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# tf.contrib.factorization.WALSModel

A model for Weighted Alternating Least Squares matrix factorization.

It minimizes the following loss function over U, V:

$$\|\sqrt W \odot (A - U V^T)\|_F^2 + \lambda (\|U\|_F^2 + \|V\|_F^2)$$

where, A: input matrix, W: weight matrix. Note that the (element-wise) square root of the weights is used in the objective function. U, V: row_factors and columnfactors matrices, $$\lambda)\: regularization. Also we assume that W is of the following special form: \( W{ij} = W_0 + R_i * Cj$$ if $$A{ij} \ne 0$$, $$W_{ij} = W_0$$ otherwise. where, $$W_0$$: unobserved_weight, $$R_i$$: row_weights, $$C_j$$: col_weights.

Note that the current implementation supports two operation modes: The default mode is for the condition where row_factors and col_factors can individually fit into the memory of each worker and these will be cached. When this condition can't be met, setting use_factors_weights_cache to False allows the larger problem sizes with slight performance penalty as this will avoid creating the worker caches and instead the relevant weight and factor values are looked up from parameter servers at each step.

Loss computation: The loss can be computed efficiently by decomposing it into a sparse term and a Gramian term, see wals.md. The loss is returned by the update_{col, row}_factors(sp_input), and is normalized as follows: _, _, unregularized_loss, regularization, sum_weights = update_row_factors(sp_input) if sp_input contains the rows $${A_i, i \in I}$$, and the input matrix A has n total rows, then the minibatch loss = unregularized_loss + regularization is

$$(\|\sqrt W_I \odot (A_I - U_I V^T)\|_F^2 + \lambda \|U_I\|_F^2) * n / |I| + \lambda \|V\|_F^2$$

The sum_weights tensor contains the normalized sum of weights $$sum(W_I) * n / |I|$$.

A typical usage example (pseudocode):

with tf.Graph().as_default(): # Set up the model object. model = tf.contrib.factorization.WALSModel(....)

# To be run only once as part of session initialization. In distributed
# training setting, this should only be run by the chief trainer and all
# other trainers should block until this is done.
model_init_op = model.initialize_op

# To be run once per worker after session is available, prior to
# the prep_gramian_op for row(column) can be run.
worker_init_op = model.worker_init

# To be run once per iteration sweep before the row(column) update
# initialize ops can be run. Note that in the distributed training
# situations, this should only be run by the chief trainer. All other
# trainers need to block until this is done.
row_update_prep_gramian_op = model.row_update_prep_gramian_op
col_update_prep_gramian_op = model.col_update_prep_gramian_op

# To be run once per worker per iteration sweep. Must be run before
# any actual update ops can be run.
init_row_update_op = model.initialize_row_update_op
init_col_update_op = model.initialize_col_update_op

# Ops to update row(column). This can either take the entire sparse
# tensor or slices of sparse tensor. For distributed trainer, each
# trainer handles just part of the matrix.
_, row_update_op, unreg_row_loss, row_reg, _ = model.update_row_factors(
sp_input=matrix_slices_from_queue_for_worker_shard)
row_loss = unreg_row_loss + row_reg
_, col_update_op, unreg_col_loss, col_reg, _ = model.update_col_factors(
sp_input=transposed_matrix_slices_from_queue_for_worker_shard,
transpose_input=True)
col_loss = unreg_col_loss + col_reg

...

# model_init_op is passed to Supervisor. Chief trainer runs it. Other
# trainers wait.
sv = tf.compat.v1.train.Supervisor(is_chief=is_chief,
...,
init_op=tf.group(..., model_init_op, ...), ...)
...

with sv.managed_session(...) as sess:
# All workers/trainers run it after session becomes available.
worker_init_op.run(session=sess)

...

while i in iterations:

# All trainers need to sync up here.
wait

# Row update sweep.
if is_chief:
row_update_prep_gramian_op.run(session=sess)
else:
wait_for_chief

# All workers run upate initialization.
init_row_update_op.run(session=sess)

# Go through the matrix.
reset_matrix_slices_queue_for_worker_shard
while_matrix_slices:
row_update_op.run(session=sess)

# All trainers need to sync up here.
wait

# Column update sweep.
if is_chief:
col_update_prep_gramian_op.run(session=sess)
else:
wait_for_chief

# All workers run upate initialization.
init_col_update_op.run(session=sess)

# Go through the matrix.
reset_transposed_matrix_slices_queue_for_worker_shard
while_transposed_matrix_slices:
col_update_op.run(session=sess)


input_rows total number of rows for input matrix.
input_cols total number of cols for input matrix.
n_components number of dimensions to use for the factors.
unobserved_weight weight given to unobserved entries of matrix.
regularization weight of L2 regularization term. If None, no regularization is done.
row_init initializer for row factor. Can be a tensor or numpy constant. If set to "random", the value is initialized randomly.
col_init initializer for column factor. See row_init for details.
num_row_shards number of shards to use for row factors.
num_col_shards number of shards to use for column factors.
row_weights Must be in one of the following three formats: None, a list of lists of non-negative real numbers (or equivalent iterables) or a single non-negative real number.

