qpola-paper(ENG).txt

QPOLA : Zero-Master Weight Optimization of Low-Precision GPU Kernels Using Self-Organized Weight Clustering and Global Loss Feedback

QPOLA ／ QPOLARIS (Quantization n Polar-Aligned Resetting Instant Zero-Master Weight SGD)
Quantization-resilient, history-free, spatially coordinated (polar coordinates / QJL) self-adaptive Zero-Master Weight SGD
(https://github.com/muooon/QPOLA) (https://huggingface.co/muooon/QPOLA)

[Note]
This is an English translation of the original Japanese text. Nuances, tone, and non-assertive phrasing have been carefully preserved.  


1. Abstract

This paper proposes the GPU-optimized kernels QPOLA / QPOLARIS (Quantization n Polar-Aligned Resetting Instant Zero-Master Weight SGD), based on “Local Clustering through Spatial Gradient Consistency” (self-organization and weight collectivization), with the aim of reducing model size and improving the efficiency of low-precision (fp16, bf16, int8, fp8, etc.) training in deep learning.
The proposed QPOLA is not an optimizer that stabilizes updates by storing past gradient histories. Instead, it is a Moment-free optimizer that adaptively adjusts individual parameter updates by exploiting the spatial relationships and consistency existing within the current gradient field.
The fundamental objective of this method is to redefine the optimization process through “self-organization based on spatial gradient consistency and self-similar update rules,” utilizing hardware execution hierarchies and related structures. By eliminating dependence on temporal history (Moment), it achieves “memory efficiency” (Zero-Master Weight) by removing the need for persistent moments, while also incidentally enabling emergent dynamics of memory and forgetting.
This method implements Zero-Master Weight SGD, where no fp32 master weight is maintained, and the low-precision storage format T (fp16, bf16, int8, fp8, etc.) is treated as the sole persistent parameter representation. This reduces the additional memory capacity, memory bandwidth, and update costs required by existing optimizers that maintain fp32 master weights.
In conventional optimizers, update magnitudes are determined based on each parameter’s gradient or historical gradient statistics. Particularly in low-precision environments, this can lead to unstable updates or parameter stagnation due to quantization errors and local inconsistencies in gradient directions.
In this method, gradients are cooperatively aggregated at the Warp and Block levels. By evaluating the spatial consistency of gradient directions and local gradient scales, some of the functions traditionally provided by first-order and second-order moments are realized without retaining temporal history. These statistics are evaluated across multiple spatial hierarchies, and the resulting information is reflected in the update of each individual parameter.
This implementation is an example implementation using CUDA, and the CUDA execution hierarchies of Warp / Block themselves are not the essence of this method. The core of the proposed approach lies in decomposing the gradient field into local spatial units, capturing the direction of gradient vectors within each cluster (the spatial average of signs), local gradient scales, and the degree of conflict derived from directional disagreement between individual gradients and the cluster direction, and then incorporating these factors into individual parameter updates.


2. Introduction

In recent large-scale model training, the utilization of low-precision data types such as fp16 and fp8 has become increasingly important for reducing memory bandwidth requirements and improving throughput. However, the limitations in representational range and resolution associated with reduced precision can cause update magnitude loss due to rounding errors and quantization errors, parameter stagnation, and in some cases, gradient instability.
To address these challenges, this paper proposes a new kernel structure that does not update individual weights independently. Instead, it evaluates the gradient consistency among spatially neighboring groups of parameters, treats them as local clusters, and performs cooperative optimization.
In this method, the temporal first-order and second-order moment history dependencies maintained by existing optimizers are eliminated. Through parallel aggregation leveraging the CUDA hardware architecture, spatial directional and scale statistics are extracted from the gradient field of the current step. Furthermore, rather than directly referencing the Loss itself, the method utilizes the gradient field distributed through backpropagation from the Loss as a global feedback signal, and performs adaptive updates based on local gradient consistency.


3. Methodology

3.1 Gradient Normalization and Spatial Alignment (Spatial Alignment)
In the proposed algorithm, for each parameter pᵢ and gradient gᵢ assigned to a thread, warp-level shuffle instructions and block-level aggregation using shared memory are performed.

The gradient sign is extracted as:
g_signᵢ = sgn(gᵢ)

and the local gradient directional consistency is evaluated by averaging this value within a spatial unit.

