Acceleration

SCS includes Anderson acceleration (AA), which can be used to speed up convergence. AA is a quasi-Newton method for the acceleration of fixed point iterations and can dramatically speed up convergence in practice, especially if higher accuracy solutions are desired. However, it can also cause severe instability of the solver and so should be used with caution. It is an open research question how to best implement AA in practice to ensure good performance across all problems and we welcome any contributions in that direction!

Mathematical details

The discussion here is taken from section 2 of our paper. Consider the problem of finding a fixed point of the function \(f: \mathbf{R}^n \rightarrow \mathbf{R}^n\), i.e.,

\[\text{Find } x \in \mathbf{R}^n \text{ such that } x = f(x).\]

In our case \(f\) corresponds to one step of Douglas-Rachford splitting and the iterate is the \(w^k\) vector, which converges to a fixed point of the DR operator. At a high level AA, from initial point \(x_0\) and max memory \(m\) (corresponding to the acceleration_lookback setting), works as follows:

\[\begin{split}\begin{array}{l} \text{For } k=0, 1, \dots \\ \quad \text{Set } m_k=\min\{m, k\} \\ \quad \text{Select weights } \alpha_j^k \text{ based on the last } m_k \text{ iterations satisfying } \sum_{j=0}^{m_k}\alpha_j^k=1 \\ \quad \text{Set } x^{k+1}=\sum_{j=0}^{m_k}\alpha_j^kf(x^{k-m_k+j}) \end{array}\end{split}\]

In other words, AA produces an iterate that is the linear combination of the last \(m_k + 1\) outputs of the map. Thus, the main challenge is in choosing the weights \(\alpha \in \mathbf{R}^{m_k+1}\). There are two ways to choose them, corresponding to type-I and type-II AA (named for the type-I and type-II Broyden updates). We shall present type-II first.

Type-II AA

Define the residual \(g: \mathbf{R}^n \rightarrow \mathbf{R}^n\) of \(f\) to be \(g(x) = x - f(x)\). Note that any fixed point \(x^\star\) satisfies \(g(x^\star) = 0\). In type-II AA the weights are selected by solving a small least squares problem.

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \|\sum_{j=0}^{m_k}\alpha_j g(x^{k-m_k+j})\|_2^2\\ \mbox{subject to} & \sum_{j=0}^{m_k}\alpha_j=1, \end{array}\end{split}\]

More explicitly, we can reformulate the above as follows:

\[\begin{array}{ll} \mbox{minimize} & \|g_k-Y_k\gamma\|_2, \end{array}\]

with variable \(\gamma=(\gamma_0,\dots,\gamma_{m_k-1}) \in \mathbf{R}^{m_k}\). Here \(g_i=g(x^i)\), \(Y_k=[y_{k-m_k}~\dots~y_{k-1}]\) with \(y_i=g_{i+1}-g_i\) for each \(i\), and \(\alpha\) and \(\gamma\) are related by \(\alpha_0=\gamma_0\), \(\alpha_i=\gamma_i-\gamma_{i-1}\) for \(1\leq i\leq m_k-1\) and \(\alpha_{m_k}=1-\gamma_{m_k-1}\).

Assuming that \(Y_k\) is full column rank, the solution \(\gamma^k\) to the above is given by \(\gamma^k=(Y_k^\top Y_k)^{-1}Y_k^\top g_k\), and hence by the relation between \(\alpha^k\) and \(\gamma^k\), the next iterate of type-II AA can be written as

\[\begin{split}\begin{align} x^{k+1}&=f(x^k)-\sum_{i=0}^{m_k-1}\gamma_i^k\left(f(x^{k-m_k+i+1})- f(x^{k-m_k+i})\right)\\ &=x^k-g_k-(S_k-Y_k)\gamma^k\\ &=x^k-(I+(S_k-Y_k)(Y_k^\top Y_k)^{-1}Y_k^\top )g_k\\ &=x^k-B_kg_k, \end{align}\end{split}\]

where \(S_k=[s_{k-m_k}~\dots~s_{k-1}]\), \(s_i=x^{i+1}-x^i\) for each \(i\), and

\[B_k=I+(S_k-Y_k)(Y_k^\top Y_k)^{-1}Y_k^\top\]

