Hybrid Precision-Scalable Reduction Tree

• Hybrid Precision-Scalable Reduction Tree is a unified datapath that supports low-precision integer inference and high dynamic range floating-point training using mode-driven accumulation.
• The design leverages early-accumulation and adaptive adder widths to minimize normalization overhead while optimizing energy efficiency, area, and accuracy.
• Hardware integration on SNAX NPU platforms demonstrates substantial energy efficiency gains and scalable performance for mixed-precision neural processing workloads.

A hybrid precision-scalable reduction tree is a datapath architecture designed to efficiently perform accumulation operations across multiple arithmetic precisions—particularly combining narrow integer formats and floating-point formats—with a single, unified tree structure compatible with next-generation neural processing unit (NPU) platforms. This construct enables both low-complexity accumulation for inference (using INT8/INT4) and high dynamic range accumulation for training (using FP8/6/4), optimizing energy efficiency, area, and accuracy by exploiting mode-driven datapath adaptations and early-accumulation logic.

1. Rationale and Design Motivation

The hybrid precision-scalable reduction tree is motivated by workload heterogeneity in continual learning applications, which necessitate NPUs supporting both low-precision inference and higher dynamic range training. Existing Microscaling (MX) MACs face a dichotomy: integer-only reduction trees yield minimal area/energy but require costly conversion logic when products are generated in floating point; conversely, accumulation in FP32 simplifies input handling but incurs up to 85% overhead in normalization and suffers quantization loss when output is recast into MX formats. The hybrid reduction tree is architected to:

• Leverage integer adder simplicity for aligned exponents (e.g., MXINT8 mode).
• Employ smaller adder width when product exponents diverge (MXFP modes).
• Perform “early-accumulation” to avoid expensive normalization in mid-tree accumulation, bounding intermediate adder width at 28 bits for L2.
• Further reduce accumulator mantissa precision (from 23 bits to 16 bits) so that addition error does not exceed intrinsic MX quantization error.

2. Architectural Structure and Core Operations

The reduction tree centers on an accumulation hierarchy with distinct levels:

• Level 1 (L1): Consists of 2-bit multiply units and small-width integer adder trees, passing through in high-precision modes and reducing in FP4 mode.
• Level 2 (L2): L2 consumes about 80% of MX MAC resources. Inputs are four 10-bit significands $m_i$ (from products), each with an exponent $e_i$, along with two shared 8-bit exponents. L2 performs:

1. $e_{\max} = \max_i e_i$.
2. For each product, $m'_i = m_i \gg (e_{\max} - e_i)$.
3. Summation: $S_2 = \sum_{i=1}^4 m'_i$, where $S_2$ maintains exponent $e_{\max}$.

• Early-Accumulation Path: A stored accumulator $\left(M_{\rm acc}, E_{\rm acc}\right)$ (mantissa width $M_{\rm acc}=16$) is aligned with $S_2$ via a 2-way multiplexer. Operands are extended by $(E_{\rm acc} - e_{\max})$ bits or vice versa, incurring a single 53-bit addition and normalization. Output mantissa is truncated back to 16 bits, keeping cumulative addition error $\epsilon_{\rm add}$ below $\epsilon_{\rm q}$ (MX quantization error).

3. Mathematical Formulation and Accuracy Relaxation

Let $p_i = m_i 2^{e_i}$ denote the four FP products, with $m_i \in [0, 2^{10})$, $e_i \in \mathbb{Z}$. The L2 alignment and addition is:

• $m'_i = m_i \gg (e_{\max} - e_i)$, $e_{\max} = \max_i e_i$
• $S_2 = \sum_{i=1}^4 m'_i$ with exponent $e_{\max}$

Stored accumulation: $A = M_{\rm acc} \times 2^{E_{\rm acc}}$ and define $\Delta = E_{\rm acc} - e_{\max}$. Operands are aligned:

• If $\Delta \geq 0$, $S_2^{\uparrow} = S_2 \ll \Delta$, $A^{\uparrow} = M_{\rm acc}$;
• Else, $S_2^{\uparrow} = S_2$, $A^{\uparrow} = M_{\rm acc} \ll (-\Delta)$.

The final accumulation: $\widetilde{S} = S_2^{\uparrow} + A^{\uparrow}$ —followed by normalization and truncation to 16-bit mantissa. Accuracy criteria is: $$\epsilon_{\rm add} = \left|\mathrm{round}_{M_r}\left(\sum p_i \right) - \sum p_i \right| \leq \epsilon_{\rm q}$$ with $M_r = 16$, validated over relevant operand distributions.

