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Free, publiclyaccessible full text available June 26, 2023

Three features are crucial for sequential forecasting and generation models: tractability, expressiveness, and theoretical backing. While neural autoregressive models are relatively tractable and offer powerful predictive and generative capabilities, they often have complex optimization landscapes, and their theoretical properties are not well understood. To address these issues, we present convex formulations of autoregressive models with one hidden layer. Specifically, we prove an exact equivalence between these models and constrained, regularized logistic regression by using semiinfinite duality to embed the data matrix onto a higher dimensional space and introducing inequality constraints. To make this formulation tractable, we approximate the constraints using a hinge loss or drop them altogether. Furthermore, we demonstrate faster training and competitive performance of these implementations compared to their neural network counterparts on a variety of data sets. Consequently, we introduce techniques to derive tractable, expressive, and theoreticallyinterpretable models that are nearly equivalent to neural autoregressive models.

Neural networks (NNs) have been extremely successful across many tasks in machine learning. Quantization of NN weights has become an important topic due to its impact on their energy efficiency, inference time and deployment on hardware. Although posttraining quantization is wellstudied, training optimal quantized NNs involves combinatorial nonconvex optimization problems which appear intractable. In this work, we introduce a convex optimization strategy to train quantized NNs with polynomial activations. Our method leverages hidden convexity in twolayer neural networks from the recent literature, semidefinite lifting, and Grothendieck’s identity. Surprisingly, we show that certain quantized NN problems can be solved to global optimality provably in polynomial time in all relevant parameters via tight semidefinite relaxations. We present numerical examples to illustrate the effectiveness of our method.

Fisher’s Linear Discriminant Analysis (FLDA) is a statistical analysis method that linearly embeds data points to a lower dimensional space to maximize a discrimination criterion such that the variance between classes is maximized while the variance within classes is minimized. We introduce a natural extension of FLDA that employs neural networks, called Neural Fisher Discriminant Analysis (NFDA). This method finds the optimal twolayer neural network that embeds data points to optimize the same discrimination criterion. We use tools from convex optimization to transform the optimal neural network embedding problem into a convex problem. The resulting problem is easy to interpret and solve to global optimality. We evaluate the method’s performance on synthetic and real datasets.

Neural networks (NNs) have been extremely successful across many tasks in machine learning. Quantization of NN weights has become an important topic due to its impact on their energy efficiency, inference time and deployment on hardware. Although posttraining quantization is wellstudied, training optimal quantized NNs involves combinatorial nonconvex optimization problems which appear intractable. In this work, we introduce a convex optimization strategy to train quantized NNs with polynomial activations. Our method leverages hidden convexity in twolayer neural networks from the recent literature, semidefinite lifting, and Grothendieck’s identity. Surprisingly, we show that certain quantized NN problems can be solved to global optimality provably in polynomial time in all relevant parameters via tight semidefinite relaxations. We present numerical examples to illustrate the effectiveness of our method.

We study training of Convolutional Neural Networks (CNNs) with ReLU activations and introduce exact convex optimization formulations with a polynomial complexity with respect to the number of data samples, the number of neurons, and data dimension. More specifically, we develop a convex analytic framework utilizing semiinfinite duality to obtain equivalent convex optimization problems for several two and threelayer CNN architectures. We first prove that twolayer CNNs can be globally optimized via an `2 norm regularized convex program. We then show that multilayer circular CNN training problems with a single ReLU layer are equivalent to an `1 regularized convex program that encourages sparsity in the spectral domain. We also extend these results to threelayer CNNs with two ReLU layers. Furthermore, we present extensions of our approach to different pooling methods, which elucidates the implicit architectural bias as convex regularizers.

We propose a randomized algorithm with quadratic convergence rate for convex optimization problems with a selfconcordant, composite, strongly convex objective function. Our method is based on performing an approximate Newton step using a random projection of the Hessian. Our first contribution is to show that, at each iteration, the embedding dimension (or sketch size) can be as small as the effective dimension of the Hessian matrix. Leveraging this novel fundamental result, we design an algorithm with a sketch size proportional to the effective dimension and which exhibits a quadratic rate of convergence. This result dramatically improves on the classical linearquadratic convergence rates of stateoftheart subsampled Newton methods. However, in most practical cases, the effective dimension is not known beforehand, and this raises the question of how to pick a sketch size as small as the effective dimension while preserving a quadratic convergence rate. Our second and main contribution is thus to propose an adaptive sketch size algorithm with quadratic convergence rate and which does not require prior knowledge or estimation of the effective dimension: at each iteration, it starts with a small sketch size, and increases it until quadratic progress is achieved. Importantly, we show that the embedding dimension remainsmore »

In the domains of dataset construction and crowdsourcing, a notable challenge is to aggregate labels from a heterogeneous set of labelers, each of whom is potentially an expert in some subset of tasks (and less reliable in others). To reduce costs of hiring human labelers or training automated labeling systems, it is of interest to minimize the number of labelers while ensuring the reliability of the resulting dataset. We model this as the problem of performing Kclass classification using the predictions of smaller classifiers, each trained on a subset of [K], and derive bounds on the number of classifiers needed to accurately infer the true class of an unlabeled sample under both adversarial and stochastic assumptions. By exploiting a connection to the classical set cover problem, we produce a nearoptimal scheme for designing such configurations of classifiers which recovers the well known onevs.one classification approach as a special case. Experiments with the MNIST and CIFAR10 datasets demonstrate the favorable accuracy (compared to a centralized classifier) of our aggregation scheme applied to classifiers trained on subsets of the data. These results suggest a new way to automatically label data or adapt an existing set of local classifiers to largerscale multiclass problems.

In secondorder optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration. Randomized sketching has emerged as a powerful technique for constructing estimates of the Hessian which can be used to perform approximate Newton steps. This involves multiplication by a random sketching matrix, which introduces a tradeoff between the computational cost of sketching and the convergence rate of the optimization algorithm. A theoretically desirable but practically much too expensive choice is to use a dense Gaussian sketching matrix, which produces unbiased estimates of the exact Newton step and which offers strong problemindependent convergence guarantees. We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching, without substantially affecting its convergence properties. This approach, called Newton LESS, is based on a recently introduced sketching technique: LEverage Score Sparsified (LESS) embeddings. We prove that NewtonLESS enjoys nearly the same problemindependent local convergence rate as Gaussian embeddings, not just up to constant factors but even down to lower order terms, for a large class of optimization tasks. In particular, this leads to a new stateoftheart convergence result for an iterative least squares solver. Finally, we extend LESS embeddings tomore »