Implicit differentiation of tensor network algorithms

• URL: http://arxiv.org/abs/2607.15030v1
• Date: Thu, 16 Jul 2026 14:13:33 GMT
• Title: Implicit differentiation of tensor network algorithms
• Abstract summary: In this work, we adopt recent advances in implicit differentiation techniques to address challenges in PEPS optimization.<n>By reformulating the core step of the gradient computation in terms of a single characteristic equation for the contraction environment, we reduce the cost of the gradient computation.<n>We also demonstrate how this approach drastically simplifies the practical implementation of stable gradient-based PEPS optimization.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The current leading approach to the variational optimization of projected entangled-pair states (PEPS) is based on automatic differentiation, which allows for a convenient evaluation of the energy gradient with respect to the local variational degrees of freedom. However, evaluating the energy gradient not only remains a major computational bottleneck of the optimization procedure, but also suffers from frequent numerical instabilities. In this work, we adopt recent advances in implicit differentiation techniques to address these challenges in PEPS optimization. By reformulating the core step of the gradient computation in terms of a single characteristic equation for the contraction environment, we reduce the cost of the gradient computation and improve its scaling with the problem size. By choosing a suitable parametrization of this characteristic equation based on the intrinsic symmetries of the contraction environment, we can directly remove instabilities from the global gradient computation that would otherwise arise from the derivatives of subroutines of the contraction algorithm. Finally, we demonstrate how this approach drastically simplifies the practical implementation of stable gradient-based PEPS optimization.

Related papers

• Natural Hypergradient Descent: Algorithm Design, Convergence Analysis, and Parallel Implementation [5.754044493040163]
  Natural Hypergradient Descent (NHGD) is a new method for solving bilevel optimization problems.<n>Our main theoretical contribution establishes high-probability error bounds and sample complexity guarantees for NHGD.<n> Empirical evaluations on representative bilevel learning tasks demonstrate the practical advantages of NHGD.
  arXiv  Detail & Related papers  (2026-02-11T14:31:33Z)
• Accelerated training of Gaussian processes using banded square exponential covariances [0.0]
  We propose a novel approach to computationally efficient GP training based on the observation that square-exponential (SE) covariance matrices contain several off-diagonal entries extremely close to zero.<n>We construct a principled procedure to eliminate those entries to produce a emphbanded-matrix approximation to the original covariance, whose inverse and determinant can be computed at a reduced computational cost.
  arXiv  Detail & Related papers  (2026-01-26T22:35:20Z)
• Revisiting the Initial Steps in Adaptive Gradient Descent Optimization [6.468625143772815]
  Adaptive gradient optimization methods, such as Adam, are prevalent in training deep neural networks across diverse machine learning tasks.<n>These methods often suffer from suboptimal generalization compared to descent gradient (SGD) and exhibit instability.<n>We introduce simple yet effective solutions: initializing the second-order moment estimation with non-zero values.
  arXiv  Detail & Related papers  (2024-12-03T04:28:14Z)
• Gradients without Backpropagation [16.928279365071916]
  We present a method to compute gradients based solely on the directional derivative that one can compute exactly and efficiently via the forward mode. We demonstrate forward descent gradient in a range of problems, showing substantial savings in computation and enabling training up to twice as fast in some cases.
  arXiv  Detail & Related papers  (2022-02-17T11:07:55Z)
• Differentiable Annealed Importance Sampling and the Perils of Gradient Noise [68.44523807580438]
  Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective. We propose a differentiable algorithm by abandoning Metropolis-Hastings steps, which further unlocks mini-batch computation.
  arXiv  Detail & Related papers  (2021-07-21T17:10:14Z)
• Adaptive Importance Sampling for Finite-Sum Optimization and Sampling with Decreasing Step-Sizes [4.355567556995855]
  We propose Avare, a simple and efficient algorithm for adaptive importance sampling for finite-sum optimization and sampling with decreasing step-sizes. Under standard technical conditions, we show that Avare achieves $mathcalO(T2/3)$ and $mathcalO(T5/6)$ dynamic regret for SGD and SGLD respectively when run with $mathcalO(T5/6)$ step sizes.
  arXiv  Detail & Related papers  (2021-03-23T00:28:15Z)
• Zeroth-Order Hybrid Gradient Descent: Towards A Principled Black-Box Optimization Framework [100.36569795440889]
  This work is on the iteration of zero-th-order (ZO) optimization which does not require first-order information. We show that with a graceful design in coordinate importance sampling, the proposed ZO optimization method is efficient both in terms of complexity as well as as function query cost.
  arXiv  Detail & Related papers  (2020-12-21T17:29:58Z)
• Channel-Directed Gradients for Optimization of Convolutional Neural Networks [50.34913837546743]
  We introduce optimization methods for convolutional neural networks that can be used to improve existing gradient-based optimization in terms of generalization error. We show that defining the gradients along the output channel direction leads to a performance boost, while other directions can be detrimental.
  arXiv  Detail & Related papers  (2020-08-25T00:44:09Z)
• Efficient Learning of Generative Models via Finite-Difference Score Matching [111.55998083406134]
  We present a generic strategy to efficiently approximate any-order directional derivative with finite difference. Our approximation only involves function evaluations, which can be executed in parallel, and no gradient computations.
  arXiv  Detail & Related papers  (2020-07-07T10:05:01Z)
• Cogradient Descent for Bilinear Optimization [124.45816011848096]
  We introduce a Cogradient Descent algorithm (CoGD) to address the bilinear problem. We solve one variable by considering its coupling relationship with the other, leading to a synchronous gradient descent. Our algorithm is applied to solve problems with one variable under the sparsity constraint.
  arXiv  Detail & Related papers  (2020-06-16T13:41:54Z)
• Proximal Gradient Algorithm with Momentum and Flexible Parameter Restart for Nonconvex Optimization [73.38702974136102]
  Various types of parameter restart schemes have been proposed for accelerated algorithms to facilitate their practical convergence in rates. In this paper, we propose an algorithm for solving nonsmooth problems.
  arXiv  Detail & Related papers  (2020-02-26T16:06:27Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.