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Time and the Value of Data. (2022). Ardalani, Newsha ; Iansiti, Marco ; Hestness, Joel ; Valavi, Ehsan.
In: Papers.
RePEc:arx:papers:2203.09118.

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  31. N (0,1) Q.E.D. Proof of Proposition 1) From our assumptions in the paper and the asymptotic efficiency of MLE [Casella and Berger (2021)], we know that limn→∞ m(d,θn) = P(d) where θn = arg maxθ Pn i=1 log (m(di,θ)). Hence, for E|log (m(di,θn))| < ∞ and using the strong law of large number we have lim n→∞ − n n X i=1 log (m(di,θn)) = H (P) + KL(P||m(d,θ∞))= H (P) + KL(P||P) = H(P) Therefore, a model that has been trained on D∞,0 should reach the loss value H (P0). Assume d(0) ∼ P0 (d) and d(t) ∼ Pt (d). Consider a model that has been trained on a dataset from time t (D∞,t) and been tested on a dataset from time 0, D∞,0. In this case, limn→∞ m d(t) ,θn = Pt (d) where θn,t = arg maxθ Pn i=1 log m d (t) i ,θ The test loss value for this model is lim n→∞ − n n
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  33. nH nH − R t t2 ψH(t)dt ⇒ fnH (t1,t2) > nH nH − R t t2 ψH(t)dt Valavi et al.: Time and the Value of Data Article submitted to ; manuscript no. (HBS Working Paper 21-016, First Draft: August 2020) 41 meaning that for all t1,t2 > 0 such that offloading is possible for the low flow rate ψL(t), such offloading is also possible for high flow rate ψH(t) and hence, the equivalent time for the high flow rate is weakly closer to the prediction time 0 compared to the equivalnet time for low flow rate. And that completes the proof.
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  47. X i=1 log m d (0) i ,θ∞,t = H (P (d)) + KL(P(d)|| m(d,θ∞,t)) = H (P0) + KL(P0||Pt) Since both H (P0) and KL(P0||Pt) are non-negative functions of distributions [Kullback and Leibler (1951),Shannon (1948)], we conclude that the loss value is higher than H (P0). Therefore, a bounded size dataset should reach the loss value H (P0) + KL(P0||Pt). Formalizing this argument, we define a neighborhood around H(P0) with the size δ > 0 and prove that with probability (1 − ), any dataset of bounded size reaches a value in the neighborhood.
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