Latency-Optimal Depth-Width Ratios

• Latency-optimal depth-width ratios are formalized architectures that balance sequential layer depth and parallel width to minimize computation latency.
• They are critical in optimizing Transformer models, binary-tree reductions, and message-passing algorithms by establishing empirical Pareto frontiers in latency versus complexity trade-offs.
• Methodologies such as empirical profiling, dynamic programming, evolutionary search, and dynamic distillation guide the selection of optimal architectural configurations under strict latency constraints.

Latency-optimal depth-width ratios formalize the architectural balancing between network depth (number of sequential computational stages) and width (parallelization within stages) under strict latency constraints. This concept is critical for hardware-efficient neural network and circuit design, particularly in Transformer-based models, deep inference pipelines, and binary-tree reductions in algorithmic graph architectures. Latency refers to the end-to-end response time for a computation, and optimizing depth-width ratios means determining the distribution of depth and width that minimizes latency for a given accuracy or complexity target.

1. Formal Definitions and Underlying Models

Depth is typically the number of sequential layers or binary operations required to transform inputs to outputs. Width refers to the parallelism at each layer, such as the hidden dimension size in neural networks or the number of gates/operations per computational stage. Latency ($l$) is the longest simple path through computation (or longest edge chain in a DAG), while complexity ($c$) is the total number of operations (e.g., internal nodes) in the computational graph.

For message-passing algorithms, the canonical binary-tree reduction problem trades off circuit size versus circuit depth. With $n$ input and $n$ output nodes, each output $y_j$ is computed from all but one input using a shared DAG structure. The size-optimal construction attains $c_n^{\min}=3n-6$ with latency $$l_{\rm co}^{\min}=\delta+\lceil\log_2(n-2^\delta)\rceil$$, and the latency-optimal construction realizes latency $l_n^{\min}=\lceil\log_2(n-1)\rceil$ at complexity $c=n\log_2(n-1)$ for $n-1=2^k$ (He et al., 2020).

SLMs and Transformer architectures generalize this by tying depth ($D$) to block count and width ($W$) to hidden size. For each block, parameters scale as $O(W^2)$, and total parameter budget $P\approx c\cdot D\cdot W^2$ determines architectural feasibility. Latency per inference is measured either empirically via look-up tables or roughly as a sum of latency per layer/block (Fu et al., 24 Nov 2025, Hou et al., 2020).

2. Analytical Depth-Width-Latency Trade-offs

Closed-form latency-optimal ratios have been established in several context-specific models:

• Message Passing and Binary Reduction: For $n$ outputs, the minimal depth is $\lceil\log_2(n-1)\rceil$, with complexity $n\log_2(n-1)$. If a shallower structure is demanded, a dynamic programming approach constructs a DAG with $c\leq n\lceil\log_2n\rceil-2$ at depth $\tau$ (He et al., 2020).
• Linear Self-Attention Regression: For data distributions modeled with random rotation structure (RRS), latency at inference scales as $P^2 N L$ where $P$ is context length, $N$ is width, and $L$ is depth. The scaling law for risk is $R(N,L)=A N^{-\alpha_N}+B L^{-\alpha_L}$ under total compute $N L=C_0$, yielding $L\propto N^\nu$, $N\sim C_0^{1/(1+\nu)}$, $L\sim C_0^{\nu/(1+\nu)}$, and a latency-optimal depth/width ratio $L/N\sim C_0^{(\nu-1)/(\nu+1)}$ (Bordelon et al., 1 Oct 2025).
• DynaBERT Adaptive Networks: The empirical latency is fitted to $$T(m_w, m_d)\approx \alpha m_d m_w^2 + \beta m_d m_w + \gamma$$, where $m_w$ and $m_d$ are fractional width and depth. Inference selects sub-network multipliers $(m_w, m_d)$ to meet latency targets while maximizing accuracy (Hou et al., 2020).

