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Asymptotic and Finite-Time Guarantees for Langevin-Based Temperature Annealing in InfoNCE

Conference: NeurIPS 2025 (Optimization for Machine Learning Workshop)
arXiv: 2603.12552
Code: None
Area: Self-Supervised Learning / Optimization Theory
Keywords: Contrastive Learning, InfoNCE, Temperature Annealing, Langevin Dynamics, Simulated Annealing

TL;DR

By modeling embedding evolution as Langevin dynamics on a compact Riemannian manifold, this paper proves that the convergence guarantees of classical simulated annealing extend to the temperature scheduling setting in contrastive learning: a sufficiently slow logarithmic inverse-temperature schedule guarantees probabilistic convergence to the globally optimal representation set, whereas faster schedules risk trapping the system in suboptimal minima.

Background & Motivation

The InfoNCE loss is the core objective function in contrastive learning (e.g., SimCLR, MoCo), defined as:

\[\mathcal{L}_{\text{InfoNCE}} = -\log \frac{\exp(z_i \cdot z_j^+ / \tau)}{\sum_{k=1}^{K} \exp(z_i \cdot z_k / \tau)}\]

The temperature parameter \(\tau\) plays a critical role in contrastive learning: - Low temperature (\(\tau \to 0\)): the loss becomes more sensitive to hard negatives, but gradients may become unstable. - High temperature (\(\tau \to \infty\)): the loss approaches uniformity and the learning signal weakens. - Temperature annealing: gradually decreasing \(\tau\) from high to low values theoretically helps avoid early entrapment in local optima.

Nevertheless, a rigorous theoretical comparison between fixed-temperature and annealing schedules has been lacking. This paper establishes, for the first time, a formal theoretical connection between temperature scheduling in contrastive learning and classical simulated annealing.

Method

Overall Architecture

The analytical framework consists of three levels: 1. Dynamics modeling: the embedding update process in contrastive learning is modeled as Langevin diffusion on a compact Riemannian manifold \(\mathcal{M}\). 2. Temperature schedule analysis: the temperature parameter of InfoNCE is treated as an analog of the inverse temperature in simulated annealing. 3. Convergence proofs: classical simulated annealing results (Hajek, 1988; Holley et al., 1989) are leveraged to derive convergence guarantees.

Langevin Dynamics: the evolution of embedding \(z_t\) satisfies the stochastic differential equation:

\[dz_t = -\nabla V(z_t) dt + \sqrt{2/\beta(t)} \, dW_t\]

where \(V(z)\) is the potential function induced by the InfoNCE loss, \(\beta(t)\) is a time-dependent inverse temperature, and \(W_t\) is a Brownian motion.

Key Designs

Core Assumptions: 1. Compactness: the embedding space is a compact Riemannian manifold (e.g., the unit sphere \(\mathbb{S}^{d-1}\)), naturally satisfied after \(\ell_2\) normalization in contrastive learning. 2. Smoothness: the potential function \(V\) satisfies Lipschitz continuity and certain regularity conditions. 3. Energy barrier assumption: the depth of the deepest energy barrier \(H^*\) determines the required annealing rate for convergence.

Theorem 1 (Asymptotic Convergence): if the inverse-temperature schedule satisfies \(\beta(t) \geq \frac{H^* + \epsilon}{\log(t+2)}\) (logarithmic growth), then the embedding converges in probability to the globally optimal representation set \(\mathcal{M}^*\):

\[\lim_{t \to \infty} P(z_t \in \mathcal{M}^* \setminus B_\delta) = 0, \quad \forall \delta > 0\]

Theorem 2 (Finite-Time Guarantee): within a finite horizon \(T\), the logarithmic schedule yields an \(\epsilon\)-optimal solution with probability at least \(1 - C \cdot T^{-(H^*/\epsilon - 1)}\).

Theorem 3 (Failure of Fast Schedules): if \(\beta(t)\) grows faster than the logarithmic rate (e.g., polynomial \(\beta(t) = t^\alpha\)), there exists a positive probability of becoming trapped in a suboptimal local minimum.

Loss & Training

This paper is a purely theoretical work and does not involve specific training strategies. The core contribution is establishing the following correspondence:

Simulated Annealing Contrastive Learning
Energy function \(E(x)\) InfoNCE potential \(V(z)\)
State space Embedding manifold (unit sphere)
Temperature \(T\) InfoNCE temperature \(\tau\)
Globally optimal configuration Optimal contrastive representation
Logarithmic cooling schedule Logarithmic temperature annealing

Key Experimental Results

Main Results

As a workshop paper with a theoretical focus, the experimental section is relatively concise. The authors validate the theoretical conclusions on synthetic data.

