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In this paper, we consider a general stochastic optimization problem which is often at the core of supervised learning, such as deep learning and linear classification. We consider a standard stochastic gradient descent (SGD) method with a fixed, large step size and propose a novel assumption on the objective function, under which this method has the improved convergence rates (to a neighborhood of the optimal solutions). We then empirically demonstrate that these assumptions hold for logistic regression and standard deep neural networks on classical data sets. Thus our analysis helps to explain when efficient behavior can be expected from the SGD method in training classification models and deep neural networks.
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Stochastic nested optimization, including stochastic compositional, min-max and bilevel optimization, is gaining popularity in many machine learning applications. While the three problems share the nested structure, existing works often treat them separately, and thus develop problem-specific algorithms and their analyses. Among various exciting developments, simple SGD-type updates (potentially on multiple variables) are still prevalent in solving this class of nested problems, but they are believed to have slower convergence rate compared to that of the non-nested problems. This paper unifies several SGD-type updates for stochastic nested problems into a single SGD approach that we term ALternating Stochastic gradient dEscenT (ALSET) method. By leveraging the hidden smoothness of the problem, this paper presents a tighter analysis of ALSET for stochastic nested problems. Under the new analysis, to achieve an $epsilon$-stationary point of the nested problem, it requires ${cal O}(epsilon^{-2})$ samples. Under certain regularity conditions, applying our results to stochastic compositional, min-max and reinforcement learning problems either improves or matches the best-known sample complexity in the respective cases. Our results explain why simple SGD-type algorithms in stochastic nested problems all work very well in practice without the need for further modifications.
The superior performance of ensemble methods with infinite models are well known. Most of these methods are based on optimization problems in infinite-dimensional spaces with some regularization, for instance, boosting methods and convex neural networks use $L^1$-regularization with the non-negative constraint. However, due to the difficulty of handling $L^1$-regularization, these problems require early stopping or a rough approximation to solve it inexactly. In this paper, we propose a new ensemble learning method that performs in a space of probability measures, that is, our method can handle the $L^1$-constraint and the non-negative constraint in a rigorous way. Such an optimization is realized by proposing a general purpose stochastic optimization method for learning probability measures via parameterization using transport maps on base models. As a result of running the method, a transport map to output an infinite ensemble is obtained, which forms a residual-type network. From the perspective of functional gradient methods, we give a convergence rate as fast as that of a stochastic optimization method for finite dimensional nonconvex problems. Moreover, we show an interior optimality property of a local optimality condition used in our analysis.
We study the implicit regularization of mini-batch stochastic gradient descent, when applied to the fundamental problem of least squares regression. We leverage a continuous-time stochastic differential equation having the same moments as stochastic gradient descent, which we call stochastic gradient flow. We give a bound on the excess risk of stochastic gradient flow at time $t$, over ridge regression with tuning parameter $lambda = 1/t$. The bound may be computed from explicit constants (e.g., the mini-batch size, step size, number of iterations), revealing precisely how these quantities drive the excess risk. Numerical examples show the bound can be small, indicating a tight relationship between the two estimators. We give a similar result relating the coefficients of stochastic gradient flow and ridge. These results hold under no conditions on the data matrix $X$, and across the entire optimization path (not just at convergence).
Modern methods for learning from data depend on many tuning parameters, such as the stepsize for optimization methods, and the regularization strength for regularized learning methods. Since performance can depend strongly on these parameters, it is important to develop comparisons between emph{classes of methods}, not just for particularly tuned ones. Here, we take aim to compare classes of estimators via the relative performance of the emph{best method in the class}. This allows us to rigorously quantify the tuning sensitivity of learning algorithms. As an illustration, we investigate the statistical estimation performance of ridge regression with a uniform grid of regularization parameters, and of gradient descent iterates with a fixed stepsize, in the standard linear model with a random isotropic ground truth parameter. (1) For orthogonal designs, we find the emph{exact minimax optimal classes of estimators}, showing they are equal to gradient descent with a polynomially decaying learning rate. We find the exact suboptimalities of ridge regression and gradient descent with a fixed stepsize, showing that they decay as either $1/k$ or $1/k^2$ for specific ranges of $k$ estimators. (2) For general designs with a large number of non-zero eigenvalues, we find that gradient descent outperforms ridge regression when the eigenvalues decay slowly, as a power law with exponent less than unity. If instead the eigenvalues decay quickly, as a power law with exponent greater than unity or exponentially, we find that ridge regression outperforms gradient descent. Our results highlight the importance of tuning parameters. In particular, while optimally tuned ridge regression is the best estimator in our case, it can be outperformed by gradient descent when both are restricted to being tuned over a finite regularization grid.
We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In the traditional analysis using a consistency property of loss functions, it is known that the expected classification error converges more slowly than the expected risk even when assuming a low-noise condition on the conditional label probabilities. Consequently, the resulting rate is sublinear. Therefore, it is important to consider whether much faster convergence of the expected classification error can be achieved. In recent research, an exponential convergence rate for stochastic gradient descent was shown under a strong low-noise condition but provided theoretical analysis was limited to the squared loss function, which is somewhat inadequate for binary classification tasks. In this paper, we show an exponential convergence of the expected classification error in the final phase of the stochastic gradient descent for a wide class of differentiable convex loss functions under similar assumptions. As for the averaged stochastic gradient descent, we show that the same convergence rate holds from the early phase of training. In experiments, we verify our analyses on the $L_2$-regularized logistic regression.
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