Optimizing Nondecomposable Data Dependent Regularizers via Lagrangian Reparameterization offers Significant Performance and Efficiency Gains

Data dependent regularization is known to benefit a wide variety of problems in machine learning. Often, these regularizers cannot be easily decomposed into a sum over a finite number of terms, e.g., a sum over individual example-wise terms. The $F_\beta$ measure, Area under the ROC curve (AUCROC) and Precision at a fixed recall (P@R) are some prominent examples that are used in many applications. We find that for most medium to large sized datasets, scalability issues severely limit our ability in leveraging the benefits of such regularizers. Importantly, the key technical impediment despite some recent progress is that, such objectives remain difficult to optimize via backpropapagation procedures. While an efficient general-purpose strategy for this problem still remains elusive, in this paper, we show that for many data-dependent nondecomposable regularizers that are relevant in applications, sizable gains in efficiency are possible with minimal code-level changes; in other words, no specialized tools or numerical schemes are needed. Our procedure involves a reparameterization followed by a partial dualization -- this leads to a formulation that has provably cheap projection operators. We present a detailed analysis of runtime and convergence properties of our algorithm. On the experimental side, we show that a direct use of our scheme significantly improves the state of the art IOU measures reported for MSCOCO Stuff segmentation dataset

Training Gaussian Boson Sampling Distributions

Gaussian Boson Sampling (GBS) is a near-term platform for photonic quantum computing. Applications have been developed which rely on directly programming GBS devices, but the ability to train and optimize circuits has been a key missing ingredient for developing new algorithms. In this work, we derive analytical gradient formulas for the GBS distribution, which can be used to train devices using standard methods based on gradient descent. We introduce a parametrization of the distribution that allows the gradient to be estimated by sampling from the same device that is being optimized. In the case of training using a Kullback-Leibler divergence or log-likelihood cost function, we show that gradients can be computed classically, leading to fast training. We illustrate these results with numerical experiments in stochastic optimization and unsupervised learning. As a particular example, we introduce the variational Ising solver, a hybrid algorithm for training GBS devices to sample ground states of a classical Ising model with high probability.Comment: 15 pages, 3 figure