Optimization of hyperparameters of Gaussian process regression (GPR) determines success or failure of the application of the method. Such optimization is difficult with sparse data, in particular in high-dimensional spaces where the data sparsity issue cannot be resolved by adding more data. We show that parameter optimization is facilitated by rectangularization of the defining equation of GPR. On the example of a 15-dimensional molecular potential energy surface we demonstrate that this approach allows effective hyperparameter tuning even with very sparse data.
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We propose Characteristic Neural Ordinary Differential Equations (C-NODEs), a framework for extending Neural Ordinary Differential Equations (NODEs) beyond ODEs. While NODEs model the evolution of the latent state as the solution to an ODE, the proposed C-NODE models the evolution of the latent state as the solution of a family of first-order quasi-linear partial differential equations (PDE) on their characteristics, defined as curves along which the PDEs reduce to ODEs. The reduction, in turn, allows the application of the standard frameworks for solving ODEs to PDE settings. Additionally, the proposed framework can be cast as an extension of existing NODE architectures, thereby allowing the use of existing black-box ODE solvers. We prove that the C-NODE framework extends the classical NODE by exhibiting functions that cannot be represented by NODEs but are representable by C-NODEs. We further investigate the efficacy of the C-NODE framework by demonstrating its performance in many synthetic and real data scenarios. Empirical results demonstrate the improvements provided by the proposed method for CIFAR-10, SVHN, and MNIST datasets under a similar computational budget as the existing NODE methods.
For the problem of maximizing a monotone, submodular function with respect to a cardinality constraint $k$ on a ground set of size $n$, we provide an algorithm that achieves the state-of-the-art in both its empirical performance and its theoretical properties, in terms of adaptive complexity, query complexity, and approximation ratio; that is, it obtains, with high probability, query complexity of $O(n)$ in expectation, adaptivity of $O(\log(n))$, and approximation ratio of nearly $1-1/e$. The main algorithm is assembled from two components which may be of independent interest. The first component of our algorithm, LINEARSEQ, is useful as a preprocessing algorithm to improve the query complexity of many algorithms. Moreover, a variant of LINEARSEQ is shown to have adaptive complexity of $O( \log (n / k) )$ which is smaller than that of any previous algorithm in the literature. The second component is a parallelizable thresholding procedure THRESHOLDSEQ for adding elements with gain above a constant threshold. Finally, we demonstrate that our main algorithm empirically outperforms, in terms of runtime, adaptive rounds, total queries, and objective values, the previous state-of-the-art algorithm FAST in a comprehensive evaluation with six submodular objective functions.
A quasi-score linearity test for continuous and count network autoregressive models is developed. We establish the asymptotic distribution of the test when the network dimension is fixed or increasing, under the null hypothesis of linearity and Pitman's local alternatives. When the parameters are identifiable, the test statistic approximates a chi-square and noncentral chi-square asymptotic distribution, respectively. These results still hold true when the parameters tested belong to the boundary of their space. When we deal with non-identifiable parameters, a suitable test is proposed and its asymptotic distribution is established when the network dimension is fixed. Since, in general, critical values of such test cannot be tabulated, the empirical computation of the p-values is implemented using a feasible bound. Bootstrap approximations are also provided. Moreover, consistency and asymptotic normality of the quasi maximum likelihood estimator is established for continuous and count nonlinear network autoregressions, under standard smoothness conditions. A simulation study and two data examples complement this work.
This paper concerns a convex, stochastic zeroth-order optimization (S-ZOO) problem. The objective is to minimize the expectation of a cost function whose gradient is not directly accessible. For this problem, traditional optimization algorithms mostly yield query complexities that grow polynomially with dimensionality (the number of decision variables). Consequently, these methods may not perform well in solving massive-dimensional problems arising in many modern applications. Although more recent methods can be provably dimension-insensitive, almost all of them require arguably more stringent conditions such as everywhere sparse or compressible gradient. In this paper, we propose a sparsity-inducing stochastic gradient-free (SI-SGF) algorithm, which provably yields a dimension-free (up to a logarithmic term) query complexity in both convex and strongly convex cases. Such insensitivity to the dimensionality growth is proven, for the first time, to be achievable when neither gradient sparsity nor gradient compressibility is satisfied. Our numerical results demonstrate a consistency between our theoretical prediction and the empirical performance.
This paper investigates why it is beneficial, when solving a problem, to search in the neighbourhood of a current solution. The paper identifies properties of problems and neighbourhoods that support two novel proofs that neighbourhood search is beneficial over blind search. These are: firstly a proof that search within the neighbourhood is more likely to find an improving solution in a single search step than blind search; and secondly a proof that a local improvement, using a sequence of neighbourhood search steps, is likely to achieve a greater improvement than a sequence of blind search steps. To explore the practical impact of these properties, a range of problem sets and neighbourhoods are generated, where these properties are satisfied to different degrees. Experiments reveal that the benefits of neighbourhood search vary dramatically in consequence. Random problems of a classical combinatorial optimisation problem are analysed, in order to demonstrate that the underlying theory is reflected in practice.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
In real word applications, data generating process for training a machine learning model often differs from what the model encounters in the test stage. Understanding how and whether machine learning models generalize under such distributional shifts have been a theoretical challenge. Here, we study generalization in kernel regression when the training and test distributions are different using methods from statistical physics. Using the replica method, we derive an analytical formula for the out-of-distribution generalization error applicable to any kernel and real datasets. We identify an overlap matrix that quantifies the mismatch between distributions for a given kernel as a key determinant of generalization performance under distribution shift. Using our analytical expressions we elucidate various generalization phenomena including possible improvement in generalization when there is a mismatch. We develop procedures for optimizing training and test distributions for a given data budget to find best and worst case generalizations under the shift. We present applications of our theory to real and synthetic datasets and for many kernels. We compare results of our theory applied to Neural Tangent Kernel with simulations of wide networks and show agreement. We analyze linear regression in further depth.
Eigendecomposition of symmetric matrices is at the heart of many computer vision algorithms. However, the derivatives of the eigenvectors tend to be numerically unstable, whether using the SVD to compute them analytically or using the Power Iteration (PI) method to approximate them. This instability arises in the presence of eigenvalues that are close to each other. This makes integrating eigendecomposition into deep networks difficult and often results in poor convergence, particularly when dealing with large matrices. While this can be mitigated by partitioning the data into small arbitrary groups, doing so has no theoretical basis and makes it impossible to exploit the full power of eigendecomposition. In previous work, we mitigated this using SVD during the forward pass and PI to compute the gradients during the backward pass. However, the iterative deflation procedure required to compute multiple eigenvectors using PI tends to accumulate errors and yield inaccurate gradients. Here, we show that the Taylor expansion of the SVD gradient is theoretically equivalent to the gradient obtained using PI without relying in practice on an iterative process and thus yields more accurate gradients. We demonstrate the benefits of this increased accuracy for image classification and style transfer.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
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