Optimizer–Quantization Interactions

• Optimizer–Quantization Interactions is a framework describing how discretization and optimization co-design shapes error accumulation, convergence, and stability in both classical and quantum models.
• Techniques such as constrained optimization with alternating learning and compression steps highlight the critical impact of operation ordering on error propagation in deep networks.
• Integrating quantization into optimizer state and mixed-precision designs enhances memory efficiency and model robustness, facilitating deployment on resource-constrained hardware.

Optimizer-Quantization Interactions characterizes the multi-faceted relationship between optimization algorithms and quantization techniques across quantum, classical deep learning, and communication systems. This interface encompasses how model parameter discretization, optimizer state quantization, task-driven training objectives, error accumulation, convergence phenomena, and even mixed-precision assignment together determine performance, stability, and efficiency in training and deploying large-scale models. The topic includes both the theoretical underpinnings of these interactions—such as composition error, non-orthogonality proofs, and propagation analysis—as well as their empirical consequences in model robustness, memory efficiency, and hardware deployment.

1. Fundamental Models of Optimizer–Quantization Interplay

There exist several principled approaches for modeling optimizer–quantization interactions. In classical neural networks, a central paradigm is to recast quantization as a constrained optimization problem, in which the weights are required to belong to a discrete set defined by the quantizer map, and the optimizer must minimize the task loss subject to this discrete constraint (Carreira-Perpiñán et al., 2017). Formally, for a neural net with weights $w$ and quantization mapping $\Delta(\Theta)$ (codebook parameters $\Theta$),

$$\min_{w} L(w) \quad \text{subject to} \quad w = \Delta(\Theta)$$

Augmented Lagrangian or quadratic-penalty schemes enable the use of standard optimizers (e.g., SGD), alternating between a "learning" (L-step) and "compression" (C-step) update:

• L-step: Optimize loss plus penalty toward the quantized weights.
• C-step: Project or quantize weights (e.g., via $k$-means for learned codebooks, or nearest-neighbor for fixed codebooks). This structure decouples gradient-based learning from the projection, forming the foundation of many modern quantization-aware optimization pipelines.

In the reinforcement learning domain, some optimizers (e.g., AlphaGrad) use global L2 normalization and non-linear gradient clipping, which naturally bound updates. These bounded updates mitigate the effects of quantization-induced value explosion or loss of resolution, particularly relevant when deploying on low-precision hardware (Sane, 22 Apr 2025).

In quantum Ising machines, optimizer–quantization interplay emerges physically: continuous variables (e.g., phases of Kerr parametric oscillators) encode binary optimization variables, and their coupling is realized via flux quantization. Optimizer robustness then hinges on both the noise characteristics of the physical device and the quantization-induced stability conferred by bifurcation (Nigg et al., 2016).

2. Error Propagation and Composition: Non-Orthogonality and Accumulation

Recent theory establishes that quantization and associated error-propagating transformations (such as sparsity) are mathematically non-orthogonal (Harma et al., 31 May 2024). For two transformations $f$ and $g$ with errors $\varepsilon_f(x)$ and $\varepsilon_g(x)$,

$$\|\varepsilon_{g \circ f}(x)\| \leq \|\varepsilon_f(x)\| + \|\varepsilon_g(x)\|$$

holds for certain ordering but can be violated if quantization precedes sparsity, leading to error amplification. The order of applying operations—e.g., sparsity on float followed by quantization (S → Q) vs. quantization first (Q → S)—critically determines total error and performance. Notably, at the dot-product level in deep networks, cross-terms arising from composition induce compounded errors outside simple summation bounds. Experiments on LLMs and ViTs confirm that S → Q yields lower overall error and better perplexity, with the ordering directly impacting optimizer convergence and gradient stability during fine-tuning.

Furthermore, theoretical frameworks such as the ABC decomposition analyze the error propagation through network layers under quantization (Vlassis et al., 27 Sep 2025). For a layer $\ell$, the squared relative error in the activation is decomposed

$R_\ell = A_\ell + B_\ell + C_\ell$

where $A_\ell$ captures the effect of error propagated from previous layers, $B_\ell$ is the local error introduced at the current layer, and $C_\ell$ is a cross-term. The dominance of propagated error (especially $A_\ell$) underscores that isolated layer metrics—such as maximum-to-mean ratio (MMR) and kurtosis—cannot reliably predict total post-quantization accuracy loss. Instead, the interaction depends on cumulative, network-level effects governed by both optimizer choice and quantizer characteristics.

3. Task-Driven Quantizer Design and Optimizer Integration

Optimization and quantization can be tightly coupled by training quantizers in an end-to-end, task-aware manner (Shlezinger et al., 2019). Modern systems employ differentiable surrogates for discretization (e.g., smooth sums of $\tanh$ or soft-to-hard activations), allowing joint optimization (by, e.g., SGD) of analog pre-processing, quantizer parameters, and end-task loss (MSE for estimation, cross-entropy for classification). This approach allocates quantization resources adaptively to high-value features rather than aiming for uniform reconstruction accuracy, and it circumvents the need for analytic signal models required in indirect rate-distortion theory.

