Progressive Mixed-Precision Search

• Progressive mixed-precision search algorithms are methods that assign heterogeneous numerical precisions per layer to achieve Pareto-optimal trade-offs between accuracy and computational efficiency.
• They utilize strategies such as greedy layerwise error-bounded search, bilevel continuous relaxation with architectural probabilities, and progressive freezing to meet strict error and cost constraints.
• Empirical results in DNN quantization and numerical optimization show these methods reduce compute, storage, and energy requirements while maintaining performance near full-precision baselines.

Progressive mixed-precision search algorithms are a class of automated methods that allocate heterogeneous numerical precisions (e.g., quantization bit-widths or floating-point types) at different layers or operations in a computational pipeline. These algorithms seek Pareto-optimal trade-offs among accuracy, computational complexity, and deployment constraints by adaptively controlling the precision assignment across the model or optimization trajectory. Progressive mixed-precision search is central in neural network quantization, scientific computing, and large-scale optimization where minimizing compute, storage, or energy under tight accuracy requirements is paramount.

1. Formalism and Problem Definition

Progressive mixed-precision search (PMPS) seeks to assign, for each module or layer $l$ in a system of $L$ modules, a precision level $p_l$ (e.g., bit-width, floating-point format) selected from a discrete set $B$. The central optimization form is:

$$\min_{p_1, \dots, p_L} \ \mathcal{C}(\mathbf{p}) \quad \text{subject to} \quad \mathcal{E}(\mathbf{p}) \leq \epsilon, \quad p_l \in B,$$

where $\mathcal{C}(\mathbf{p})$ denotes cost (e.g., mean bit-width, inference latency, or the total bit operations), and $\mathcal{E}(\mathbf{p})$ encodes an error metric (e.g., quantization error, task loss), with $\epsilon$ a user-imposed tolerance.

MixQuant (Kloberdanz et al., 2023) operationalizes this for DNN weights: for each layer $l$, one selects the smallest $b_l$ such that its mean-squared quantization error (QE) satisfies $QE_l(b_l) \leq \alpha \, QE_l(8)$, for user-supplied quality factor $\alpha \geq 1$.

Progressive search also encompasses bilevel or constraint-regularized neural architecture search (e.g., EBS (Li et al., 2020), BP-NAS (Yu et al., 2020), SSPS (Sun et al., 2021)) and trust-region numerical optimization with precision-adaptive subproblems (e.g., TROPHY (Clancy et al., 2022), mixed-precision Krylov methods (Suzuki et al., 27 May 2025)). The commonality is the systematic, often layer-wise or block-wise, determination of precisions to meet global objectives.

2. Methodological Classes

Several concrete algorithmic strategies instantiate progressive mixed-precision search:

2.1 Greedy Layerwise Error-Bounded Search

MixQuant (Kloberdanz et al., 2023) performs a single forward pass over all layers, assigning to each weight tensor $W_l$ the minimal bit-width $b_l$ (from $B = \{2,3,\dots,8\}$) such that

$$QE_l(b_l) = \frac{1}{n_l}\lVert W_l - \widetilde{W}_l^{(b_l)} \rVert^2 \leq \alpha \cdot QE_l(8).$$

This progressive per-layer search guarantees no more quantization error than an 8-bit uniform baseline (when $\alpha = 1.0$), frequently yielding $b_l < 8$ and reduced average model width with minimal accuracy loss.

2.2 Bilevel Continuous Relaxation with Architectural Probabilities

The Efficient Bitwidth Search (EBS) approach (Li et al., 2020) introduces learnable "strength" vectors ${\bf r}^\ell, {\bf s}^\ell$ per layer, where Softmax(${\bf r}$) yields a weighting over candidate bit-widths. The surrogate (aggregated) weights and activations are convex sums over quantization branches. A continuous relaxation makes the expected FLOPs along these soft paths differentiable, enabling joint optimization:

$$\min_{r,s} \ L_{val}(W^*(r,s), r, s) + \lambda \max(0, \mathrm{FLOPs}(r,s) - \mathrm{FLOPs}_{\text{target}})$$

subject to

$W^*(r,s) = \arg\min_W L_{train}(W, r, s).$

Post-search, the most probable precision per layer is selected, yielding a discrete, deployable mixed-precision model.

2.3 Soft Constraint and Progressive Freezing

BP-NAS (Yu et al., 2020) imposes a differentiable "soft barrier" to enforce complexity constraints, e.g., total bit operations $\mathcal{B}(\mathcal{N};MP) \leq B_{\max}$. The loss includes a log-log barrier penalty, which is nearly flat inside the feasible region but diverges near the constraint boundary, driving the search within valid domains throughout training. The Prob-1 regularizer ensures each softmax per layer converges to one-hot, collapsing the architecture to discrete assignments.

Sequential Single Path Search (SSPS) (Sun et al., 2021) progressively fixes layers as their bit-path selection entropy drops. At "decision epochs," the most certain layer is frozen to its argmax path, exponentially shrinking the search space as training proceeds.

2.4 Adaptive Precision in Optimization Trajectories

In numerical optimization, e.g., trust-region methods, progressive mixed-precision is realized by dynamically escalating the working precision only when required by the state of the optimization. TROPHY (Clancy et al., 2022) begins at the minimal viable precision (e.g., half or single floating-point), increasing precision only when the model's predicted reduction becomes unreliable, as measured by a residual error estimate $\theta_k$. Outer iterations in higher precision are invoked only if lower precision produces insufficient reduction.

Nested Krylov methods (Suzuki et al., 27 May 2025) for linear solvers embed a multi-level hierarchy: GMRES in high precision, calling inner Krylov or Richardson iterations in reduced precision (down to fp16), with precision shields to prevent spurious progress in very low mantissa regimes. The schedule is fixed but exploits the rapidity of coarse-precision iterates for the majority of computation.

