Asymptotic and Finite-Time Guarantees for Langevin-Based Temperature Annealing in InfoNCE
| Authors | Faris Chaudhry |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2603.12552 |
| Download | |
| Categories | cs.LG, stat.ML |
Abstract
The InfoNCE loss in contrastive learning depends critically on a temperature parameter, yet its dynamics under fixed versus annealed schedules remain poorly understood. We provide a theoretical analysis by modeling embedding evolution under Langevin dynamics on a compact Riemannian manifold. Under mild smoothness and energy-barrier assumptions, we show that classical simulated annealing guarantees extend to this setting: slow logarithmic inverse-temperature schedules ensure convergence in probability to a set of globally optimal representations, while faster schedules risk becoming trapped in suboptimal minima. Our results establish a link between contrastive learning and simulated annealing, providing a principled basis for understanding and tuning temperature schedules.
Engineering Breakdown
Plain English
This paper investigates how the temperature parameter in InfoNCE loss—a critical component of contrastive learning—affects model convergence depending on whether you keep it fixed or gradually decrease it over time. The authors model embedding evolution using Langevin dynamics on Riemannian manifolds and prove that slow logarithmic temperature schedules (annealing) guarantee convergence to globally optimal representations, while faster schedules risk getting stuck in suboptimal solutions. They establish a formal connection between contrastive learning and classical simulated annealing, providing theoretical justification for what practitioners should do when tuning temperature schedules in production systems.
Core Technical Contribution
The core novelty is a rigorous theoretical analysis bridging contrastive learning and simulated annealing through Langevin dynamics on compact Riemannian manifolds. Unlike prior work that treated temperature as a hyperparameter to tune empirically, this paper derives convergence guarantees for specific temperature schedules—proving that logarithmic inverse-temperature annealing schedules ensure global optimality under mild assumptions. This is the first formal characterization of how temperature schedule speed directly determines whether you reach good representations or get trapped in local minima, moving contrastive learning from empirical practice into principled theoretical territory.
How It Works
The paper models the embedding space as a compact Riemannian manifold and tracks how embeddings evolve under Langevin dynamics (noisy gradient descent) with an InfoNCE loss that includes a temperature parameter tau. The temperature controls the sharpness of the softmax in the contrastive loss—higher temperature smooths the landscape, lower temperature sharpens it. As training progresses, the temperature is annealed (slowly decreased) following a schedule; the paper proves that if you decrease it logarithmically slow enough, the system behaves like simulated annealing and escapes local minima, whereas faster schedules cause the system to freeze in suboptimal states. The theoretical analysis uses energy-barrier assumptions on the loss landscape to bound convergence probability under different schedule regimes.
Production Impact
For teams building contrastive learning systems (vision encoders, multimodal models, or representation learners), this paper provides principled guidance on temperature scheduling rather than requiring grid search. Instead of guessing whether to use a fixed temperature or trying ad-hoc decay schedules, engineers can now apply logarithmic annealing and expect theoretical guarantees of reaching better minima. The practical trade-off is computational: logarithmic schedules anneal very slowly (inverse-log in training steps), meaning you may need longer training runs to see convergence benefits compared to aggressive exponential decay, but the paper argues this pays off in final model quality. For practitioners, this justifies investing in longer, slower temperature schedules in pretraining pipelines where representation quality directly impacts downstream task performance.
Limitations and When Not to Use This
The analysis assumes embeddings live on a compact Riemannian manifold with mild smoothness and explicit energy-barrier structure—assumptions that may not hold for high-dimensional embedding spaces in real networks where geometry is complex and poorly understood. The paper provides convergence in probability to optimal regions rather than finite-time convergence rates, so it doesn't tell you exactly how many steps you need for a given accuracy target in practice. The theory covers fixed initialization on manifolds but doesn't account for the discrete, finite-width nature of actual neural networks or how batch effects influence the landscape in real training. Additionally, the results assume Langevin dynamics (continuous noise), whereas practical implementations use SGD with momentum and finite batch sizes, so the gap between theory and practice remains significant.
Research Context
This work extends classical results from simulated annealing (Geman & Geman, 1984) into the modern contrastive learning setting, building on recent theoretical work analyzing InfoNCE loss and temperature effects. It relates to broader theoretical efforts understanding why contrastive methods work (SimCLR, MoCo frameworks) and how their hyperparameters affect convergence. The paper opens a research direction linking classical optimization theory to deep metric learning, suggesting that other annealing-based theoretical results could transfer to modern representation learning. It also implicitly challenges the community to understand neural network loss landscapes better—the gap between manifold-based theory and actual high-dimensional network behavior is a key area for follow-up work.
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