tick.solver.AdaGrad¶

class tick.solver.AdaGrad(step: float = 0.01, epoch_size: int = None, rand_type: str = 'unif', tol: float = 1e-10, max_iter: int = 100, verbose: bool = True, print_every: int = 10, record_every: int = 1, seed: int = -1)[source]¶

Adaptive stochastic gradient descent solver

For the minimization of objectives of the form

\[\frac 1n \sum_{i=1}^n f_i(w) + g(w),\]

where the functions \(f_i\) have smooth gradients and \(g\) is prox-capable and separable, namely

\[g(w) = \sum_{j=1}^d g_j(w_j)\]

where \(g_j\) are prox-capable scalar functions of a single coordinate \(w_j\) of the vector of weights \(w \in \mathbb R^d\). Function \(f = \frac 1n \sum_{i=1}^n f_i\) corresponds to the model.loss method of the model (passed with set_model to the solver) and \(g\) corresponds to the prox.value method of the prox (passed with the set_prox method). The given prox must be, as explained above, separable. One iteration of AdaGrad corresponds to the following iteration applied epoch_size times:

\[\begin{split}\begin{align*} &\text{for } j=1, \ldots, d \; \text{ do the following:} \\ & \quad g_j \gets ( \nabla f_i(w) )_j \\ & \quad d_j \gets d_j + g_j^2 \\ & \quad w_j \gets w_j - \frac{\eta}{\sqrt{d_j + 10^{-6}}} \; g_j \\ & \quad w_j \gets \mathrm{prox}_{\eta_j g_j}(w_j) \end{align*}\end{split}\]

where \(i\) is sampled at random (strategy depends on rand_type) at each iteration, where \(\eta\) that can be tuned with step. The seed of the random number generator for generation of samples \(i\) can be seeded with seed. The iterations stop whenever tolerance tol is achieved, or after max_iter epochs (namely max_iter\(\times\) epoch_size). The obtained solution \(w\) is returned by the solve method, and is also stored in the solution attribute of the solver.

Parameters

step : float, default=1e-2

Step-size parameter, the most important parameter of the solver. A try-an-improve approach should be used.

tol : float, default=1e-10

The tolerance of the solver (iterations stop when the stopping criterion is below it)

max_iter : int, default=100

Maximum number of iterations of the solver, namely maximum number of epochs (by default full pass over the data, unless epoch_size has been modified from default)

rand_type : {‘unif’, ‘perm’}, default=’unif’

How samples are randomly selected from the data

• if 'unif' samples are uniformly drawn among all possibilities

• if 'perm' a random permutation of all possibilities is generated and samples are sequentially taken from it. Once all of them have been taken, a new random permutation is generated

verbose : bool, default=True

If True, solver verboses history, otherwise nothing is displayed, but history is recorded anyway

print_every : int, default=10

Print history information every time the iteration number is a multiple of print_every. Used only is verbose is True

record_every : int, default=1

Save history information every time the iteration number is a multiple of record_every

seed : int, default=-1

The seed of the random sampling. If it is negative then a random seed (different at each run) will be chosen.

epoch_size : int, default given by model

Epoch size, namely how many iterations are made before updating the variance reducing term. By default, this is automatically tuned using information from the model object passed through set_model.

Attributes

model : Model

The model used by the solver, passed with the set_model method

prox : Prox

Proximal operator used by the solver, passed with the set_prox method

solution : numpy.array, shape=(n_coeffs,)

Minizer found by the solver

history : dict-like

A dict-type of object that contains history of the solver along iterations. It should be accessed using the get_history method

time_start : str

Start date of the call to solve()

time_elapsed : float

Duration of the call to solve(), in seconds

time_end : str

End date of the call to solve()

dtype : {'float64', 'float32'}, default=’float64’

Type of the arrays used. This value is set from model and prox dtypes.

References

• J. Duchi, E. Hazan, Y. Singer, Adaptive Subgradient Methods for Online Learning and Stochastic Optimization, Journal of Machine Learning Research (2011)

__init__(step: float = 0.01, epoch_size: int = None, rand_type: str = 'unif', tol: float = 1e-10, max_iter: int = 100, verbose: bool = True, print_every: int = 10, record_every: int = 1, seed: int = -1)[source]¶

Initialize self. See help(type(self)) for accurate signature.

get_history(key=None)¶

Returns history of the solver

Parameters

key : str, default=None

• If None all history is returned as a dict

• If str, name of the history element to retrieve

Returns

output : list or dict

• If key is None or key is not in history then output is a dict containing history of all keys

• If key is the name of an element in the history, output is a list containing the history of this element

objective(coeffs, loss: float = None)¶

Compute the objective function

Parameters

coeffs : np.array, shape=(n_coeffs,)

Point where the objective is computed

loss : float, default=`None`

Gives the value of the loss if already known (allows to avoid its computation in some cases)

Returns

output : float

Value of the objective at given coeffs

set_model(model)[source]¶

Set model in the solver

Parameters

model : Model

Sets the model in the solver. The model gives the first order information about the model (loss, gradient, among other things)

Returns

output : Solver

The Solver with given model

set_prox(prox: tick.prox.base.prox.Prox)¶

Set proximal operator in the solver

Parameters

prox : Prox

The proximal operator of the penalization function

Returns

output : Solver

The solver with given prox

Notes

In some solvers, set_model must be called before set_prox, otherwise and error might be raised

solve(x0=None, step=None)¶

Launch the solver

Parameters

x0 : np.array, shape=(n_coeffs,), default=`None`

Starting point of the solver

step : float, default=`None`

Step-size or learning rate for the solver. This can be tuned also using the step attribute

Returns

output : np.array, shape=(n_coeffs,)

Obtained minimizer for the problem, same as solution attribute