Timeline for Bounds on difference between "logsumexp" and variance?
Current License: CC BY-SA 4.0
10 events
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| Apr 15, 2019 at 15:08 | comment | added | dohmatob | Ok. Thanks once again. | |
| Apr 15, 2019 at 14:03 | comment | added | Iosif Pinelis | @dohmatob : I am glad that you found this answer to be of use. As for your further question, I think it is a good one; however, addressing it will probably require a full-blown paper (at least), rather than a typical MO answer. | |
| Apr 15, 2019 at 6:04 | comment | added | dohmatob | Put another way, let $Z_1,\ldots,Z_n$ be an i.i.d sample from the distribution of $Z$, and define $\|f(Z)\|_{\alpha,n} := (\frac{1}{n}\sum_{i=1}^n|f(Z_i)|^\alpha)^{1/\alpha}$. Finally, define the random variable $\hat{C}_{n,\alpha}^\delta := \inf_{\theta} \left(\theta + \frac{1}{\delta^{1/\alpha}}\|(Z-\theta)_+ \|_{\alpha,n}\right)$. Is there a reasonable bound of the form (for example!): $P(Z > \hat{C}_{n,\alpha}^\delta - \epsilon(n)) \le \delta$, where $\epsilon_n \rightarrow 0$ very fast (say $\epsilon_n = \mathcal O(1/\sqrt{n})$, etc.). | |
| Apr 15, 2019 at 5:41 | comment | added | dohmatob | Question: Do you think one can consider an empirical version of the norms $\|Z\|_\alpha$ in the definition of $C_{Z;\alpha}^\delta$ and still get a similar bound (perhaps with some correction terms) ? I ask this because in the case of SVP, there is such an empirical version called the empirical Bennet inequality (see theorem 4 www0.cs.ucl.ac.uk/staff/M.Pontil/reading/svp-final.pdf). Thanks in advance! | |
| Apr 15, 2019 at 5:41 | vote | accept | dohmatob | ||
| Apr 15, 2019 at 5:36 | comment | added | dohmatob | Really nice answer; thanks. This answer and the reference paper are particularly interesting because my question is originally motivated by considerations in risk-averse decision-making. Also the classical CVaR also seems to correspond to your $Q_1(Z;1-\delta)$, and therefore FWIW, you have a corollary of the form $P(Z \ge CVaR_Z^{1-\delta}) \le \delta$. | |
| Apr 14, 2019 at 19:05 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 14, 2019 at 18:01 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 14, 2019 at 17:55 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 14, 2019 at 17:46 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |