Gradient codes use data replication to mitigate the effect of straggling machines in distributed machine learning. Approximate gradient codes consider codes where the data replication factor is too low to recover the full gradient exactly. Our work is motivated by the challenge of designing approximate gradient codes that simultaneously work well in both the adversarial and random straggler models. We introduce novel approximate gradient codes based on expander graphs. We analyze the decoding error both for random and adversarial stragglers, when optimal decoding coefficients are used. With random stragglers, our codes achieve an error to the gradient that decays exponentially in the replication factor. With adversarial stragglers, the error is smaller than any existing code with similar performance in the random setting. We prove convergence bounds in both settings for coded gradient descent under standard assumptions. With random stragglers, our convergence rate improves upon rates obtained via black-box approaches. With adversarial stragglers, we show that gradient descent converges down to a noise floor that scales linearly with the adversarial error to the gradient. We demonstrate empirically that our codes achieve near-optimal error with random stragglers and converge faster than algorithms that do not use optimal decoding coefficients.