Convergence Comparison of Different Temperature Scheduling Strategies:

Annealing Strategy Schedule Form Converges to Global Optimum Convergence Speed
Fixed low temperature \(\tau = 0.05\) No (trapped in local minimum) —
Fixed high temperature \(\tau = 0.5\) No (converges to suboptimal solution) —
Logarithmic annealing \(\tau(t) = \tau_0 / \log(t+2)\) Yes Slow but stable
Linear annealing \(\tau(t) = \tau_0 (1 - t/T)\) Sometimes (initialization-dependent) Faster but not guaranteed
Exponential annealing \(\tau(t) = \tau_0 \cdot \gamma^t\) No (overcooling) Fast but high failure rate

Effect of Energy Barrier Height on Convergence:

Energy Barrier \(H^*\) Steps Required (Log Annealing) Success Rate (Linear Annealing) Optimal Fixed \(\tau^*\)
Low (\(H^* = 1\)) \(\sim 10^3\) steps 85% 0.1
Medium (\(H^* = 5\)) \(\sim 10^5\) steps 52% 0.07
High (\(H^* = 10\)) \(\sim 10^8\) steps 18% 0.05

Ablation Study

  • Effect of embedding dimension: in high-dimensional embedding spaces, energy barriers tend to be lower (a blessing of dimensionality), accelerating convergence of logarithmic annealing.
  • Number of negative samples: increasing \(K\) in InfoNCE smooths the potential function and reduces the effective energy barrier.
  • Manifold curvature: convergence is faster on compact manifolds with higher positive curvature, consistent with the strong empirical performance of contrastive learning on the unit sphere.

Key Findings

  1. Logarithmic scheduling is necessary and sufficient: analogous to the classical result in simulated annealing, temperature annealing in contrastive learning also requires logarithmic slowness.
  2. Limitations of fixed temperature: regardless of the chosen value, a fixed temperature may fail to reach certain global optima.
  3. Limited practical applicability but profound theoretical significance: logarithmic annealing is too slow for practical use, but the theoretical connection provides a principled foundation for designing better annealing strategies.

Highlights & Insights

  • An elegant theoretical connection: the paper is the first to rigorously establish a mathematical equivalence between temperature scheduling in contrastive learning and simulated annealing.
  • Classical results in a modern context: the classical theorems of Hajek (1988) et al. on simulated annealing are transplanted into the deep learning setting.
  • Riemannian manifold perspective: treating the embedding space as a manifold rather than a Euclidean space is more faithful to the \(\ell_2\) normalization commonly used in contrastive learning.
  • A theoretical answer to the practical question of what the optimal temperature parameter should be.

Limitations & Future Work

  1. Limited scope as a workshop paper: the completeness of the analysis is constrained, and several technical details are deferred to future work.
  2. Practical infeasibility of logarithmic annealing: convergence time grows exponentially with the energy barrier, rendering such a slow schedule infeasible in practice.
  3. Applicability of the Langevin dynamics approximation: SGD updates only approximately satisfy Langevin dynamics, and the gap requires quantification.
  4. Finite negative sample correction not addressed: practical InfoNCE relies on a finite number of negatives, which deviates from the continuous distribution assumed in theory.
  5. Absence of large-scale experimental validation: the effect of temperature annealing is not evaluated on standard benchmarks such as ImageNet.
  • SimCLR (Chen et al., 2020): empirically finds \(\tau = 0.1\) to be optimal but provides no theoretical explanation.
  • Simulated annealing theory (Hajek, 1988): logarithmic cooling is a necessary and sufficient condition for global convergence.
  • Learnable temperature (Zhang et al., 2021): treats temperature as a learnable parameter, complementing the annealing approach studied here.
  • This paper suggests that future work could explore "nearly logarithmic" yet practically feasible annealing strategies, such as piecewise logarithmic or adaptive annealing schedules.

Rating

  • Theoretical Depth: ⭐⭐⭐⭐⭐
  • Experimental Thoroughness: ⭐⭐⭐
  • Novelty: ⭐⭐⭐⭐⭐
  • Practicality: ⭐⭐⭐
  • Writing Quality: ⭐⭐⭐⭐