Trainable Inverse Temperature & Bias

• Trainable inverse temperature and bias are parameterization strategies that adjust distribution sharpness and model offsets for enhanced calibration and learning dynamics.
• They actively improve optimization by adapting sensitivity in loss functions, regularizing via temperature scaling, and shifting decision thresholds with bias terms.
• These strategies are applied in contrastive learning, Bayesian inference, and energy-based models to promote robust generalization and efficient sampling.

Trainable inverse temperature and bias refer to parameterization strategies in statistical models and machine learning systems that adapt, fit, or optimize the "temperature" (often appearing as an inverse temperature parameter β or scaling coefficient) and explicit additive bias terms. These parameters govern sensitivity and shape in probability distributions, loss functions, and regularization, with direct implications for generalization, optimization dynamics, phase transitions, and calibration. Across Bayesian inference, energy-based models, contrastive learning, and deep neural networks, trainable inverse temperature and bias provide powerful levers for controlling uncertainty, enabling phase transitions, improving sampling, and optimizing model representations.

1. Theoretical Foundations of Inverse Temperature and Bias

Inverse temperature (β) is foundational in statistical mechanics, where it tunes the trade-off between energy and entropy in Boltzmann distributions. Many modern statistical and machine learning frameworks lift this concept: inverse temperature or its reciprocal (i.e., temperature) regulates distribution sharpness, controls regularization strength, and mediates learning dynamics.

• In Bayesian inference, "On the correspondence between thermodynamics and inference" (LaMont et al., 2017) establishes a principled analogy where inverse temperature β is mapped to sample size N: large N concentrates the posterior, analogous to "cooling" a physical system. Bias terms (often additive constants in energy or score functions) serve to shift model outputs and can represent regularization offsets, intercepts, or modality-specific adjustments.
• In transformer architectures and modern Hopfield networks, the inverse temperature modulates the peakedness of the Softmax operator, governing phase transitions between global and pattern-specific minima (Koulischer et al., 2023).

Mathematically, β enters as a scaling parameter in exponentials:

• Boltzmann distribution: $p(x) = \exp(-\beta E(x)) / Z(\beta)$
• Softmax: $$\text{Softmax}(z; \beta) = \exp(\beta z_i) / \sum_j \exp(\beta z_j)$$
• Contrastive loss (Sigmoid): $$L(\theta, \phi; t, b) = \sum_i \log(1 + \exp(-t\langle U_i, V_i\rangle + b)) + \sum_{i \neq j} \log(1 + \exp(t\langle U_i, V_j\rangle - b))$$ (Bangachev et al., 23 Sep 2025)

Bias terms shift thresholds, margins, and partition points in models, and their trainability allows models to tailor alignment between subspaces and modalities.

2. Bayesian Inference, Objective Priors, and Learning Capacity

Within Bayesian frameworks, the inverse temperature (often appearing as β or as a scaling factor in the likelihood) is not merely a numerical curiosity. It determines how concentrated posteriors are and is intimately tied to model capacity and generalization.

• The learning capacity $C$ is introduced as a thermodynamic analogue (heat capacity), defined by $C \equiv -N^2 \partial_N^2(N F)$, where $F$ is a "free energy" analogue. For regular models, $C \approx K/2$ ($K$ = parameter count), reflecting equipartition; in singular models, $C$ drops, capturing the "frozen out" effective dimensionality (LaMont et al., 2017).
• The generalized principle of indifference (GPI) for setting objective priors allocates equal weight to every statistically distinguishable model at resolution N (sample size or effective inverse temperature). This adapts bias and temperature implicitly, resolving paradoxes and allowing learning in high-dimensional spaces.

Trainable temperature/bias thus provides a principled way to navigate the bias-variance trade-off: tuning β alters posterior concentration and adapts learning capacity dynamically according to model singularity, prior information, and sample size.

3. Regularization, Loss Landscapes, and Deep Neural Network Training

Inverse temperature and bias appear prominently in regularized learning systems and loss function design.

• In cross-entropy and Softmax settings, "Temperature check: theory and practice for training models with softmax-cross-entropy losses" (Agarwala et al., 2020) demonstrates that generalization and training dynamics are highly sensitive to the inverse temperature β, while initial logit scale is far less influential. Key findings:
  • Early learning dynamics collapse against rescaling the effective learning rate (η_eff ∼ β²), while nonlinear transition time scales inversely with β.
  • Tuning β within $10^{-2}$ to $10^1$ across architectures (e.g., WRN, ResNet-50, GRU) robustly enhances generalization, with optimal values architecture-dependent.
  • Lower β yields faster departure from linear regime and sometimes better peak performance at the cost of reduced stability.
• Trainable bias terms, when included in loss landscapes or as explicit offsets in representations, provide the flexibility to anchor decision boundaries, synchronize modalities, or counteract systematic trends.
• Layer-wise temperature balancing (learning-rate scheduling as temperature proxy) via TempBalance (Zhou et al., 2023) adapts effective temperature per layer using heavy-tailed self-regularization theory. The PL exponent α diagnostic ranks the "hot/cold" state of layers, leading to adaptive adjustment of bias and temperature (learning rate), mitigating model bias and promoting robust generalization.

