Bitwidth-Aware Analytical Modeling

• Bitwidth-aware analytical modeling is a framework that employs optimization theory and sensitivity analysis to assign precise numerical bitwidths while balancing quantization error and hardware constraints.
• It integrates mathematical programming, noise-injection models, and heuristic search methods to optimize digital system performance under stringent resource limits.
• The approach enables significant improvements in accuracy, area, and power efficiency across applications such as DNN quantization, FPGA design, and processing-in-memory platforms.

Bitwidth-aware analytical modeling encompasses a suite of methodologies and frameworks for systematically analyzing, predicting, and optimizing the assignment of numerical bitwidths in digital computing systems. These approaches enable fine-grained control over precision, resource consumption, and application-level performance, and are critical across domains such as deep neural network (DNN) quantization, FPGA design, block floating-point architectures, and processing-in-memory (PIM) platforms. Unlike heuristic or uniform bitwidth selection, bitwidth-aware analytical modeling employs mathematical tools—often rooted in optimization theory, sensitivity analysis, or game theory—to capture accurate trade-offs between application error, hardware cost, and operational constraints.

1. Mathematical Foundations in Bitwidth Assignment

The central goal is to assign bitwidths to weights, activations, or signal representations such that requirements for task accuracy and system-level constraints (e.g., area, energy, latency, memory footprint) are jointly optimized. This is formalized as a mathematical programming problem, typically of the form:

$$\min_Q\ L_{\text{val}}(W^*, Q) \quad \text{subject to} \quad W^* = \arg\min_W L_{\text{train}}(W, Q), \quad \Omega(Q) \leq \Omega_0$$

where $Q$ encodes layerwise or elementwise bitwidth policy, $L_{\text{val}}$ and $L_{\text{train}}$ are validation and training losses, and $\Omega(Q)$ is a resource function (e.g., BOPs, model size, FLOPs, power) constrained by $\Omega_0$ (Kang et al., 5 Aug 2025).

In block floating-point DNN operators, this problem generalizes to include block size and tile layouts:

$$\min_{\text{SE}, \text{BS}, I, P}\ O(\text{SE}, \text{BS}, I, P) = \text{Acc}_{\text{loss}}(\text{SE}, \text{BS}) + \alpha \cdot \text{Perf}_{\text{loss}}(\text{SE}, \text{BS}, I, P)$$

subject to hardware and memory constraints (Xu et al., 2024).

When quantization is applied per-weight or per-group, the assignment problem takes the form of convex or discrete optimization under total bit budgets and/or explicit error minimization (Zhang et al., 6 Jun 2025).

2. Analytical Models of Quantization-Induced Error

Bitwidth-aware models require calculating how quantization at a given bitwidth impacts downstream accuracy or quality. The principal models are:

• Second-order loss models: For post-training quantization, the expected increase in task loss when a parameter $w_{ij}$ is quantized to $R_{ij}$ bits is

$L_{ij}(R_{ij}) \approx c_{ij}\,2^{-2R_{ij}}$

where $c_{ij}$ combines quantization range and Hessian-based sensitivity, specifically,

$$c_{ij} = \frac{(w_{ij}^{\max}-w_{ij}^{\min})^2}{12\,[H_F^{-1}]_{n_{ij},n_{ij}}}$$

as in BAQ (Zhang et al., 6 Jun 2025).

• Noise-injection models: For block floating-point DNNs, the variance due to quantization in block $x$ is

$$\sigma^2_{\ell,x}(\mathrm{SE}, \mathrm{BS}) \simeq 2^{-2(q_b-\mathrm{SE}_{\ell,x})} \cdot E_\Gamma[2^{2\Gamma}]/12$$

with total predicted accuracy drop a linear function of these variances weighted by sensitivity coefficients $S_{\ell,x}$ (Xu et al., 2024).

• Interval and SMT range analysis: In FPGA pipelines, interval propagation and Satisfiability Modulo Theory (SMT)–based constraint solving accurately estimate the range at each dataflow stage, from which the minimal integer bit count $\alpha$ is:

$$\alpha = \lceil \log_2(\max\{|r_\text{min}|,|r_\text{max}|\} + 1)\rceil + 1$$

enabling overflow- and underflow-free fixed-point allocation (Benara et al., 2018).

3. Optimization Methodologies for Bitwidth Allocation

Analytical models underpin various optimization paradigms tailored to the granularity and structure of the system:

• Convex programming: Formulating bit allocation as minimization of expected quantization loss subject to a total bit budget, with closed-form “water-filling” solutions. This leads to “equal-loss” structures, where all assigned bits yield equal marginal impact on loss (Zhang et al., 6 Jun 2025).
• Shapley value assignment: Modeling the bitwidth selection as a cooperative game, where value functions $V(S)$ evaluate the task accuracy under subsets of bitwidth choices. The Shapley value $\phi_i$ quantifies the marginal contribution of each operation averaged across all possible subset orderings:

$$\phi_i(V) = \frac{1}{|N|} \sum_{S \subseteq N \setminus \{i\}} \frac{V(S \cup \{i\}) - V(S)}{\binom{|N|-1}{|S|}}$$

(Kang et al., 5 Aug 2025).

