We consider stochastic approximation with block-coordinate stepsizes and propose adaptive stepsize rules that aim to minimize the expected distance from the next iterate to an optimal point. These stepsize rules employ online estimates of the second moment of the search direction along each block coordinate. The popular Adam algorithm can be interpreted as a particular heuristic for such estimation. By leveraging a simple conditional estimator, we derive a new method that obtains comparable performance as Adam but requires less memory and fewer hyper-parameters. We prove that this family of methods converges almost surely to a small neighborhood of the optimal point, and the radius of the neighborhood depends on the bias and variance of the second-moment estimator. Our analysis relies on a simple aiming condition that assumes neither convexity nor smoothness, thus has broad applicability.

The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. Despite its far-reaching application, there is almost no work on applying stochastic approximation to learning problems with constraints. The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. We propose that interior-point methods are a natural solution. We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization.

We describe a new algorithmic framework for inference in probabilistic models, and apply it to inference for latent Dirichlet allocation. Our framework adopts the methodology of variational inference, but unlike existing variational methods such as mean field and expectation propagation it is not restricted to tractable classes of approximating distributions. Our approach can also be viewed as a sequential Monte Carlo (SMC) method, but unlike existing SMC methods there is no need to design the artificial sequence of distributions. Notably, our framework offers a principled means to exchange the variance of an importance sampling estimate for the bias incurred through variational approximation. Experiments on a challenging inference problem in population genetics demonstrate improvements in stability and accuracy over existing methods, and at a comparable cost.

Deep neural networks (DNNs) have shown their success as high-dimensional function approximators in many applications; however, training DNNs can be challenging in general. DNN training is commonly phrased as a stochastic optimization problem whose challenges include non-convexity, non-smoothness, insufficient regularization, and complicated data distributions. Hence, the performance of DNNs on a given task depends crucially on tuning hyperparameters, especially learning rates and regularization parameters. In the absence of theoretical guidelines or prior experience on similar tasks, this requires solving many training problems, which can be time-consuming and demanding on computational resources. This can limit the applicability of DNNs to problems with non-standard, complex, and scarce datasets, e.g., those arising in many scientific applications. To remedy the challenges of DNN training, we propose slimTrain, a stochastic optimization method for training DNNs with reduced sensitivity to the choice hyperparameters and fast initial convergence. The central idea of slimTrain is to exploit the separability inherent in many DNN architectures; that is, we separate the DNN into a nonlinear feature extractor followed by a linear model. This separability allows us to leverage recent advances made for solving large-scale, linear, ill-posed inverse problems. Crucially, for the linear weights, slimTrain does not require a learning rate and automatically adapts the regularization parameter. Since our method operates on mini-batches, its computational overhead per iteration is modest. In our numerical experiments, slimTrain outperforms existing DNN training methods with the recommended hyperparameter settings and reduces the sensitivity of DNN training to the remaining hyperparameters.

With the increasing penetration of high-frequency sensors across a number of biological and physical systems, the abundance of the resulting observations offers opportunities for higher statistical accuracy of down-stream estimates, but their frequency results in a plethora of computational problems in data assimilation tasks. The high-frequency of these observations has been traditionally dealt with by using data modification strategies such as accumulation, averaging, and sampling. However, these data modification strategies will reduce the quality of the estimates, which may be untenable for many systems. Therefore, to ensure high-quality estimates, we adapt stochastic approximation methods to address the unique challenges of high-frequency observations in data assimilation. As a result, we are able to produce estimates that leverage all of the observations in a manner that avoids the aforementioned computational problems and preserves the statistical accuracy of the estimates.

The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. Despite its far-reaching application, there is almost no work on applying stochastic approximation to learning problems with constraints. The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. We propose that interior-point methods are a natural solution. We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization.

We describe a new algorithmic framework for inference in probabilistic models, and apply it to inference for latent Dirichlet allocation. Our framework adopts the methodology of variational inference, but unlike existing variational methods such as mean field and expectation propagation it is not restricted to tractable classes of approximating distributions. Our approach can also be viewed as a sequential Monte Carlo (SMC) method, but unlike existing SMC methods there is no need to design the artificial sequence of distributions. Notably, our framework offers a principled means to exchange the variance of an importance sampling estimate for the bias incurred through variational approximation. Experiments on a challenging inference problem in population genetics demonstrate improvements in stability and accuracy over existing methods, and at a comparable cost.

The need for parameter estimation with massive data has reinvigorated interest in iterative estimation procedures. Stochastic approximations, such as stochastic gradient descent, are at the forefront of this recent development because they yield simple, generic, and extremely fast iterative estimation procedures. Such stochastic approximations, however, are often numerically unstable. As a consequence, current practice has turned to proximal operators, which can induce stable parameter updates within iterations. While the majority of classical iterative estimation procedures are subsumed by the framework of Robbins and Monro (1951), there is no such generalization for stochastic approximations with proximal updates. In this paper, we conceptualize a general stochastic approximation method with proximal updates. This method can be applied even in situations where the analytical form of the objective is not known, and so it generalizes many stochastic gradient procedures with proximal operators currently in use. Our theoretical analysis indicates that the proposed method has important stability benefits over the classical stochastic approximation method. Exact instantiations of the proposed method are challenging, but we show that approximate instantiations lead to procedures that are easy to implement, and still dominate classical procedures by achieving numerical stability without tradeoffs. This last advantage is akin to that seen in deterministic proximal optimization, where the framework is typically impossible to instantiate exactly, but where approximate instantiations lead to new optimization procedures that dominate classical ones.

The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. Despite its far-reaching application, there is almost no work on applying stochastic approximation to learning problems with constraints. The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. We propose that interior-point methods are a natural solution. We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization.

We describe a new algorithmic framework for inference in probabilistic models, and apply it to inference for latent Dirichlet allocation. Our framework adopts the methodology of variational inference, but unlike existing variational methods such as mean field and expectation propagation it is not restricted to tractable classes of approximating distributions. Our approach can also be viewed as a sequential Monte Carlo (SMC) method, but unlike existing SMC methods there is no need to design the artificial sequence of distributions. Notably, our framework offers a principled means to exchange the variance of an importance sampling estimate for the bias incurred through variational approximation. Experiments on a challenging inference problem in population genetics demonstrate improvements in stability and accuracy over existing methods, and at a comparable cost.