4. Contrastive Representation Learning: Sigmoid Loss, Margin, and Modality Gap

Trainable inverse temperature and bias assume pivotal roles in contrastive learning, specifically in recent SigLIP (Sigmoid Loss in CLIP) models (Bangachev et al., 23 Sep 2025):

• The Sigmoid contrastive loss admits global minimizers as $(m,b)_{\text{rel}}$–constellations: geometric configurations parameterized by margin m and relative bias $b_{\text{rel}} = b/t$.
• Trainability of t (inverse temperature) amplifies margin effects: higher t sharpens the penalty on violations, driving the loss to zero exponentially for good $(m,b)$–constellations. Bias b shifts the center of the margin, controlling the threshold for alignment between modalities.
• The modality gap—alignment where embeddings for images and text occupy linearly separable subspaces—is a product of proper tuning of t and b, shown both theoretically and in experiments.

Reparameterizing the loss using explicit relative bias $b_{\text{rel}}$ enhances training dynamics and allows synchronization via adapters when one encoder is fixed.

5. Energy-Based Models, Annealing, and Physical Analogies

Sampling and inference in energy-based models (EBMs), physical simulation, and variational learning are deeply sensitive to inverse temperature and bias.

• In Boltzmann generators for molecular systems, temperature-annealed training (Schopmans et al., 31 Jan 2025) initializes flow-based models at high temperature (β low), then utilizes reweighting-based annealing to progressively adapt distribution to lower temperature (higher β). This staged process uses trainable β (explicit or implicit conditioning), enabling accurate sampling without mode collapse.
• Deep generative modelling of canonical ensembles (Li et al., 29 Apr 2024) incorporates temperature as a differentiable argument in explicit density models. The model $q_\theta(x,\beta)$ directly approximates the Boltzmann distribution over a continuous β range, allowing efficient and precise estimation of the free energy and its derivatives at arbitrary temperatures, which is essential for studying phase transitions.
• Diabatic quantum annealing (Kim et al., 11 Sep 2025) leverages analytic scheduling controls to produce Boltzmann samples at programmable inverse temperature. Systematic temperature misalignment in hardware is corrected via analytical rescaling.

These procedures highlight that trainable temperature (and bias, via reweighting or offset coupling) is fundamental for overcoming barriers in multimodal distribution sampling, quantifying uncertainty, and calibrating simulators in both classical and quantum domains.

6. Inverse Problems, Variational Inference, and Uncertainty Calibration

Trainable inverse temperature and bias extend to inverse problems and Bayesian optimization.

• In variational inference for inverse problems, scaling the ELBO by temperature T (Laves et al., 2021) introduces a mechanism for regularization adjustment. Bayesian optimization jointly tunes posterior temperature and prior scale, achieving robust performance and reliable uncertainty calibration in sparse-view CT reconstruction.
• In invertible generative models for imaging (Asim et al., 2019), the penalty on latent norm ($\|z\|_2^2$) directly acts as an "inverse temperature," balancing data fidelity and likelihood. The theoretical error bounds clarify the joint effect of measurement count, regularization strength (temperature), and bias in the recovery process.

Optimization and calibration of posterior temperature and bias are seen to alleviate overconfidence, mitigate bias, and enhance predictive quality and uncertainty quantification.

7. Thermodynamic Computing and Gradient-Based Parameter Adjustment

Trainable bias and effective inverse temperature underpin the design and training of thermodynamic computers (Whitelam, 18 Sep 2025):

• Device dynamics are governed by overdamped Langevin equations, with energy potentials parameterized by couplings and local biases. Training proceeds by maximizing the probability of generating target trajectories (teacher activations) via gradient descent on the Onsager–Machlup loss, adjusting both biases and couplings.
• Though temperature T is externally set, its impact emerges in dynamic "gain" and can be considered in parameter scaling strategies. Gradient-based adjustment of bias and coupling parameters aligns device output with desired computations, realizing energy advantages of up to seven orders of magnitude compared to digital implementations.

Trainable inverse temperature and bias thus provide essential control for efficient, low-power, physical computation aligned with high-level task dynamics.

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In sum, trainable inverse temperature and bias constitute critical degrees of freedom for tuning generalization, regularization, phase behavior, and calibration in contemporary machine learning, statistical inference, physical modeling, and hardware systems. Across architectures, the capacity to adapt these parameters—often by gradient-driven optimization or Bayesian search—yields robust performance, efficient computation, and deeper alignment between model and data structure.