Multinomial

Versioned name: Multinomial-13

Category: Generation

Short description: Multinomial operation generates a sequence of class indices sampled from the multinomial distribution based on the input class probabilities.

Detailed description: Multinomial operation generates a sequence of class indices sampled from the multinomial distribution. In this context, the probs values represent the probabilities associated with each class within the multinomial distribution. When executing this operation, it randomly selects a class based on these probabilities. Subsequently, the index of the chosen class in the probs array is appended to the output sequence in the corresponding batch.

Algorithm formulation:

Note

The following notation denotes a range of real numbers between a and b.

\[[a, b] => { x \in \mathbb{R}, a <= x <= b }\]

Given a list of probabilities x1, x2, …, xn:

• If log_probs is true:

  • For each probability x, replace it with a value \(e^{x}\).

• Create an array - discrete CDF (Cumulative Distribution Function) - the cumulative sum of those probabilities, ie. create an array of values where the ith value is the sum of the probabilities x1, …, xi.

• Divide the created array by its maximum value to normalize the cumulative probabilities between the real values in the range [0, 1]. This array is, by definition of CDF, sorted in ascending order, hence the maximum value is the last value of the array.

• Randomly generate a sequence of double-precision floating point numbers in the range [0, 1].

• For each generated number, assign the class with the lowest index for which the cumulative probability is less or equal to the generated value.

• If with_replacement is False (sampling without replacement):

  • Assume a class with index i has been selected - then every CDF value starting at i-th index should be lowered by the original probability of the selected class. This effectively sets the probability of sampling the given class to 0.

  • Afterwards, divide the CDF by its last (maximum) value to normalize the cumulative probabilities between the real values in the range [0, 1].

• Convert the output indices to convert_type.

• Return output indices.

Example computations:

Example 1 - simple 2D tensor with one batch

• Let probs = [[0.1, 0.5, 0.4]], num_samples = 5, log_probs = false, with_replacement = true

• CDF of probs = [[0.1, 0.1 + 0.5, 0.1 + 0.5 + 0.4]] = [[0.1, 0.6, 1]]

• Randomly generated floats = [[0.2, 0.4, 0.6, 0.8, 1]]

• Assigned classes = [[1, 1, 1, 2, 2]]

Example 2 - 2D tensor, log probabilities

• Let probs = [[-1, 1, 2], [50, 1, 21]], num_samples = 10, log_probs = true, with_replacement = true

• Exponentiated probs = [[0.36, 2.71, 7.38], [5184705528587072464087.45, 2.71, 1318815734.48]]

• CDF of probs, per batch = [[0.36, 3.07, 10.45], [5184705528587072464087.45, 5184705528587072464090.16, 5184705528588391279824.64]]

• Normalized CDF = [[0.03, 0.29, 1], [1.0, 1.0, 1.0]]

• Randomly generated floats = [[0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1], [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]]

• Assigned classes = [[1, 1, 2, 2, 2, 2, 2, 2, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

Example 3 - 2D tensor, without replacement

• Let probs = [[0.1, 0.5, 0.4]], num_samples = 2, log_probs = false, with_replacement = false

• CDF of probs = [[0.1, 0.6, 1]]

• Randomly generated floats = [[0.3, 0.2]]

• In a loop:

  • For a value of 0.3, a class with idx 1 is selected

  • Therefore, in CDF, for every class starting with idx 1 subtract the probability of class at idx 1 = probs[1] = 0.5

  • CDF = [[0.1, 0.6 - 0.5, 1.0 - 0.5]] = [[0.1, 0.1, 0.5]]

  • Normalize CDF by dividing by last value: CDF = [[0.2, 0.2, 1.0]]

  • Take the next randomly generated float, here 0.2, and repeat until all random samples have assigned classes. Notice that for sampled values <= 0.2, only the class with idx 0 can be selected, since the search stops at the index with the first value satisfying sample value <= CDF probability

• Assigned classes = [[1, 2]]

Attributes:

• convert_type

  • Description: the type of the output. Determines generation algorithm and affects resulting values. Output numbers generated for different values of convert_type may not be equal.

