Quantized Confidence: Shrinking Neural Networks Without Losing Certainty

Author: Denis Avetisyan

New research demonstrates that Bayesian Neural Networks can be dramatically compressed for efficient deployment without sacrificing their crucial ability to estimate prediction uncertainty.

A Bayesian Neural Network extends the standard neural network paradigm by replacing deterministic parameters with probability distributions, allowing for a mathematically rigorous treatment of uncertainty and enabling probabilistic inference.

Multi-level quantization of Stochastic Variational Inference-based Bayesian Neural Networks preserves both accuracy and uncertainty estimation down to 4-bit precision.

While Bayesian Neural Networks (BNNs) offer principled uncertainty quantification, their computational demands hinder deployment on resource-limited platforms. This limitation is addressed in ‘Uncertainty-Preserving QBNNs: Multi-Level Quantization of SVI-Based Bayesian Neural Networks for Image Classification’, which introduces a multi-level quantization framework for BNNs, achieving up to 8x memory reduction with minimal impact on both accuracy and calibrated uncertainty estimation. The authors demonstrate that these networks can be effectively quantized down to 4-bit precision, preserving crucial distinctions between aleatoric and epistemic uncertainty. Could this work unlock the potential for truly low-precision “Bayesian Machines” and enable robust, uncertainty-aware AI on the edge?

The Inherent Uncertainty of Conventional Deep Learning

Conventional deep learning systems, despite achieving remarkable performance on numerous tasks, often operate as ‘black boxes’ incapable of expressing the confidence level associated with their predictions. This deficiency presents a substantial risk in applications where reliability is paramount, such as medical diagnosis, autonomous driving, and financial modeling. A model’s inability to recognize when it doesn’t know – to quantify its own uncertainty – can lead to overconfident, yet incorrect, outputs. Consequently, even highly accurate models can make catastrophic errors when presented with ambiguous or out-of-distribution data, as they lack the mechanisms to signal potential unreliability. The implications extend beyond simple prediction error; without a measure of uncertainty, it becomes difficult to trust model decisions, hindering deployment in safety-critical scenarios and demanding the development of more robust and transparent AI systems.

The reliability of artificial intelligence systems hinges on their ability to not only make predictions but also to indicate how confident they are in those predictions. Accurate uncertainty estimation is paramount, particularly when systems encounter ambiguous or noisy data – scenarios common in real-world applications like medical diagnosis or autonomous driving. A model that confidently asserts an incorrect answer can be far more dangerous than one that admits its lack of certainty. This need for robustness drives the development of techniques that allow AI to effectively say “I don’t know,” preventing potentially catastrophic errors and fostering greater trust in these increasingly prevalent technologies. Without a reliable measure of confidence, AI remains vulnerable to unexpected inputs and can produce results that, while statistically plausible, are practically unreliable, hindering its deployment in safety-critical contexts.

Bayesian Neural Networks (BNNs) represent a theoretically sound method for gauging prediction uncertainty, differing from standard neural networks which often provide overconfident results without indicating the reliability of their outputs. Unlike their deterministic counterparts, BNNs treat model weights as probability distributions rather than single values, allowing the network to express its own epistemic uncertainty – essentially, “how much does the model know?” However, this principled approach comes at a considerable cost. Calculating predictions within a BNN requires marginalizing over the entire distribution of weights – a computationally intensive process. Traditional methods, such as Markov Chain Monte Carlo (MCMC), are often too slow for practical applications, while variational inference, though faster, can introduce approximations that compromise the accuracy of uncertainty estimates. Consequently, despite their theoretical advantages, the high computational burden of BNNs has historically limited their widespread adoption, fueling ongoing research into efficient approximations and scalable implementations.

The pursuit of computationally feasible Bayesian Neural Networks (BNNs) is actively fueling investigations into model compression techniques, with quantization emerging as a particularly promising avenue. Quantization reduces the precision of the network’s weights and activations – for example, representing values with 8-bit integers instead of 32-bit floating-point numbers – thereby decreasing both memory footprint and computational demands. This allows for the deployment of BNNs on resource-constrained devices and facilitates faster inference times, crucial for real-time applications. While a reduction in precision can potentially impact accuracy, ongoing research focuses on minimizing this trade-off through techniques like post-training quantization and quantization-aware training, where the network learns to maintain performance despite the lower precision. The ultimate goal is to unlock the benefits of well-calibrated uncertainty estimation offered by BNNs without incurring prohibitive computational costs, broadening their applicability across diverse fields.

Using Joint Quantization on MNIST variants, this analysis demonstrates that while in-domain accuracy gradually decreases with more aggressive quantization, uncertainty disentanglement suffers a substantial decline even at low bit-widths like 3-bit precision.

Quantization: A Pragmatic Approach to Bayesian Inference

Neural network quantization diminishes the memory footprint and computational demands of Bayesian Neural Networks (BNNs) by representing weights and activations with lower numerical precision. Traditionally, neural networks utilize 32-bit floating-point numbers; quantization reduces this to 8-bit integers or even lower. This reduction in precision directly translates to smaller model sizes, requiring less storage space, and fewer operations during both training and inference. For BNNs, where maintaining accurate uncertainty estimates is crucial, quantization presents a unique challenge; however, the gains in efficiency are substantial, enabling deployment on resource-constrained devices and facilitating faster experimentation. The reduction in bit-width impacts both the forward pass – reducing multiply-accumulate operations – and memory bandwidth requirements, contributing to significant performance improvements.

Quantization-Aware Training (QAT) and Post-Training Quantization (PTQ) represent the two dominant strategies for reducing the precision of neural network weights and activations. PTQ applies quantization to a fully trained, floating-point model, offering simplicity but potentially greater accuracy loss. QAT, conversely, incorporates quantization directly into the training process, simulating the effects of reduced precision during backpropagation. This allows the network to adapt and compensate for the quantization, generally resulting in higher accuracy compared to PTQ, but at the cost of requiring retraining. The choice between QAT and PTQ involves a trade-off: PTQ prioritizes ease of implementation, while QAT aims to maximize accuracy in a quantized model. Both methods typically involve converting weights and activations from floating-point representations, such as 32-bit floating point, to lower-precision integer formats like 8-bit integer.

Quantization of Bayesian Neural Networks (BNNs) introduces challenges to accurate uncertainty estimation due to the reduced numerical precision. While quantization reduces computational cost and model size, it can diminish the sensitivity of the network to input perturbations, potentially leading to underestimation of predictive variance. This is because lower precision weights and activations limit the expression of epistemic uncertainty – uncertainty stemming from a lack of knowledge. Specifically, the discretization of weights can reduce the entropy of the approximate posterior distribution, biasing uncertainty estimates. Therefore, strategies such as maintaining higher precision for key layers involved in uncertainty calculation or employing techniques like noise injection during quantization-aware training are often necessary to preserve the quality of uncertainty estimates in quantized BNNs.

Stochastic Variational Inference (SVI) provides a scalable approach to approximate Bayesian inference in neural networks, and is particularly advantageous when applied to quantized Bayesian Neural Networks (BNNs). Unlike Markov Chain Monte Carlo (MCMC) methods, SVI utilizes stochastic optimization to find a variational distribution $q(\mathbf{w})$ that minimizes the Kullback-Leibler (KL) divergence from the true posterior $p(\mathbf{w}|\mathbf{D})$, where $\mathbf{w}$ represents the model weights and $\mathbf{D}$ the observed data. The stochastic nature of SVI, achieved through mini-batch updates, allows for efficient computation, crucial given the increased computational demands often associated with Bayesian methods. Furthermore, the ability to utilize gradient-based optimization makes SVI well-suited to the low-precision arithmetic resulting from quantization, mitigating some of the accuracy loss typically observed when reducing numerical precision.

Logarithmic quantization of the standard deviation parameter minimizes error amplification during backpropagation, particularly for small values, outperforming linear quantization which suffers from substantially higher relative errors due to the 1/σ² term in log-probability calculations.

Refining Uncertainty: Fine-Grained Variational Quantization

Variational Parameter Quantization and Sampled Parameter Quantization are techniques designed to reduce the computational and storage demands of Stochastic Variational Inference (SVI). Variational Parameter Quantization directly reduces the precision of the parameters defining the variational distribution, such as the means and standard deviations of the approximate posterior. This reduces the number of bits required to store and update these parameters during training. Sampled Parameter Quantization focuses on reducing the size of the parameter vectors used in the SVI optimization process, effectively reducing the number of parameters updated in each iteration. Both methods offer targeted compression by selectively quantizing specific parameters within the SVI framework, allowing for a trade-off between model size/speed and accuracy.

Joint quantization in variational inference combines the compression of variational parameters and sampled parameters to achieve higher compression rates than either technique used independently. However, this approach necessitates careful design considerations to mitigate potential accuracy loss. The combined quantization process introduces increased error accumulation, requiring strategies such as optimized quantization levels, adaptive step sizes, or noise injection to preserve model performance. Specifically, the interaction between quantized parameters and sampled values must be considered during the design of the quantization scheme to avoid introducing bias or instability into the inference process. Failure to address these interactions can lead to significant degradation in the quality of the approximate posterior and reduced predictive performance.

