Scaled-Integer Range Analysis (SIRA)

• SIRA is a static analysis method that tracks tensor intervals and affine quantization metadata to optimize FPGA neural network accelerators.
• It propagates bounds through quantized network graphs, enabling tailored bitwidth adaptation, scale/bias aggregation, and conversion of elementwise operations into thresholds.
• Experimental results report resource reductions of 17% for LUTs, 66% for DSPs, and 22% for accumulator bitwidths compared to conservative datatype bounds.

Scaled-Integer Range Analysis (SIRA) is a static analysis technique for quantized neural networks that employs interval arithmetic to determine the range, scale, and bias for tensors, with the explicit goal of optimizing FPGA dataflow neural network accelerators (Umuroglu et al., 29 Aug 2025). In this formulation, SIRA is not merely a method for checking overflow margins; it is a graph-level analysis of affine quantization structure and tensor bounds that enables tailored bitwidth adaptation for accumulators and downstream operations, aggregation of scales and biases, and conversion of consecutive elementwise operations to thresholding operations. The method is integrated into the open-source FINN framework, and the corresponding work reports average reductions of 17% for LUTs, 66% for DSPs, and 22% for accumulator bitwidths with SIRA optimizations (Umuroglu et al., 29 Aug 2025).

1. Definition and representational basis

SIRA tracks each tensor through a quantized neural network as both a value interval and an affine quantization relation. The interval component records conservative bounds $[\underline{v}, \overline{v}]$, while the scaled-integer component records the relationship

$v = s_v \cdot v_{int} + b_v$

where $v$ is the full-precision value, $v_{int}$ is the integer quantized value with known range, $s_v$ is the scale, and $b_v$ is the bias (Umuroglu et al., 29 Aug 2025). For each tensor, SIRA records the possible value range, integer range, scale, and bias, and these properties can vary in granularity as global, per-channel, or per-tensor.

This representation is significant because it exposes information that a datatype-only view does not capture. The 2025 formulation states explicitly that aggressive quantization can expose non-matrix-multiply operations as significant performance and resource bottlenecks on embedded systems, so the analysis must cover the full inference computation rather than only matrix multiplications. In that sense, SIRA generalizes range analysis from a local arithmetic check into a hardware-directed description of how quantized tensors behave across the graph.

A central component is the quantizer model

$$\mathcal{Q}(x) = s \cdot (\text{round}(x / s) + z),$$

after which the output becomes a scaled-integer with given scale and bias, and SIRA records the allowed integer interval (Umuroglu et al., 29 Aug 2025). This makes the analysis directly compatible with affine quantization pipelines in which the integer domain, scale, and zero-point-like bias jointly determine implementation cost.

2. Interval propagation through quantized neural network graphs

SIRA is implemented as a graph pass traversing the QNN computation graph, for example in QONNX format, while maintaining a mapping from tensor names to their range, scale, and bias information (Umuroglu et al., 29 Aug 2025). The propagation algorithm proceeds node by node, invoking custom handlers for key operator types so that both interval bounds and affine quantization metadata are updated as far as the operator semantics permit.

For elementwise monotonic functions such as ReLU and elementwise add or mul, interval propagation is derived from the extrema of the input intervals. For constant-weighted dot products such as MatMul and Conv, the interval arithmetic uses the sign of weights: minimization uses the maximum of an input interval for negative weights and the minimum for positive weights, with the converse choice for maximization (Umuroglu et al., 29 Aug 2025). This is the mechanism by which SIRA derives safe bounds for accumulations and downstream tails.

The scale-and-bias propagation rules are more selective than the interval rules. For addition, if one input is a scaled-integer and the other is a constant, bias is adjusted directly; if both inputs are scaled-integers with matching or integer-multiple scales, the integer components are summed in the lower-scale domain, with the combined bias added afterward. For multiplication, scale and bias are propagated in the constant case, but general scaled-integer times scaled-integer propagation is not supported unless the constant case applies. For matrix multiplication and convolution, if weights and inputs are both scaled-integers, the output scale is the product of input scales, provided there is no scale mixing within dot products, while the output bias is $W \cdot b_X$ when $b_W = 0$ (Umuroglu et al., 29 Aug 2025).

