Classifications

• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
  • G06F7/38—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation
  • G06F7/48—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using non-contact-making devices, e.g. tube, solid state device; using unspecified devices
  • G06F7/499—Denomination or exception handling, e.g. rounding or overflow
  • G06F7/49942—Significance control
  • G06F7/49947—Rounding
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
  • G06F7/58—Random or pseudo-random number generators
  • G06F7/588—Random number generators, i.e. based on natural stochastic processes
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N20/00—Machine learning

Definitions

• reduced-precision numerical representations may be used in training machine learning models.
• the weights in neural networks may have a reduced-precision format, and thus require less computational resources for processing.
• some operations may still (albeit temporarily) produce wider-precision numerical representations.
• stochastic rounding is performed instead of trivial truncation.
• the stochastic rounding of wider-precision numerical representations avoids introducing bias and therefore, improves the accuracy of the resultant machine learning models.
• stochastic rounding on a wider-precision decimal rounds the value up or down with a probability proportional to the least-significant decimals that are to be dropped from the wider-precision decimal. Accordingly, the value of 37.25 would be rounded up to 38 with a 25% probability, and rounded down to 37 with a 75% probability.
• One approach for implementing stochastic rounding is by executing multiple instructions that yield the result of the rounding.
• the software program may contain the appropriate command(s) for stochastic rounding, which during the compilation of such a program, would yield multiple instructions to be executed by a hardware processor.
• the multiple instructions incur high overhead when processed: multiple processor cycles, potential multiple memory lookups, and pipeline stalls, among others.
• FIG. 1 is a block diagram that depicts an example circuit for summation with stochastic rounding, in an embodiment.
• FIG. 2 is a block diagram that depicts example circuits of carry-save adder (CSA) logic components, in an embodiment.
• CSA carry-save adder
• FIG. 3 is a block diagram that depicts a circuit for accumulating a stream of input numbers and stochastically rounding such an accumulation, in an embodiment.
• FIG. 4 is a block diagram that depicts a circuit for summing multiple streams of input numbers and stochastically rounding such a summation, in an embodiment.
• FIG. 5 is a block diagram that depicts a circuit that stochastically rounds a sum based on its sign, in an embodiment.
• FIG. 6 is a block diagram depicting an example circuit for stochastic rounding of a signed summation, in an embodiment.
• FIG. 7 is a flowchart depicting a process for stochastically rounding a sum of two or more numbers, in an embodiment.
• FIG. 8 is a flowchart depicting a process for stochastically rounding a signed sum of two or more numbers, in an embodiment.
• a random number is generated and added to the summation of two or more numbers to generate a stochastically rounded sum of the two or more numbers.
• a random number is generated and added, in an embodiment.
• the random number is added at the same time as the other addends.
• the randomly-generated number has the same width as the difference between the desired lower-precision width and the higher precision width.
• the stochastically rounded sum is generated by retrieving the lower-precision width of the most significant digits and discarding the rest of the result.
• carry-save adder (CSA) logic is used to add a random number value to presented input values to generate a stochastically rounded sum of input values.
• the term“carry-save adder (CSA) logic” refers to a circuit that has three or more inputs and produces a carry output and a carry-less partial sum output.
• the carry-less partial sum output is the sum of all the inputs, without considering carry values generated while adding the corresponding digits.
• the unaccounted carry values for the corresponding digits are aggregated as the carry output of the CSA logic.
• a carry-save adder is an example of CSA logic.
• the circuits and techniques described herein may refer to a binary carry-save adder for purposes of explanation, any other CSA logic may be effectively substituted.
• circuits and techniques should not be interpreted as being limited to carry- save adders, and may be implemented by any CSA logic.
• CSA logic is coupled to adder logic to generate the full sum of input values.
• adder logic refers to a circuit that has two or more inputs and produces a full sum of the values presented at the two or more inputs (including any generated carries).
• the circuits and techniques described herein may refer to a carry completing adder for purposes of explanation, as the adder logic, such circuits and techniques should not be interpreted as being limited to full adders but rather to any adder logic.
