CONTINUAL FINITE-SUM MINIMIZATION UNDER THE POLYAK-ŁOJASIEWICZ CONDITION
Keywords: Continual Learning, Finite Sum Minimization
Abstract: Given functions $f_1,\ldots,f_n$ where $f_i:\mathcal{D}\mapsto \mathbb{R}$, \textit{continual finite-sum minimization} (CFSM) asks for an $\epsilon$-optimal sequence $\hat{x}_1,\ldots,\hat{x}_n \in \mathcal{D}$ such that
$$\sum_{j=1}^i f_j(x_i)/i - \min_{x \in \mathcal{D}}\sum_{j=1}^if_j(x)/i \leq \epsilon$$
In this work, we develop a new CFSM framework under the Polyak-Łojasiewicz condition (PL), where each prefix-sum function $\sum_{j=1}^i f_j(x)/i$ satisfies the PL condition, extending the recent result on CFSM with strongly convex functions. We present a new first-order method that under the PL condition producing an $\epsilon$-optimal sequence with overall $\mathcal{O}(n/\sqrt{\epsilon})$ first-order oracles (FOs), where an FO corresponds to the computation of a single gradient $\nabla f_j(x)$ at a given $x \in \mathcal{D}$ for some $j \in [n]$. Our method also improves upon the $\mathcal{O}(n^2 \log (1/\epsilon))$ FO complexity of state-of-the art variance reduction methods as well as upon the $\mathcal{O}(n/\epsilon)$ FO complexity of $\mathrm{StochasticGradientDescent}$. We experimentally evaluate our method in continual learning and the unlearning settings, demonstrating the potential of the CFSM framework in non-convex, deep learning problems.
Supplementary Material: zip
Primary Area: optimization
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Submission Number: 10980
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