The analysis is intentionally conservative. The reported validation states that empirical data confirms SIRA’s interval bounds always enclose observed values, although often conservative, which is necessary for correctness (Umuroglu et al., 29 Aug 2025). A common misconception is therefore that SIRA computes exact tensor distributions. It does not. Its guarantees are framed as safe, conservative bounds and algebraically propagated affine metadata.

3. Optimization mechanisms enabled by SIRA

The principal contribution of SIRA is that the propagated range, scale, and bias information can be exploited to restructure hardware implementations. Three optimizations are described explicitly: tailored bitwidth adaptation, scale/bias aggregation, and conversion of elementwise tails to thresholding operations (Umuroglu et al., 29 Aug 2025).

Tailored bitwidth adaptation targets accumulation, identified in the paper as a resource bottleneck at low bitwidths. Given a signed integer output interval $[z_{min}, z_{max}]$, SIRA chooses the accumulator width by

$v = s_v \cdot v_{int} + b_v$0

where the additional $v = s_v \cdot v_{int} + b_v$1 accounts for the sign bit (Umuroglu et al., 29 Aug 2025). This yields much smaller accumulators than conservative datatype-bound widths that ignore value distribution and constants in weights. The reduced accumulator width then propagates downstream, shrinking datapaths, FIFOs, and the implementation cost of subsequent non-matrix layers.

Scale/bias aggregation collapses contiguous subgraphs of affine operations so that all scale and bias computations are aggregated into a single representative scale and bias at the boundary, usually before a non-linear activation (Umuroglu et al., 29 Aug 2025). All scale and bias contributions are traced and collapsed into one Mul/Add per output channel, and intermediate operations are replaced with identity. This does not change the logical computation, but it repositions and compresses it into a form more suitable for code generation and resource minimization.

The thresholding transformation targets layer tails composed of per-channel or elementwise operations such as scaling, bias, activation, and output quantization. SIRA converts such a tail into a single thresholding function of the form

$v = s_v \cdot v_{int} + b_v$2

The thresholds $v = s_v \cdot v_{int} + b_v$3 are computed by evaluating the subgraph’s end-to-end mapping over the full input integer range and accounting for all fused operations (Umuroglu et al., 29 Aug 2025). The stated benefit is that potentially expensive elementwise floating-point operations collapse into one highly efficient integer operation, particularly for few-bit activations.

4. FINN integration and implementation trade-offs

SIRA is integrated into the FINN flow as a shared optimization pass via QONNX (Umuroglu et al., 29 Aug 2025). After the analysis, code generation replaces affine layer tails with one of two implementation styles. A composite elementwise meta-kernel is used for high bitwidths or when thresholding is inapplicable; a threshold kernel is used when multi-threshold conversion is favorable. The threshold kernel was previously implemented via parallel comparators and is now implemented via a binary search for scalability.

The paper supplements these transformations with analytical models that guide the choice between composite and thresholded tails. For thresholding kernels, threshold count is given as $v = s_v \cdot v_{int} + b_v$4, where $v = s_v \cdot v_{int} + b_v$5 is the number of channels, and memory grows as $v = s_v \cdot v_{int} + b_v$6 bits (Umuroglu et al., 29 Aug 2025). The reported crossover analysis states that thresholding is almost always more efficient at small bitwidths, specifically for activations below 4 bits, whereas at higher bitwidths, or for per-channel thresholding with many channels, composite tails can be preferable because threshold count grows exponentially.

