[Paper Review] Large-scale Stochastic Optimization of NDCG Surrogates for Deep Learning with Provable Convergence

Zi-Hao Qiu, Quanqi Hu+3 more|arXiv (Cornell University)|Feb 24, 2022

Recommender Systems and Techniques4 citations

TL;DR

This paper proposes the first stochastic optimization algorithms for NDCG and top-K NDCG surrogates in deep learning with provable convergence guarantees. By formulating NDCG optimization as a finite-sum coupled compositional problem and top-K NDCG as a bilevel compositional problem, the authors develop efficient stochastic algorithms with per-iteration complexity scaling only with mini-batch size, achieving O(1/ε⁴) convergence rate and outperforming prior methods on ranking benchmarks.

ABSTRACT

NDCG, namely Normalized Discounted Cumulative Gain, is a widely used ranking metric in information retrieval and machine learning. However, efficient and provable stochastic methods for maximizing NDCG are still lacking, especially for deep models. In this paper, we propose a principled approach to optimize NDCG and its top-$K$ variant. First, we formulate a novel compositional optimization problem for optimizing the NDCG surrogate, and a novel bilevel compositional optimization problem for optimizing the top-$K$ NDCG surrogate. Then, we develop efficient stochastic algorithms with provable convergence guarantees for the non-convex objectives. Different from existing NDCG optimization methods, the per-iteration complexity of our algorithms scales with the mini-batch size instead of the number of total items. To improve the effectiveness for deep learning, we further propose practical strategies by using initial warm-up and stop gradient operator. Experimental results on multiple datasets demonstrate that our methods outperform prior ranking approaches in terms of NDCG. To the best of our knowledge, this is the first time that stochastic algorithms are proposed to optimize NDCG with a provable convergence guarantee. Our proposed methods are implemented in the LibAUC library at https://libauc.org/.

Motivation & Objective

• To address the lack of efficient, provably convergent stochastic optimization methods for maximizing NDCG in deep learning.
• To overcome the high per-iteration cost of existing NDCG optimization methods, which scale with the total number of items n.
• To develop a principled stochastic optimization framework for both standard NDCG and top-K NDCG surrogates with theoretical convergence guarantees.
• To improve practical effectiveness in deep learning through initial warm-up and stop gradient strategies.

Proposed method

• Formulates NDCG surrogate optimization as a finite-sum coupled compositional optimization (FCCO) problem.
• Develops a novel stochastic algorithm for FCCO with O(1/ε⁴) convergence rate, improving upon prior O(1/ε⁵) bounds.
• Proposes a bilevel compositional optimization formulation for top-K NDCG, incorporating multiple lower-level selection problems.
• Introduces a smoothing technique for non-smooth selection operators in top-K NDCG optimization.
• Employs a mini-batch sampling strategy over lower-level problems to reduce per-iteration complexity.
• Incorporates practical strategies: initial warm-up for better initialization and stop gradient operator to simplify top-K NDCG optimization.

Experimental results

Research questions

• RQ1 Can stochastic optimization for NDCG be achieved with per-iteration complexity independent of the total number of items n?
• RQ2 Is it possible to develop a provably convergent stochastic algorithm for NDCG surrogate optimization in non-convex deep learning settings?
• RQ3 How can the top-K NDCG optimization problem, involving a non-differentiable selection operator, be effectively formulated and solved?
• RQ4 What practical strategies enhance the convergence and performance of NDCG optimization in deep learning?
• RQ5 Can the proposed methods achieve better NDCG performance than existing state-of-the-art ranking approaches?

Key findings

• The proposed stochastic algorithm for NDCG surrogate optimization achieves an O(1/ε⁴) convergence rate, improving upon the O(1/ε⁵) rate of prior work.
• The algorithm's per-iteration complexity scales only with mini-batch size B, not with the total number of items n, enabling large-scale deep learning applications.
• For top-K NDCG, the bilevel compositional formulation with smoothed selection enables efficient stochastic optimization with the same O(1/ε⁴) convergence rate.
• The initial warm-up strategy significantly improves the quality of the initial solution, leading to faster convergence.
• The stop gradient operator simplifies the optimization of the top-K NDCG surrogate and enhances training stability.
• Empirical results on learning to rank and recommender system tasks show consistent improvements in NDCG over prior methods.