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Towards an Expressivity-Normalized Energy-Demand Comparison of ANNs and SNNs
Miriam Kranzlmüller, Pascal Esser, Gitta Kutyniok
TL;DR
SNN energy advantages are difficult to assess because operation counts omit memory and temporal overhead. The paper develops a hardware-informed, expressivity-normalized comparison of ReLU ANNs and integrate-and-fire SNNs, finding that SNN efficiency depends on width, sparsity, and depth-scaling regimes.
Problem
Operation-count comparisons can overlook memory access, data movement, and repeated multi-timestep computation when evaluating ANN and SNN inference energy.
Method
The framework combines a hardware-informed energy model with theoretical representational-region bounds under matched expressive capacity.
Results
SNN expressivity-normalized efficiency is characterized by explicit thresholds in network width, spike sparsity, and ANN depth scaling.
Takeaways & Limitations
SNNs are more efficient at comparable theoretical expressivity only in specific architectural and sparsity regimes where sparse computation offsets temporal overhead.
Takeaways & Limitations
The comparison uses general upper bounds on representational capacity that may not be attained in practice, and comparable time-series bounds remain an open direction.
Abstract
from arXiv · showhide
Spiking neural networks (SNNs) are often regarded as energy-efficient alternatives to artificial neural networks (ANNs), yet their advantage depends critically on both network architecture and data properties. We develop an analytical framework to compare fully-connected ReLU ANNs and integrate-and-fire SNNs for time-series data with respect to their theoretical energy efficiency at matched expressive capacity. By relating an inference-energy model to theoretical bounds on representational expressivity, we derive an expressivity-normalized efficiency ratio and explicit thresholds in network width, spike sparsity, and ANN depth scaling. Our analysis characterizes the regimes in which event-driven computation offsets the temporal overhead of SNNs, providing capacity-aware principles for designing energy-efficient temporal networks. It shows that ANNs exceed SNNs in expressivity-normalized efficiency only in specific regimes.
1 Introduction
SNNs offer sparse, event-driven processing for temporal data, but their energy advantage depends on a fair comparison that includes memory, data movement, and temporal overhead. This work compares ANNs and SNNs at matched expressive capacity using theoretical region counts.
- SNNs update neuron states only when incoming spikes drive them, making them suitable for temporal and spatio-temporal data.
- Dense matrix operations make conventional ANNs computationally costly in resource-constrained settings.
- Memory access, data movement, and repeated multi-timestep computation can substantially affect inference cost beyond arithmetic operation counts.
- The framework compares fully connected ReLU ANNs and integrate-and-fire SNNs under matched expressive capacity using induced linear or constant regions as a capacity proxy.
- SNN efficiency is conditional: sparse event-driven computation offsets temporal overhead only in specific architectural and sparsity regimes.
2 A Hardware-informed Energy Model
The hardware-informed model estimates inference energy from arithmetic operations and memory accesses while accounting for architecture, sequence length, and activation sparsity. For SNNs, estimated energy grows linearly with sequence length and depends on sparsity.
- Energy Demands: Figure 1 depicts relative energy demand from arithmetic operations and memory accesses for the specified network configuration.
- Energy Demands: Inference energy is modeled as the sum of costs for multiplications, additions, activations, multi-bit accesses, and single-bit accesses.
- Energy Demands: Memory access accounts for the largest portion of computational energy in the adopted model, while bit-proportional scaling is an idealized approximation.
- ANN Energy: The ANN energy expression combines multiply-accumulate costs across input, hidden, and output connections with layer-wise activation and memory-access costs.
- SNN Energy: The SNN avoids multiplications but processes inputs over multiple timesteps, producing energy that scales as O(Tw(m + (1 − S)(Lw + n))).
3 Energy-Expressivity Trade-Off
The comparison matches ANN and SNN theoretical expressivity and defines an energy-efficiency ratio, then derives width, sparsity, and depth-scaling thresholds for SNN advantage. These thresholds show when event-driven sparsity can offset temporal processing overhead under the stated assumptions.
- Expressivity matching: Expressivity is compared through upper bounds on the number of linear or constant regions, with the SNN capacity limit determined by its temporal structure.The framework treats constant and linear regions as equally significant, particularly for classification tasks.
- Efficiency ratio: The ratio η(α) normalizes inference energy by logarithmic expressivity, and an SNN is more efficient when η(α) < 1.The comparison relates a single-layer SNN with sequence length T to an ANN of depth LANN = αT.
- Critical width: For αT > 1, widths above the unique critical width wcrit yield strictly better expressivity-normalized energy efficiency for the SNN.The threshold is the positive root derived from the quadratic inequality in Theorem 1.
- Critical sparsity: SNN efficiency is guaranteed when activation sparsity exceeds Scrit for fixed network dimensions, sequence length, and depth scaling.The critical sparsity is given explicitly by the derived threshold expression.
