Quantized Decision Tree Inference

• Quantized decision tree inference is a method that maps continuous values to a finite set, enhancing scalability and efficiency.
• It employs strategies like range-based quantization, fixed-point arithmetic, and Algebraic Decision Diagrams to balance model size and error.
• This approach supports hardware acceleration, privacy-preserving analytics, and quantum-inspired techniques, broadening its applicability.

Quantized decision tree inference refers to the class of methodologies in which the computation within decision tree models—including node thresholds, leaf outputs, splits, and intermediate statistics—is performed using values mapped into a discretized set (usually integer or low-bitwidth representations), subject to constraints on memory, compute, and inference accuracy. This paradigm is leveraged to address scalability, efficiency on resource-constrained hardware, and integration with specialized domains such as privacy-preserving cryptography and quantum computing. Approaches span range-based quantization, hashing-based bucketing, fixed-point conversion, algebraic decision diagrams, and quantum state representations.

1. Principles and Methodologies in Quantized Decision Tree Inference

Quantization of decision tree inference typically proceeds via a mapping $Q: T \rightarrow Q$ from a domain $T$ (real, continuous, or high-precision values) to a finite codomain $Q$ consisting of a reduced number of discrete values. For decision trees used in probabilistic graphical model inference, this causes states of a potential to share the same numerical value, introducing context-specific independence exploitable by compact representations such as Algebraic Decision Diagrams (ADDs). The process is formalized by $F_Q(w) = Q(w)$ for each $w$ in the range of $F$, reducing leaf-node diversity and, consequently, shrinking the size of the tree representation (Gogate et al., 2012).

Quantization is cast as a multi-objective optimization problem, balancing representation size and error:

• Size constraint: restrict the ADD or tree node count below a bound $k$.
• Error constraint: for all quantizations under the size bound, select the mapping that minimizes a divergence measure $D(F, F_Q)$ (e.g., KL-divergence, mean-squared error, absolute error).

Heuristics such as min-error (minimizing approximation error for a target leaf count), min-merge (directly merging leaves to minimize size), or combined min-error-merge, facilitate practical solution of the quantization problem.

Beyond range-based quantization, other strategies include dynamic quantization of numerical values via hashing (assigning values to buckets indexed by $h = \lfloor x / r \rfloor$ for some quantization radius $r$) with robust incremental mean/variance estimation per bucket (Mastelini et al., 2020), and fixed-point integer quantization ($q(x) = \text{round}(x / \Delta)$ for scaling factor $\Delta$) supporting hardware or cryptographic constraints (Frery et al., 2023, Bart et al., 21 May 2025).

2. Structured Representations: Algebraic Decision Diagrams and Compact Tree Structures

Algebraic Decision Diagrams (ADDs) are canonical, reduced, directed acyclic graphs that encode real-valued Boolean functions, sharing structural similarity with decision trees. Each internal node is labeled by a Boolean variable, with child arcs representing true/false assignments; leaf nodes store real values (Gogate et al., 2012). Reduction rules coalesce isomorphic subgraphs or identical children, leading to highly compact representations that leverage context-specific independence.

Quantization refines ADDs (or more generally decision tree structures) by consolidating leaf values, allowing further reduction and removal of redundancy. Quantized ADDs exhibit strictly smaller or equal size relative to their unquantized counterparts, as formalized by:

Proposition: Let $F_Q$ denote the quantization of $F$ with respect to $Q$. Then size(ADD($F_Q$)) $\leq$ size(ADD($F$)).

In online scenarios, compact representations can be maintained using hashing-based bucketing at each node, with low memory cost per feature and constant $O(1)$ update cost per instance (Mastelini et al., 2020).

3. Quantization in Decision Tree Training, Splitting, and Inference Algorithms

In classical training, quantized thresholds discretize the split candidates (for continuous features, using sorted unique values $w_1^f < w_2^f < \cdots < w_{u(f)}^f$ and midpoints for splits) (Mazumder et al., 2022). Quantile-based or equal-width binning can further reduce the candidate threshold space, facilitating scalable optimization.

In specialized scenarios:

• Gradient-Driven Quantized Training: Models such as quantized GBDT use quantized gradient and Hessian values in histogram construction and split gain evaluation. For a $B$-bit quantization of gradient $g_i$, the formula $\delta_g = (\max_i |g_i|)/[2^{B-1} - 1]$ is used, followed by stochastic rounding ($\hat{g}_i = \text{Round}(g_i / \delta_g)$) (Shi et al., 2022). This enables integer operations and hardware acceleration, achieving speedups of up to $2\times$ over baseline while maintaining accuracy.
• Branch-and-Bound over Quantized Spaces: Quant-BnB implements optimization for optimal tree construction by quantizing threshold intervals, maintaining alive lists of candidate subspaces, and recursively pruning subregions with lower bounds exceeding global optima (Mazumder et al., 2022). This process is robust for both regression and classification, and enables efficient search for optimal shallow trees.
• Integer-Only Inference: Frameworks such as InTreeger convert both comparisons and probability accumulation into integer arithmetic. Leaf probabilities $p$ are expressed as $p_\text{int} = \lfloor p \times 2^{32}/n \rfloor$, where $n$ is the number of ensemble trees, allowing inference free of floating-point operations and hence suitable for embedded, edge, or ultra-low-power devices (Bart et al., 21 May 2025).
• Online Regression with Quantizer Observer: Dynamical quantization via hashing (quantization observer) enables effective split decisions within Hoeffding Trees with $O(1)$ monitoring and sub-linear split-query costs, compared to prior $O(\log n)$ methods (Mastelini et al., 2020).
• Neural-Network Reformulation: Ensemble-trees are recast as fully-connected neural networks with quantized, one-hot encoded inputs, simplifying the architecture and enabling GPU acceleration (Saberian et al., 2019).