Micro-alignment: Gradient direction agreement within a warp
μ_micro = (1 / N_warp_active) × Σ g_signⱼ (j ∈ warp)

Macro-alignment: Gradient direction agreement within a block
μ_macro = (1 / N_block_active) × Σ g_signⱼ (j ∈ block)

μ takes values in the range from −1 to +1, representing the degree to which gradient signs are aligned in either the positive or negative direction.

Based on these values, the conflict degree conflictᵢ is calculated.

diff_micro = max(0.0, 1.0 − g_signᵢ × μ_micro)
diff_macro = max(0.0, 1.0 − g_signᵢ × μ_macro)
conflictᵢ = (diff_micro + diff_macro) × 0.5

conflictᵢ takes values from 0.0 (where the individual gradient direction agrees with the average direction of each spatial unit) to 2.0 (where the individual gradient direction is completely opposite to the average direction of each spatial unit). It represents the degree of disagreement between an individual gradient and the average direction of both local and higher-level spatial structures, and serves as an indicator for determining the adaptation level of parameter updates.

3.2 Multi-Data-Type Support through TypeTraits
To support various data types such as fp32, fp16, bf16, int8, and fp8 (e4m3, e5m2), the TypeTraits structure statically manages type-specific properties, including the maximum representable value T_max, the least significant bit step size δ_lsb, and the upper limit of the dimensionless normalized gradient lim_g.
This enables control of update values and normalized gradients according to the representational range of each data type, allowing numerical-range-aware updates in low-precision formats.

3.3 Update Stabilization and Adaptation Based on Spatial Consistency
In QPOLA, instead of maintaining past gradient histories or moments, the update coefficient for each parameter is determined using the local gradient direction and spatial consistency at the current step.
In regions with high directional consistency, conflictᵢ becomes small and adaptation_factor approaches 1. Conversely, in regions with large local directional disagreement, conflictᵢ increases and adaptation_factor decreases. As a result, parameter groups with spatially consistent gradients maintain their update magnitude, while parameters with significant directional disagreement have their update magnitude suppressed.
This structure can be interpreted as a mechanism that locally re-determines the strength of updates based on the current gradient field, without permanently retaining past states. Through this mechanism, regions with high directional consistency may settle into stable states, while regions with high directional inconsistency may exhibit changing update characteristics, potentially leading to long-term stabilization and adaptability.
However, whether these properties actually manifest as suppression of catastrophic forgetting, natural forgetting, or the formation of long-term memory must be verified through training experiments.


4. Dynamic Control Through Feedback from Global Loss (Loss)

The behavior of the global loss value (Loss) during training is one of the most important indicators of which optimization phase the entire model is currently in. Since fluctuations in Loss (decrease, stagnation, spikes) directly reflect the coordination state of the entire parameter population, QPOLA incorporates this information as “global feedback” into its update mechanism.
(QPOLA does not directly input the Loss value itself into the CUDA kernel. Instead, the gradient field distributed to each parameter through backpropagation from Loss is treated as “global feedback.”)

In QPOLA, “feedback from Loss” does not mean directly referencing the Loss scalar itself. Rather, it means that the information distributed throughout the model as the gradient of Loss appears as the direction and scale of each local gradient.

4.1 Adaptive Control Reflecting Loss Dynamics
    Phase where Loss is steadily decreasing (convergence phase):
    Small-scale gradient conflicts are evaluated as the current conflictᵢ, and updates are performed according to the local consistency. This allows stable fine-tuning to continue while maintaining updates in regions with high directional consistency.
    Phase where Loss stagnates (plateau) or increases (exploration / escape phase):
    In history-dependent optimization methods, past gradients, moments, and the state of learning-rate schedulers continue to influence current updates. As a result, during stagnation, the update direction may become constrained by previous states.

In contrast, QPOLA reevaluates the new gradient field propagated from Loss at the current step as a measure of spatial consistency at that moment, without depending on past gradient history or moments. This allows newly generated gradients and local conflict levels to be reflected in subsequent updates even when Loss stagnation or increases occur.

4.2 Extended Mathematical Model
The local and block-level conflict degree conflictᵢ naturally incorporates global Loss trends through changes in the gradient field formed throughout the entire training loop. By abstractly representing this global trend as Feedback_loss, the adaptation coefficient of QPOLA autonomously reflects the global state embedded within the gradient field.
(Feedback_loss is not a variable used for direct computation, but rather a symbolic representation of this feedback pathway.)