• Monte Carlo approximation: For large action spaces (e.g., many candidate bitwidths), stochastic estimation of Shapley values via random permutation rollouts achieves tractable complexity while providing statistical error bounds (Kang et al., 5 Aug 2025).
• Heuristic and greedy search: In practical hardware flows, e.g., PolyMage for FPGAs, interval and profile-driven bitwidth inference is combined with greedy backward search to set fractional widths per stage, balancing error and hardware efficiency (Benara et al., 2018).
• Mixed-integer programming and enumeration: For block floating-point quantization, enumeration over candidate exponent/ mantissa splits, block sizes, and tilings is used, with algorithmic pruning for tractability (Xu et al., 2024).

4. Hardware and System Cost Modeling

Accurate bitwidth modeling incorporates the direct mapping of bit allocation to hardware resource and performance metrics:

• Combinational logic cost: In FPGA implementations, cost models relate adders and multipliers to bitwidth:
  • Adders: $\text{LUT}_{\rm add} \sim c_1 \cdot W$ (linear in total width $W$)
  • Multipliers: $\text{LUT}_{\rm mul} \sim c_3 \cdot W^2$ (quadratic)
  • Power consumption and routing overhead scale similarly (Benara et al., 2018).
• Block floating-point cost: Mantissa and shared exponent bits contribution is amortized across block size, making block size and exponent allocation pivotal in controlling both quantization noise and memory transfer cost (Xu et al., 2024).
• Processing-in-memory cycle count: In Bitlet, operation complexity, alignment, and transfer are parameterized as functions of bitwidth $n$:

$$C_\text{total}(n) = [\alpha n + \beta n^2] + [A n + \mathrm{ROW}] + 2\frac{\lceil n/T\rceil T}{B \, CT}$$

capturing both compute and data movement phases (Korgaonkar et al., 2019).

5. Policy Selection, Validation, and Empirical Performance

Bitwidth-aware analytical modeling frameworks utilize the above optimization and cost models to derive deployable bitwidth assignments, subject to experimental validation:

• Policy extraction: Winner-take-all selection (per-layer argmax of Shapley/score) or knapsack solutions (maximizing total contribution under resource constraint) are standard. In BAQ, optimal or column-wise rounded bitwidth assignment is computed from closed-form equal-loss expressions (Zhang et al., 6 Jun 2025Kang et al., 5 Aug 2025).
• Validation metrics: Key performance indices include model accuracy (top-1, PSNR, AAE), resource use (LUTs, FFs, power, BOPs, data movement), and search/compilation time. Notably, SMPQ (a Shapley-based method) demonstrates higher empirical translation between predicted contribution and realized test accuracy (Kendall’s $\tau=0.494$) compared to gradient-based methods (near zero) (Kang et al., 5 Aug 2025).
• Comparative gains: Bitwidth-aware analytical models achieve 2–6× area and 1.5–4× power improvements over uniform-precision floating-point benchmarks in FPGA pipelines, and up to 56× reduction in LLM perplexity at fixed average bitwidth in BAQ. In block floating-point schemes, accuracy is preserved with marked energy reduction against baseline equal-bitwidth allocations (Benara et al., 2018Zhang et al., 6 Jun 2025Xu et al., 2024).

6. Algorithmic and Practical Considerations

Bitwidth-aware analytical modeling is tightly linked to implementation-specific features and practical constraints:

• Sampling complexity: Monte Carlo or enumeration-based estimators require $O(M\cdot|N|)$ forward passes, but typically stabilize at small sample sizes (e.g., $M \approx 10$ suffices in layered DNNs) (Kang et al., 5 Aug 2025).
• Compiler integration: Frameworks such as PolyMage automate interval/affine/SMT analyses, precision search, and HDL generation, enabling practical use in production hardware flows (Benara et al., 2018).
• Workload/architecture adaptation: The optimal bitwidth policy is highly sensitive to layerwise or blockwise sensitivity, block size, tiling, and underlying hardware constraints. Sweet spots (e.g., BFP block size 4–8, exponent bits 3–5 in 8-bit words) reflect practical balancing of quantization error vs. data movement (Xu et al., 2024).
• Scalability and generalization: Channel/ groupwise quantization, hardware-in-the-loop adaptation (e.g., latency, energy in $V(S)$), and structure-aware sampling are open areas for scaling existing models (Kang et al., 5 Aug 2025).

7. Broader Implications and Limitations

Bitwidth-aware analytical modeling provides a unified lens for design-space exploration across hardware and model compression, explaining precision–resource–accuracy trade-offs at scale. Limitations include: sampling or solver scalability for very large networks, potential conservatism in worst-case range-based analyses, dependency on accurate sensitivity estimation (e.g., second-order/Hessian proxies), and the need for empirical calibration in non-ideal or highly correlated workloads.

Future research directions include tighter integration of application-level metrics (e.g., hardware latency, energy), extension beyond per-layer to groupwise/channelwise models, and efficient structure-exploiting algorithms for large design spaces (Kang et al., 5 Aug 2025Xu et al., 2024Zhang et al., 6 Jun 2025Benara et al., 2018Korgaonkar et al., 2019).