DualHash: A Stochastic Primal-Dual Algorithm with Theoretical Guarantee for Deep Hashing

TL;DR

This work tackles the core challenge of deep hashing with W-type regularization by formulating a two-block finite-sum problem and applying Fenchel duality to convert the nonconvex regularization into a convex dual, enabling closed-form proximal updates. The authors propose DualHash, a stochastic primal-dual algorithm with two practical variants: DualHash-StoM (momentum-based) and DualHash-StoRM (variance-reduced via STORM), achieving non-asymptotic convergence guarantees of $igO( ext{eps}^{-4})$ and $igO( ext{eps}^{-3})$ oracle complexity, respectively. The method shows superior retrieval performance and lower quantization error across CIFAR-10, NUS-WIDE, and ImageNet-100, demonstrating robustness to hyperparameters and scalability to large datasets. The results provide a rigorous optimization framework for nonconvex, nonsmooth regularization in deep hashing, with potential applicability to neural network quantization and sparse regularized learning.

Abstract

Deep hashing converts high-dimensional feature vectors into compact binary codes, enabling efficient large-scale retrieval. A fundamental challenge in deep hashing stems from the discrete nature of quantization in generating the codes. W-type regularizations, such as $||z|-1|$, have been proven effective as they encourage variables toward binary values. However, existing methods often directly optimize these regularizations without convergence guarantees. While proximal gradient methods offer a promising solution, the coupling between W-type regularizers and neural network outputs results in composite forms that generally lack closed-form proximal solutions. In this paper, we present a stochastic primal-dual hashing algorithm, referred to as DualHash, that provides rigorous complexity bounds. Using Fenchel duality, we partially transform the nonconvex W-type regularization optimization into the dual space, which results in a proximal operator that admits closed-form solutions. We derive two algorithm instances: a momentum-accelerated version with $\mathcal{O}(\varepsilon^{-4})$ complexity and an improved $\mathcal{O}(\varepsilon^{-3})$ version using variance reduction. Experiments on three image retrieval databases demonstrate the superior performance of DualHash.

Figures (14)

• Figure 1: The V-type (left) versus W-type (right) regularizations.
• Figure 2: Comparison between deep learning for hashing and classification
• Figure 3: W-type regularization function $h(z) = \lambda ||z| - 1|$ and its conjugate function $h^*(x)$.
• Figure 4: Training loss and the mAP results of DualHash and baselines with 64 bits on CIFAR-10 and NUS-WIDE datasets.
• Figure 5: Retrieval performance of DualHash and baselines on CIFAR-10 and NUS-WIDE datasets. (a), (c): AP@topK with 64 bits; (b), (d): AP@r(=2) with different numbers of bits.

Theorems & Definitions (42)

• Remark 4.1
• Remark 4.2
• Lemma 1
• Proof 1
• Remark 5.1
• Lemma 2
• Proof 2
• Theorem 1
• Theorem 2
• Remark 5.2