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1. Extracting Dual Solutions via Primal Optimizers

   https://doi.org/10.4230/LIPICS.ITCS.2025.29  

   Carmon, Yair; Jambulapati, Arun; O'Carroll, Liam; Sidford, Aaron (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We provide a general method to convert a "primal" black-box algorithm for solving regularized convex-concave minimax optimization problems into an algorithm for solving the associated dual maximin optimization problem. Our method adds recursive regularization over a logarithmic number of rounds where each round consists of an approximate regularized primal optimization followed by the computation of a dual best response. We apply this result to obtain new state-of-the-art runtimes for solving matrix games in specific parameter regimes, obtain improved query complexity for solving the dual of the CVaR distributionally robust optimization (DRO) problem, and recover the optimal query complexity for finding a stationary point of a convex function. 

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2. Solving Non-smooth Constrained Programs with Lower Complexity than O(1/ε): A Primal-Dual Homotopy Smoothing Approach

   Wei, Xiaohan; Yu, Hao; Ling, Qing; Neely, Michael J. (December 2018, 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.)

   We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex. The best known iteration complexity solving such a non-smooth problem is O(ε−1). In this paper, we show that by leveraging a local error bound condition on the dual function, the proposed algorithm can achieve a better primal convergence time of O 􏰕ε−2/(2+β) log2(ε−1)􏰖, where β ∈ (0, 1] is a local error bound parameter. As an example application of the general algorithm, we show that the distributed geometric median problem, which can be formulated as a constrained convex program, has its dual function non-smooth but satisfying the aforementioned local error bound condition with β = 1/2, therefore enjoying a convergence time of O 􏰕ε−4/5 log2(ε−1)􏰖. This result improves upon the O(ε−1) convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm. 

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3. Strong Duality Relations in Nonconvex Risk-Constrained Learning

   https://doi.org/10.1109/CISS59072.2024.10480186  

   Kalogerias, Dionysis; Pougkakiotis, Spyridon (March 2024, IEEE)

   We establish strong duality relations for functional two-step compositional risk-constrained learning problems with multiple nonconvex loss functions and/or learning constraints, regardless of nonconvexity and under a minimal set of technical assumptions. Our results in particular imply zero duality gaps within the class of problems under study, both extending and improving on the state of the art in (risk-neutral) constrained learning. More specifically, we consider risk objectives/constraints which involve real-valued convex and positively homogeneous risk measures admitting dual representations with bounded risk envelopes, generalizing expectations and including popular examples, such as the conditional value-at-risk (CVaR), the meanabsolute deviation (MAD), and more generally all real-valued coherent risk measures on integrable losses as special cases. Our results are based on recent advances in risk-constrained nonconvex programming in infinite dimensions, which rely on a remarkable new application of J. J. Uhl’s convexity theorem, which is an extension of A. A. Lyapunov’s convexity theorem for general, infinite dimensional Banach spaces. By specializing to the risk-neutral setting, we demonstrate, for the first time, that constrained classification and regression can be treated under a unifying lens, while dispensing certain restrictive assumptions enforced in the current literature, yielding a new state-of-the-art strong duality framework for nonconvex constrained learning. 

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4. A Primal-Dual Subgradient Approach for Fair Meta Learning

   Zhao, Chen; Chen, Feng; Wang, Zhuoyi; Khan, Latifur (March 2021, Proceedings of the IEEE International Conference on Data Mining (ICDM 2020))

   null (Ed.)

   The problem of learning to generalize on unseen classes during the training step, also known as few-shot classification, has attracted considerable attention. Initialization based methods, such as the gradient-based model agnostic meta-learning (MAML) [1], tackle the few-shot learning problem by “learning to fine-tune”. The goal of these approaches is to learn proper model initialization so that the classifiers for new classes can be learned from a few labeled examples with a small number of gradient update steps. Few shot meta-learning is well-known with its fast-adapted capability and accuracy generalization onto unseen tasks [2]. Learning fairly with unbiased outcomes is another significant hallmark of human intelligence, which is rarely touched in few-shot meta-learning. In this work, we propose a novel Primal-Dual Fair Meta-learning framework, namely PDFM, which learns to train fair machine learning models using only a few examples based on data from related tasks. The key idea is to learn a good initialization of a fair model’s primal and dual parameters so that it can adapt to a new fair learning task via a few gradient update steps. Instead of manually tuning the dual parameters as hyperparameters via a grid search, PDFM optimizes the initialization of the primal and dual parameters jointly for fair meta-learning via a subgradient primal-dual approach. We further instantiate an example of bias controlling using decision boundary covariance (DBC) [3] as the fairness constraint for each task, and demonstrate the versatility of our proposed approach by applying it to classification on a variety of three real-world datasets. Our experiments show substantial improvements over the best prior work for this setting. 

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5. Faster Matchings via Learned Duals

   Dinitz, Michael; Im, Sungjin; Lavastida, Thomas; Moseley, Benjamin; Vassilvitskii, Sergei (January 2021, Advances in neural information processing systems)

   A recent line of research investigates how algorithms can be augmented with machine-learned predictions to overcome worst case lower bounds. This area has revealed interesting algorithmic insights into problems, with particular success in the design of competitive online algorithms. However, the question of improving algorithm running times with predictions has largely been unexplored. We take a first step in this direction by combining the idea of machine-learned predictions with the idea of "warm-starting" primal-dual algorithms. We consider one of the most important primitives in combinatorial optimization: weighted bipartite matching and its generalization to b-matching. We identify three key challenges when using learned dual variables in a primal-dual algorithm. First, predicted duals may be infeasible, so we give an algorithm that efficiently maps predicted infeasible duals to nearby feasible solutions. Second, once the duals are feasible, they may not be optimal, so we show that they can be used to quickly find an optimal solution. Finally, such predictions are useful only if they can be learned, so we show that the problem of learning duals for matching has low sample complexity. We validate our theoretical findings through experiments on both real and synthetic data. As a result we give a rigorous, practical, and empirically effective method to compute bipartite matchings. 

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