3. Empirical Design Guidelines and Observed Pareto Frontiers

Experimental results consistently report that the deepest possible models under a parameter constraint maximize accuracy, but latency targets can shift the Pareto frontier. For SLMs on NVIDIA H100:

Model	Depth $D$	Width $W$	Latency (s)	Accuracy (%)
Nemotron-Flash-1B	12	2048	14.45	49.6
Nemotron-Flash-3B	18	3072	28.7	61.0
Qwen3-1.7B	–	–	38.2	55.5
Qwen3-0.6B	–	–	27.6	44.1

For DynaBERT on SST-2:

$(m_w, m_d)$	Params (M)	Latency (ms, GPU)	Accuracy (%)
(1.0, 1.0)	110	18.0	93.2
(0.75, 0.75)	57	10.6	93.1
(0.25, 0.5)	15	5.5	91.6

These results demonstrate the existence of a "sweet spot" depth-width pair for any fixed latency: shallower-wider models are favored for low-latency operation, while deeper-thinner variants are optimal within looser latency budgets (Fu et al., 24 Nov 2025, Hou et al., 2020).

4. Algorithmic and Architectural Methodologies

Key approaches for discovering latency-optimal ratios include:

• Empirical Latency Profiling: Real-device latency is measured for network blocks of various widths to produce LUTs. Model configurations are selected by sweeping $D, W$ pairs and applying fitted augmentation laws for loss prediction (Fu et al., 24 Nov 2025).
• Evolutionary Search for Hybrid Operators: Mutation and selection of block operator types (e.g., Attention, Mamba2, DeltaNet), guided by early training proxy metrics, efficiently optimize architectures under latency constraints (Fu et al., 24 Nov 2025).
• Dynamic Distillation and Masking: DynaBERT uses staged distillation and head/neuron importance-based network rewiring to permit sub-network carving with adaptive $m_d, m_w$. Sub-networks are selected post-training via their fitted latency (Hou et al., 2020).
• Dynamic Programming for Complexity-Latency Boundaries: For binary-tree reductions, a DP combination over substructures yields candidates for every allowed depth, conjectured to be complexity-optimal (He et al., 2020).

5. Applications and Domain-Specific Considerations

Latency-optimal depth-width ratios have domain-specific relevance:

• Small LLMs: SLMs deployed on real-world devices, where latency constraints are paramount, benefit from search methodologies that factor operator choices and depth-width scaling rather than only parameter count. Hybrid operator composition further improves both latency and accuracy (Fu et al., 24 Nov 2025).
• Message-Passing and Graph Algorithms: DAG design for node computations in parallel algorithms often faces depth-complexity trade-offs strictly governed by the output structure and input omission conditions (He et al., 2020).
• Transformer Compression and Edge Deployment: Dynamic adjustment of depth and width—via once-for-all teacher-distilled models—allows efficient deployment across heterogeneous hardware, enabling tailored accuracy-latency operation (Hou et al., 2020).
• In-context Regression and Scaling Law Analysis: Theoretical models show that the optimal architecture balance depends on statistical properties of data such as covariance spectrum decay (capacity exponent $\nu$) in random rotation regimes, and that depth yields diminishing returns in isotropic or fixed-structured scenarios (Bordelon et al., 1 Oct 2025).

6. Open Problems, Limitations, and Generalizations

The main gaps lie in theoretical optimality for intermediate complexity-latency trade-offs and in circuit models allowing gates with higher fan-in or explicit width constraints per layer. While binary-tree structures (fan-in=2) are well characterized, bounded-width circuit trade-offs remain open. The DP constructions are conjectured but not universally proven optimal for all $(n,\tau)$ (He et al., 2020). Extensions to other operator sets, block designs, and real-time hardware environments are under active investigation (Fu et al., 24 Nov 2025). Moreover, the role of capacity exponents in practical data regimes requires further empirical clarification (Bordelon et al., 1 Oct 2025).

7. Practical Selection Guidelines

Recommended procedure for practitioners:

1. Measure block-level latency on target hardware to construct LUTs.
2. Fit the augmented scaling law $L(D,W)=L_0 + a D^{-\alpha} + b W^{-\beta}$ on pilot architectures.
3. Profile $(D,W)$ pairs to select the configuration yielding minimal loss under latency $\leq$ budget.
4. Employ evolutionary search or dynamic distillation for further accuracy-latency frontier optimization.
5. Incorporate advanced normalization or meta-token strategies if allowed by application (Fu et al., 24 Nov 2025, Hou et al., 2020).

This systematic methodology yields architectures that occupy the empirical Pareto frontier for latency versus accuracy or complexity in diverse computational settings.