Logarithmic quantization offers improved performance when applied to parameters naturally expressed in log-space, notably standard deviations ($\sigma$) within variational inference. This technique leverages the observation that parameters governing scales, like standard deviations, often span several orders of magnitude. By quantizing the logarithm of the parameter rather than the parameter itself, logarithmic quantization provides a more uniform distribution of quantization error. This is because equal-sized quantization steps in log-space translate to approximately equal percentage changes in the original parameter space, reducing the impact of quantization on model accuracy, particularly for small parameter values where precision is critical. The method effectively allocates more quantization levels to smaller values and fewer to larger values, resulting in a more efficient representation and reduced information loss compared to linear quantization for scale parameters.

A comprehensive evaluation of a model’s reliability necessitates distinguishing between different sources of uncertainty. Aleatoric uncertainty, inherent to the data itself and irreducible even with perfect knowledge, can be quantified through measures like $Softmax$ entropy, reflecting the noise within the observations. However, equally important is epistemic uncertainty, arising from the model’s lack of knowledge – essentially, what the model doesn’t know. This is often assessed using Mutual Information, which gauges how much the model’s predictions change with variations in its own parameters. By quantifying both aleatoric and epistemic uncertainty, researchers gain a more nuanced understanding of a model’s limitations and potential failure points, thereby enabling a more robust assessment of its overall trustworthiness and informing strategies for improvement.

JQA leverages uniform quantization generally, but employs logarithmic quantization for standard deviation parameters to enable precise representation across a broad range of values.

The Impact of Precision: Disentangling Uncertainty

Rectified Linear Unit (ReLU) activation functions, while computationally efficient, present challenges when combined with the precision reduction of quantization techniques. Studies reveal that as model weights and activations are compressed into fewer bits, ReLU’s ability to effectively separate aleatoric from epistemic uncertainty diminishes. This is because ReLU outputs zero for negative inputs, creating a “dead neuron” effect that can obscure the model’s confidence and hinder its ability to accurately represent both data uncertainty – inherent noise – and model uncertainty – a lack of knowledge. Consequently, the model’s uncertainty estimates become less reliable, potentially leading to overconfident predictions and reduced robustness, particularly at extremely low bit-widths like 2-bit quantization where the separation of these uncertainty types essentially collapses. This highlights a trade-off between computational efficiency and the quality of uncertainty estimation when employing ReLU with quantized neural networks.

Research demonstrates that employing SoftPlus activation functions within neural networks yields notably improved disentanglement of uncertainty compared to traditional ReLU functions. This enhancement is critical for generating more reliable uncertainty estimates, allowing for a more nuanced understanding of a model’s confidence in its predictions. While ReLU can struggle to effectively separate aleatoric – inherent noise in the data – from epistemic uncertainty – stemming from a lack of knowledge – SoftPlus facilitates a clearer distinction. This separation is quantified through metrics like softmax entropy and mutual information, revealing that SoftPlus-based models maintain a more robust representation of both uncertainty types, even under conditions of aggressive quantization. Consequently, the utilization of SoftPlus contributes to a system’s ability to identify not what it predicts, but how sure it is about those predictions, a crucial capability for safety-critical applications and reliable decision-making.

Investigations into model performance under increasingly constrained precision reveal critical limitations with quantization. While 3-bit quantization still allows for reasonable performance, a noticeable degradation in accuracy begins to emerge. However, the most significant issue arises at 2-bit quantization, where the model’s ability to distinguish between different types of uncertainty-specifically, aleatoric and epistemic-completely breaks down. This ‘collapse of uncertainty separation’ indicates the model can no longer reliably assess its own confidence, rendering its predictions potentially misleading and highlighting a fundamental precision barrier for deploying these models in safety-critical applications. The findings suggest that maintaining even a modest level of precision is vital for preserving meaningful uncertainty estimates and ensuring robust model behavior, particularly when dealing with complex or ambiguous data.

Using SoftPlus activation consistently yields better uncertainty calibration in Dirty-MNIST with 7-bit SPQ quantization, as evidenced by clearer clustering and separation of aleatoric and epistemic uncertainty compared to ReLU.

Towards Analog Bayesian Machines: A Future of Efficient Inference

Analog Bayesian Machines represent a paradigm shift in neural network design, prioritizing energy efficiency through the deliberate use of low-precision computation and inherent stochasticity. Unlike conventional digital computers that rely on precise calculations, these machines embrace approximation and randomness, mirroring the probabilistic nature of Bayesian inference. By operating with reduced bit-widths-often just a few bits-and employing physical stochasticity, such as thermal noise or transistor variations, Analog Bayesian Machines drastically reduce energy consumption. This approach isn’t about sacrificing accuracy; rather, it’s about recognizing that many real-world tasks don’t require the absolute precision of $32$-bit floating-point numbers. The resulting architecture offers the potential for orders-of-magnitude improvements in energy efficiency, opening doors to deploying complex Bayesian neural networks on resource-constrained devices and enabling truly pervasive intelligence.