• When set to None, w_ij = unobserved_weight, which simplifies to ALS. Note that col_weights must also be set to "None" in this case.
• If it is a list of lists of non-negative real numbers, it needs to be in the form of [[w_0, w_1, ...], [w_k, ... ], [...]], with the number of inner lists matching the number of row factor shards and the elements in each inner list are the weights for the rows of the corresponding row factor shard. In this case, w_ij = unobserved_weight + row_weights[i] * col_weights[j].
• If this is a single non-negative real number, this value is used for all row weights and (w_ij) = unobserved_weight + row_weights * col_weights[j]. Note that it is allowed to have row_weights as a list while col_weights a single number or vice versa.
col_weights See row_weights.
use_factors_weights_cache When True, the factors and weights will be cached on the workers before the updates start. Defaults to True. Note that the weights cache is initialized through worker_init, and the row/col factors cache is initialized through initialize_{col/row}_update_op. In the case where the weights are computed outside and set before the training iterations start, it is important to ensure the worker_init op is run afterwards for the weights cache to take effect.
use_gramian_cache When True, the Gramians will be cached on the workers before the updates start. Defaults to True.
use_scoped_vars When True, the factor and weight vars will also be nested in a tf.name_scope.

col_factors Returns a list of tensors corresponding to column factor shards.
col_update_prep_gramian_op Op to form the gramian before starting col updates.

Must be run before initialize_col_update_op and should only be run by one trainer (usually the chief) when doing distributed training.

col_weights Returns a list of tensors corresponding to col weight shards.
initialize_col_update_op Op to initialize worker state before starting column updates.
initialize_op Returns an op for initializing tensorflow variables.
initialize_row_update_op Op to initialize worker state before starting row updates.
row_factors Returns a list of tensors corresponding to row factor shards.
row_update_prep_gramian_op Op to form the gramian before starting row updates.

Must be run before initialize_row_update_op and should only be run by one trainer (usually the chief) when doing distributed training.

row_weights Returns a list of tensors corresponding to row weight shards.
worker_init Op to initialize worker state once before starting any updates.

Note that specifically this initializes the cache of the row and column weights on workers when use_factors_weights_cache is True. In this case, if these weights are being calculated and reset after the object is created, it is important to ensure this ops is run afterwards so the cache reflects the correct values.

## Methods

### project_col_factors

View source

Projects the column factors.

This computes the column embedding (v_j) for an observed column (a_j) by solving one iteration of the update equations.

Args
sp_input A SparseTensor representing a set of columns. Please note that the row indices of this SparseTensor must match the model row feature indexing while the column indices are ignored. The returned results will be in the same ordering as the input columns.
transpose_input If true, the input will be logically transposed and the columns corresponding to the transposed input are projected.
projection_weights The column weights to be used for the projection. If None then 1.0 is used. This can be either a scaler or a rank-1 tensor with the number of elements matching the number of columns to be projected. Note that the row weights will be determined by the underlying WALS model.

Returns
Projected column factors.

### project_row_factors

View source

Projects the row factors.

This computes the row embedding (u_i) for an observed row (a_i) by solving one iteration of the update equations.

Args
sp_input A SparseTensor representing a set of rows. Please note that the column indices of this SparseTensor must match the model column feature indexing while the row indices are ignored. The returned results will be in the same ordering as the input rows.
transpose_input If true, the input will be logically transposed and the rows corresponding to the transposed input are projected.
projection_weights The row weights to be used for the projection. If None then 1.0 is used. This can be either a scaler or a rank-1 tensor with the number of elements matching the number of rows to be projected. Note that the column weights will be determined by the underlying WALS model.

Returns
Projected row factors.

### scatter_update

View source

Helper function for doing sharded scatter update.

### update_col_factors

View source

Args
sp_input A SparseTensor representing a subset of columns of the full input. Please refer to comments for update_row_factors for restrictions.
transpose_input If true, the input will be logically transposed and the columns corresponding to the transposed input are updated.

Returns
A tuple consisting of the following elements:
new_values New values for the column factors.
update_op An op that assigns the newly computed values to the column factors.
unregularized_loss A tensor (scalar) that contains the normalized minibatch loss corresponding to sp_input, without the regularization term. If spinput contains the columns $${A{:, j}, j \in J}$$, and the input matrix A has m total columns, then the unregularized loss is: $$\|\sqrt W_J \odot (A_J - U V_J^T)\|_F^2 * m / |I|$$ The total loss is unregularized_loss + regularization.
regularization A tensor (scalar) that contains the normalized regularization term for the minibatch loss corresponding to sp_input. If spinput contains the columns $${A{:, j}, j \in J}$$, and the input matrix A has m total columns, then the regularization term is: $$\lambda \|V_J\|_F^2) * m / |J| + \lambda \|U\|_F^2$$.
sum_weights The sum of the weights W_J corresponding to sp_input, normalized by a factor of $$m / |J|$$. The root weighted squared error is: \sqrt(unregularized_loss / sum_weights).

### update_row_factors

View source

Args
sp_input A SparseTensor representing a subset of rows of the full input in any order. Please note that this SparseTensor must retain the indexing as the original input.
transpose_input If true, the input will be logically transposed and the rows corresponding to the transposed input are updated.

Returns
A tuple consisting of the following elements:
new_values New values for the row factors.
update_op An op that assigns the newly computed values to the row factors.
unregularized_loss A tensor (scalar) that contains the normalized minibatch loss corresponding to sp_input, without the regularization term. If spinput contains the rows $${A{i, :}, i \in I}$$, and the input matrix A has n total rows, then the unregularized loss is: $$\|\sqrt W_I \odot (A_I - U_I V^T)\|_F^2 * n / |I|$$ The total loss is unregularized_loss + regularization.
regularization A tensor (scalar) that contains the normalized regularization term for the minibatch loss corresponding to sp_input. If spinput contains the rows $${A{i, :}, i \in I}$$, and the input matrix A has n total rows, then the regularization term is: $$\lambda \|U_I\|_F^2) * n / |I| + \lambda \|V\|_F^2$$.
sum_weights The sum of the weights W_I corresponding to sp_input, normalized by a factor of $$n / |I|$$. The root weighted squared error is: \sqrt(unregularized_loss / sum_weights).

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