Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but are solutions of an ordinary differential equation (ODE); however, these networks are still optimized via discrete methods (e.g. gradient descent). In this paper, we explore a different direction: namely, we propose a novel framework for learning in which the parameters themselves are solutions of ODEs. By viewing the optimization process as the evolution of a port-Hamiltonian system, we can ensure convergence to a minimum of the objective function. Numerical experiments have been performed to show the validity and effectiveness of the proposed methods.

We study the multi-channel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel $\mathbf a$ and multiple sparse inputs $\{\mathbf x_i\}_{i=1}^p$ from their circulant convolution $\mathbf y_i = \mathbf a \circledast \mathbf x_i $ ($i=1,\cdots,p$). We formulate the task as a nonconvex optimization problem over the sphere. Under mild statistical assumptions of the data, we prove that the vanilla Riemannian gradient descent (RGD) method, with random initializations, provably recovers both the kernel $\mathbf a$ and the signals $\{\mathbf x_i\}_{i=1}^p$ up to a signed shift ambiguity. In comparison with state-of-the-art results, our work shows significant improvements in terms of sample complexity and computational efficiency. Our theoretical results are corroborated by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods on both synthetic and real datasets.

A grouping problem involves partitioning a set of items into mutually disjoint groups or clusters according to some guiding decision criteria and imperative constraints. Grouping problems have many relevant applications and are computationally difficult. In this work, we present a general weight learning based optimization framework for solving grouping problems. The central idea of our approach is to formulate the task of seeking a solution as a real-valued weight matrix learning problem that is solved by first order gradient descent. A practical implementation of this framework is proposed with tensor calculus in order to benefit from parallel computing on GPU devices. To show its potential for tackling difficult problems, we apply the approach to two typical and well-known grouping problems (graph coloring and equitable graph coloring). We present large computational experiments and comparisons on popular benchmarks and report improved best-known results (new upper bounds) for several large graphs.

Stochastic Gradient Langevin Dynamics, the "unadjusted Langevin algorithm", and Adaptive Langevin Dynamics (also known as Stochastic Gradient Nosé-Hoover dynamics) are examples of existing thermodynamic parameterization methods in use for machine learning, but these can be substantially improved. We find that by partitioning the parameters based on natural layer structure we obtain schemes with rapid convergence for data sets with complicated loss landscapes. We describe easy-to-implement hybrid partitioned numerical algorithms, based on discretized stochastic differential equations, which are adapted to feed-forward neural networks, including LaLa (a multi-layer Langevin algorithm), AdLaLa (combining the adaptive Langevin and Langevin algorithms) and LOL (combining Langevin and Overdamped Langevin); we examine the convergence of these methods using numerical studies and compare their performance among themselves and in relation to standard alternatives such as stochastic gradient descent and ADAM. We present evidence that thermodynamic parameterization methods can be (i) faster, (ii) more accurate, and (iii) more robust than standard algorithms incorporated into machine learning frameworks, in particular for data sets with complicated loss landscapes. Moreover, we show in numerical studies that methods based on sampling excite many degrees of freedom.

Novel coordinate descent (CD) methods are proposed for minimizing nonconvex functions consisting of three terms: (i) a continuously differentiable term, (ii) a simple convex term, and (iii) a concave and continuous term. First, by extending randomized CD to nonsmooth nonconvex settings, we develop a coordinate subgradient method that randomly updates block-coordinate variables by using block composite subgradient mapping. This method converges asymptotically to critical points with proven sublinear convergence rate for certain optimality measures. Second, we develop a randomly permuted CD method with two alternating steps: linearizing the concave part and cycling through variables. We prove asymptotic convergence to critical points and sublinear complexity rate for objectives with both smooth and concave parts. Third, we extend accelerated coordinate descent (ACD) to nonsmooth and nonconvex optimization to develop a novel randomized proximal DC algorithm whereby we solve the subproblem inexactly by ACD. Convergence is guaranteed with at most a few number of ACD iterations for each DC subproblem, and convergence complexity is established for identification of some approximate critical points. Fourth, we further develop the third method to minimize certain ill-conditioned nonconvex functions: weakly convex functions with high Lipschitz constant to negative curvature ratios. We show that, under specific criteria, the ACD-based randomized method has superior complexity compared to conventional gradient methods. Finally, an empirical study on sparsity-inducing learning models demonstrates that CD methods are superior to gradient-based methods for certain large-scale problems.