In the context of mixed-precision quantization, recent work extends quantizer optimization to combinatorial assignments, formulating globally optimal layerwise bitwidth allocation as a binary quadratic program whose parameters (sensitivity vector and interaction matrix) are estimated via Shapley-based progressive quantization (Zhao et al., 18 Sep 2025). This process explicitly accounts for, and seeks to minimize, deleterious inter-layer error propagation as part of the optimization.

4. Optimizer State Quantization: Memory Efficiency and Stability

Quantizing optimizer state (e.g., running moment accumulators in Adam) is essential for large-scale models where optimizer state can dominate memory consumption. Approaches incorporating block-wise dynamic quantization—where state tensors are partitioned, normalized per-block, and quantized independently—enable both memory and computational savings while preserving optimizer behavior (Dettmers et al., 2021). Dynamic range expansion and per-group quantization strategies further improve the quantization of momentum and variance by aligning the distributions more closely with the quantizer format, especially in very low-precision regimes (e.g., FP8 training in COAT (Xi et al., 25 Oct 2024)).

With careful design, these schemes can achieve comparable loss and generalization as 32-bit training, as shown in large LLM and computer vision applications, while enabling scaling to smaller hardware. However, not all optimizer states are equally robust: second-moment estimates are particularly sensitive to quantization noise, with symmetrical low-precision schemes sometimes resulting in zero "clustering" and catastrophic weight updates (Chitsaz et al., 16 Jul 2024).

5. Convergence, Recurrence, and Training Dynamics Under Quantization

Training with discrete weights and activations alters the optimization landscape. Projected gradient or "fake gradient" methods using the straight-through estimator (STE) drive not to true convergence, but to recurrence: the parameter sequence recurrently revisits the global optimum or a set of near-optimal quantized states (Long et al., 2020). This cyclic or oscillatory dynamic emerges as an inherent property of quantized systems, subject to the properties of the coarse gradient and the "closeness" of the teacher parameters to a quantized configuration. Convergence guarantees (in the sense of stationarity) remain valid for methods that alternate SGD with discrete projection for constrained quantization settings, provided the quantization constraint is integrated into the augmented Lagrangian (Carreira-Perpiñán et al., 2017).

For optimizers with stateless, bounded updates (e.g., AlphaGrad), normalization and inherent value clamping render the method naturally compatible with low-precision arithmetic, potentially leading to stable training without needing auxiliary state that would itself require quantization (Sane, 22 Apr 2025).

6. Interaction-Driven Mixed-Precision and Resource-Optimal Quantization

Interaction-aware quantization methods recognize that optimal bit allocation across layers requires accounting for joint intervention effects rather than per-layer sensitivity alone (Zhao et al., 18 Sep 2025). Progressive quantization and cooperative game-theoretic analysis (e.g., Shapley value computation) yield sensitivity and interaction matrices used in binary quadratic programs or integer linear program (ILP) relaxations, subject to constraints such as global memory, latency, or power budgets.

Fractional-bit quantizers and fine-grained allocation, validated against information-theoretic rate–distortion bounds for Gaussianized weights, form the basis for deploying LLMs and foundation models on edge platforms (Lee et al., 24 Sep 2025). These approaches exploit the approximate linearity of the quantization-induced loss, using sensitivity coefficients akin to curvature information in second-order optimization, and they are tightly integrated into end-to-end deployment and optimization pipelines.

7. Empirical Findings: Optimizer Choice, Scaling Laws, and Robustness

Large-scale studies demonstrate that the optimizer’s effect on quantization robustness is not reliably predicted by conventional outlier metrics such as MMR or kurtosis (Vlassis et al., 27 Sep 2025). Empirically, optimizers that control spectral change and alignment—thereby attenuating error propagation—achieve higher parameter efficiency and lower loss under quantization-aware training (QAT). Models trained with Shampoo, for example, retain the highest parameter efficiency as measured by scaling laws: $L = \frac{A'}{(N \cdot \rho)^{\alpha}} + E$ with $\rho_{4bit}$ highest for Shampoo among tested optimizers.

These results emphasize that optimal performance post-quantization cannot be inferred solely from full-precision validation scores or isolated layer statistics, but requires careful optimizer-quantization co-design and a holistic understanding of error dynamics, sparsity/quantization ordering, and the propagation of quantization-induced perturbations through the network.

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This comprehensive treatment demonstrates that optimizer-quantization interactions are central to the design, analysis, and deployment of both classical and quantum optimization systems. The theme unifies physical design constraints, discrete mathematics, combinatorial optimization, resource allocation, and deep learning stability under a singular framework where the interactions are nontrivial, nonlinear, and governing for end-to-end performance.