We show that, although solution trajectories of gradient-based algorithms may vary depending on the learning task, they behave almost monotonically when projected onto an appropriate orthonormal function basis. Such projection gives rise to a basis function decomposition of the solution trajectory. Theoretically, we use our proposed basis function decomposition to establish the convergence of gradient descent (GD) on several representative learning tasks. In particular, we improve the convergence of GD on symmetric matrix factorization and provide a completely new convergence result for the orthogonal symmetric tensor decomposition. Empirically, we illustrate the promise of our proposed framework on realistic deep neural networks (DNNs) across different architectures, gradient-based solvers, and datasets. Our key finding is that gradient-based algorithms monotonically learn the coefficients of a particular orthonormal function basis of DNNs defined as the eigenvectors of the conjugate kernel after training.

Continual learning aims to alleviate catastrophic forgetting when handling consecutive tasks under non-stationary distributions. Gradient-based meta-learning algorithms have shown the capability to implicitly solve the transfer-interference trade-off problem between different examples. However, they still suffer from the catastrophic forgetting problem in the setting of continual learning, since the past data of previous tasks are no longer available. In this work, we propose a novel efficient meta-learning algorithm for solving the online continual learning problem, where the regularization terms and learning rates are adapted to the Taylor approximation of the parameter's importance to mitigate forgetting. The proposed method expresses the gradient of the meta-loss in closed-form and thus avoid computing second-order derivative which is computationally inhibitable. We also use Proximal Gradient Descent to further improve computational efficiency and accuracy. Experiments on diverse benchmarks show that our method achieves better or on-par performance and much higher efficiency compared to the state-of-the-art approaches.

As more practical and scalable quantum computers emerge, much attention has been focused on realizing quantum supremacy in machine learning. Existing quantum ML methods either (1) embed a classical model into a target Hamiltonian to enable quantum optimization or (2) represent a quantum model using variational quantum circuits and apply classical gradient-based optimization. The former method leverages the power of quantum optimization but only supports simple ML models, while the latter provides flexibility in model design but relies on gradient calculation, resulting in barren plateau (i.e., gradient vanishing) and frequent classical-quantum interactions. To address the limitations of existing quantum ML methods, we introduce Quark, a gradient-free quantum learning framework that optimizes quantum ML models using quantum optimization. Quark does not rely on gradient computation and therefore avoids barren plateau and frequent classical-quantum interactions. In addition, Quark can support more general ML models than prior quantum ML methods and achieves a dataset-size-independent optimization complexity. Theoretically, we prove that Quark can outperform classical gradient-based methods by reducing model query complexity for highly non-convex problems; empirically, evaluations on the Edge Detection and Tiny-MNIST tasks show that Quark can support complex ML models and significantly reduce the number of measurements needed for discovering near-optimal weights for these tasks.

Decision trees provide a rich family of highly non-linear but efficient models, due to which they continue to be the go-to family of predictive models by practitioners across domains. But learning trees is challenging due to their discrete decision boundaries. The state-of-the-art (SOTA) techniques resort to (a) learning \textit{soft} trees thereby losing logarithmic inference time; or (b) using methods tailored to specific supervised learning settings, requiring access to labeled examples and loss function. In this work, by leveraging techniques like overparameterization and straight-through estimators, we propose a unified method that enables accurate end-to-end gradient based tree training and can be deployed in a variety of settings like offline supervised learning and online learning with bandit feedback. Using extensive validation on standard benchmarks, we demonstrate that our method provides best of both worlds, i.e., it is competitive to, and in some cases more accurate than methods designed \textit{specifically} for the supervised settings; and in bandit settings, where most existing tree learning techniques are not applicable, our models are still accurate and significantly outperform the applicable SOTA methods.

Implicit neural networks have become increasingly attractive in the machine learning community since they can achieve competitive performance but use much less computational resources. Recently, a line of theoretical works established the global convergences for first-order methods such as gradient descent if the implicit networks are over-parameterized. However, as they train all layers together, their analyses are equivalent to only studying the evolution of the output layer. It is unclear how the implicit layer contributes to the training. Thus, in this paper, we restrict ourselves to only training the implicit layer. We show that global convergence is guaranteed, even if only the implicit layer is trained. On the other hand, the theoretical understanding of when and how the training performance of an implicit neural network can be generalized to unseen data is still under-explored. Although this problem has been studied in standard feed-forward networks, the case of implicit neural networks is still intriguing since implicit networks theoretically have infinitely many layers. Therefore, this paper investigates the generalization error for implicit neural networks. Specifically, we study the generalization of an implicit network activated by the ReLU function over random initialization. We provide a generalization bound that is initialization sensitive. As a result, we show that gradient flow with proper random initialization can train a sufficient over-parameterized implicit network to achieve arbitrarily small generalization errors.

The NP-hard problem of optimizing a shallow ReLU network can be characterized as a combinatorial search over each training example's activation pattern followed by a constrained convex problem given a fixed set of activation patterns. We explore the implications of this combinatorial aspect of ReLU optimization in this work. We show that it can be naturally modeled via a geometric and combinatoric object known as a zonotope with its vertex set isomorphic to the set of feasible activation patterns. This assists in analysis and provides a foundation for further research. We demonstrate its usefulness when we explore the sensitivity of the optimal loss to perturbations of the training data. Later we discuss methods of zonotope vertex selection and its relevance to optimization. Overparameterization assists in training by making a randomly chosen vertex more likely to contain a good solution. We then introduce a novel polynomial-time vertex selection procedure that provably picks a vertex containing the global optimum using only double the minimum number of parameters required to fit the data. We further introduce a local greedy search heuristic over zonotope vertices and demonstrate that it outperforms gradient descent on underparameterized problems.