We consider stochastic gradient descent algorithms for minimizing a non-smooth, strongly-convex function. Several forms of this algorithm, including suffix averaging, are known to achieve the optimal $O(1/T)$ convergence rate in expectation. We consider a simple, non-uniform averaging strategy of Lacoste-Julien et al. (2011) and prove that it achieves the optimal $O(1/T)$ convergence rate with high probability. Our proof uses a recently developed generalization of Freedman's inequality. Finally, we compare several of these algorithms experimentally and show that this non-uniform averaging strategy outperforms many standard techniques, and with smaller variance.

We keep the BERT encoder unfrozen so that all weights are updated with every iteration. Given the number of trainable parameters it's useful to train the model on multiple GPUs in parallel. I used 4 Tesla K80's for about 4500 training samples. Just remember that to access any model attribute, you can access it using modelName.module.attribute I used Stochastic Gradient Descent with momentum as the optimizer and found that cycling both the learning rates and momentum really helped to get the training and validation losses down.

A versatile and effective approach to meta-learning is to infer a gradient-based up-date rule directly from data that promotes rapid learning of new tasks from the same distribution. Current methods rely on backpropagating through the learning process, limiting their scope to few-shot learning. In this work, we introduce Warped Gradient Descent (WarpGrad), a family of modular optimisers that can scale to arbitrary adaptation processes. WarpGrad methods meta-learn to warp task loss surfaces across the joint task-parameter distribution to facilitate gradient descent, which is achieved by a reparametrisation of neural networks that interleaves warp layers in the architecture. These layers are shared across task learners and fixed during adaptation; they represent a projection of task parameters into a meta-learned space that is conducive to task adaptation and standard backpropagation induces a form of gradient preconditioning. WarpGrad methods are computationally efficient and easy to implement as they rely on parameter sharing and backpropagation. They are readily combined with other meta-learners and can scale both in terms of model size and length of adaptation trajectories as meta-learning warp parameters do not require differentiation through task adaptation processes. We show empirically that WarpGrad optimisers meta-learn a warped space where gradient descent is well behaved, with faster convergence and better performance in a variety of settings, including few-shot, standard supervised, continual, and reinforcement learning.

Stochastic Gradient Langevin Dynamics, the "unadjusted Langevin algorithm", and Adaptive Langevin Dynamics (also known as Stochastic Gradient Nos\'{e}-Hoover dynamics) are examples of existing thermodynamic parameterization methods in use for machine learning, but these can be substantially improved. We find that by partitioning the parameters based on natural layer structure we obtain schemes with rapid convergence for data sets with complicated loss landscapes. We describe easy-to-implement hybrid partitioned numerical algorithms, based on discretized stochastic differential equations, which are adapted to feed-forward neural networks, including LaLa (a multi-layer Langevin algorithm), AdLaLa (combining the adaptive Langevin and Langevin algorithms) and LOL (combining Langevin and Overdamped Langevin); we examine the convergence of these methods using numerical studies and compare their performance among themselves and in relation to standard alternatives such as stochastic gradient descent and ADAM. We present evidence that thermodynamic parameterization methods can be (i) faster, (ii) more accurate, and (iii) more robust than standard algorithms incorporated into machine learning frameworks, in particular for data sets with complicated loss landscapes. Moreover, we show in numerical studies that methods based on sampling excite many degrees of freedom. The equipartition property, which is a consequence of their ergodicity, means that these methods keep in play an ensemble of low-loss states during the training process. By drawing parameter states from a sufficiently rich distribution of nearby candidate states, we show that the thermodynamic schemes produce smoother classifiers, improve generalization and reduce overfitting compared to traditional optimizers.

Abstract: This paper describes neural-fortran, a parallel Fortran framework for neural networks and deep learning. It features a simple interface to construct feed-forward neural networks of arbitrary structure and size, several activation functions, and stochastic gradient descent as the default optimization algorithm. Neural-fortran also leverages the Fortran 2018 standard collective subroutines to achieve data-based parallelism on shared- or distributed-memory machines. First, I describe the implementation of neural networks with Fortran derived types, whole-array arithmetic, and collective sum and broadcast operations to achieve parallelism. Second, I demonstrate the use of neural-fortran in an example of recognizing hand-written digits from images.