• Non-limiting examples of a full adder are a ripple-carry adder and a carry-lookahead adder.
• FIG. 1 is a block diagram that depicts an example circuit for summation with stochastic rounding, in an embodiment.
• the example circuit includes accumulator register 105, CSA 110 and full adder 120.
• CSA 110 has three inputs, X31-X0, Y15-Y0, and Z15-Z0.
• the binary numbers presented at each of those inputs are summed by CSA 110 and full adder 120.
• the CSA 1 lO’s carry output, C31-0 is coupled with full adder l20’s inputs U31-U0
• the CSA 1 lO’s partial sum output, S31-S0 is coupled with full adder l20’s inputs V31-V0, respectively.
• CSA 110 is configured to perform a partial summation (carry-less summation) of corresponding bits of inputs X31-X0, Y15-Y0 and Z15-Z0.
• the partial sum is provided at output S31-S0.
• Aggregation of carries for each corresponding input bits’ addition is provided at output C31-C0.
• Full adder 120 performs carry-complete addition of binary values presented at inputs U31-U0 and V31-V0.
• the full sum generated by adder 120 is provided at output 31-0.
• CSA inputs X31-X0 are coupled to accumulator register 105. Accordingly, the number stored in accumulator register 105 is presented as an input to CSA 110 at inputs X31-X0. The accumulated number is added to a 16-bit input number that is presented at CSA 110 inputs Y15-Y0. The other input to CSA 110 is a l6-bit random number, which is presented at CSA 110 inputs Z15-Z0.
• the random number is generated by a linear-feedback shift register.
• the linear-feedback shift register is coupled to the random number input of CSA logic.
• the linear-feedback shift register may be of any bit-length and generate a pseudo random number of such a bit length.
• a register is coupled to the random number input of CSA logic.
• the random number is generated and stored in the register, prior to computing the stochastic rounding of a sum.
• a CSA logic is optimized for one or more input bits for which no random number input is presented.
• the random number presented at a CSA logic input has less bit-width than at least one other input to the CSA logic.
• Circuit components of a CSA logic that have no random number input may be eliminated or optimized. Doing so improves the density, power consumption, and performance of the circuit.
• an input of the CSA logic is an accumulated input.
• An accumulated input is wider than the other inputs of the CSA logic because at the accumulated input, an intermediate result of a previous operation is presented.
• the previous operation may have resulted in additional bit(s), which are accommodated by the wider-width accumulated input.
• the accumulated input may be coupled to an accumulator register that is wider in width and into which the intermediate result of previous operations is stored.
• the non-accumulated input of a CSA logic has less bit-width than at least the accumulated logic of the CSA logic.
• Circuit components of a CSA logic that have no non-accumulated number input (or are otherwise always set to a zero-value for the non- accumulated input) are eliminated or optimized similarly to the components without random number inputs, in an embodiment.
• FIG. 2 is a block diagram that depicts example circuits of CSA logic components, in an embodiment.
• an accumulated input of the CSA logic is a 32-bit input
• a non-accumulated input is a l6-bit input
• a random number input is an 8-bit input.
• Circuit 200 is a sample logic circuit that yields the least significant bit of a carry output, CO, and the least significant bit of a partial sum output, SO. The output is produced by presenting the least significant bit of the accumulated input at input A0, the least significant bit of the non- accumulated input at input B0, and the least significant bit of the random number input at input CO.
• Circuit 220 is a sample logic that yields the eighth bit of a carry output, C8, and the eighth bit of a partial sum output, S8. Rather than using a circuit logic as one depicted for the least significant bits, circuit 200, circuit 220 is optimized. The optimization is due to the lack of random number input, C, because the random number input is lower-width input of 8- bits and thus has meaningful inputs only from bits 0 to 7. Because of the lack of the random number input, equivalent logic gates of 204, 208 and 210 of circuit 200 are eliminated in circuit 220. Logic gate 222, equivalent to logic gate 202 in circuit 200, and logic gate 228, equivalent to logic gate 208 in circuit 200, are used to produce the eighth-bit partial sum, S8, and the eighth-bit carry output, C8.