The experimental results provide concrete resource reductions. Across quantized neural network workloads on modern Xilinx FPGAs, SIRA optimizations yield average reductions of 17% for LUTs, 66% for DSPs, and 22% for accumulator bitwidths compared to conservative datatype bounds (Umuroglu et al., 29 Aug 2025). Microbenchmarks further report that composite tails using fixed-point logic consumed only 21–58% of the LUTs of floating-point baselines at higher bitwidths. These results situate SIRA as a hardware optimization framework driven by static analysis rather than by isolated operator-level heuristics.

5. Relation to earlier and adjacent range-analysis work

SIRA belongs to a broader family of static analyses that reason about numeric ranges, but its formal emphasis is distinctive. An earlier line of work on parametric program analysis introduced parametric strategy iteration, an algorithm that determines the precise least solution of systems of integer equations depending on surplus parameters and allows construction of parametric integer interval analysis as well as parametric analysis of differences of integer variables (Gawlitza et al., 2014). That work represents solutions as piecewise affine functions over parameter regions using region trees, and for a fixed, small number of parameters, the required operations are polynomial-time. A plausible implication is that SIRA’s graph-level interval reasoning for quantized networks occupies a different point in the design space: it prioritizes scalable affine-quantization-aware propagation and hardware transformations over exact least solutions of parametric integer systems.

Adjacent quantization research uses similar language around range, scale, and scaled integer representations, but with a different objective. The work on Adaptive Block-Scaled Data Types analyzes block scaling, representable ranges, and mean squared error for formats such as INT4, FP4, and IF4, using relations such as $v = s_v \cdot v_{int} + b_v$7 and $v = s_v \cdot v_{int} + b_v$8 (Cook et al., 30 Mar 2026). Its explanatory material states that scaled-integer range analysis examines how block scaling and format selection affect dynamic range and quantization error, and that adaptive selection can choose between FP4 and INT4 per block to minimize error. This suggests an adjacent, inference-time quantization interpretation of “scaled-integer range analysis,” centered on local distribution matching and error minimization rather than static graph analysis for FPGA compilation.

Another neighboring research direction is range refinement typing. Ranger focuses on integer range types, uses bidirectional typing, modular backward dataflow analysis, flow-sensitive smart casts, monotonicity analysis in loops, and abstract interpretation over intervals (Aebi et al., 1 Jul 2026). This is a different verification setting, but it underscores that range reasoning appears both as a graph optimization problem in quantized neural networks and as a language-level correctness problem in program verification.

6. Limitations, misconceptions, and future directions

SIRA’s current scope is explicitly bounded. The 2025 formulation assumes static, constant scale and bias; generalization to dynamic or asymmetric quantization is identified as future work (Umuroglu et al., 29 Aug 2025). Non-monotonic or non-thresholdable activations cannot be fused into thresholds, although they can still benefit from scale/bias aggregation. The authors also point to tighter range analysis beyond intervals, including polyhedral or symbolic methods, as a possible direction for improvement.

These caveats address several common misconceptions. First, SIRA is not a general-purpose quantization method; it is a static analysis and optimization technique tailored to quantized neural network inference graphs and their FPGA realization. Second, thresholding is not universally optimal. The analytical guidance states that thresholding is preferred for layer tails when activation bitwidth is at most 4 bits, especially for per-tensor scaling, but composite elementwise tails are recommended for higher bitwidths, many channels, or cases where thresholding is not supported (Umuroglu et al., 29 Aug 2025). Third, the safety of the method derives from conservative interval bounds, not from exact value-set characterization.

Within its stated domain, SIRA provides a unified static analysis for the range, scale, and bias of all tensors, enabling precise, aggressive resource optimization across both matrix-multiply and non-matrix-multiply layers (Umuroglu et al., 29 Aug 2025). Its significance lies in treating quantized inference graphs as analyzable affine-integer systems whose bounds can drive code generation, operator fusion, and hardware specialization. That perspective connects SIRA to a wider landscape of interval arithmetic, integer range analysis, and affine reasoning, while preserving a specific focus on FPGA dataflow neural network accelerators and the non-matrix-multiply bottlenecks that emerge after aggressive quantization.