Colleagues, coming in at #7 on our Top 10 Countdown is Data Structures, Algorithms, and Machine Learning Optimization. Learn to use "Big O" notation to characterize the time efficiency and space efficiency of a given algorithm, enabling you to select or devise the most sensible approach for tackling a particular machine learning problem with the hardware resources available to you, get acquainted with the entire range of the most widely-used Python data structures, including list-, dictionary-, tree-, and graph-based structures, develop a working understanding of all of the essential algorithms for working with data, including those for searching, sorting, hashing, and traversing, discover how the statistical and machine learning approaches to optimization differ, and why you would select one or the other for a given problem you're solving, understand exactly how the extremely versatile (stochastic) gradient descent optimization algorithm works and how to apply it and learn "fancy" optimizers that are available for advanced machine learning approaches (e.g., deep learning) and when you should consider using them. Training modules include: 1) Data Structures and Algorithms, 2) "Big O" Notation, 3) List-Based Data Structures, 4) Searching and Sorting, 5) Sets and Hashing, 6) Trees, 7) Graphs, 8) Machine Learning Optimization, and 9) Fancy Deep Learning Optimizers. Much career success, Lawrence E. Wilson -- Artificial Intelligence Academy (share & subscribe)

Sparse inversion and classification problems are ubiquitous in modern data science and imaging. They are often formulated as non-smooth minimisation problems. In sparse inversion, we minimise, e.g., the sum of a data fidelity term and an L1/LASSO regulariser. In classification, we consider, e.g., the sum of a data fidelity term and a non-smooth Ginzburg--Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such problems. Splitting techniques are much more useful: here, the target function is partitioned into a sum of two subtarget functions -- each of which can be efficiently optimised. Splitting proceeds by performing optimisation steps alternately with respect to each of the two subtarget functions. In this work, we study splitting from a stochastic continuous-time perspective. Indeed, we define a differential inclusion that follows one of the two subtarget function's negative subdifferential at each point in time. The choice of the subtarget function is controlled by a binary continuous-time Markov process. The resulting dynamical system is a stochastic approximation of the underlying subgradient flow. We investigate this stochastic approximation for an L1-regularised sparse inversion flow and for a discrete Allen-Cahn equation minimising a Ginzburg--Landau energy. In both cases, we study the longtime behaviour of the stochastic dynamical system and its ability to approximate the underlying subgradient flow at any accuracy. We illustrate our theoretical findings in a simple sparse estimation problem and also in low- and high-dimensional classification problems.

We study the application of variance reduction (VR) techniques to general non-convex stochastic optimization problems. In this setting, the recent work STORM [Cutkosky-Orabona '19] overcomes the drawback of having to compute gradients of "mega-batches" that earlier VR methods rely on. There, STORM utilizes recursive momentum to achieve the VR effect and is then later made fully adaptive in STORM+ [Levy et al., '21], where full-adaptivity removes the requirement for obtaining certain problem-specific parameters such as the smoothness of the objective and bounds on the variance and norm of the stochastic gradients in order to set the step size. However, STORM+ crucially relies on the assumption that the function values are bounded, excluding a large class of useful functions. In this work, we propose META-STORM, a generalized framework of STORM+ that removes this bounded function values assumption while still attaining the optimal convergence rate for non-convex optimization. META-STORM not only maintains full-adaptivity, removing the need to obtain problem specific parameters, but also improves the convergence rate's dependency on the problem parameters. Furthermore, META-STORM can utilize a large range of parameter settings that subsumes previous methods allowing for more flexibility in a wider range of settings. Finally, we demonstrate the effectiveness of META-STORM through experiments across common deep learning tasks. Our algorithm improves upon the previous work STORM+ and is competitive with widely used algorithms after the addition of per-coordinate update and exponential moving average heuristics.

Training large neural networks is time consuming. To speed up the process, distributed training is often used. One of the largest bottlenecks in distributed training is communicating gradients across different nodes. Different gradient compression techniques have been proposed to alleviate the communication bottleneck, including topK gradient sparsification, which truncates the gradient to the largest K components before sending it to other nodes. While some authors have investigated topK gradient sparsification in the parameter-server framework by applying topK compression in both the worker-to-server (uplink) and server-to-worker (downlink) direction, the currently accepted belief says that adding extra compression degrades the convergence of the model. We demonstrate, on the contrary, that adding downlink compression can potentially improve the performance of topK sparsification: not only does it reduce the amount of communication per step, but also, counter-intuitively, can improve the upper bound in the convergence analysis. To show this, we revisit non-convex convergence analysis of topK stochastic gradient descent (SGD) and extend it from the unidirectional to the bidirectional setting. We also remove a restriction of the previous analysis that requires unrealistically large values of K. We experimentally evaluate bidirectional topK SGD against unidirectional topK SGD and show that models trained with bidirectional topK SGD will perform as well as models trained with unidirectional topK SGD while yielding significant communication benefits for large numbers of workers.