• Circuit 230 is a further optimization of circuit 220 based on the lack of both the random number input and non-accumulated number input starting at the 16 th -bit input of the sample CSA logic. With the further optimization, no logic gates are used to produce the carry output and partial sum output for the bits 16 through 31.
• the partial sum output of CSA logic and the carry output of CSA logic are coupled to respective intermediate registers.
• the intermediate registers themselves are coupled to the input of a full adder and in a feedback path to the at least two inputs of the CSA logic.
• the remaining input(s) of the CSA logic are presented with respective input stream(s) of numbers to be summed.
• an input number is partially summed with the previous cycle’s partial sum
• the carry output is stored in the intermediate registers
• a new partial sum and new carry output are stored in the intermediate registers.
• a random number is presented at the input stream input of the CSA logic.
• the random number is presented based on multiplexing an output from a register containing the random number or from a linear-feedback shift register.
• the final accumulated carry and partial sum are accumulated together with the random number.
• a number of the resulting accumulation’s most significant bits are the stochastically rounded sum of the presented input stream.
• one of the intermediate registers is initialized to a random number.
• the random number is accumulated with the first number of the input stream.
• most significant bits of the output represent the stochastically rounded sum of the input stream.
• FIG. 3 is a block diagram that depicts a circuit for accumulating a stream of input numbers and stochastically rounding such an accumulation, in an embodiment.
• the example circuit includes accumulator registers 315 and 325, CSA 310 and full adder 320.
• CSA 310 has two 32-bit inputs, X31-X0 and Y31-Y0, and one 16-bit input, Z15-Z0.
• the binary numbers presented at each of those inputs are summed by CSA 310 and full adder 320.
• the CSA 3 lO’s carry output, C31-C0 is coupled to register 315, which itself is coupled with full adder 320’s input U31-U0.
• the CSA 1 lO’s partial sum output, S31- S0 is coupled with register 325, which itself is coupled with full adder 320’s input V31-V0, respectively.
• Registers 315 and 325 are respectively coupled to input X31-X0 and Y31-Y0, respectively, in a feedback loop in this example. Registers 315 and 325 are initialized to zero.
• the l6-bit input number is multiplexed in by multiplexer 330 to be presented at inputs Z15-Z0.
• CSA 310 evaluates and sums the input number with the previously evaluated carry output and partial sum presented from registers 315 and 325. For example, in the first cycle, register 315 and 325 contain zero, thus the evaluation by CSA 310 results in storing the input number in register 325 as a partial sum output with zero values. Register 315 continues to store zero because adding zero to an input number yields no carry output. In the next cycle, the previous input is presented again at input Y31-0 due to feedback coupling of register 325 with CSA 310.
• the newly presented input number is accumulated with the previous input number to yield a new partial sum to be stored in register 325 and a carry output of the sum to be stored in register 315.
• the cycle is repeated until all the input numbers in the input stream are presented at the Z15-0 input of CSA 310.
• a control signal for 16-bit multiplexer 330 selects the random number input to be presented at Z15-0 input of CSA 310.
• the random number is partially summed at CSA 310 with the accumulated carry value and accumulated partial sum value presented from registers 315 and 325 at inputs X31-0 and Y31-0, respectively.
• the resulting partial sum and the resulting carry output at respective registers 315 and 325 include a random number and are added at full adder 320.
• the most significant bits 31-16 of full adder 320’s output represent the stochastically rounded sum of the input stream, while the rest of the output bits are discarded.
• a circuit simultaneously sums and stochastically rounds the sum of three or more input numbers.
• the circuit uses CSA logic that includes Wallace tree adder logic.
• The“Wallace tree adder logic” term refers herein to a circuit that has four or more inputs and produces a carry output and a carry -less partial sum output of input numbers presented at the inputs.
• a Wallace Tree adder is an example of Wallace tree adder logic.
