Fast quantum state reconstruction via accelerated non-convex programming

We propose a new quantum state reconstruction method that combines ideas from compressed sensing, non-convex optimization, and acceleration methods. The algorithm, called Momentum-Inspired Factored Gradient Descent (\texttt{MiFGD}), extends the applicability of quantum tomography for larger systems. Despite being a non-convex method, \texttt{MiFGD} converges \emph{provably} to the true density matrix at a linear rate, in the absence of experimental and statistical noise, and under common assumptions. With this manuscript, we present the method, prove its convergence property and provide Frobenius norm bound guarantees with respect to the true density matrix. From a practical point of view, we benchmark the algorithm performance with respect to other existing methods, in both synthetic and real experiments performed on an IBM's quantum processing unit. We find that the proposed algorithm performs orders of magnitude faster than state of the art approaches, with the same or better accuracy. In both synthetic and real experiments, we observed accurate and robust reconstruction, despite experimental and statistical noise in the tomographic data. Finally, we provide a ready-to-use code for state tomography of multi-qubit systems.

GIST: Distributed Training for Large-Scale Graph Convolutional Networks

The graph convolutional network (GCN) is a go-to solution for machine learning on graphs, but its training is notoriously difficult to scale in terms of both the size of the graph and the number of model parameters. These limitations are in stark contrast to the increasing scale (in data size and model size) of experiments in deep learning research. In this work, we propose GIST, a novel distributed approach that enables efficient training of wide (overparameterized) GCNs on large graphs. GIST is a hybrid layer and graph sampling method, which disjointly partitions the global model into several, smaller sub-GCNs that are independently trained across multiple GPUs in parallel. This distributed framework improves model performance and significantly decreases wall-clock training time. GIST seeks to enable large-scale GCN experimentation with the goal of bridging the existing gap in scale between graph machine learning and deep learning.

Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets

We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one projections of an approximately low-rank PSD matrix, we characterize the radius of the set of PSD matrices that satisfy the measurements. This result yields a sampling rate to guarantee singleton solution sets when the true matrix is exactly low-rank, such that the choice of the objective function or the algorithm to be used is inconsequential in its recovery. We discuss applications of this contribution and compare it to recent literature regarding implicit regularization for similar problems. We demonstrate practical implications of this result by applying conic projection methods for PSD matrix recovery without incorporating low-rank regularization.

On Continuous Local BDD-Based Search for Hybrid SAT Solving

We explore the potential of continuous local search (CLS) in SAT solving by proposing a novel approach for finding a solution of a hybrid system of Boolean constraints. The algorithm is based on CLS combined with belief propagation on binary decision diagrams (BDDs). Our framework accepts all Boolean constraints that admit compact BDDs, including symmetric Boolean constraints and small-coefficient pseudo-Boolean constraints as interesting families. We propose a novel algorithm for efficiently computing the gradient needed by CLS. We study the capabilities and limitations of our versatile CLS solver, GradSAT, by applying it on many benchmark instances. The experimental results indicate that GradSAT can be a useful addition to the portfolio of existing SAT and MaxSAT solvers for solving Boolean satisfiability and optimization problems.

On Generalization of Adaptive Methods for Over-parameterized Linear Regression

Over-parameterization and adaptive methods have played a crucial role in the success of deep learning in the last decade. The widespread use of over-parameterization has forced us to rethink generalization by bringing forth new phenomena, such as implicit regularization of optimization algorithms and double descent with training progression. A series of recent works have started to shed light on these areas in the quest to understand -- why do neural networks generalize well? The setting of over-parameterized linear regression has provided key insights into understanding this mysterious behavior of neural networks. In this paper, we aim to characterize the performance of adaptive methods in the over-parameterized linear regression setting. First, we focus on two sub-classes of adaptive methods depending on their generalization performance. For the first class of adaptive methods, the parameter vector remains in the span of the data and converges to the minimum norm solution like gradient descent (GD). On the other hand, for the second class of adaptive methods, the gradient rotation caused by the pre-conditioner matrix results in an in-span component of the parameter vector that converges to the minimum norm solution and the out-of-span component that saturates. Our experiments on over-parameterized linear regression and deep neural networks support this theory.

