel—separated by a disordered phase, shaped by the interplay of local majority and outcome-based updates, the degree of memory in agent adaptation, and the update-scheduling paradigm. The critical exponents and coarsening dynamics situate these transitions firmly within the generalized voter-model universality class, rendering these findings relevant for statistical learning, social dynamics, and theoretical biology (Bhat et al., 2011).