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Register a new userStochastic gradient descent for hybrid quantum-classical optimization
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Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with $k$ measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of $k$. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases $k$-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain doubly stochastic gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.
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Conditional Stochastic Optimization (CSO) covers a variety of applications ranging from meta-learning and causal inference to invariant learning. However, constructing unbiased gradient estimates in CSO is challenging due to the composition structure. As an alternative, we propose a biased stochastic gradient descent (BSGD) algorithm and study the bias-variance tradeoff under different structural assumptions. We establish the sample complexities of BSGD for strongly convex, convex, and weakly convex objectives, under smooth and non-smooth conditions. We also provide matching lower bounds of BSGD for convex CSO objectives. Extensive numerical experiments are conducted to illustrate the performance of BSGD on robust logistic regression, model-agnostic meta-learning (MAML), and instrumental variable regression (IV).
Epoch gradient descent method (a.k.a. Epoch-GD) proposed by Hazan and Kale (2011) was deemed a breakthrough for stochastic strongly convex minimization, which achieves the optimal convergence rate of $O(1/T)$ with $T$ iterative updates for the {it objective gap}. However, its extension to solving stochastic min-max problems with strong convexity and strong concavity still remains open, and it is still unclear whether a fast rate of $O(1/T)$ for the {it duality gap} is achievable for stochastic min-max optimization under strong convexity and strong concavity. Although some recent studies have proposed stochastic algorithms with fast convergence rates for min-max problems, they require additional assumptions about the problem, e.g., smoothness, bi-linear structure, etc. In this paper, we bridge this gap by providing a sharp analysis of epoch-wise stochastic gradient descent ascent method (referred to as Epoch-GDA) for solving strongly convex strongly concave (SCSC) min-max problems, without imposing any additional assumption about smoothness or the functions structure. To the best of our knowledge, our result is the first one that shows Epoch-GDA can achieve the optimal rate of $O(1/T)$ for the duality gap of general SCSC min-max problems. We emphasize that such generalization of Epoch-GD for strongly convex minimization problems to Epoch-GDA for SCSC min-max problems is non-trivial and requires novel technical analysis. Moreover, we notice that the key lemma can also be used for proving the convergence of Epoch-GDA for weakly-convex strongly-concave min-max problems, leading to a nearly optimal complexity without resorting to smoothness or other structural conditions.
Despite the strong theoretical guarantees that variance-reduced finite-sum optimization algorithms enjoy, their applicability remains limited to cases where the memory overhead they introduce (SAG/SAGA), or the periodic full gradient computation they require (SVRG/SARAH) are manageable. A promising approach to achieving variance reduction while avoiding these drawbacks is the use of importance sampling instead of control variates. While many such methods have been proposed in the literature, directly proving that they improve the convergence of the resulting optimization algorithm has remained elusive. In this work, we propose an importance-sampling-based algorithm we call SRG (stochastic reweighted gradient). We analyze the convergence of SRG in the strongly-convex case and show that, while it does not recover the linear rate of control variates methods, it provably outperforms SGD. We pay particular attention to the time and memory overhead of our proposed method, and design a specialized red-black tree allowing its efficient implementation. Finally, we present empirical results to support our findings.
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.
Understanding the bottlenecks in implementing stochastic gradient descent (SGD)-based distributed support vector machines (SVM) algorithm is important in training larger data sets. The communication time to do the model synchronization across the parallel processes is the main bottleneck that causes inefficiency in the training process. The model synchronization is directly affected by the mini-batch size of data processed before the global synchronization. In producing an efficient distributed model, the communication time in training model synchronization has to be as minimum as possible while retaining a high testing accuracy. The effect from model synchronization frequency over the convergence of the algorithm and accuracy of the generated model must be well understood to design an efficient distributed model. In this research, we identify the bottlenecks in model synchronization in parallel stochastic gradient descent (PSGD)-based SVM algorithm with respect to the training model synchronization frequency (MSF). Our research shows that by optimizing the MSF in the data sets that we used, a reduction of 98% in communication time can be gained (16x - 24x speed up) with respect to high-frequency model synchronization. The training model optimization discussed in this paper guarantees a higher accuracy than the sequential algorithm along with faster convergence.
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