The Complexity of Finding Local Optima in Contrastive Learning
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2025
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| _version_ | 1866911166072094720 |
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| author | Yan, Jingming Luo, Yiyuan Chatziafratis, Vaggos Panageas, Ioannis Shahkar, Parnian Stavroulakis, Stelios |
| author_facet | Yan, Jingming Luo, Yiyuan Chatziafratis, Vaggos Panageas, Ioannis Shahkar, Parnian Stavroulakis, Stelios |
| contents | Contrastive learning is a powerful technique for discovering meaningful data representations by optimizing objectives based on $\textit{contrastive information}$, often given as a set of weighted triplets $\{(x_i, y_i^+, z_{i}^-)\}_{i = 1}^m$ indicating that an "anchor" $x_i$ is more similar to a "positive" example $y_i$ than to a "negative" example $z_i$. The goal is to find representations (e.g., embeddings in $\mathbb{R}^d$ or a tree metric) where anchors are placed closer to positive than to negative examples. While finding $\textit{global}$ optima of contrastive objectives is $\mathsf{NP}$-hard, the complexity of finding $\textit{local}$ optima -- representations that do not improve by local search algorithms such as gradient-based methods -- remains open. Our work settles the complexity of finding local optima in various contrastive learning problems by proving $\mathsf{PLS}$-hardness in discrete settings (e.g., maximize satisfied triplets) and $\mathsf{CLS}$-hardness in continuous settings (e.g., minimize Triplet Loss), where $\mathsf{PLS}$ (Polynomial Local Search) and $\mathsf{CLS}$ (Continuous Local Search) are well-studied complexity classes capturing local search dynamics in discrete and continuous optimization, respectively. Our results imply that no polynomial time algorithm (local search or otherwise) can find a local optimum for various contrastive learning problems, unless $\mathsf{PLS}\subseteq\mathsf{P}$ (or $\mathsf{CLS}\subseteq \mathsf{P}$ for continuous problems). Even in the unlikely scenario that $\mathsf{PLS}\subseteq\mathsf{P}$ (or $\mathsf{CLS}\subseteq \mathsf{P}$), our reductions imply that there exist instances where local search algorithms need exponential time to reach a local optimum, even for $d=1$ (embeddings on a line). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Complexity of Finding Local Optima in Contrastive Learning Yan, Jingming Luo, Yiyuan Chatziafratis, Vaggos Panageas, Ioannis Shahkar, Parnian Stavroulakis, Stelios Machine Learning Computational Complexity Optimization and Control Contrastive learning is a powerful technique for discovering meaningful data representations by optimizing objectives based on $\textit{contrastive information}$, often given as a set of weighted triplets $\{(x_i, y_i^+, z_{i}^-)\}_{i = 1}^m$ indicating that an "anchor" $x_i$ is more similar to a "positive" example $y_i$ than to a "negative" example $z_i$. The goal is to find representations (e.g., embeddings in $\mathbb{R}^d$ or a tree metric) where anchors are placed closer to positive than to negative examples. While finding $\textit{global}$ optima of contrastive objectives is $\mathsf{NP}$-hard, the complexity of finding $\textit{local}$ optima -- representations that do not improve by local search algorithms such as gradient-based methods -- remains open. Our work settles the complexity of finding local optima in various contrastive learning problems by proving $\mathsf{PLS}$-hardness in discrete settings (e.g., maximize satisfied triplets) and $\mathsf{CLS}$-hardness in continuous settings (e.g., minimize Triplet Loss), where $\mathsf{PLS}$ (Polynomial Local Search) and $\mathsf{CLS}$ (Continuous Local Search) are well-studied complexity classes capturing local search dynamics in discrete and continuous optimization, respectively. Our results imply that no polynomial time algorithm (local search or otherwise) can find a local optimum for various contrastive learning problems, unless $\mathsf{PLS}\subseteq\mathsf{P}$ (or $\mathsf{CLS}\subseteq \mathsf{P}$ for continuous problems). Even in the unlikely scenario that $\mathsf{PLS}\subseteq\mathsf{P}$ (or $\mathsf{CLS}\subseteq \mathsf{P}$), our reductions imply that there exist instances where local search algorithms need exponential time to reach a local optimum, even for $d=1$ (embeddings on a line). |
| title | The Complexity of Finding Local Optima in Contrastive Learning |
| topic | Machine Learning Computational Complexity Optimization and Control |
| url | https://arxiv.org/abs/2509.16898 |