4. Mode-Dependent Control and Functional Adaptation

Precision mode control is realized via a 2-bit signal from the MAC’s finite state machine (FSM):

• MXINT8 mode: Common exponent; L2 alignment is bypassed, no FP logic, pure integer adder tree for 16-bit fixed-point products.
• MXFP8/6/4 modes: L2 alignment, 28-bit adder, and early-accumulation are gated in. L1 reduction adapts based on mode, following PS-MX_MAC topology.

The FSM directly clocks functional units on or off according to the active precision mode. This enables dynamic adaptation for mixed-precision workloads, optimizing both resource usage and computational efficiency.

5. Hardware Integration and System-Level Considerations

The hybrid tree is instantiated in the SNAX NPU platform:

• MX Tensor Core: 8×8 array of MAC units, controlled by a lightweight FSM (with three control/status registers, CSRs) for precision mode, accumulation depth, and tile size selection.
• SIMD Quantization Unit: Converts 64 FP MAC sums to MX format at array output.
• Data Supply: Dynamic data streamers use runtime channel gating: 1 channel for INT8, up to 4 channels for FP modes, with a programmable address generation unit (AGU) for MX layout-specific memory mapping.
• Control Integration: RISC-V Snitch cores program CSRs and synchronize with MAC FSM via valid/ready handshakes.

MAC pipeline stages encompass multiply/exponent sum, L1 add or bypass, and L2 align/add plus early-accumulation and normalization. The system achieves utilization of 94–99% on benchmark tasks like ResNet-18 and Vision Transformer with batch 32.

6. Performance Metrics, Area, and Design Trade-Offs

Operating at 500 MHz in GF 22FDX process:

Mode	Throughput (GOPS)	Energy Efficiency (GOPS/W)	Speedup over PS-MX_MAC
MXINT8	64	657	1.59× (vs. 412)
MXFP8/6	256	1438–1675	3.05–3.21× (vs. 472–521)
MXFP4	512	4065	1.13× (vs. 3597)

• Cycle counts: 1 output/cycle for FP4; 2 for FP8/6; 8 for INT8.
• Area: MX tensor core occupies ~0.18 mm² (29.5% of 0.60 mm² total NPU area); hybrid tree adds ~10% area in FP modes compared to integer-only tree.
• Latency: FP modes incur an extra cycle for alignment/normalization; INT8 mode is zero overhead.
• Accuracy vs. energy: Reducing accumulator mantissa (23→16 bits) improves normalizer energy by ~12% with negligible effect on task error.
• Future prospects include per-layer mantissa width adaptation, use of approximate adders for greater energy savings, and fused quantization/normalization units.

7. Algorithmic Generation and Scalability of Reduction Trees

Recent advances treat arithmetic tree construction (adders/multipliers) as combinatorial optimization over compressor and prefix trees (Lai et al., 10 May 2024). A single-player game formulation ("MultGame") uses PPO reinforcement learning for compressor-tree phase and Monte-Carlo Tree Search (MCTS) for prefix-phase.

• State: Bit vector $b \in \mathbb{N}^{2N}$; action is selection of half/full adder in minimal index column.
• Prefix-tree: Upper triangular binary matrix $A \in \{0,1\}^{(2N)\times(2N)}$.
• RL rewards are based on synthesized delay/area; tree regularization penalizes non-reducing adders.
• Compressor and prefix sequences are parameterizable, supporting any $N$ (8–128 bits); module generation is scalable by compile-time width specification.

Numerical benchmarks demonstrate Pareto-superiority: up to 54% speed improvement vs. RL-MUL and default tool flow across 8–64 bit multipliers, with consistent area reductions.

8. Context and Impact

The hybrid precision-scalable reduction tree innovates at the intersection of hardware design and algorithmic optimization for mixed-precision neural workloads. It delivers significant improvements—up to 3.2× energy efficiency gains over prior designs—while maintaining the flexibility required by modern MX standards and continual learning applications. Integration with scalable tree-generation methodologies (Lai et al., 10 May 2024) indicates that future reduction trees can be auto-generated to minimize latency and area as arithmetic demands and technology nodes evolve. The architectural principles established by the SNAX NPU and MX tensor core are likely to inform subsequent research in both hardware-accelerated AI and general-purpose mixed-precision computation.