KL-Annealing emerges as a powerful regularization strategy when training Bayesian Neural Networks (BNNs), proving remarkably resilient even when those networks operate with severely quantized weights and activations. This technique introduces a gradually increasing Kullback-Leibler (KL) divergence penalty during training, effectively preventing the posterior distributions from collapsing to sharp, overconfident predictions. By carefully balancing the trade-off between model fit and distributional spread, KL-Annealing encourages well-calibrated uncertainty estimates. Crucially, its efficacy extends to the quantized domain-where traditional optimization methods often struggle-allowing for the creation of highly efficient BNNs without sacrificing the benefits of robust uncertainty quantification. This makes it a particularly valuable tool for deploying Bayesian methods on resource-constrained devices, enabling reliable predictions even with limited computational power and memory.

Uniform quantization serves as a fundamental benchmark in the pursuit of energy-efficient Bayesian neural networks (BNNs). This technique simplifies the representation of neural network weights and activations by mapping a continuous range of values to a discrete set of levels with equal spacing. While potentially introducing significant quantization error, uniform quantization establishes a clear performance floor against which more complex quantization schemes – such as those employing non-uniform step sizes or learned quantization levels – can be evaluated. By providing a straightforward and easily implementable baseline, researchers can objectively assess the benefits of increased algorithmic complexity in terms of accuracy, energy consumption, and robustness. Essentially, the simplicity of uniform quantization allows for focused analysis of whether advanced techniques truly justify their computational overhead, driving innovation in the field of efficient uncertainty quantification and deployment of BNNs in resource-constrained environments.

The progression of uncertainty quantification is increasingly reliant on the creation of Bayesian Neural Networks (BNNs) engineered for practical deployment, demanding both computational efficiency and resilience to real-world data variations. Current research prioritizes minimizing the energy footprint of these networks – crucial for edge computing and resource-constrained devices – without sacrificing their ability to accurately estimate predictive uncertainties. This necessitates exploring novel architectures and training methodologies that embrace low-precision computation and stochasticity. Successful development in this area promises to unlock the full potential of BNNs in critical applications, ranging from autonomous robotics and medical diagnosis to financial modeling and climate prediction, where reliable uncertainty estimates are paramount for informed decision-making and risk mitigation. The pursuit of robust and efficient BNNs represents a significant step towards truly intelligent systems capable of operating reliably in complex and unpredictable environments.

The Dirty-MNIST dataset, comprising MNIST, Ambiguous-MNIST, and Fashion-MNIST examples, was used to evaluate in-domain prediction quality and both aleatoric and epistemic uncertainty.

The pursuit of quantized Bayesian Neural Networks, as detailed in this work, echoes a fundamental mathematical principle. The researchers demonstrate a commitment to preserving not merely the result of computation, but the inherent understanding of its limitations-the uncertainty. This aligns perfectly with the spirit of rigorous proof. As Paul Erdős once stated, “A mathematician knows a lot of things, but he doesn’t know everything.” This sentiment encapsulates the core of Bayesian methods and is powerfully manifested in the paper’s success in maintaining accurate uncertainty estimates even with aggressive quantization. The ability to approach lower precision – to let N approach infinity in terms of model compression – without sacrificing the validity of uncertainty quantification is a testament to the elegance of the approach and the careful consideration of invariant properties within the system.

What’s Next?

The demonstrated resilience of Bayesian Neural Networks to aggressive quantization is, predictably, not a free lunch. While maintaining calibrated uncertainty estimates at such low precision is commendable, it merely shifts the burden of proof. The current work addresses what can be quantized, not why it works. A satisfying solution would reveal the underlying mathematical invariants preserved during this process – if it feels like magic, one hasn’t revealed the invariant. The exploration of alternative quantization schemes, beyond uniform approaches, remains conspicuously absent; a more nuanced mapping of weights to low-precision representations might yield even greater compression without sacrificing epistemic rigor.

Furthermore, the focus on image classification, while practical, represents a narrow slice of the potential application space. The generalization of these findings to other data modalities – time series, natural language – and more complex model architectures warrants investigation. The interplay between quantization and other model compression techniques, such as pruning and knowledge distillation, is similarly underexplored. A truly elegant solution will not simply shrink the model, but fundamentally simplify it, revealing a core computational efficiency.

Ultimately, the pursuit of low-precision Bayesian inference isn’t merely about deploying models on edge devices. It’s about forcing a deeper understanding of the information content within neural networks. If a model can perform well with only four bits per weight, what were those discarded bits truly representing? The answer, one suspects, lies not in empirical observation, but in a more formal theory of neural computation.

Original article: https://arxiv.org/pdf/2512.10602.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/