Source-linked AI summary
Random feedback weights support learning in deep neural networks
Timothy P. Lillicrap, Daniel Cownden, Douglas B. Tweed, Colin J. Akerman
TL;DR
Deep networks require upstream synapses to use downstream errors, but backpropagation depends on transporting precise synaptic-weight information. The paper studies feedback alignment, which uses random feedback matrices, and reports analytical conditions and connections to pseudobackprop and Gauss–Newton updates. Its experiments and analyses support feedback alignment as a simpler mechanism that can transmit useful error information to deep neurons, while the guaranteed-success conditions remain incompletely characterized.
Problem
How can upstream synapses in deep networks be adjusted from downstream errors when backpropagation’s precise weight transport is biologically implausible?
Method
The paper combines feedback-alignment learning with analytical results on error reduction, pseudoinverse-based updates, and their relation to Gauss–Newton optimization.
Results
Feedback alignment reduces error under an established class of conditions, and its hidden-unit updates can be related to pseudobackprop through forward-weight alignment.
Takeaways & Limitations
The results provide a basis for transmitting useful error information to neurons deep within a network without exact weight transport.
Takeaways & Limitations
The paper cannot yet sharply delineate the learning problems for which feedback alignment is guaranteed to work.
Abstract
from arXiv · showhide
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.
Methods Summary
The study trains feedforward networks on linear approximation, handwritten-digit classification, and nonlinear function approximation using backpropagation, feedback alignment, and related methods.
- Methods Summary: Three tasks evaluated learning algorithms: linear function approximation, MNIST digit classification, and nonlinear function approximation.The networks included 30–20–10 linear models, a 784–1000–10 sigmoidal MNIST model, and three- or four-layer tanh networks.
- Methods Summary: Feedback alignment updates hidden weights with random feedback matrices, while backpropagation uses transposed forward weights.For the linear task, feedback matrices had elements drawn uniformly from [−0.5, 0.5].
- Methods Summary: MNIST training used 60,000 images and evaluated errors on a held-aside test set of 10,000 images.Both algorithms used η = 10−3 and weight decay α = 10−6.
Full Methods
The experiments used fixed or manually selected settings, repeated random draws, matched stopping times, and task-specific datasets to compare learning algorithms.
- Full Methods: Performance comparisons used straightforward procedures, including fixed learning rates and simple hyperparameter selection.The authors note that more complicated schemes might improve performance, but chose simple procedures to isolate the main ideas.
- Full Methods: Linear-function experiments fixed the target matrix and data sequence across algorithms.Inputs were sampled from N(0, I), and network weights were initialized uniformly in [−0.01, 0.01].
- Full Methods: MNIST experiments reused forward-matrix scales and learning rates across backpropagation and feedback alignment after manual tuning.A new random feedback matrix was drawn for each of 20 simulations, and sparse-weight experiments removed 50% of W and B elements.
- Full Methods: Nonlinear function-approximation experiments used fixed data sequences, new random network parameters for each of 20 simulations, and 5,000 held-out test points.Performance was evaluated with normalized squared error on the test set.
- Full Methods: Training ended after the same number of iterations for each algorithm, with termination chosen after backpropagation made negligible training-error gains.Deep-network feedback matrices were sampled from uniform distributions with manually selected scale parameters.
Supplementary Figures.
Supplementary Figure S1 compares receptive fields of randomly selected MNIST hidden units before and after learning with three learning variants.
- Supplementary Figures.: Figure S1 shows receptive fields for 100 randomly selected hidden units at initialization and after three learning variants.Each receptive field represents connection strengths from the 28×28 MNIST pixels.
- Supplementary Figures.: White denotes strong positive pixel connections, whereas black denotes strong negative connections.Gray levels encode the strength of each pixel-to-hidden-unit connection.
Introduction to analytic results.
The paper presents three analytical results addressing when feedback alignment reduces error, how pseudobackprop approximates Gauss–Newton optimization, and how alignment may relate to that modification.
- Introduction to analytic results.: The first analytical result gives conditions guaranteeing that feedback alignment reduces a network’s function error to zero.The stated class includes cases requiring downstream synaptic-weight modifications.
- Introduction to analytic results.: The second result shows that replacing W^T with W^+ in backpropagation yields an update rule approximating Gauss–Newton optimization.The third result examines a possible connection between feedback alignment and this Gauss–Newton modification.
Proof #1: Condition for alignment to reduce error to zero.
This proof establishes that feedback alignment can reduce training error to zero for a class of linear networks under specific dynamical and matrix conditions. The result includes cases requiring useful downstream synaptic modifications.
- The guarantee applies only to a restricted class of problems, although that class includes cases where downstream synaptic weights must be modified.The paper does not sharply delimit the full space of problems for which feedback alignment is guaranteed.
- The network maps inputs through a hidden layer using matrices A and W, and training seeks functional equivalence to the linear target T.The output is generated from input x through hidden activity and synaptic weight matrices.
- Under the normal-input and batch-learning assumptions, discrete updates simplify into continuous-time learning dynamics as the learning rate becomes small.The analysis takes the large-batch limit and then considers η approaching zero.
- Theorem 1 guarantees that feedback alignment reduces network error to zero when its learning dynamics and matrix conditions hold.The proof uses a Lyapunov-style argument together with Barbalat’s lemma.
- The proof shows that a nonnegative Lyapunov quantity decreases, its derivative converges to zero, and the resulting error matrix E becomes identically zero.The argument establishes boundedness and convergence conditions before concluding ||E|| = 0.
Proof #2: Gauss-Newton modification of backprop
This proof analyzes replacing backpropagation’s transpose feedback matrix with the Moore–Penrose pseudoinverse. It shows that the resulting update approximates Gauss–Newton optimization and can align with feedback-alignment updates under restrictive conditions.
- Replacing W^T with W^+ yields a backpropagation variant that approximates Gauss–Newton optimization for hidden-unit updates.The pseudoinverse both satisfies the required error-direction condition and prescribes second-order updates.
- Gauss–Newton optimization minimizes squared error by iteratively improving a parameter estimate using local curvature information.For small error, the relevant Hessian expression is approximated through the pseudoinverse.
- Pseudobackprop replaces W^T in the hidden-unit update and produces a change in h proportional to the Gauss–Newton adjustment.With a suitable learning rate, the proportionality factor can equal 1; otherwise the step size can be kept controlled.
- The authors interpret feedback alignment as an approximate implementation of pseudobackprop because learning encourages W to act like B^+ on error vectors.They also note that pseudobackprop itself may be of limited practical interest because computing the pseudoinverse is expensive.
- Under the stated single-example dynamics and initialization conditions, feedback-alignment hidden-unit updates are always a positive scalar multiple of pseudobackprop updates.The proof assumes a fixed full-rank random feedback matrix and suitably chosen learning rates.