Various ways of writing Nesterov’s acceleration

Posted on 2020-06-04

Nestorov’s acceleration is usually presented in the following form (AGM1)

\[\begin{split} \begin{aligned} y_{t+1} &\leftarrow x_{t}-\frac{1}{\beta} \nabla f\left(x_{t}\right) \\ x_{t+1} &\leftarrow\left(1-\frac{1-\lambda_{t}}{\lambda_{t+1}}\right) y_{t+1}+\frac{1-\lambda_{t}}{\lambda_{t+1}} y_{t} \end{aligned} \end{split}\]

where \(\beta\) is the gradient Lipschitz parameter of \(f\).

The update is known to be notoriously opaque. There are alternatives ways of rewriting the first-order scheme, however, two of which we will cover here.

Acceleration from primal dual perspective

We want to rewrite AGM1 as the following scheme (called AGM2)

\[\begin{split} \begin{aligned} y_{t+1} &\leftarrow x_{t}-\frac{1}{\beta} \nabla f\left(x_{t}\right) \\ z_{t+1} &\leftarrow z_{t}-\eta_{t} \nabla f\left(x_{t}\right) \\ x_{t+1} &\leftarrow\left(1-\tau_{t+1}\right) y_{t+1}+\tau_{t+1} z_{t+1} \end{aligned} \end{split}\]

where \(f\) is \(\beta\)-smooth, \(\eta_{t}=\frac{t+1}{2 \beta}\) and \(\tau_{t}=\frac{2}{t+2}\). This is a linear coupling of a small step size update (\(y_{t+1}\)) and a large step size update (\(z_{t+1}\)).

Derivation Notice that the definition for \(y_{t+1}\) is already the same which leaves us with verifying the update for \(x_{t+1}\) and \(z_{t+1}\). Let’s start by assuming that the correct update for \(x_t\) exists

(6)\[x_{t}:=\left(1-\tau_{t}\right) y_{t}+\tau_{t} z_{t}\]

This leaves us with checking whether this leads to the right definition of \(z_t\). That is, we need to show that \(z_{t+1} - z_{t}=-\eta_{t} \nabla f\left(x_{t}\right)\) according to the AGM2 update. First we make a guess about the relationship between parameters,

\[ \tau_t = \frac{1}{\lambda_t} \]

Using this after isolating \(z_t\) in (6) gives us,

\[\begin{split}\begin{aligned} z_{t}&=\frac{1}{\tau_{t}} x_{t}-\left(\frac{1}{\tau_{t}}-1\right) y_{t}\\ &= \lambda_t x_{t}-\left(\lambda_t-1\right) y_{t}. \end{aligned}\end{split}\]

Expanding \(z_{t+1}-z_{t}\)

\[\begin{aligned} z_{t+1}-z_{t}&=\left(\lambda_{t+1}\left(x_{t+1}-y_{t+1}\right)+y_{t+1}\right)-\left(\lambda_{t}\left(x_{t}-y_{t}\right)+y_{t}\right) \end{aligned}\]

Using definition of \(x_t\) in AGM1,

\[ \lambda_{t+1}\left(x_{t+1}-y_{t+1}\right)-\left(1-\lambda_{t}\right)\left(y_{t}-y_{t+1}\right)=0. \]

We can cancel terms by subtracting it

\[ z_{t+1}-z_{t} = \lambda_{t} y_{t+1}-\lambda_{t} x_{t}. \]

We can further develop by using the update for \(y_{t+1}\),

\[ \lambda_{t} y_{t+1}-\lambda_{t} x_{t} = -\frac{\lambda_{t}}{\beta} \nabla f\left(x_{t}\right). \]

What is \(\frac{\lambda_{t}}{\beta}\)? If it is \(\frac{\lambda_{t}}{\beta}=\eta_{t}\) then we are done.

\[ \frac{\lambda_t}{\beta} = \frac{1}{\tau_t\beta} = \frac{t+2}{2\beta} = \eta_{t+1} \]

So we are weirdly enough off by one since it should be \(\eta_t\)…

Source: Bansal and Gupta [2017] and p. 468 of Drori and Teboulle [2014].

As momentum

NAG can be written as a momentum scheme where the gradient is queried at “a future point”.

\[\begin{split} \begin{aligned} v_{t+1} &\leftarrow \alpha_t v_{t}-\frac{1}{\beta} \nabla f\left(y_{t}+\alpha_t v_{t}\right) \\ y_{t+1} &\leftarrow y_{t}+v_{t+1} \end{aligned} \end{split}\]

Derivation We start with AGM1

\[\begin{split} \begin{aligned} y_{t+1} &= x_{t}-\frac{1}{\beta} \nabla f\left(x_{t}\right) \\ x_{t+1} &=\left(1-\frac{1-\lambda_{t}}{\lambda_{t+1}}\right) y_{t+1}+\frac{1-\lambda_{t}}{\lambda_{t+1}} y_{t}\\ &= y_{t+1}+\frac{\lambda_{t}-1}{\lambda_{t+1}} (y_{t+1} - y_{t}) \end{aligned} \end{split}\]

Now, naturally define the momentum as \(v_{t+1} = y_{t+1} - y_t\) and the parameter \(\alpha_{t+1}=\frac{\lambda_{t}-1}{\lambda_{t+1}}\) for convenience. This lets us write the above as,

\[ x_{t+1} = y_{t+1} + \alpha_{t+1}v_{t+1} \]

This definition can be used in the update for \(y_{t+1}\)

\[y_{t+1} = y_t + \alpha_tv_t-\frac{1}{\beta} \nabla f\left(y_t + \alpha_tv_t\right)\]

Finally using this form of \(y_{t+1}\) in the definition for \(v_{t+1}\) we obtain the desired momentum update

\[ v_{t+1} = y_{t+1} - y_t = \alpha_tv_t-\frac{1}{\beta} \nabla f\left(y_t + \alpha_tv_t\right) \]

Simply writing \(y_{t+1}\) using the definition of \(v_{t+1}\) completes the scheme

\[\begin{split}\begin{aligned} v_{t+1} &= \alpha_tv_t-\frac{1}{\beta} \nabla f\left(y_t + \alpha_tv_t\right)\\ y_{t+1} &= y_t + v_{k+1} \end{aligned}\end{split}\]

Source: [Sutskever et al., 2013].

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BG17

Nikhil Bansal and Anupam Gupta. Potential-function proofs for first-order methods. arXiv preprint arXiv:1712.04581, 2017. URL: https://arxiv.org/pdf/1712.04581.pdf.

DT14

Yoel Drori and Marc Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451–482, 2014. URL: https://link.springer.com/content/pdf/10.1007/s10107-013-0653-0.pdf.

SMDH13

Ilya Sutskever, James Martens, George Dahl, and Geoffrey Hinton. On the importance of initialization and momentum in deep learning. In International conference on machine learning, 1139–1147. PMLR, 2013. URL: http://proceedings.mlr.press/v28/sutskever13.pdf.