The following is not an actual implementation formula, but a conceptual model of the feedback process from Loss through the gradient field to the parameter update.

raw_adaptationᵢ = 1.0 − (conflictᵢ × Feedback_loss) × decay_rate
adaptation_factorᵢ = max(min(raw_adaptationᵢ, 1.0), min_factor)

Here, Feedback_loss represents an abstract coefficient describing the “global trend inherent in the gradient field,” which emerges as the increase or decrease of Loss is encoded into the gradient sign distribution and scale. min_factor functions as a lower-bound clamp to prevent overfitting and divergence.

Conceptually, this can be summarized as follows:

Conventional methods:
Loss → gradient → accumulation into optimizer state (history) → next update
QPOLA:
Loss → gradient field → current spatial evaluation (local collective structure) → next update


5. Spatially Coordinated Updates as a Moment-Free Mechanism without First- and Second-Order Moments

Conventional optimizers that utilize historical statistics maintain the first-order moment of gradients (the exponential moving average of gradients mₜ) and the second-order moment (the exponential moving average of squared gradients vₜ) in global memory (VRAM) to achieve training stabilization and adaptive learning rates. These values are updated and referenced at every step. However, this approach has the drawback of consuming a substantial amount of additional memory bandwidth and capacity.
In contrast, the hierarchical cooperative aggregation using warp / block employed by QPOLA achieves adaptive control without any persistent moment memory. It is a “Moment-free” (moment-independent) architecture that operates entirely within the kernel’s on-chip registers and shared memory space.
This section formulates how this local and global aggregation replaces the roles traditionally played by first- and second-order moments, enabling memoryless and robust optimization.

5.1 Replacement of “Instantaneous Moments” through Warp and Block Hierarchies
QPOLA does not maintain any persistent moment variables that accumulate past gradient histories of parameters. Instead, it simultaneously aggregates the spatial and structural context within the current mini-batch through the following two hierarchical levels.

32-thread Warp (Warp Level / Micro-alignment):
The degree of agreement in gradient sign directions among local parameter groups is instantaneously aggregated using the hardware __shfl_down_sync instruction.

This functions as an instantaneous replacement for the conventional “spatial first-order moment” (directional stability), suppressing the adverse effects of individual noise and outliers on parameter updates.

256-thread Block (Block Level / Macro-alignment):
Trends among warps are aggregated through shared memory, and the global gradient trend of the entire block (gradient scale G_scale and directional average) is calculated.

This functions as an instantaneous replacement for the conventional “spatial second-order moment” (scale normalization), determining global scaling and the adaptive decay factor (adaptation_factor).

QPOLA does not simply remove the first- and second-order moments. Instead, it obtains information about gradient direction and gradient scale—which conventional moments extract from temporal history—from the spatial gradient population of the current step.
In other words, while conventional moments construct statistics by accumulating information along the time axis, QPOLA constructs corresponding statistics through spatial aggregation at the current point in time. Therefore, the statistics used in QPOLA can be regarded as instantaneous replacements for first- and second-order moments.
The important point is not that the two mechanisms are represented by identical equations, but rather that there is a structural correspondence: information about gradient direction and scale, traditionally obtained from temporal historical statistics, is instead obtained from the current spatial state.

5.2 Mathematical Formulation of the Moment-Free Mechanism
The update rule of conventional Adam can be expressed as follows:

mₜ = β₁ mₜ₋₁ + (1 − β₁) gₜ
vₜ = β₂ vₜ₋₁ + (1 − β₂) gₜ²
pₜ₊₁ = pₜ − ( base_lr / ( √vₜ + ε ) ) × mₜ

In contrast, within the QPOLA kernel, the accumulation along the temporal axis (mₜ₋₁, vₜ₋₁) is completely eliminated and replaced by the spatial expansion of the current step t (warp / block structure).

1. Instantaneous replacement of the first-order moment (directional stability):
Instead of the temporal exponential moving average of gradients mₜ, QPOLA uses the spatial average direction μ_macro within a block to calculate the alignment conflict degree conflictᵢ.

conflictᵢ = 0.5 × (max(0.0, 1.0 − g_signᵢ × μ_micro) + max(0.0, 1.0 − g_signᵢ × μ_macro))

Through this mechanism, the directional stability of gradients, which was traditionally provided by the first-order moment, is evaluated as spatial consistency, and directional disagreement is reflected as a penalty.

