Approximate bi-level optimization (ABLO) consists of (outer-level) optimization problems, involving numerical (inner-level) optimization loops. While ABLO has many applications across deep learning, it suffers from time and memory complexity proportional to the length $r$ of its inner optimization loop. To address this complexity, an earlier first-order method (FOM) was proposed as a heuristic that omits second derivative terms, yielding significant speed gains and requiring only constant memory. Despite FOM's popularity, there is a lack of theoretical understanding of its convergence properties. We contribute by theoretically characterizing FOM's gradient bias under mild assumptions. We further demonstrate a rich family of examples where FOM-based SGD does not converge to a stationary point of the ABLO objective. We address this concern by proposing an unbiased FOM (UFOM) enjoying constant memory complexity as a function of $r$. We characterize the introduced time-variance tradeoff, demonstrate convergence bounds, and find an optimal UFOM for a given ABLO problem. Finally, we propose an efficient adaptive UFOM scheme.

翻译：近似双级优化(ABLO)由(外)优化问题组成,涉及数字(内)优化循环。虽然ABLO在深层学习中有许多应用,但它有时间和记忆的复杂性,与其内部优化循环的长度成正比。为解决这一复杂问题,建议采用较早的一级优化方法(FOM)作为一种杂交法,省略第二衍生术语,产生显著的速度增益,并只需要不断记忆。尽管FOM受到欢迎,但对其趋同特性缺乏理论上的理解。我们在轻度假设下对FOM的梯度偏差作了理论上的定性。我们进一步展示了基于FOM的SGD没有与ABLO目标的固定点趋同的大量实例。我们提出一个无偏重的FOM(UOM),将保持恒定的记忆复杂性作为美元函数,以解决这一关切。我们提出了一种不带偏见的FOMM(UFOM),将长期记忆的复杂性视为一种函数。我们描述引入的时间差交易的特点,展示汇合关系,并为一个特定ABLO问题找到一个最佳的UM。我们提议一个高效的适应性UOM计划。