# On Training Implicit Models

Source: [On Training Implicit Models](<https://www.sota2.com/research/paper/68fa6b9c1e209131cea2f4fb>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T23:53:05.688Z

Authors: Zhengyang Geng, Xin-Yu Zhang, Shaojie Bai, Yisen Wang, Zhouchen Lin

Published: 2021-11-09T14:40:24.000Z

## Abstract

This paper focuses on training implicit models of infinite layers. Specifically, previous works employ implicit differentiation and solve the exact gradient for the backward propagation. However, is it necessary to compute such an exact but expensive gradient for training? In this work, we propose a novel gradient estimate for implicit models, named phantom gradient, that 1) forgoes the costly computation of the exact gradient; and 2) provides an update direction empirically preferable to the implicit model training. We theoretically analyze the condition under which an ascent direction of the loss landscape could be found, and provide two specific instantiations of the phantom gradient based on the damped unrolling and Neumann series. Experiments on large-scale tasks demonstrate that these lightweight phantom gradients significantly accelerate the backward passes in training implicit models by roughly 1.7 times, and even boost the performance over approaches based on the exact gradient on ImageNet.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2111.05177>)

[PDF](<https://arxiv.org/pdf/2111.05177.pdf>)