### Version¶

• domain: ai.onnx.preview.training

• since_version: 1

• function: False

• support_level: SupportType.COMMON

• shape inference: True

This version of the operator has been available since version 1 of domain ai.onnx.preview.training.

### Summary¶

Compute one iteration of ADAGRAD, a stochastic gradient based optimization algorithm. This operator can conduct the optimization of multiple tensor variables.

Let’s define the behavior of this operator. As you can imagine, ADAGRAD requires some parameters:

• The initial learning-rate “R”.

• The update count “T”. That is, the number of training iterations conducted.

• A L2-norm regularization coefficient “norm_coefficient”.

• A learning-rate decay factor “decay_factor”.

• A small constant “epsilon” to avoid dividing-by-zero.

At each ADAGRAD iteration, the optimized tensors are moved along a direction computed based on their estimated gradient and accumulated squared gradient. Assume that only a single tensor “X” is updated by this operator. We need the value of “X”, its gradient “G”, and its accumulated squared gradient “H”. Therefore, variables in this operator’s input list are sequentially “R”, “T”, “X”, “G”, and “H”. Other parameters are given as attributes because they are usually constants. Also, the corresponding output tensors are the new value of “X” (called “X_new”), and then the new accumulated squared gradient (called “H_new”). Those outputs are computed from the given inputs following the pseudo code below.

Let “+”, “-”, “*”, and “/” are all element-wise arithmetic operations with numpy-style broadcasting support. The pseudo code to compute those outputs is:

// Compute a scalar learning-rate factor. At the first update of X, T is generally // 0 (0-based update index) or 1 (1-based update index). r = R / (1 + T * decay_factor);

// Add gradient of 0.5 * norm_coefficient * ||X||_2^2, where ||X||_2 is the 2-norm. G_regularized = norm_coefficient * X + G;

// Compute new accumulated squared gradient. H_new = H + G_regularized * G_regularized;

// Compute the adaptive part of per-coordinate learning rate. Note that Sqrt(…) // computes element-wise square-root. H_adaptive = Sqrt(H_new) + epsilon

// Compute the new value of “X”. X_new = X - r * G_regularized / H_adaptive;

If one assign this operators to optimize multiple inputs, for example, “X_1” and “X_2”, the same pseudo code may be extended to handle all tensors jointly. More specifically, we can view “X” as a concatenation of “X_1” and “X_2” (of course, their gradient and accumulate gradient should be concatenated too) and then just reuse the entire pseudo code.

Note that ADAGRAD was first proposed in http://jmlr.org/papers/volume12/duchi11a/duchi11a.pdf. In that reference paper, this operator is a special case of the Figure 1’s composite mirror descent update.

### Attributes¶

• decay_factor - FLOAT (default is '0.0'):

The decay factor of learning rate after one update.The effective learning rate is computed by r = R / (1 + T * decay_factor). Default to 0 so that increasing update counts doesn’t reduce the learning rate.

• epsilon - FLOAT (default is '0.0'):

Small scalar to avoid dividing by zero.

• norm_coefficient - FLOAT (default is '0.0'):

Regularization coefficient in 0.5 * norm_coefficient * ||X||_2^2. Default to 0, which means no regularization.

### Inputs¶

Between 3 and 2147483647 inputs.

• R (heterogeneous) - T1:

The initial learning rate.

• T (heterogeneous) - T2:

The update count of “X”. It should be a scalar.

The current values of optimized tensors, followed by their respective gradients, followed by their respective accumulated squared gradients.For example, if two tensor “X_1” and “X_2” are optimized, The input list would be [“X_1”, “X_2”, gradient of “X_1”, gradient of “X_2”, accumulated squared gradient of “X_1”, accumulated squared gradient of “X_2”].

### Outputs¶

Between 1 and 2147483647 outputs.

Updated values of optimized tensors, followed by their updated values of accumulated squared gradients. For example, if two tensor “X_1” and “X_2” are optimized, the output list would be [new value of “X_1,” new value of “X_2” new accumulated squared gradient of “X_1”, new accumulated squared gradient of “X_2”].

### Type Constraints¶

• T1 in ( tensor(double), tensor(float) ):

Constrain input types to float scalars.

• T2 in ( tensor(int64) ):

Constrain input types to 64-bit integer scalars.

• T3 in ( tensor(double), tensor(float) ):

Constrain input and output types to float tensors.