[Paper Review] Random feedback weights support learning in deep neural networks

Timothy Lillicrap, Daniel Cownden+2 more|arXiv (Cornell University)|Nov 2, 2014

Advanced Memory and Neural Computing20 references88 citations

TL;DR

This paper proposes Feedback Alignment, a biologically plausible deep learning algorithm that replaces the weight-tied error backpropagation in backpropagation with fixed random feedback weights. Despite using random matrices instead of transposed synaptic weights, the method achieves performance comparable to backpropagation on various tasks, demonstrating that random feedback weights can still provide effective credit assignment in deep networks.

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.

Motivation & Objective

• To address the biological implausibility of weight transport in backpropagation, where neurons must share precise synaptic weight information across layers.
• To develop a learning algorithm that enables deep networks to learn efficiently without requiring knowledge of forward weights in the feedback path.
• To demonstrate that random feedback weights can still deliver useful error signals for credit assignment in deep networks.
• To provide a mechanism for how upstream synapses in the brain could be updated based on downstream error signals, resolving a longstanding question in neuroscience.

Proposed method

• Replace the transpose of forward weights $W^T$ in backpropagation with a fixed random matrix $B$ in the feedback path, so that hidden unit updates are computed as $\Delta \mathbf{h}_{\text{FA}} = B\mathbf{e}$.
• Use the condition that the angle between the feedback alignment update and the true backpropagation update remains within 90 degrees, ensuring the learning signal points in a useful direction.
• Demonstrate that even with fixed random feedback weights, the network can learn to align its internal representations to minimize error, as long as the random matrix $B$ has a non-zero inner product with the true gradient direction.
• Show that the algorithm is stable and converges by proving that hidden unit updates under feedback alignment are always a positive scalar multiple of those under pseudobackpropagation, ensuring consistent learning direction.
• Implement and test the algorithm on linear and nonlinear tasks, comparing performance to backpropagation, reinforcement learning, and shallow learning methods.
• Use theoretical analysis to prove convergence under simplified dynamics, showing that the learning signal remains aligned with the true gradient direction over time.

Experimental results

Research questions

• RQ1 Can deep neural networks learn effectively when error signals are propagated using fixed random feedback weights instead of transposed synaptic weights?
• RQ2 Is the performance of such a network comparable to standard backpropagation in terms of speed and accuracy?
• RQ3 Does the use of random feedback weights still allow for effective credit assignment in deep networks?
• RQ4 Can the network learn to align its internal representations even when the feedback weights are not related to the forward weights?
• RQ5 What are the theoretical conditions under which random feedback weights can still produce useful learning signals?

Key findings

• Feedback Alignment with fixed random feedback weights achieves performance comparable to standard backpropagation on both linear and nonlinear tasks, as measured by normalized squared error (NSE).
• The angle between the update directions of Feedback Alignment and backpropagation remains consistently below 90 degrees, indicating that the learning signal is aligned with the true gradient direction.
• The method is robust to the choice of random feedback matrix $B$, as long as $\mathbf{e}^T W B \mathbf{e} > 0$ on average, ensuring the feedback signal lies in a useful direction.
• Theoretical analysis shows that hidden unit updates under Feedback Alignment are always a positive scalar multiple of those under pseudobackpropagation, guaranteeing consistent learning direction.
• The algorithm converges under simplified dynamics, proving that learning can occur without weight transport, even when $W$ and $A$ are initialized to zero.
• The method outperforms reinforcement learning and shallow learning mechanisms in terms of learning speed and scalability, while preserving the representational power of deep networks.