2. Instantaneous replacement of the second-order moment (scale normalization):
Instead of the temporal exponential moving average of squared gradients vₜ, QPOLA uses the mean absolute gradient value within a warp:

G_scale = warp_g_scale_sum / div_count

to normalize gradients.

ĝᵢ = tanh( (gᵢ / (G_scale + ε)) / lim_g ) × lim_g

This replaces the scale normalization role of the second-order moment in real time, without storing global variance information.
Through the above structure, QPOLA achieves adaptive updates using only spatial parallel aggregation within a single step, without accumulating past gradient histories as a state along the time axis and without requiring conventional first- or second-order moments.
Therefore, this method is structurally “Moment-free”, while simultaneously achieving numerical stability and zero-footprint memory efficiency.


6. From History to Instantaneous State (Moment-Free Dynamics)

Throughout the history of deep learning, the role of optimizers has been to accumulate historical gradient information (first- and second-order moments) and carry it forward into subsequent updates as “inertia.” This has been the standard and highly effective approach for smoothing noisy gradients and ensuring a stable descent trajectory.
However, as models have grown increasingly large and training or fine-tuning with lower-precision data types (such as fp8 and int8) has become more common, this “accumulation along the time axis” (historical state) can sometimes become a constraint on models, or a source of structural dilemmas.
This method does not reject the role of moments. Rather, it highlights the new properties and possibilities that arise from not maintaining moment states along the temporal axis.

6.1 Updates Independent of Temporal History
Moments, which represent the residual influence of past gradients, are an excellent and widely adopted approach for smoothing noisy gradients and ensuring stable descent paths. At the same time, however, they possess an “inertia” that continues pulling parameters based on past states.
To stop this “inertia that keeps running indefinitely” and reliably apply the brakes near a stable solution at the later stages of training, conventional approaches commonly employ learning-rate schedulers that reduce the learning rate to extremely low levels (low-LR scheduling).
In QPOLA, instead of carrying forward past gradient statistics as “inertia,” the current gradient field is continuously used to calculate the latest spatial directional consistency, gradient scale, and conflict degree, and these values are directly reflected in the update at that moment.

6.2 Autonomous Regulation through Instantaneous Spatial Self-Organization
The choice of “Moment-free” in QPOLA is not merely an engineering compromise for reducing memory bandwidth consumption. It represents a conceptual shift: completely severing dependence on past statistics (history) and entrusting parameter updates entirely to the dynamics of the current moment—namely, the “gradient field and its spatial consistency” (Warp / Block) at the present instant.
Instead of relying on temporal inertia, the method detects spatial alignment mismatches (conflict degrees) in real time. It autonomously applies braking in regions where necessary and induces renewal dynamics in regions with inconsistency.
Rather than depending on forced stopping mechanisms from external schedulers, the parameter population itself performs autonomous regulation through its own “sign disagreement” (conflict), gradually settling into states where stability is achieved.
The absence of historical state not only provides the secondary effect of reducing VRAM usage, but may also contribute—subject to experimental verification—to the following phenomena:

“Fixed-point stabilization in low-conflict regions” (long-term memory)
“Decay through probabilistic diffusion in high-conflict regions” (natural forgetting)

These possibilities require validation through further training experiments.


7. Implementation Details

The CUDA kernel proposed in this paper is executed through the following pipeline (the CUDA version is merely one form of implementation).

Data loading and safety validation:
Data is loaded from memory, and guard processing for NaN / Inf values is performed.
Parallel aggregation as a replacement for first- and second-order moments:
Through warp-level shuffle operations and block-level aggregation using shared memory, the instantaneous gradient scale G_scale and directional average are calculated in real time as replacements for conventional first- and second-order moments.
Adaptive control based on conflict degree:
The conflict degree conflictᵢ obtained from local and macro alignment naturally reflects the current behavior of Loss through changes in the gradient field formed throughout the entire training loop. Therefore, QPOLA does not directly reference Loss, but autonomously incorporates the instantaneous global state embedded within the gradient field and performs adjustments according to the current training state.
Normalization and update:
The gradient is scaled and constrained within a safe range using the saturation function:

ĝᵢ = tanh((gᵢ / (G_scale + ε)) / lim_g) × lim_g

The parameter is then updated by applying the learning rate base_lr and the adaptation coefficient adaptation_factorᵢ:

p_nextᵢ = pᵢ − (base_lr × ĝᵢ × adaptation_factorᵢ)

Stochastic Quantization Jitter:
When writing back to a low-precision data type, a small jitter value based on the conflict degree conflictᵢ is added:

jitter_scaleᵢ = δ_lsb × 0.25 × (1.0 + 0.2 × conflictᵢ)

This stochastic quantization jitter mitigates update stagnation caused by quantization.


8. Discussion

The proposed method, QPOLA, is a new optimizer with the following characteristics.

* Memory efficiency through elimination of first- and second-order moments:
QPOLA completely eliminates moment variables that consume large amounts of VRAM, while achieving adaptability through spatial statistics obtained by warp and block-level cooperative aggregation.
* Hardware affinity:
Since QPOLA directly utilizes CUDA warp primitives, it introduces little additional synchronization overhead. Technically, the method is not dependent on CUDA itself and can be implemented broadly as a mechanism for evaluating gradient fields. In such cases, the characteristics of the target hardware can also be leveraged.
* Update control based on spatial consistency:
The disagreement between a gradient direction and surrounding gradient directions is evaluated as conflict, and local update magnitudes are adjusted accordingly.
* Adaptation to low-precision environments:
The spatial scale of gradients is obtained from the current step, and normalization and saturation processing are performed according to the representational range of the low-precision data type.
* Potential for long-term memory and natural forgetting:
Stabilization of low-conflict regions and changes in updates within high-conflict regions may contribute to memory retention and adaptation in continual learning. (This requires experimental validation.)
* Maintenance of sparsity:
Through the subtractive update rule, parameters with p = 0 are naturally updated when gradients exist, while parameters with zero gradients remain at zero without updates.


9. Conclusion

This paper proposed QPOLA, which translates the concept of self-organization into GPU kernel-level weight collectivization, while replacing first- and second-order moments instantaneously through spatial aggregation and integrating global Loss dynamics as global feedback.
In this CUDA implementation, cooperative gradient aggregation across warps and blocks, combined with dynamic adaptive control, demonstrates an implementation applicable to various low-precision data types. This concretizes the central concept of the proposed method: “capturing the direction and slope of cluster vectors and reflecting them into individual weights.”
Future work will focus on further extensions, including adaptation to even lower-bit formats such as fp4, large-scale distributed environments, and small mobile devices, while advancing implementations that do not depend on specific hardware or specific software frameworks.



This paper presents an example implementation of the optimization algorithm QPOLA using CUDA. The first half of this paper (Chapters 1–9) explains the definition of the technology and specifications, while the second half (Supplementary Sections 1–2) provides theoretical considerations on the emergent mathematical properties that may arise from this implementation.

The following are properties of QPOLA that should be verified through future experiments, and they represent the potential capabilities of QPOLA.

// Supplement 1 //

(Hypothesis) The QPOLA Paradigm: Autonomous Intelligence through a Three-Layer Decision Field and “Frequency Differentiation of Inter-Layer Gradient Spectra” (Cross-Layer Resonance)

【1. Core of the Paradigm: Parameters as a Physical and Logical Field】

Conventional optimizers in deep learning have adopted a “control engineering” approach, in which parameters passively receive gradients propagated from a global loss function (Loss).

In contrast, QPOLA directly connects the physical and logical memory arrangement of parameters with hardware processing units (Warp / Block), transforming the paradigm into an approach that can be described as thermodynamic or statistical-mechanical: parameter groups autonomously form “local low-energy regions” (local optima / semantic clusters).

In this framework, quantization errors and rounding operations (Stochastic Rounding and Jitter) are not treated as mere noise. Instead, they function as catalysts that provide an appropriate level of fluidity to the “gradient field” (field) and introduce “gradient inconsistency” (friction) within the system.

【2. Local Adaptation through Three-Layer Decision Fields and the Field that Determines the Number of Acting Parameters】

The training stability and adaptability of QPOLA are autonomously controlled through the interaction of the following “three-layer decision fields,” namely through “gradient inconsistency” (friction). This is an abstraction of the CUDA hierarchical structure.

