Random feedback weights support learning in deep neural networks

TL;DR

A surprisingly simple algorithm is presented, which assigns blame by multiplying error signals by random synaptic weights, and it is shown that a network can learn to extract useful information from signals sent through these random feedback connections, in essence, the network learns to learn.

Abstract

The brain processes information through many layers of neurons. This deep architecture is representationally powerful, but it complicates learning by making it hard to identify the responsible neurons when a mistake is made. In machine learning, the backpropagation algorithm assigns blame to a neuron by computing exactly how it contributed to an error. To do this, it multiplies error signals by matrices consisting of all the synaptic weights on the neuron's axon and farther downstream. This operation requires a precisely choreographed transport of synaptic weight information, which is thought to be impossible in the brain. Here we present a surprisingly simple algorithm for deep learning, which assigns blame by multiplying error signals by random synaptic weights. We show that a network can learn to extract useful information from signals sent through these random feedback connections. In essence, the network learns to learn. We demonstrate that this new mechanism performs as quickly and accurately as backpropagation on a variety of problems and describe the principles which underlie its function. Our demonstration provides a plausible basis for how a neuron can be adapted using error signals generated at distal locations in the brain, and thus dispels long-held assumptions about the algorithmic constraints on learning in neural circuits.

Figures (5)

• Figure 1: Random feedback weights can deliver useful teaching signals to preceding layers of a neural network. a, The backprop learning algorithm is powerful, but requires biologically implausible transport of individual synaptic weight information. For backprop, neurons must know each other’s synaptic weights, e.g. the three coloured synapses on the feedback cell at bottom must have weights equal to those of the corresponding coloured synapses on three cells in the forward path. On a computer it is simple to use the synaptic weights in both forward and backward computations, but synapses in the brain communicate information unidirectionally. b, Implemented in a computer, backprop computes how to change hidden unit activities by multiplying the error vector, $\mathbf{e}=\mathbf{y}^{*}-\mathbf{y}$, by the transpose matrix of the forward weights, i.e., $\Delta\mathbf{h}_{\mathrm{BP}}=W^{T}\mathbf{e}$. c, Our feedback alignment method uses the counterintuitive observation that learning is still effective if $W^{T}$ is replaced by a matrix of fixed random weights, $B$, so that $\Delta\mathbf{h}_{\mathrm{\mathrm{FA}}}=B\mathbf{e}$. d, Four algorithms learn to mimic a linear function: 'shallow' learning (light gray), reinforcement learning (dark gray), backprop (black), and feedback alignment (green). NSE is normalized squared error. e, Angle between the hidden-unit update vector prescribed by feedback alignment and that prescribed by backprop, i.e., $\Delta\mathbf{h}_{\mathrm{FA}}\measuredangle\Delta\mathbf{h}_{\mathrm{BP}}$. Error bars are two standard deviations for a sliding window of 10 examples.
• Figure 2: Feedback alignment solves nonlinear, real-world problems. a, A 784--1000--10 network of logistic units learns to recognize handwritten digits. Representative performance curves for backprop (black), feedback alignment (green), and shallow learning (light grey) on 10,000 test images. b, Angle between the hidden-unit update made by feedback alignment and that prescribed by backprop, i.e., $\Delta\mathbf{h}_{\mathrm{FA}}\measuredangle\Delta\mathbf{h}_{\mathrm{BP}}$. Error bars are one standard deviation around the time-averaged mean. c, Feedback alignment can train deeper layers via random weights, e.g. $B_{1}$ and $B_{2}$. d, Normalized squared error curves, each an average over 20 trials, for a nonlinear function-approximation task; three-layer network trained with shallow learning (grey), backprop (black), and feedback alignment (green); four-layer network trained with backprop (magenta) and feedback alignment (blue).
• Figure 3: Network dynamics underlying feedback alignment. a, Three-neuron network learning to match a linear function, $\mathbf{y}^{*}=T\mathbf{x}$, with $T=1$ and $B$ 'randomly' chosen to be 1. b, Vector flow field (small arrows) demonstrates the evolution of $W_{0}$ and $W$ during feedback alignment. Thick lines are solution manifolds (i.e. $W_{0}W=1=T$) where: $\mathbf{e}WB\mathbf{e}>0$ (grey), $\mathbf{e}WB\mathbf{e}<0$ (black), or unstable solutions (dashed black). There is a small region of weight space (shaded grey) from which the system travels to the "bad" hyperbola at lower left, but this is simply avoided by starting near 0. Large arrow traces the trajectory for the initial condition $W_{0}=W=0$.
• Figure 4: If $W_{0}$ and $W$ start small then $W$ learns to act like a local pseudoinverse of $B$. a, Each trace is a single run of feedback alignment learning with the elements of $W_{0}$ and $W$ drawn uniformly from $\left[-\omega,\omega\right]$, where $\omega=[0.0001,0.001,0.01,0.05,0.1,0.125,0.15,0.2,0.25]$, corresponding to blue through red, respectively. Loss is normalized squared error (NSE). b-c, Angle between the hidden unit changes prescribed by feedback alignment versus backprop (panel b) and versus pseudobackprop (panel c).
• Figure S1: Receptive fields for 100 randomly selected hidden units shown at the beginning of learning (top left) and for the three learning variants discussed in the main text. Grey scale indicates the strength of connection from each of $28\times 28$ pixels in MNIST images (white denotes strong positive, black denotes strong negative).

Theorems & Definitions (10)

• Theorem 1
• proof
• Lemma 1: Barbalat's Lemma
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• proof
• Theorem 2
• proof