Observe that \(B_k\) minimizes \(\|B_k-I\|_F\) subject to the inverse multi-secant condition \(B_kY_k=S_k\), and hence can be regarded as an approximate inverse Jacobian of \(g\). The update of \(x^k\) can then be considered as a quasi-Newton-type update, with \(B_k\) being a generalized second (or type-II) Broyden’s update of \(I\) satisfying the inverse multi-secant condition.

Type-I AA

In the same spirit, we define type-I AA, in which we find an approximate Jacobian of \(g\) minimizing \(\|H_k-I\|_F\) subject to the multi-secant condition \(H_kS_k=Y_k\). Assuming that \(S_k\) is full column rank, we obtain (by symmetry) that

\[H_k=I+(Y_k-S_k)(S_k^\top S_k)^{-1}S_k^\top\]

and the update scheme is defined as

\[x^{k+1}=x^k-H_k^{-1}g_k\]

assuming \(H_k\) to be invertible. A direct application of the Woodbury matrix identity shows that

\[H_k^{-1}=I+(S_k-Y_k)(S_k^\top Y_k)^{-1}S_k^\top\]

where again we have assumed that \(S_k^\top Y_k\) is invertible. Notice that this explicit formula of \(H_k^{-1}\) is preferred in that the most costly step, inversion, is implemented only on a small \(m_k\times m_k\) matrix.

In SCS

In SCS both types of acceleration are available, though by default type-I is used since it tends to have better performance. If you wish to use AA then set the acceleration_lookback setting to a non-zero value (10 works well for many problems and is the default). This setting corresponds to \(m\), the maximum number of SCS iterates that AA will use to extrapolate to the new point.

To enable type-II acceleration then set acceleration_lookback to a negative value, the sign is interpreted as switching the AA type (this is mostly so that we can test it without fully exposing it the user).

The setting acceleration_interval controls how frequently AA is applied. If acceleration_interval \(=k\) for some integer \(k \geq 1\) then AA is applied every \(k\) iterations (AA simply ignores the intermediate iterations). This has the benefit of making AA \(k\) times faster and approximating a \(k\) times larger memory, as well as improving numerical stability by ‘decorrelating’ the data. On the other hand, older iterates might be stale. More work is needed to determine the optimal setting for this parameter, but 10 appears to work well in practice and is the default.

The details about how the linear systems are solved and updated is abstracted away into the AA package (eg, QR decomposition, SVD decomposition etc). Exactly how best to solve and update the equations is still open.

Regularization

By default we also add a small amount of regularization to the matrices that are being inverted in the above expressions, ie, in the type-II update

\[(Y_k^\top Y_k)^{-1} \text{ becomes } (Y_k^\top Y_k + \epsilon I)^{-1}\]

for some small \(\epsilon > 0\), and similarly for the type-I update

\[(S_k^\top Y_k)^{-1} \text{ becomes } (S_k^\top Y_k + \epsilon I)^{-1}\]

which is equivalent to adding regularization to the \(S_k^\top S_k\) matrix before using the Woodbury matrix identity. The regularization ensures the matrices are invertible and helps stability. In practice type-I tends to require more regularization than type-II for good performance. The regularization shrinks the AA update towards the update without AA, since if \(\epsilon\rightarrow\infty\) then \(\gamma^\star = 0\) and the AA step reduces to \(x^{k+1} = f(x^k)\). Note that the regularization can be folded into the matrices by appending \(\sqrt{\epsilon} I\) to the bottom of \(S_k\) or \(Y_k\), which is useful when using a QR or SVD decomposition to solve the equations.

Max \(\gamma\) norm

As the algorithm converges to the fixed point the matrices to be inverted can become ill-conditioned and AA can become unstable. In this case the \(\gamma\) vector can become very large. As a simple heuristic we reject the AA update and reset the AA state whenever \(\|\gamma\|_2\) is greater than max_weight_norm (eg, something very large like \(10^{10}\)).

Safeguarding

We also apply a safeguarding step to the output of the AA step. Explicitly, let \(x^k\) be the current iteration and let \(x_\mathrm{AA} = x^{k+1}\) be the output of AA. We reject the AA step if

\[\|x_\mathrm{AA} - f(x_\mathrm{AA}) \|_2 > \zeta \|x^k - f(x^k) \|_2\]

where \(\zeta\) is the safeguarding tolerance factor (safeguard_factor) and defaults to 1. In other words we reject the step if the norm of the residual after the AA step is larger than some amount (eg, if it increases the residual from the previous iterate). After rejecting a step we revert the iterate to \(x^k\) and reset the AA state.

Relaxation

In some works relaxation has been shown to improve performance. Relaxation replaces the final step of AA by mixing the map inputs and outputs as follows:

\[x^{k+1} = \beta \sum_{j=0}^{m_k}\alpha_j^k f(x^{k-m_k+j}) + (1-\beta) \sum_{j=0}^{m_k}\alpha_j^k x^{k-m_k+j}\]

where \(\beta\) is the relaxation parameter, and \(\beta=1\) recovers vanilla AA. This can be computed using the matrices defined above using

\[x^{k+1} = \beta (f(x^k) - (S_k - Y_k) \gamma^k) + (1-\beta) (x^k - S_k \gamma^k)\]

Anderson acceleration API

For completeness, we document the full Anderson acceleration API below.

Typedefs

typedef scs_float aa_float
typedef scs_int aa_int
typedef struct ACCEL_WORK AaWork

Functions

AaWork *aa_init(aa_int dim, aa_int mem, aa_int type1, aa_float regularization, aa_float relaxation, aa_float safeguard_factor, aa_float max_weight_norm, aa_int verbosity)

Initialize Anderson Acceleration, allocates memory.

Parameters
  • dim – the dimension of the variable for AA

  • mem – the memory (number of past iterations used) for AA

  • type1 – if True use type 1 AA, otherwise use type 2

  • regularization – type-I and type-II different, for type-I: 1e-8 works well, type-II: more stable can use 1e-12 often

  • relaxation – float in [0,2], mixing parameter (1.0 is vanilla)

  • safeguard_factor – factor that controls safeguarding checks larger is more aggressive but less stable

  • max_weight_norm – float, maximum norm of AA weights

  • verbosity – if greater than 0 prints out various info

Returns

pointer to AA workspace

aa_float aa_apply(aa_float *f, const aa_float *x, AaWork *a)

Apply Anderson Acceleration. The usage pattern should be as follows:

  • for i = 0 .. N:

    • if (i > 0): aa_apply(x, x_prev, a)

    • x_prev = x.copy()

    • x = F(x)

    • aa_safeguard(x, x_prev, a) // optional but helps stability

    Here F is the map we are trying to find the fixed point for. We put the AA before the map so that any properties of the map are maintained at the end. Eg if the map contains a projection onto a set then the output is guaranteed to be in the set.

Parameters
  • f – output of map at current iteration, overwritten with AA output

  • x – input to map at current iteration

  • a – workspace from aa_init

Returns

(+ or -) norm of AA weights vector. If positive then update was accepted and f contains new point, if negative then update was rejected and f is unchanged

aa_int aa_safeguard(aa_float *f_new, aa_float *x_new, AaWork *a)

Apply safeguarding.

This step is optional but can improve stability.

Parameters
  • f_new – output of map after AA step

  • x_new – AA output that is input to the map

  • a – workspace from aa_init

Returns

0 if AA step is accepted otherwise -1, if AA step is rejected then this overwrites f_new and x_new with previous values

void aa_finish(AaWork *a)

Finish Anderson Acceleration, clears memory.

Parameters

a – AA workspace from aa_init

void aa_reset(AaWork *a)

Reset Anderson Acceleration.

Resets AA as if at the first iteration, reuses original memory allocations.

Parameters

a – AA workspace from aa_init