Stochastic Multi-level Composition Optimization Algorithms with Level-Independent Convergence Rates

• URL: http://arxiv.org/abs/2008.10526v5
• Date: Mon, 14 Feb 2022 05:55:45 GMT
• Title: Stochastic Multi-level Composition Optimization Algorithms with Level-Independent Convergence Rates
• Authors: Krishnakumar Balasubramanian, Saeed Ghadimi, Anthony Nguyen
• Abstract summary: We study smooth multi-level composition optimization problems, where the objective function is a nested composition of $T$ functions. We show that the first algorithm, which is a generalization of citeGhaRuswan20 to the $T$ level case, can achieve a sample complexity of $mathcalO (1/epsilon$6) This is the first time that such an online algorithm designed for the (un) multi-level setting, obtains the same sample complexity under standard assumptions.
• Score: 12.783783498844022
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 \cite{GhaRuswan20} to the $T$ level case, can achieve a sample complexity of $\mathcal{O}(1/\epsilon^6)$ 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 $\mathcal{O}(1/\epsilon^4)$. {\color{black}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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