Approximate computing techniques — Hinglish notes

6.5.16 · Hardware › Advanced & Emerging Architectures

YEH KAAM KARTA HI KYUN HAI?

Resilience ke teen structural reasons hain:

Perceptual limits — insaanon ki finite resolution hoti hai (audio, video, images).
Noisy/redundant input — sensor data pehle se hi noisy hota hai; extra tiny error noise floor mein kho jaata hai.
Statistical aggregation / self-healing — sums, averages, aur iterative algorithms (gradient descent, PageRank) per-operation error ko absorb kar lete hain.

Formula Aggregation error kyun chhupaata hai (scratch se derive kiya)

Maan lo hum $N$ numbers ka sum karte hain, aur har addition ek independent error $ε_{i}$ inject karta hai jiska mean $0$ aur standard deviation $σ$ hai. Total error hai $E = \sum_{i = 1}^{N} ε_{i} .$ Kyunki errors independent hain aur mean zero hai:

\operatorname{Var}(E)=\sum \operatorname{Var}(\varepsilon_i)=N\sigma^2.$$ Toh **typical error $\sqrt{N}\,\sigma$ ki tarah badhta hai**, jabki **signal (sum) $N$ ki tarah badhta hai**. *Relative* error isliye shrink karta hai: $$\frac{\text{error}}{\text{signal}}\sim\frac{\sqrt N\,\sigma}{N\mu}=\frac{\sigma}{\mu\sqrt N}\xrightarrow{N\to\infty}0.$$ **Yeh kyun important hai:** aggregation jitna bada hoga, utna zyada approximation afford kar sakte ho. Yahi approximate hardware ka mathematical license hai.

KAISE: techniques ka toolbox

Approximation stack ki har layer par hoti hai.

Definition Main techniques

Precision scaling (bit-width reduction): kam bits use karo (jaise FP32 → FP16 → INT8). Kam bits ⇒ chhhote adders/multipliers ⇒ kam energy aur area.
Approximate arithmetic circuits: ek adder/multiplier design karo jo rare cases mein galat ho lekin bahut chhhota ho (jaise ek truncated multiplier, ya ek approximate full-adder jo carry chain drop kar de).
Loop perforation: loop ke kuch iterations skip karo (jaise har doosra pixel process karo).
Memoization / load-value approximation: recompute karne ki jagah ek previous/similar result reuse karo.
Voltage overscaling (VOS): safe voltage se neeche run karo; occasional timing errors aate hain, lekin power $\propto V^{2}$ hoti hai toh badi savings milti hai.
Approximate memory: DRAM refresh rate kam karo ya kam SRAM cells rakho → non-critical data mein kuch bit flips.

Figure — Approximate computing techniques

Deep dive 1 — Precision scaling itna kyun bachata hai

Deep dive 2 — Voltage overscaling ek $V^{2}$ jackpot kyun hai (aur ek $V$ risk bhi)

Dynamic power hai $P = α C V^{2} f$ . $V$ ko 20% drop karo ⇒ power $1 - 0. 8^{2} = 36%$ drop hoti hai. Lekin circuit delay $\propto V / (V - V_{t h})^{β}$ badhti hai jab $V$ girta hai, toh fixed clock par kuch paths apni deadline miss karte hain → timing errors. VOS kehta hai: un rare errors ko least-significant bits par tolerate karo, badi power win lo.

Worked examples

Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bache ko explain karo

Socho tum crayons se picture bana rahe ho. Agar tum lines se thodi si bahar ho, picture phir bhi zabardast lagti hai — lekin tune bahut tezi se finish kiya aur kam crayon use kiya. Computers bhi yahi trick kar sakte hain: photos, music, ya "kya yeh dog hai?" jaisi cheezein guess karne ke liye, woh math thoda sloppy karte hain purpose se. Sloppy math faster hai aur kam battery use karta hai, aur tum bata bhi nahi sakte ki answer thoda off hai. Clever part yeh hai ki kahaan sloppy hona hai yeh choose karo — kabhi important cheezoon par nahi (jaise page number), sirf un parts par jo koi notice nahi karta.

Active-recall flashcards

#flashcards/hardware

Approximate computing mein core trade-off kya hai?

Output quality ki ek chhhoti, bounded loss ke badle energy, speed, ya area mein bade fayde.

Summing/averaging approximation kyun tolerate kar sakta hai?

N

independent zero-mean errors ke liye, total error

N σ

ki tarah badhta hai jabki sum

N

ki tarah, toh relative error

\to 0

Bit-width aadha karne se multiplier energy ~4× kyun kam hoti hai?

n \times n

array multiplier

\approx n^{2}

full-adder cells use karta hai, aur energy

\propto n^{2} V^{2}

;

n \to n /2

(1/2)^{2} = 1/4

milta hai.

Voltage overscaling kya hai aur yeh powerful kyun hai?

Safe voltage se neeche run karna; power

\propto V^{2}

hoti hai toh badi savings, non-critical bits par rare timing errors ki cost par.

Loop perforation kya hai?

Kuch loop iterations skip karna (jaise har doosra pixel) aur reuse/interpolate karna, redundancy exploit karke speedup ke liye.

Program ke kaun se hisse KABHI approximate nahi karne chahiye?

Control flow, loop bounds, pointers/addresses, aur FP exponents — wahaan errors crashes ya catastrophic magnitude errors cause karte hain.

"Error averages out" assumption FAIL kab hoti hai?

Jab errors biased (non-zero mean) hon; systematic error tab

N

ki tarah badhti hai,

N

ki tarah nahi.

Approximation error bound karne ke liye ek quality metric batao.

PSNR (images), classification accuracy (ML), ya relative/absolute error.

Approximation bug ke barabar kyun nahi hai?

Yeh engineered aur bounded hoti hai — tum choose karte ho error kahaan jaaye aur quality metric guarantee karo; bug uncontrolled hota hai.

Load-value/memoization approximation kya hai?

Ek previous ya similar computed value reuse karna recomputing ki jagah, jab inputs kaafi close hon.

Connections

Precision and Number Formats (FP32, FP16, INT8)
Dynamic Power P = alpha C V^2 f
DRAM Refresh and Memory Reliability
Neural Network Quantization
Ripple-Carry vs Array Multipliers
Error-Correcting Codes (bilkul ulta philosophy — error hatane ke liye energy kharchna)
Dark Silicon and Energy-Efficient Architectures