4. Trade-offs and Performance Considerations

Quantization provides substantial gains in model size, computational cost, and resource utilization, but introduces approximation errors that must be controlled. Empirical results demonstrate:

• Quantized partition functions and bounds (ABQ/IABQ) outperform conventional mini-bucket elimination, TRW, and BoxProp schemes on benchmark contexts with context-specific independence, with tight upper/lower partition function bounds (Gogate et al., 2012).
• Integer-only inference yields $2.1\times$ speedup and $21\%$ lower energy consumption against floating-point baselines on ARMv7 and RISC-V platforms, with non-impacting error margins ($\sim 10^{-8}$–$10^{-10}$ in probability) (Bart et al., 21 May 2025).
• Histogram computation time in quantized GBDT training can be reduced by up to $3.8\times$ on GPU platforms, with minimal accuracy loss for quantizations down to $2$–$3$ bits (Shi et al., 2022).
• In branch-and-bound optimal tree construction, quantization facilitates global optimization with computation scaling linearly in sample size and with improved test error versus greedy heuristics (Mazumder et al., 2022).

The choice of quantization technique (binning, fixed-point, stochastic rounding, or heuristic merging) impacts error, memory footprint, and computational latency. Deciding the optimal quantization scheme is domain and task dependent.

5. Implications for Hardware, Quantum, and Privacy-Preserving Inference

Quantization is instrumental in the deployment of decision tree models on constrained or specialized hardware:

• FPGAs/ASICs: TreeLUT implements quantized GBDTs using LUTs, combining bias, shifted leaves, and scaling into $w_\text{tree}$-bit values, resulting in fully unrolled LUT-based architectures with reduced area-delay products versus DNN/other LUT-based decision tree designs. Benchmark results include high throughput, low latency, and competitive accuracy ($97\%$ on MNIST, $92$–$93\%$ on NID) (Khataei et al., 2 Jan 2025).
• Fully Homomorphic Encryption (FHE): Privacy-preserving tree inference is enabled by quantizing all model components (features, thresholds, leaves) into integers and replacing control flow with tensorized computations, using TFHE’s programmable bootstrapping for secure comparisons and table lookups. Quantization bitwidths are constrained by correctness and noise parameters; 6 bits proved effective in experiments (Frery et al., 2023).
• Quantum and Quantum-Inspired Trees: Quantum decision trees use state superpositions, amplitude encoding, and quantum circuit traversals, with quantized outcomes determined by measurement probabilities or quantum splitting criteria, such as fidelity between quantum states constructed from feature-class correlations (Kak, 2017, Heese et al., 2021, Sharma et al., 2023, Li et al., 17 Feb 2025). Quantum algorithms (e.g., Des-q) utilize amplitude estimation, SWAP tests, and quantum clustering for piecewise linear splits and rapid retraining, achieving poly-logarithmic complexity in data size (Kumar et al., 2023). Quantum circuit approaches can realize probabilistic traversals with constant classical memory, and quantum splitting measures yield more balanced trees and improved classification metrics compared to classical splits.

6. Extensions: Uncertainty, Interpretability, and Advanced Inference

Handling uncertain, noisy, or probabilistic data is facilitated by quantized inference protocols. Indecision Trees generalize splitting by propagating sample probability masses along all branches, using soft aggregation schemes for entropy and information gain, allowing inference under measurement uncertainty and yielding distributions over possible labels rather than hard classifications (Kent et al., 2022). Tree structures can be deconstructed into logical chains and probabilistic argument sets for integration into advanced reasoning systems, supporting explainable AI in regulated domains.

Quantized approaches also support end-to-end tree training via gradient descent with straight-through estimators, as in DGT models, where the non-differentiable quantized step function is approximated for backpropagation, enabling dense parameter updates and robust training in supervised and bandit feedback settings (Karthikeyan et al., 2021).

7. State-of-the-Art Comparisons and Application Domains

Quantized decision tree inference exhibits strong advantages versus traditional floating-point or greedy methodology:

Method	Latency/Area Reduction	Accuracy Impact	Key Quantization Component
TreeLUT FPGA	≥10× lower ADP	Competitive	Bit-shifted, scaled leaf/thresholds
Quantized GBDT	2×–3.8× speedup	Near lossless	Gradient/Hessian quantization with SR
InTreeger (Integer-only)	2× speedup/21% energy	Negligible loss	Fixed-point probabilities, threshold casting
Quant-BnB Branch-and-Bound	Linear scaling in n	Lower test error	Quantile-based binning

These strategies have demonstrated utility in domains including tabular data modeling, real-time data streaming, hardware deployment (FPGA/ASIC), privacy-preserving analytics, quantum machine learning, and robust/uncertain sensory environments.

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Quantized decision tree inference encompasses a spectrum of algorithmic and systems-level strategies that convert high-precision, continuous decision tree operations into discrete, bounded-error computations for scalable, efficient, and interpretable machine learning. Its utility spans probabilistic graphical inference, hardware acceleration, privacy-preserving analytics, quantum computation, and uncertainty-aware reasoning, supported by strong theoretical bounds and empirical validation across multiple technological fronts.