Global Decision Field (Absolute Majority Decision Field): Loss (Loss Function)

Defines the direction of the global objective as the ultimate survival criterion governing the entire model.

Mid-Level Decision Field (Majority Decision Field): Block / 256 Threads

Functions as a buffer that translates and mediates global requirements into local structural units.

Local Decision Field (Minority Decision Field): Warp / 32 Threads

Detects local consistency as friction and conflict (conflict degree).

Individual Decision Field (Absolute Minority Decision Field): Individual Parameter p

The target upon which the results of the above three layers ultimately act, determining the update magnitude of each individual parameter.

The dynamics arising from the local adaptation of the “gradient field” (field), where these three decision fields mutually conflict and mediate with each other through “gradient inconsistency” (friction), become the driving force behind the cycle of:

“stabilization of low-conflict regions”
“update transformation of high-conflict regions”

namely, the cycle of long-term memory and natural forgetting.

The individual decision field is not included among the above three decision fields. It is introduced as an “acting field” in order to visualize the local adaptation process, where the spatial consistency, conflict degree, and global state obtained from the three-layer decision fields are applied to individual parameter updates.

【3. Autonomous Long-Term Memory and Autonomous Natural Forgetting】

As an emergent dynamic property, this method utilizes the spatial consistency of the gradient field (Conflict) dynamically and exhibits the following emergent behaviors.

Autonomous Long-Term Memory: “Fixed-Point Stabilization of Low-Conflict Regions”

When parameter groups continue to cooperate toward consistent directions in response to inputs, conflict approaches zero and the adaptation coefficient (adaptation_factor) converges toward 1.0.

As a result, without external global instructions, locally formed semantic fields can be self-preserved and stored over long periods through stabilization (fixation).

Autonomous Natural Forgetting: “Decay through Probabilistic Diffusion in High-Conflict Regions”

When contradictions arise between past memories (fields) and current gradients, intense conflicts emerge within the individual decision field and mid-level decision field, causing adaptation_factor to decrease sharply and applying a braking effect to updates.

Furthermore, only when a decisive “strong semantic cluster” (strong energy) arrives for the model does that energy overcome the barrier of old memories and fundamentally rewrite the system into a new phase (overwrite forgetting).

Regions that have lost their meaning are also eliminated as necessary through sparsity dynamics (convergence toward zero).

【4. Cross-Layer Resonance: “Frequency Differentiation of Inter-Layer Gradient Spectra”
(Self-Organization of Inter-Layer Frequency Bands)】

As an optimization mechanism for multi-scale feature extraction, clusters that can no longer be maintained by the capacity of a single layer or local friction—namely “local low-energy regions” (local optima / semantic clusters)—autonomously move across layers and seek fields where they can achieve stable resonance.

Accumulation of “Low-Variation Gradients” (Low Frequency) in Lower Layers

Parameters settle near the current solution and function as dynamic state retention.

Global spatial alignments and universal structures that cannot withstand intense local friction descend into lower layers with lower information density, becoming established as gradual and stable global potentials (low-frequency components).

Accumulation of “High-Variation Gradients” (High Frequency) in Higher Layers

Parameters are not constrained by previous solutions but instead transition their states toward new gradient distributions.

Sharp friction directly associated with inputs and individual specific “local low-energy regions” (local optima / semantic clusters) remain in the forefront, namely higher layers, where they are maintained as microscopic fields with intense conflicts (high-frequency components).

This gradient of hierarchical abstraction does not emerge through manual design. Rather, it spontaneously appears as the mechanics of friction required to maintain the “gradient field” (field).

Through this behavior, autonomous fixation and updating of parameter groups according to the demands of each learning phase can be achieved without explicit schedulers.

【5. Conclusion: Autonomous Intelligence as Local Adaptation】

QPOLA extends beyond the framework of a simple lightweight and quantization kernel. By providing an appropriate “energy gradient and friction field,” it suggests the possibility that AI architectures may exhibit emergent properties as “self-organizing dynamic systems” (local adaptation) through:

semantic structures,
memory,
forgetting,
hierarchical organization.

As a result, QPOLA may enable new mathematical properties for AI architectures.