• the circuits and techniques described herein may refer to a Wallace tree adder for purposes of explanation, any other Wallace tree adder logic may be effectively substituted. Therefore, such circuits and techniques should not be interpreted as being limited to a Wallace tree adder, but are applicable to any Wallace tree adder logic.
• FIG. 4 is a block diagram that depicts a circuit for summing multiple streams of input numbers and stochastically rounding such a summation, in an embodiment.
• the circuit uses a Wallace tree adder.
• a Wallace tree adder includes a network of coupled CSA adders that simultaneously accumulate multiple input numbers presented at the inputs of the Wallace tree adder to yield an accumulated carry output and an accumulated partial sum output.
• Wallace tree adder 410 has ten inputs, two of the inputs, C and S, are coupled to carry output accumulator register 415 and partial sum output accumulator register 425, similar to FIG. 3. Each CSA of Wallace tree adder reduces the number of inputs to one less number of outputs. The output of such a CSA is then coupled to another CSA’s input to further lessen the number of outputs, until there are only two outputs, carry output P and partial sum output Q.
• Wallace tree adder 4l0’s outputs P and Q are coupled to inputs of CSA 413.
• the third input of CSA 413 is coupled to multiplexer 430.
• a random number input or a zero value is selected to be presented at the input of CSA 413.
• Presenting a zero value allows for streams of numbers at inputs of Xi- Xs to be accumulated at CSA 4l3’s output registers 415 and 425.
• mux 430 selects the random number input for a random number to be added to the final accumulation, in one embodiment. In other embodiments, the random number input is selected by mux 430 in any other accumulation cycle.
• CSA 413 The outputs of CSA 413 are coupled to carry output register 415 and partial sum output register 425, which themselves are coupled to full adder 420.
• Full adder 420 evaluates the full addition of the carry output and the partial sum of CSA 413 as accumulated in output registers 415 and 425. Accordingly, full adder 420 yields carry complete results of summations of multiple streams of numbers.
• a number of most significant bits of full adder 420’s output are the stochastically rounded sum of the input stream of numbers presented at the inputs of Wallace tree adder 410.
• a random number is subtracted (or stated otherwise, a negative random number is added). For example, a sample sum of two or more numbers is -37.25. When stochastically rounding such a negative sum, the result is -37 with a 75% probability and is -38 with a 25% probability. When a random number is generated in the range from -1 to 0 (-1 not included), the random number has a 75% probability to be in the range from -0.75 to 0 (-0.75 not included) and a 25% probability to be from -1 to -0.75 (-1 not included).
• a positive random number is generated (e.g. using the techniques described herein) and then converted to a negative number.
• the positive random number is converted to a negative number of the same width by negating each bit of the positive number to yield one’s complement negative number.
• a positive random number’s each bit is negated and then a value of one is added to convert the positive random number to the two’s complement of the random number.
• the most significant bit of a randomly generated positive number is negated to yield a negative random number albeit not equal in absolute value to the positive random number.
• the sum is generated without stochastic rounding and then based on the sign of the sum (positive or negative) a random number is added with or without conversion to a negative random number.
• FIG. 5 is a block diagram that depicts a circuit that stochastically rounds a sum based on its sign, in an embodiment.
• the sample circuit of FIG. 5 includes full adder 540 and a l6-bit XOR gate 550.
• a 32-bit sum is an input U31-0 to full adder 540 to generate stochastic rounding of the input sum.
• Full adder 540 also exposes input CIN for carry-in input to the full adder.
• the most significant bit of the input sum, U31, is also coupled to each bit of XOR gate 550’s l6-bit input.
• the other 16-bit input of the XOR gate is coupled to positive random number input.
• the output of XOR gate 550 is coupled to full adder 540’s inputs V15- V0.
• each of the random number bits is XORed with a value one.
• Such an XOR operation yields a one’s complement of the positive random number.
• the one’s complement random number is then presented as an input to full adder 540 at inputs V15-V0 to be added with the sum at input U31-U0.
• the most significant bit of the sum, U31 is coupled to input CIN of full adder 540, thus adding a value of one to the summation of the sum and the negative random number.
