ECCC-Report TR23-142https://eccc.weizmann.ac.il/report/2023/142Comments and Revisions published for TR23-142en-usSat, 27 Jul 2024 14:59:11 +0300
Revision 2
| Provable Advantage in Quantum PAC Learning |
Wilfred Salmon,
Sergii Strelchuk,
Tom Gur
https://eccc.weizmann.ac.il/report/2023/142#revision2We revisit the problem of characterising the complexity of Quantum PAC learning, as introduced by Bshouty and Jackson [SIAM J. Comput.
1998, 28, 1136–1153]. Several quantum advantages have been demonstrated in this setting, however, none are generic: they apply to particular concept classes and typically only work when the distribution that generates the data is known. In the general case, it was recently shown by Arunachalam and de Wolf [JMLR, 19 (2018) 1-36] that quantum PAC learners can only achieve constant factor advantages over classical PAC learners.
We show that with a natural extension of the definition of quantum PAC learning used by Arunachalam and de Wolf, we can achieve a generic advantage in quantum learning. To be precise, for any concept class $\mathcal{C}$ of VC dimension $d$, we show there is an $(\epsilon, \delta)$-quantum PAC learner with sample complexity
\[
O\left(\frac{1}{\sqrt{\epsilon}}\left[d+ \log(\frac{1}{\delta})\right]\log^9(1/\epsilon)\right).
\]
Up to polylogarithmic factors, this is a square root improvement over the classical learning sample complexity. We show the tightness of our result by proving an $\Omega(d/\sqrt{\epsilon})$ lower bound that matches our upper bound up to polylogarithmic factors.Sat, 27 Jul 2024 14:59:11 +0300https://eccc.weizmann.ac.il/report/2023/142#revision2
Revision 1
| Provable Advantage in Quantum PAC Learning |
Wilfred Salmon,
Sergii Strelchuk,
Tom Gur
https://eccc.weizmann.ac.il/report/2023/142#revision1We revisit the problem of characterising the complexity of Quantum PAC learning, as introduced by Bshouty and Jackson [SIAM J. Comput.
1998, 28, 1136–1153]. Several quantum advantages have been demonstrated in this setting, however, none are generic: they apply to particular concept classes and typically only work when the distribution that generates the data is known. In the general case, it was recently shown by Arunachalam and de Wolf [JMLR, 19 (2018) 1-36] that quantum PAC learners can only achieve constant factor advantages over classical PAC learners.
We show that with a natural extension of the definition of quantum PAC learning used by Arunachalam and de Wolf, we can achieve a generic advantage in quantum learning. To be precise, for any concept class $\mathcal{C}$ of VC dimension $d$, we show there is an $(\epsilon, \delta)$-quantum PAC learner with sample complexity
\[
O\left(\frac{1}{\sqrt{\epsilon}}\left[d+ \log(\frac{1}{\delta})\right]\log^9(1/\epsilon)\right).
\]
Up to polylogarithmic factors, this is a square root improvement over the classical learning sample complexity. We show the tightness of our result by proving an $\Omega(d/\sqrt{\epsilon})$ lower bound that matches our upper bound up to polylogarithmic factors.Sat, 27 Jul 2024 14:59:07 +0300https://eccc.weizmann.ac.il/report/2023/142#revision1
Paper TR23-142
| Provable Advantage in Quantum PAC Learning |
Wilfred Salmon,
Sergii Strelchuk,
Tom Gur
https://eccc.weizmann.ac.il/report/2023/142We revisit the problem of characterising the complexity of Quantum PAC learning, as introduced by Bshouty and Jackson [SIAM J. Comput.
1998, 28, 1136–1153]. Several quantum advantages have been demonstrated in this setting, however, none are generic: they apply to particular concept classes and typically only work when the distribution that generates the data is known. In the general case, it was recently shown by Arunachalam and de Wolf [JMLR, 19 (2018) 1-36] that quantum PAC learners can only achieve constant factor advantages over classical PAC learners.
We show that with a natural extension of the definition of quantum PAC learning used by Arunachalam and de Wolf, we can achieve a generic advantage in quantum learning. To be precise, for any concept class $\mathcal{C}$ of VC dimension $d$, we show there is an $(\epsilon, \delta)$-quantum PAC learner with sample complexity
\[
O\left(\frac{1}{\sqrt{\epsilon}}\left[d+ \log(\frac{1}{\delta})\right]\log^9(1/\epsilon)\right).
\]
Up to polylogarithmic factors, this is a square root improvement over the classical learning sample complexity. We show the tightness of our result by proving an $\Omega(d/\sqrt{\epsilon})$ lower bound that matches our upper bound up to polylogarithmic factors.Thu, 21 Sep 2023 23:52:23 +0300https://eccc.weizmann.ac.il/report/2023/142