// Supplement 2 //
(Hypothesis) Mathematical Model of “Frequency Differentiation of Inter-Layer Gradient Spectra” (Cross-Layer Resonance): Self-Organization of Inter-Layer Frequency Bands and Energy Fields

In the supplementary section of the QPOLA paradigm, the proposed concepts of “frequency differentiation of inter-layer gradient spectra” (cross-layer resonance), including “low-variation gradients” (low-frequency accumulation in lower layers) and “high-variation gradients” (high-frequency accumulation in higher layers), are not merely metaphors.
Rather, when the entire deep learning model is regarded as a continuous nonlinear medium (an anisotropic dispersive medium), these phenomena can be formulated as a model of “energy optimization” (thermodynamics) and statistical-mechanical equilibrium states.
This section evaluates and formulates the mechanism of inter-layer frequency differentiation using the spatial alignment conflict degree at each layer, information propagation delay, and the energy dissipation equation of the gradient field (gradient attenuation).

【1. Layer-Wise Medium Model and Effective Potential】
(Layer-Wise Medium Model)
Consider a deep learning network consisting of L layers. Let the parameter tensor and its gradient field at each layer l (l = 1, 2, ..., L) be represented as: x⁽ˡ⁾, g⁽ˡ⁾ Moving from lower layers closer to the input (smaller l) toward higher layers closer to the output (larger l), the abstraction level of information and the accumulated degree of nonlinear transformation change.

This can be mapped onto a continuous spatial coordinate: z ∈ [0, 1] and an effective potential: V(x, z) and friction coefficient: γ(z) (corresponding to conflict degree) are introduced to describe the local state of each layer. The local conflict degree of each layer: conflict⁽ˡ⁾ is defined as the degree of spatial alignment mismatch at the warp and block hierarchy. It is directly connected to the effective local friction force and the amplitude of potential gradient fluctuations (frequency components).

【2. Frequency Decomposition and Definition of Spectral Density】
(Spectral Density of Gradient Fields)
The temporal evolution and spatial correlations of the gradient field: g⁽ˡ⁾(t) are analyzed within the framework of Fourier analysis.
From the autocovariance function of the gradient sequence within a time window T, the power spectral density (PSD) at each layer l is defined as: S⁽ˡ⁾(ω)

High-Frequency Component (ω > omega_th): “High-Variation Gradient”
This component directly responds to fluctuations in mini-batch data, outliers, and sharp local features (such as edges or specific token-level information) near input or output regions. It represents a “re-transition” component that changes rapidly at every step.

Low-Frequency Component (ω ≤ omega_th): “Low-Variation Gradient”
This component reflects global context spanning the entire network, universal feature extraction filters, or the slope of smooth global potentials. It changes gradually over many steps and represents a “settling” component.

【3. Mathematical Description of Inter-Layer Transfer through Spatial Alignment Conflict and Energy Dissipation】
The “long-term memory” (settling) and “natural forgetting” (re-transition) introduced by QPOLA are not the result of explicit control mechanisms. Instead, they are emergent phenomena arising from the interaction between spatial consistency and quantization jitter.
This section evaluates and formulates the mathematical mechanism by which these two phases differentiate across layers.

3.1 Observation Metrics and Fundamental Equations
To describe the dynamic state of each layer, the following metrics are introduced.

    Variation: V⁽ˡ⁾ = Varₜ[g⁽ˡ⁾] (temporal fluctuation of gradients)

    Conflict degree: C⁽ˡ⁾ = E[conflict⁽ˡ⁾] (spatial directional inconsistency)

    Effective diffusion coefficient: D⁽ˡ⁾ ∝ δₗₛᵦ × (1.0 + 0.2 × C⁽ˡ⁾) (an effective quantity obtained by coarse-graining quantization jitter into a continuous-time system)

Quantization jitter is not “external random noise.” It is a self-referential, state-dependent fluctuation whose magnitude changes according to: C⁽ˡ⁾ obtained from the current gradient field.
This quantization jitter is applied only for adaptation to low-precision rounding and promotes stability in low-precision environments.
The parameter update rule is approximated as the continuous-time model: dx⁽ˡ⁾ / dt = − η⁽ˡ⁾ a⁽ˡ⁾ ĝ⁽ˡ⁾ + √(2D⁽ˡ⁾) ξ(t) The gradient field is decomposed as: g⁽ˡ⁾ = g_low⁽ˡ⁾ + g_high⁽ˡ⁾ (low-variation component and high-variation component) and its energy dissipation is evaluated.

