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Register a new userDiscretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows
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We compare the discretize-optimize (Disc-Opt) and optimize-discretize (Opt-Disc) approaches for time-series regression and continuous normalizing flows (CNFs) using neural ODEs. Neural ODEs are ordinary differential equations (ODEs) with neural network components. Training a neural ODE is an optimal control problem where the weights are the controls and the hidden features are the states. Every training iteration involves solving an ODE forward and another backward in time, which can require large amounts of computation, time, and memory. Comparing the Opt-Disc and Disc-Opt approaches in image classification tasks, Gholami et al. (2019) suggest that Disc-Opt is preferable due to the guaranteed accuracy of gradients. In this paper, we extend the comparison to neural ODEs for time-series regression and CNFs. Unlike in classification, meaningful models in these tasks must also satisfy additional requirements beyond accurate final-time output, e.g., the invertibility of the CNF. Through our numerical experiments, we demonstrate that with careful numerical treatment, Disc-Opt methods can achieve similar performance as Opt-Disc at inference with drastically reduced training costs. Disc-Opt reduced costs in six out of seven separate problems with training time reduction ranging from 39% to 97%, and in one case, Disc-Opt reduced training from nine days to less than one day.
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We introduce a learning-based framework to optimize tensor programs for deep learning workloads. Efficient implementations of tensor operators, such as matrix multiplication and high dimensional convolution, are key enablers of effective deep learning systems. However, existing systems rely on manually optimized libraries such as cuDNN where only a narrow range of server class GPUs are well-supported. The reliance on hardware-specific operator libraries limits the applicability of high-level graph optimizations and incurs significant engineering costs when deploying to new hardware targets. We use learning to remove this engineering burden. We learn domain-specific statistical cost models to guide the search of tensor operator implementations over billions of possible program variants. We further accelerate the search by effective model transfer across workloads. Experimental results show that our framework delivers performance competitive with state-of-the-art hand-tuned libraries for low-power CPU, mobile GPU, and server-class GPU.
Normalizing flows transform a simple base distribution into a complex target distribution and have proved to be powerful models for data generation and density estimation. In this work, we propose a novel type of normalizing flow driven by a differential deformation of the Wiener process. As a result, we obtain a rich time series model whose observable process inherits many of the appealing properties of its base process, such as efficient computation of likelihoods and marginals. Furthermore, our continuous treatment provides a natural framework for irregular time series with an independent arrival process, including straightforward interpolation. We illustrate the desirable properties of the proposed model on popular stochastic processes and demonstrate its superior flexibility to variational RNN and latent ODE baselines in a series of experiments on synthetic and real-world data.
A normalizing flow is an invertible mapping between an arbitrary probability distribution and a standard normal distribution; it can be used for density estimation and statistical inference. Computing the flow follows the change of variables formula and thus requires invertibility of the mapping and an efficient way to compute the determinant of its Jacobian. To satisfy these requirements, normalizing flows typically consist of carefully chosen components. Continuous normalizing flows (CNFs) are mappings obtained by solving a neural ordinary differential equation (ODE). The neural ODEs dynamics can be chosen almost arbitrarily while ensuring invertibility. Moreover, the log-determinant of the flows Jacobian can be obtained by integrating the trace of the dynamics Jacobian along the flow. Our proposed OT-Flow approach tackles two critical computational challenges that limit a more widespread use of CNFs. First, OT-Flow leverages optimal transport (OT) theory to regularize the CNF and enforce straight trajectories that are easier to integrate. Second, OT-Flow features exact trace computation with time complexity equal to trace estimators used in existing CNFs. On five high-dimensional density estimation and generative modeling tasks, OT-Flow performs competitively to state-of-the-art CNFs while on average requiring one-fourth of the number of weights with an 8x speedup in training time and 24x speedup in inference.
The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem, and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters, in contrast to the prediction error of the parameters. This loss function was recently introduced in Elmachtoub and Grigas (2017) and referred to as the Smart Predict-then-Optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out-of-sample, in the context of the SPO loss. Since the SPO loss is non-convex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points, but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions, and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation.
We introduce a novel end-to-end approach for learning to cluster in the absence of labeled examples. Our clustering objective is based on optimizing normalized cuts, a criterion which measures both intra-cluster similarity as well as inter-cluster dissimilarity. We define a differentiable loss function equivalent to the expected normalized cuts. Unlike much of the work in unsupervised deep learning, our trained model directly outputs final cluster assignments, rather than embeddings that need further processing to be usable. Our approach generalizes to unseen datasets across a wide variety of domains, including text, and image. Specifically, we achieve state-of-the-art results on popular unsupervised clustering benchmarks (e.g., MNIST, Reuters, CIFAR-10, and CIFAR-100), outperforming the strongest baselines by up to 10.9%. Our generalization results are superior (by up to 21.9%) to the recent top-performing clustering approach with the ability to generalize.
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