- Critical depth scaling: A depth scaling factor above αcrit is required for the ANN comparison to overcome the SNN’s temporal overhead under fixed configuration parameters.The critical thresholds are valid provided their denominators remain positive.
- Threshold interpretation: These corollaries identify configurations and sparsity regimes in which SNNs outperform ANNs in efficiency relative to expressiveness, assuming fixed shared energy costs.Figure 2 tracks critical sparsity, width, and depth-scaling values under parameter shifts.
4 Discussion
The empirical analysis compares effective regions and energy demand in randomly initialized ANNs and SNNs, revealing a gap between theoretical bounds and realized capacity as width changes.
- Theory versus practice: Scaling sparsity and depth produces results comparable to the theoretical thresholds, whereas shifting width and sequence length reveals significant disparities.
- Empirical analysis: Figure 3 compares effective regions and energy demand for randomly initialized ANNs and SNNs, alongside actual region counts under shifting width.The base parameters are m = 1, n = 100, w = 4, T = 3, S = 0.5, and α = 1.
- Theory versus practice: At α = 1, the two architectures have inherently identical theoretical upper bounds for representational regions.
- Theory versus practice: Actual ANN region counts scale significantly faster with increasing width than actual SNN region counts.This empirical contrast motivates average-case bounds or high-probability guarantees as a future direction.
A.1 Neural Networks
The paper defines fully connected ReLU ANNs as single-pass feed-forward networks and compares them with integrate-and-fire SNNs that maintain membrane states across timesteps.
- Artificial neural networks: A fully connected feed-forward ANN Φ : R^m → R^n has L + 1 layers, ReLU activations, input dimension m, output dimension n, and fixed hidden width w.L is the number of layers after the input layer.
- Spiking neural networks: An SNN processes an input sequence over T timesteps, producing binary output spikes sℓ(t), membrane potentials uℓ(t), and an output vector y.
- Spiking neural networks: The SNN formulation includes initial membrane potentials, leaky terms βℓ, thresholds θℓ, and an output encoding D for the membrane-potential time series.
- Model assumption: The analysis focuses on integrate-and-fire neurons with β = 1.
A.2 Energy Derivation
The energy derivation estimates inference demand from operation counts and memory accesses for dense ANN computation and sparsity-dependent, event-driven SNN computation.
- Energy estimation: Inference energy is estimated by quantifying the total operations required by each network rather than relying on direct hardware measurements.The approach follows existing operation-count-based analyses.
- Artificial neural networks: For an ANN layer, the feed-forward process requires nℓ·nℓ−1 multiplications, with cumulative additions accounting for weighted-input accumulation and bias terms.
- Artificial neural networks: Standard ANN MAC operations require three reads for input, weight, and partial sum, plus one write for the updated partial sum.The derivation also includes bias loading and saving and activation processing.
- Spiking neural networks: SNN computation processes input spikes concurrently at each timestep and omits weights associated with non-spiking neurons.This models sparsity-dependent computation as analogous to an ANN linear layer.
- Spiking neural networks: SNN additions include accumulation from spiking inputs, bias addition, membrane-potential updates, and potential membrane resets.The derivation also counts activations, memory accesses, updated membrane potentials, and spike outputs.
A.3 Energy-Expressivity Trade-Off
The analysis derives an expressivity-normalized energy inequality and uses it to identify critical sparsity and depth-scaling thresholds. These thresholds specify when the SNN becomes more efficient than the ANN under the comparison setup.
- Energy comparison: The expressivity-normalized comparison is reduced to an energy-difference inequality under matched parameter choices.The derivation uses LANN = αT, LSNN = 1, and mANN = mSNNT.
- Critical sparsity threshold: A critical sparsity threshold Scrit specifies when SNN sparsity guarantees η(α) < 1.The threshold is obtained by reordering the comparison inequality.
- Critical depth scaling threshold: A critical depth-scaling threshold αcrit specifies when α > αcrit yields η(α) < 1.The threshold reflects the ANN depth scaling needed to exceed the SNN’s temporal overhead.
- Energy components: The energy expression includes memory, accumulation, and differential-operation terms in addition to MAC-related terms.The denominator is interpreted as the SNN’s savings per additional layer.
A.4 Exact Number of Regions
Exact region counting is computationally difficult, so the analysis uses an upper bound based on binary neuron states. A depth-first search examines possible state combinations.
- Computational difficulty: Exact counting of linear or constant regions is a complex combinatorial problem.Depending on the definition, the problem is already P#- or NP-hard for one-hidden-layer ReLU networks.
- Region upper bound: The analysis bounds the maximum number of regions by K = 2^#neurons because each neuron has two possible states.The states are active/inactive for ReLU neurons and spike/no-spike for integrate-and-fire neurons.
- Enumeration procedure: A depth-first search enumerates possible combinations of neuron states leading to region configurations.