In this paper, we study smooth stochastic multi-level composition optimization problems, where the objective function is a nested composition of T functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an \epsilon-stationary point of the problem. We show that the first algorithm, which is a generalization of 22 to the T level case, can achieve a sample complexity of O(1/\epsilon^6) by using mini-batches of samples in each iteration. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to O(1/\epsilon^4). This modification also removes the requirement of having a mini-batch of samples in each iteration  . To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multi-level setting, obtains the same sample complexity of the smooth single-level setting, under mild assumptions on the stochastic first-order oracle.
In this paper, we study smooth stochastic multi-level composition optimization problems, where the objective function is a nested composition of T functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an \epsilon-stationary point of the problem. We show that the first algorithm, which is a generalization of \mbox{%DIFAUXCMD GhaRuswan20 to the T level case, can achieve a sample complexity of O(1/\epsilon^6) by using mini-batches of samples in each iteration. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to O(1/\epsilon^4). This modification not only removes the requirement of having a mini-batch of samples in each iteration , but also makes the algorithm parameter-free and easy to implement . To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multi-level setting, obtains the same sample complexity of the smooth single-level setting, under standard assumptions (unbiasedness and boundedness of the second moments) on the stochastic first-order oracle.
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22
\mbox{%DIFAUXCMD GhaRuswan20
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This modification also
This modification not only
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, but also makes the algorithm parameter-free and easy to implement
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mild assumptions
standard assumptions (unbiasedness and boundedness of the second moments)