3. Quantization Error, Constraint Handling, and Progressive Scheduling

All PMPS methods are predicated on careful measurement or bounding of the error or cost associated with each allowed precision.

• Quantization Error: MixQuant uses mean squared error of dequantized weights relative to full precision; EBS and BP-NAS encode accuracy into the main loss or as constraint terms, using task loss or bit-operation cost.
• Constraint Handling: BP-NAS employs a soft interior-point constraint penalty scaled by a dynamic barrier parameter $\mu$, ensuring models remain inside the bit-operations budget during the entire search. SSPS incorporates explicit bit-width and operation cost targets into its loss.
• Progressive Scheduling: In SSPS, decision certainty (measured by the entropy of architectural logits) triggers layer freezing, which collapses the search space in a schedule tied to the certainty evolution. In TROPHY and progressive coordinate minimization (Salgia et al., 2020), the precision (or optimization tolerance) is shrunk geometrically according to the optimization trajectory or regret minimization bounds.

4. Computational Efficiency and Hardware Realizability

PMPS algorithms are constructed to minimize search, memory, and deployment overhead.

• Memory Efficiency: EBS (Li et al., 2020) and SSPS (Sun et al., 2021) only materialize a single meta-weight per layer (as opposed to DARTS/DNAS-style supernets with one copy per branch per bit-width), leading to $O(1)$ memory growth with the number of candidate precisions.
• Search Speed: MixQuant achieves per-layer greedy search in $O(L|B|)$ time, typically sub-second for entire networks (Kloberdanz et al., 2023). EBS achieves >$10\times$ reduction in GPU memory and search time versus supernet alternatives.
• Deployability: The binary decomposition scheme of EBS (Li et al., 2020) enables any per-layer quantization tuple to map onto standard hardware by decomposing $k$-bit tensors into binary stacks aligned with hardware popcount and shift primitives.
• Precision Hierarchies: Nested Krylov methods (Suzuki et al., 27 May 2025) demonstrate empirically that fp16 solves, when nested deeply, do not harm outer convergence if iteration counts are limited — confirmable via the inexact Krylov theory.

5. Empirical Performance and Applications

PMPS has demonstrated substantial gains across several benchmarks and domains:

• DNN Quantization: On ImageNet, MixQuant combined with BRECQ (4-bit average) yielded Top-1 improvements of 0.3–1.0% over BRECQ alone, and on ResNet-50, retained >99.8% of int8 accuracy with a mean 6–7 bit allocation (Kloberdanz et al., 2023). EBS (Li et al., 2020) achieved 69.4% Top-1 at 4.93$\times$ FLOPs reduction (ResNet-18/ImageNet), outperforming uniform 3b/3b.
• Search Efficiency: EBS search consumed only 7.3 GB and 22 s per ResNet-18 configuration, compared to >72 GB and 100 s for DNAS approaches. SSPS (Sun et al., 2021) achieved 2–4$\times$ search speedup and 50% memory reduction relative to DARTS/DNAS.
• Optimization Algorithms: TROPHY reduced effective "adjusted calls" by up to 73% in large-scale optimization without losing convergence (Clancy et al., 2022). Nested Krylov methods with fp16 innermost solvers yielded up to 2.42$\times$ speedups versus double-precision on CPU without compromising convergence (Suzuki et al., 27 May 2025).
• Object Detection and Classification: BP-NAS (Yu et al., 2020) and SSPS (Sun et al., 2021) surpassed prior automated and manual mixed-precision quantization methods on COCO and ImageNet benchmarks under stringent bit-operation budgets.

6. Connections, Limitations, and Future Directions

Progressive mixed-precision search encompasses both fully discrete (greedy, progressive freezing) and continuous (softmax relaxed) NAS paradigms, as well as precision-adaptive control in numerical optimization. Its major limitations are constraint to per-layer search granularity (coarser than block-wise or operator-wise for very fine-grained hardware tuning), reliance on error proxies (such as MSE for weights, not task loss), and, in some methods, the need for post-search fine-tuning steps.

Open research directions include finer granularity (sub-layer or tensor-element), dynamic run-time adaptation, richer constraint models (e.g., energy, thermal, communication), and end-to-end co-design with hardware-aware simulators. The nested precision model in numerical optimization may also be extended to distributed or communication-minimized settings, and further analysis of rounding-error accumulation in deep mixed-precision pipelines remains a topic of ongoing investigation.

7. Representative Algorithms: Method Comparison

Algorithm	Problem Domain	Precision Assignment	Constraint Handling
MixQuant (Kloberdanz et al., 2023)	DNN quantization (PTQ)	Layerwise, greedy	Per-layer QE relative to baseline
EBS (Li et al., 2020)	DNN quantization (search)	Layerwise, bilevel relax	FLOPs penalty (softmax-relaxed)
BP-NAS (Yu et al., 2020)	DNN quantization (NAS)	Layerwise/block, softmax	Soft barrier for BOPs budget
SSPS (Sun et al., 2021)	DNN quantization (NAS)	Progressive, entropy-freezing	Direct constraint loss
TROPHY (Clancy et al., 2022)	Numerical optimization	Iterationwise, adaptive	Reduction reliability test
F3R (Suzuki et al., 27 May 2025)	Krylov linear solvers	Levelwise, fixed schedule	No explicit constraint

Each paradigm exploits a progressively or adaptively staged precision allocation, rooted in error or cost metrics and progressing by layer, block, or optimization-trajectory step. Compared to static, one-size-fits-all quantization, PMPS methods yield finer trade-offs and more efficient hardware/energy utilization across applications.