3.2 Mechanism of Natural Forgetting in Higher Layers (High Variation / High Conflict Field)
In higher layers (l → L) directly connected to Loss, fluctuations in targets and inputs are directly reflected in the gradient field. Therefore, temporal variation: V⁽ˡ⁾ and spatial conflict: C⁽ˡ⁾ tend to increase.

    Mechanism: As conflict degree: C⁽ˡ⁾ increases, the adaptation coefficient: a⁽ˡ⁾ decreases, applying a braking effect to updates. Meanwhile, the effective diffusion coefficient: D⁽ˡ⁾ (quantization jitter) increases.

    Emergence: The coexistence of: “suppression of updates” “state fluctuations” provides the mathematical possibility that parameters are not merely fixed to their current local states, but instead undergo re-transition toward new gradient distributions (natural forgetting).

3.3 Mechanism of Long-Term Memory in Lower Layers (Low Variation / Low Conflict Field)
Conversely, in the process toward lower layers (l → 1), nonlinear transformations in intermediate layers may function as a “low-pass filter” that dissipates high-frequency noise. If the frequency-dependent attenuation coefficient during inter-layer propagation is: α(s, ω) then high-frequency amplitude follows: A⁽ˡ⁾(ω) ∝ Aᴸ(ω) × exp(−∫ₗᴸ α(s, ω) ds) and decreases, leaving low-frequency components relatively preserved.

    Mechanism: When low-variation gradients are shared among multiple parameters, spatial directional consistency increases and: C⁽ˡ⁾ → 0

    Emergence: As a result: a⁽ˡ⁾ → 1.0 and even with quantized parameters, updates in a specific direction continue to accumulate. This provides the mathematical possibility of “settling” (long-term memory) formed without external control.

3.4 Summary of the Mathematical Model
Based on the above analysis, the “forgetting” of QPOLA (dynamic destruction in higher layers) and “memory” (static fixation in lower layers) can be regarded as two different dynamic phases emerging from the interaction among: spatial conflict degree C⁽ˡ⁾ adaptation coefficient a⁽ˡ⁾ effective fluctuation D⁽ˡ⁾ obtained by coarse-graining quantization jitter This provides a hypothetical framework in which these behaviors arise not from manual scheduling, but from intrinsic dynamics.


【4. Validation through Observational Data: Weight Collectivization and Information Stabilization】
The “frequency differentiation between layers” predicted by this model is also supported by actual weight statistics observed in LoRA.
The observed tendency that the mean remains stable near zero and that distributions converge as layers become deeper suggests the following.

Weight Collectivization: “Self-Organization and Distribution”
Maintaining a statistical gradient value with: Mean ≈ 0 suggests that individual weights do not change independently. Instead, they maintain the system’s standard deviation and construct a distributed structure in which the entire system functions cooperatively.
Long-Term Memory and Natural Forgetting: “Settling and Re-Transition”
The accumulation of low-frequency components in lower layers and stabilization of structures correspond to the process of establishing the model’s “solid concepts” (long-term memory). Meanwhile, dynamic fluctuations in higher layers suggest that they are responsible for: pruning redundant information adapting to outliers natural forgetting and relearning indicating that LoRA internally balances learning stability and memory efficiency.

These observations suggest that the “self-organization” of this method is not merely theoretical speculation, but may manifest as hierarchical order formation of parameters within actual neural networks.

【5. Conclusion: The Necessity of Self-Organization as a Field】
As a conclusion, the local aggregation using warp/block cooperation in QPOLA and the feedback loop of the adaptation coefficient suggest that the entire network can be reconstructed as: “a nonlinear dissipative medium with an energy gradient.”
Even without manually designing learning rates or frequency characteristics for each layer, the hardware hierarchy and stochastic jitter feedback loop itself may generate mathematical behavior equivalent to physical laws (the balance between diffusion and dissipation in thermodynamics).
Through this mechanism, there is a possibility that: “low-variation gradients” (low-frequency accumulation in lower layers) “high-variation gradients” (high-frequency accumulation in higher layers) are spontaneously differentiated into hierarchical structures. This suggests that cross-layer frequency organization may emerge naturally from the intrinsic dynamics of the system.