6.5.16 · D1Approximate computing techniques
Approximate computing kehta hai: ek perfect answer zyada energy, time, aur chip-area leta hai ek "good enough" answer ke comparison mein — toh jin tasks mein koi fark nahi pata (photos, sound, AI guesses), hum deliberately thoda galat compute karte hain taaki bahut zyada savings mile. Parent page par jo bhi hai woh sirf do sawaalon ka careful jawab hai: kahan hum galat afford kar sakte hain, aur kitna ?
Yeh page assume karta hai ki tumne kuch bhi nahi dekha. Parent note padhne se pehle tumhe kuch tools chahiye. Hum har ek ko ek picture se banate hain, batate hain kyun topic ko woh chahiye, aur tabhi uska symbol use karte hain.
Definition Bit aur bit-widthn bit ek single yes/no switch hai — yeh 0 1 Bit-width , jo n kitne aise switches hum ek saath jodte hain ek number store karne ke liye hai.
Ek row of light switches ki picture karo. n 2 n n 2 n
n Kam switches matlab physically chota, sasta, lower-energy circuit. Precision scaling (FP32 → FP16 → INT8) ka poora idea hai "chhota n n
In bit-patterns ke real number formats mein kaise badlte hain iske liye dekho: Precision and Number Formats (FP32, FP16, INT8) .
Saare switches equally matter nahi karte. Number 101 4 + 0 + 1 = 5 left wala 1 4 right wala 1 1
Most significant bit (MSB) sabse left wala switch hai — yeh value ka sabse bada chunk control karta hai. Least significant bit (LSB) sabse right wala hai — yeh sabse chhoti , finest detail control karta hai.
Intuition Topic ko yeh distinction kyun chahiyeLSB mein galat hona ek number ko thodi si amount se badalta hai; MSB mein galat hona use double ya half kar sakta hai. Approximate computing apni saari deliberate errors LSBs mein dalta hai aur MSBs ko kabhi nahi chhedta. Exactly isliye parent warn karta hai: "exponent ko kabhi approximate mat karo" — exponent ek super-MSB ki tarah hai jo magnitude control karta hai. In bits ke physically kahan rehne ke liye dekho Ripple-Carry vs Array Multipliers .
Parent ka aggregation proof char Greek-flavoured symbols use karta hai. Aao har ek ko samjhte hain.
ε ε "ek chhoti error" matlab hai — jo answer humne compute kiya aur exact answer ke beech ka gap. Agar exact = 5 4.9 ε = − 0.1
μ E [ ⋅ ] μ average (typical value) hai. E [ X ] expected value of X E [ ε ] = 0 "average par, error zero hai" — kabhi zyada, kabhi kam, cancel out ho jaata hai.
Definition Standard deviationσ σ kitni spread out hain — average se typical distance. Chhota σ σ
E σ "errors average out ho jaati hain" . Yeh sentence tab tak sense nahi banta jab tak tum nahi keh sako average = 0 (E [ ε ] = 0 spread = σ . Yeh do knobs exactly decide karte hain ki ek million chhoti errors cancel hongi ya pile up hongi.
E [ ε ] = 0 Kyun sahi lagta hai: zero average zero error jaisa laagta hai. Fix: har individual ε i zero nahi hai — sirf bahut sari ka average zero hai. Individual errors alive aur well hain; woh bas random directions mein point karte hain.
Definition Sigma-sum symbol∑ i = 1 N ε i = ε 1 + ε 2 + ⋯ + ε N . "counter i 1 ε i N Yeh bas ek compact "yeh sab add karo" instruction hai.
N N N random directions mein σ N N N total error N σ signal (seedha sum) N Yeh approximate hardware ke liye mathematical permission slip hai.
Linked idea: yahi "errors aggregate hone par shrink hoti hain" reason hai ki Neural Network Quantization INT8 mein survive karta hai.
∝ A ∝ B "A B — B A shape par focus kar sakein.
V 2 V voltage hai (electrical "push" jo chip ko milta hai). V 2 V × V V 0.8 V 2 0.64 20% 36%
Is equation ki full derivation: Dynamic Power P = alpha C V^2 f . Area/energy version (E ∝ n 2 V 2 Ripple-Carry vs Array Multipliers par rely karta hai.
Approximation tab hi allowed hai jab hum measure kar sakein ki yeh kitna bura hua. Woh measuring stick quality metric hai.
Definition Teen metrics jo tumhe milenge
Relative error : ∣ exact ∣ ∣ approx − exact ∣ 2% Accuracy : ek classifier ke liye, inputs ka fraction jo woh abhi bhi correctly label karta hai.PSNR (peak signal-to-noise ratio, decibels/dB mein): bada number = cleaner image. ≈ 40
number kyun chahiye, feeling nahi
"Mujhe theek lagta hai" engineering nahi hai. Ek quality metric "good enough" ko ek hard threshold ("PSNR ≥ 40 prove kar sakte hain ki error bounded rehti hai. Woh word bounded is field ka poora promise hai.
Definition "Arrow" kya kehta haix N → ∞ 0 "jaise N x 0 Yeh ek single value nahi, ek trend ki destination describe karta hai.
Intuition Topic ko limit kyun chahiyeRelative-error formula μ N σ exactly zero nahi hota, lekin jaise hum zyada aggregate karte hain yeh udhar head karta hai. Limit arrow hume crisp punchline batane deta hai: jitna zyada aggregate karo, utni zyada sloppiness afford kar sakte ho.
Number formats FP32 FP16 INT8
Where to place error safely
Error epsilon mean mu spread sigma
Expectation E and variance
Proportional to and V squared
Power law P = alpha C V^2 f
Quality metrics PSNR accuracy
Yeh foundations samajhne ke baad related destinations: DRAM Refresh and Memory Reliability , Error-Correcting Codes , Dark Silicon and Energy-Efficient Architectures , aur parent 6.5.16 Approximate computing techniques (Hinglish) .
Right side cover karo. Agar tum har ek ka jawab de sako, tum parent note ke liye ready ho.
n Kitne switch-wires ek number banate hain — circuit ki size, energy, aur precision ka master dial.
Kaunse bits corrupt karna safe hai aur kaunse forbidden hain? LSBs (chhoti value) approximate karna safe hai; MSBs aur exponents (badi value / magnitude) forbidden hain.
E [ ε ] = 0 Error bahut sari operations mein average ho kar zero ho jaati hai — yeh nahi ki koi single error zero hai.
σ Errors ki typical spread (scatter) unke average ke around.
N N N Random-direction errors partly cancel ho jaate hain, random walk ki tarah, toh bheed sirf lagbhag
N steps drift karti hai.
A ∝ B A B
V 2 Power voltage squared par depend karti hai, toh ek chhota voltage cut ek outsized power saving deta hai.
Quality metric kis liye hai, ek word mein? "Good enough" ko ek measurable, bounded, provable threshold banana.
x N → ∞ 0 Ek trend ki destination jaise N x 0