  • Range of values: “i32”, “i64”.

  • Type: string

  • Required: Yes

• with_replacement

  • Description: controls whether to sample with replacement (classes can be sampled multiple times).

  • Range of values: true, false

    • true - class indices can be sampled multiple times.

    • false - class indices will not repeat in the output and the size of probs’ class_size dimension is required to be larger or equal to num_samples value. Might affect performance.

  • Type: bool

  • Required: Yes

• log_probs

  • Description: allows to control whether inputs should be treated as probabilities or unnormalized log probabilities.

  • Range of values: true, false

    • true - set values in inputs are unnormalized log probabilities that can be any real number.

    • false - probabilities in inputs are expected to be non-negative, finite and have a non-zero-sum.

  • Type: bool

  • Required: Yes

• global_seed

  • Description: global seed value.

  • Range of values: non-negative integers

  • Type: unsigned int 64-bit

  • Default value: 0

  • Required: No

• op_seed

  • Description: operational seed value.

  • Range of values: non-negative integers

  • Type: unsigned int 64-bit

  • Default value: 0

  • Required: No

Inputs:

• 1: probs - A 2D tensor of type T_IN and shape [batch_size, class_size] with probabilities. Allowed values depend on the log_probs attribute. The values are internally normalized to have values in the range of [0, 1] with the sum of all probabilities in the given batch equal to 1. Required.

• 2: num_samples - A scalar or 1D tensor with a single element of type T_SAMPLES specifying the number of samples to draw from the multinomial distribution. Required.

Outputs:

• 1: output- A tensor with type specified by the attribute convert_type and shape [batch_size, num_samples].

Types

• T_IN: any supported floating-point type.

• T_SAMPLES: 32-bit or 64-bit integers.

Example 1: 2D input tensor with one batch.

 <layer ... name="Multinomial" type="Multinomial">
     <data convert_type="f32", with_replacement="true", log_probs="false", global_seed="234", op_seed="148"/>
     <input>
         <port id="0" precision="FP32">  <!-- probs value: [[0.1, 0.5, 0.4]] -->
             <dim>1</dim> <!-- batch size of 2 -->
             <dim>3</dim>
         </port>
         <port id="1" precision="I32"/> <!-- num_samples value: 5 -->
     </input>
     <output>
         <port id="3" precision="I32" names="Multinomial:0">
             <dim>1</dim> <!--dimension depends on input batch size -->
             <dim>5</dim> <!--dimension depends on num_samples -->
         </port>
     </output>
 </layer>

Example 2: 2D input tensor with multiple batches.

 <layer ... name="Multinomial" type="Multinomial">
     <data convert_type="f32", with_replacement="true", log_probs="true", global_seed="234", op_seed="148"/>
     <input>
         <port id="0" precision="FP32">  <!-- probs value: [[-1, 1, 2], [50, 1, 21]] -->
             <dim>2</dim> <!-- batch size of 2 -->
             <dim>3</dim>
         </port>
         <port id="1" precision="I32"/> <!-- num_samples value: 10 -->
     </input>
     <output>
         <port id="3" precision="I32" names="Multinomial:0">
             <dim>2</dim> <!--dimension depends on input batch size -->
             <dim>10</dim> <!--dimension depends on num_samples -->
         </port>
     </output>
 </layer>

Example 3: 2D input tensor without replacement.

 <layer ... name="Multinomial" type="Multinomial">
     <data convert_type="f32", with_replacement="false", log_probs="false", global_seed="234", op_seed="148"/>
     <input>
         <port id="0" precision="FP32">  <!-- probs value: [[0.1, 0.5, 0.4]] -->
             <dim>2</dim> <!-- batch size of 2 -->
             <dim>3</dim>
         </port>
         <port id="1" precision="I32"/> <!-- num_samples value: 2 -->
     </input>
     <output>
         <port id="3" precision="I32" names="Multinomial:0">
             <dim>2</dim> <!-- batch size of 2 -->
             <dim>2</dim> <!-- 2 unique samples of classes -->
         </port>
     </output>
 </layer>