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1. From Optimization Dynamics to Generalization Bounds via {\L}ojasiewicz Gradient Inequality

   Fusheng Liu; Haizhao Yang; Soufiane Hayou; Qianxiao Li (October 2022, Transactions on machine learning research)

   Yiming Ying (Ed.)

   Optimization and generalization are two essential aspects of statistical machine learning. In this paper, we propose a framework to connect optimization with generalization by analyz- ing the generalization error based on the optimization trajectory under the gradient flow algorithm. The key ingredient of this framework is the Uniform-LGI, a property that is generally satisfied when training machine learning models. Leveraging the Uniform-LGI, we first derive convergence rates for gradient flow algorithm, then we give generalization bounds for a large class of machine learning models. We further apply our framework to three distinct machine learning models: linear regression, kernel regression, and two-layer neural networks. Through our approach, we obtain generalization estimates that match or extend previous results. 

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2. Certifying the True Error: Machine Learning in Coq with Verified Generalization Guarantees

   https://doi.org/10.1609/aaai.v33i01.33012662  

   Bagnall, Alexander; Stewart, Gordon (July 2019, Proceedings of the AAAI Conference on Artificial Intelligence)

   We present MLCERT, a novel system for doing practical mechanized proof of the generalization of learning procedures, bounding expected error in terms of training or test error. MLCERT is mechanized in that we prove generalization bounds inside the theorem prover Coq; thus the bounds are machine checked by Coq’s proof checker. MLCERT is practical in that we extract learning procedures defined in Coq to executable code; thus procedures with proved generalization bounds can be trained and deployed in real systems. MLCERT is well documented and open source; thus we expect it to be usable even by those without Coq expertise. To validate MLCERT, which is compatible with external tools such as TensorFlow, we use it to prove generalization bounds on neural networks trained using TensorFlow on the extended MNIST data set. 

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3. High-Dimensional Optimization in Adaptive Random Subspaces

   Lacotte, Jonathan; Pilanci, Mert; Pavone, Marco (December 2019, Advances in neural information processing systems)

   We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly outperforms the oblivious sampling method, which is the common choice in the recent literature. The adaptive subspace can be efficiently generated by a correlated random matrix ensemble whose statistics mimic the input data. We prove that the improvement in the relative error of the solution can be tightly characterized in terms of the spectrum of the data matrix, and provide probabilistic upper-bounds. We then illustrate the consequences of our theory with data matrices of different spectral decay. Extensive experimental results show that the proposed approach offers significant speed ups in machine learning problems including logistic regression, kernel classification with random convolution layers and shallow neural networks with rectified linear units. Our analysis is based on convex analysis and Fenchel duality, and establishes connections to sketching and randomized matrix decompositions. 

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4. Testing Group Fairness via Optimal Transport Projections

   Si, Nian; Murthy, Karthyek; Blanchet, Jose; Nguyen, Viet Anh (January 2021, Proceedings of the 38th International Conference on Machine Learning)

   Meila, Marina and (Ed.)

   InProceedings{pmlr-v139-si21a, title = {}, author = {}, booktitle = {}, pages = {9649--9659}, We have developed a statistical testing framework to detect if a given machine learning classifier fails to satisfy a wide range of group fairness notions. Our test is a flexible, interpretable, and statistically rigorous tool for auditing whether exhibited biases are intrinsic to the algorithm or simply due to the randomness in the data. The statistical challenges, which may arise from multiple impact criteria that define group fairness and which are discontinuous on model parameters, are conveniently tackled by projecting the empirical measure to the set of group-fair probability models using optimal transport. This statistic is efficiently computed using linear programming, and its asymptotic distribution is explicitly obtained. The proposed framework can also be used to test for composite fairness hypotheses and fairness with multiple sensitive attributes. The optimal transport testing formulation improves interpretability by characterizing the minimal covariate perturbations that eliminate the bias observed in the audit. 

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5. Coresets for Data-efficient Training of Machine Learning Models

   Mirzasoleiman, R.; Bilmes, J.; Leskovec, J. (April 2020, International Conference on Machine Learning (ICML))

   Incremental gradient (IG) methods, such as stochastic gradient descent and its variants are commonly used for large scale optimization in machine learning. Despite the sustained effort to make IG methods more data-efficient, it remains an open question how to select a training data subset that can theoretically and practically perform on par with the full dataset. Here we develop CRAIG, a method to select a weighted subset (or coreset) of training data that closely estimates the full gradient by maximizing a submodular function. We prove that applying IG to this subset is guaranteed to converge to the (near)optimal solution with the same convergence rate as that of IG for convex optimization. As a result, CRAIG achieves a speedup that is inversely proportional to the size of the subset. To our knowledge, this is the first rigorous method for data-efficient training of general machine learning models. Our extensive set of experiments show that CRAIG, while achieving practically the same solution, speeds up various IG methods by up to 6x for logistic regression and 3x for training deep neural networks. 

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