• Such an addition effectively makes the negative random number a two’s complement of the presented positive random number.
• a number of most significant bits of the result from the output of full adder 540, such as output 31-16 are selected as the stochastically rounded negative sum.
• a circuit for stochastic rounding of sum includes two sub circuits for stochastic rounding of sums, the two sub-circuits being different in the sign of the random number input.
• One sub-circuit has a positive random number at the random number input, the other has a negative random number at the random number input.
• the stochastic rounded sum is calculated using both sub-circuits, one or both of the most significant bits of the respective results are used to select which of the two stochastic rounded sum results to select as the output result of the circuit. For example, if the most significant bit of the result of the positive random number sub-circuit is positive then the result is selected as the output of the circuit. If the result of the positive random number sub-circuit is negative then the result of the negative random number sub-circuit is used.
• FIG. 6 is a block diagram depicting an example circuit for stochastic rounding of a signed summation, in an embodiment.
• the example circuit includes sub-circuits 610A and 61 OB.
• Sub-circuits 610A and B have the same circuit that sums input numbers and presents a stochastically rounded number.
• Sub-circuits 610A and B may include any of the circuits discussed herein.
• the input numbers presented as an input to sub-circuits 610A and B are the same. However, the random number input is inverted for sub-circuit B by inverter 640.
• Inverter 640 is coupled to only most significant bit of the random number input. Such an inversion (complete or most significant bit (MSB) only), converts a positive random number presented at the random number input into a different negative, yet still random, number.
• MSB most significant bit
• the signs of outputs of sub-circuits 610A and 610B determine which of sub-circuit’s output reflects the correct stochastic rounding.
• multiplexer 630 is coupled to the most significant bits of sub-circuit 610B. The multiplexer 630 selects the sub-circuit 6lOA’s output when the sub-circuits 6lOB’s sign is negative and selects the sub-circuit 6lOB’s output when the sub-circuits 6lOB’s sign is positive.
• FIG. 7 is a flowchart depicting a process for stochastically rounding a sum of two or more numbers, in an embodiment.
• two or more input numbers are received for summation.
• a random number is received for stochastic rounding.
• the process as depicted in FIG. 7 may be performed iteratively, the random number is received at step 710 during a single iteration rather than at every iteration of the process. This ensures that the random number is added only once during an iterative accumulation.
• the random number may be generated by a linear-feedback shift register or any other methodology at step 700.
• Step 715 a half addition is performed on the input numbers and the random number.
• the digits are summed without propagating any carry to generate a partial sum output, and the generated carries are aggregated as a separate carry output.
• Steps 705-715 are performed by one or more carry-save adders, in an embodiment.
• the partial sum and the carry output may be stored at step 720 and provided as feedback as input numbers at step 705. Such a feedback generates accumulative partial summation of one or more input streams of numbers.
• the carry output and the partial sum are received, and at step 730, a full addition (including carry propagation) of the partial sum output and the carry output is performed generating full addition of input number(s) and random number as an output. Steps 725-730 are performed by an adder logic, in an embodiment.
• a number of most significant bits of the full sum output are selected as a stochastically rounded result of the summation of the two or more numbers.
• the selected number of significant bits depends on the number of bits used for the random number.
• FIG. 8 is a flowchart depicting a process for stochastically rounding a signed sum of two or more numbers, in an embodiment.
• step 805 two or more input numbers are received for summation.
• step 810 a full addition (including carry propagations) of the two or more numbers is performed to generate an intermediate signed sum. Based on the most significant bit, the sign of the intermediate sum is determined at step 815.
• a random number is converted to a negative random number.
• the random number may be converted based on negating all its bits or by negating only the most significant bit of the random number. If the intermediate sum is positive, as indicated by the most significant bit being clear, then the random number is used without any conversion.
• step 825 the random number is received, and at step 830, an addition is performed of the random number with the intermediate sum generated at step 810. At step 830, the full sum is generated.
• a number of most significant bits of the full sum output are selected as stochastically rounded result of the summation of the two or more numbers.
• the number of significant bits depends on the number of bits used for the random number input.

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