Low-precision training is a promising way of decreasing the time and energy cost of training machine learning models. Previous work has analyzed low-precision training algorithms, such as low-precision stochastic gradient descent, and derived theoretical bounds on their convergence rates. These bounds tend to depend on the dimension of the model $d$ in that the number of bits needed to achieve a particular error bound increases as $d$ increases. In this paper, we derive new bounds for low-precision training algorithms that do not contain the dimension $d$, which lets us better understand what affects the convergence of these algorithms as parameters scale. Our methods also generalize naturally to let us prove new convergence bounds on low-precision training with other quantization schemes, such as low-precision floating-point computation and logarithmic quantization.

Low-precision training is a promising way of decreasing the time and energy cost of training machine learning models. Previous work has analyzed low-precision training algorithms, such as low-precision stochastic gradient descent, and derived theoretical bounds on their convergence rates. These bounds tend to depend on the dimension of the model d in that the number of bits needed to achieve a particular error bound increases as d increases. In this paper, we derive new bounds for low-precision training algorithms that do not contain the dimension d, which lets us better understand what affects the convergence of these algorithms as parameters scale. Our methods also generalize naturally to let us prove new convergence bounds on low-precision training with other quantization schemes, such as low-precision floating-point computation and logarithmic quantization.

Low-precision training is a promising way of decreasing the time and energy cost of training machine learning models. Previous work has analyzed low-precision training algorithms, such as low-precision stochastic gradient descent, and derived theoretical bounds on their convergence rates. These bounds tend to depend on the dimension of the model d in that the number of bits needed to achieve a particular error bound increases as d increases. In this paper, we derive new bounds for low-precision training algorithms that do not contain the dimension d, which lets us better understand what affects the convergence of these algorithms as parameters scale. Our methods also generalize naturally to let us prove new convergence bounds on low-precision training with other quantization schemes, such as low-precision floating-point computation and logarithmic quantization.

Low-precision training is a promising way of decreasing the time and energy cost of training machine learning models. Previous work has analyzed low-precision training algorithms, such as low-precision stochastic gradient descent, and derived theoretical bounds on their convergence rates. These bounds tend to depend on the dimension of the model $d$ in that the number of bits needed to achieve a particular error bound increases as $d$ increases. In this paper, we derive new bounds for low-precision training algorithms that do not contain the dimension $d$, which lets us better understand what affects the convergence of these algorithms as parameters scale. Our methods also generalize naturally to let us prove new convergence bounds on low-precision training with other quantization schemes, such as low-precision floating-point computation and logarithmic quantization.

Low-precision computation has been gaining a lot of traction in machine learning. Companies have even started developing new hardware architectures that natively support and accelerate low-precision operations including Microsoft's Project Brainwave and Google's TPU. Even though using low precision can have a lot of systems benefits, low-precision methods have been used primarily for inference--not for training. Previous low-precision training algorithms suffered from a fundamental tradeoff: when calculations use fewer bits, more round-off error is added, which limits training accuracy. According to conventional wisdom, this tradeoff limits practitioners' ability to deploy low-precision training algorithms in their systems. But is this tradeoff really fundamental?