ImCLR: Implicit Contrastive Learning for Image Classification

Contrastive learning is an effective method for learning visual representations. In most cases, this involves adding an explicit loss function to encourage similar images to have similar representations, and different images to have different representations. Inspired by contrastive learning, we introduce a clever input construction for Implicit Contrastive Learning (ImCLR), primarily in the supervised setting: there, the network can implicitly learn to differentiate between similar and dissimilar images. Each input is presented as a concatenation of two images, and the label is the mean of the two one-hot labels. Furthermore, this requires almost no change to existing pipelines, which allows for easy integration and for fair demonstration of effectiveness on a wide range of well-accepted benchmarks. Namely, there is no change to loss, no change to hyperparameters, and no change to general network architecture. We show that ImCLR improves the test error in the supervised setting across a variety of settings, including 3.24% on Tiny ImageNet, 1.30% on CIFAR-100, 0.14% on CIFAR-10, and 2.28% on STL-10. We show that this holds across different number of labeled samples, maintaining approximately a 2% gap in test accuracy down to using only 5% of the whole dataset. We further show that gains hold for robustness to common input corruptions and perturbations at varying severities with a 0.72% improvement on CIFAR-100-C, and in the semi-supervised setting with a 2.16% improvement with the standard benchmark $\Pi$-model. We demonstrate that ImCLR is complementary to existing data augmentation techniques, achieving over 1% improvement on CIFAR-100 and 2% improvement on Tiny ImageNet by combining ImCLR with CutMix over either baseline, and 2% by combining ImCLR with AutoAugment over either baseline.

Bayesian Coresets: An Optimization Perspective

Bayesian coresets have emerged as a promising approach for implementing scalable Bayesian inference. The Bayesian coreset problem involves selecting a (weighted) subset of the data samples, such that posterior inference using the selected subset closely approximates posterior inference using the full dataset. This manuscript revisits Bayesian coresets through the lens of sparsity constrained optimization. Leveraging recent advances in accelerated optimization methods, we propose and analyze a novel algorithm for coreset selection. We provide explicit convergence rate guarantees and present an empirical evaluation on a variety of benchmark datasets to highlight our proposed algorithm's superior performance compared to state of the art on speed and accuracy.

FourierSAT: A Fourier Expansion-Based Algebraic Framework for Solving Hybrid Boolean Constraints

The Boolean SATisfiability problem (SAT) is of central importance in computer science. Although SAT is known to be NP-complete, progress on the engineering side, especially that of Conflict-Driven Clause Learning (CDCL) and Local Search SAT solvers, has been remarkable. Yet, while SAT solvers aimed at solving industrial-scale benchmarks in Conjunctive Normal Form (CNF) have become quite mature, SAT solvers that are effective on other types of constraints, e.g., cardinality constraints and XORs, are less well studied; a general approach to handling non-CNF constraints is still lacking. In addition, previous work indicated that for specific classes of benchmarks, the running time of extant SAT solvers depends heavily on properties of the formula and details of encoding, instead of the scale of the benchmarks, which adds uncertainty to expectations of running time. To address the issues above, we design FourierSAT, an incomplete SAT solver based on Fourier analysis of Boolean functions, a technique to represent Boolean functions by multilinear polynomials. By such a reduction to continuous optimization, we propose an algebraic framework for solving systems consisting of different types of constraints. The idea is to leverage gradient information to guide the search process in the direction of local improvements. Empirical results demonstrate that FourierSAT is more robust than other solvers on certain classes of benchmarks.

Optimal Mini-Batch Size Selection for Fast Gradient Descent

This paper presents a methodology for selecting the mini-batch size that minimizes Stochastic Gradient Descent (SGD) learning time for single and multiple learner problems. By decoupling algorithmic analysis issues from hardware and software implementation details, we reveal a robust empirical inverse law between mini-batch size and the average number of SGD updates required to converge to a specified error threshold. Combining this empirical inverse law with measured system performance, we create an accurate, closed-form model of average training time and show how this model can be used to identify quantifiable implications for both algorithmic and hardware aspects of machine learning. We demonstrate the inverse law empirically, on both image recognition (MNIST, CIFAR10 and CIFAR100) and machine translation (Europarl) tasks, and provide a theoretic justification via proving a novel bound on mini-batch SGD training.