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1. THB-Diff: a GPU-accelerated differentiable programming framework for THB-splines

   https://doi.org/10.1007/s00366-023-01929-1  

   Moola, Ajith; Balu, Aditya; Krishnamurthy, Adarsh; Pawar, Aishwarya (December 2023, Engineering with Computers)

   We have developed a differentiable programming framework for truncated hierarchical B-splines (THB-splines), which can be used for several applications in geometry modeling, such as surface fitting and deformable image registration, and can be easily integrated with geometric deep learning frameworks. Differentiable programming is a novel paradigm that enables an algorithm to be differentiated via automatic differentiation, i.e., using automatic differentiation to compute the derivatives of its outputs with respect to its inputs or parameters. Differentiable programming has been used extensively in machine learning for obtaining gradients required in optimization algorithms such as stochastic gradient descent (SGD). While incorporating differentiable programming with traditional functions is straightforward, it is challenging when the functions are complex, such as splines. In this work, we extend the differentiable programming paradigm to THB-splines. THB-splines offer an efficient approach for complex surface fitting by utilizing a hierarchical tensor structure of B-splines, enabling local adaptive refinement. However, this approach brings challenges, such as a larger computational overhead and the non-trivial implementation of automatic differentiation and parallel evaluation algorithms. We use custom kernel functions for GPU acceleration in forward and backward evaluation that are necessary for differentiable programming of THB-splines. Our approach not only improves computational efficiency but also significantly enhances the speed of surface evaluation compared to previous methods. Our differentiable THB-splines framework facilitates faster and more accurate surface modeling with local refinement, with several applications in CAD and isogeometric analysis. 

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2. Slice Sampling Reparameterization Gradients

   Zoltowski, David; Cai, Diana; Adams, Ryan P. (January 2021, Advances in neural information processing systems)

   Many probabilistic modeling problems in machine learning use gradient-based optimization in which the objective takes the form of an expectation. These problems can be challenging when the parameters to be optimized determine the probability distribution under which the expectation is being taken, as the na\"ive Monte Carlo procedure is not differentiable. Reparameterization gradients make it possible to efficiently perform optimization of these Monte Carlo objectives by transforming the expectation to be differentiable, but the approach is typically limited to distributions with simple forms and tractable normalization constants. Here we describe how to differentiate samples from slice sampling to compute \textit{slice sampling reparameterization gradients}, enabling a richer class of Monte Carlo objective functions to be optimized. Slice sampling is a Markov chain Monte Carlo algorithm for simulating samples from probability distributions; it only requires a density function that can be evaluated point-wise up to a normalization constant, making it applicable to a variety of inference problems and unnormalized models. Our approach is based on the observation that when the slice endpoints are known, the sampling path is a deterministic and differentiable function of the pseudo-random variables, since the algorithm is rejection-free. We evaluate the method on synthetic examples and apply it to a variety of applications with reparameterization of unnormalized probability distributions. 

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3. Entropy-Penalized Semidefinite Programming

   https://doi.org/10.24963/ijcai.2019/157  

   Krechetov, Mikhail; Marecek, Jakub; Maximov, Yury; Takac, Martin (August 2019, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence)

   Low-rank methods for semi-definite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schatten-norm penalties, which are difficult to implement in practice due to high computational efforts. In this paper, we propose Entropy-Penalized Semi-Definite Programming (EP-SDP), which provides a unified framework for a broad class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit an efficient numerical algorithm, having (almost) linear time complexity of the gradient computation; this makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems. 

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4. Linearly Constrained Bilevel Optimization: A Smoothed Implicit Gradient Approach

   Prashant Khanduri * 1 Ioannis Tsaknakis * 2 Yihua Zhang 3 Jia Liu 4 Sijia Liu 3 Jiawei Zhang 5 Mingyi Hong (August 2023, International Conference on Machine Learning)

   This work develops analysis and algorithms for solving a class of bilevel optimization problems where the lower-level (LL) problems have linear constraints. Most of the existing approaches for constrained bilevel problems rely on value function-based approximate reformulations, which suffer from issues such as non-convex and non-differentiable constraints. In contrast, in this work, we develop an implicit gradient-based approach, which is easy to implement, and is suitable for machine learning applications. We first provide an in-depth understanding of the problem, by showing that the implicit objective for such problems is in general non-differentiable. However, if we add some small (linear) perturbation to the LL objective, the resulting implicit objective becomes differentiable almost surely. This key observation opens the door for developing (deterministic and stochastic) gradient-based algorithms similar to the state-of-the-art ones for unconstrained bi-level problems. We show that when the implicit function is assumed to be stronglyconvex, convex, and weakly-convex, the resulting algorithms converge with guaranteed rate. Finally, we experimentally corroborate the theoretical findings and evaluate the performance of the proposed framework on numerical and adversarial learning problems. 

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5. A Two-Step Fixed-Point Proximity Algorithmfor a Class of Non-differentiable Optimization Models in Machine Learning

   Zheng Li, Guohui Song (September 2019, Journal of scientific computing)

   Sparse learning models are popular in many application areas. Objective functions in sparse learning models are usually non-smooth, which makes it difficult to solve them numerically. We develop a fast and convergent two-step iteration scheme for solving a class of non-differentiable optimization models motivated from sparse learning. To overcome the difficulty of the non-differentiability of the models, we first present characterizations of their solutions as fixed-points of mappings involving the proximity operators of the functions appearing in the objective functions. We then introduce a two-step fixed-point algorithm to compute the solutions. We establish convergence results of the proposed two-step iteration scheme and compare it with the alternating direction method of multipliers (ADMM). In particular, we derive specific two-step iteration algorithms for three models in machine learning: l1-SVM classification, l1-SVM regression, and the SVM classification with the group LASSO regularizer. Numerical experiments with some synthetic datasets and some benchmark datasets show that the proposed algorithm outperforms ADMM and the linear programming method in computational time and memory storage costs. 

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