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e044e4eacd87edd699e9623c98e94c22c71cc760 | subsection | 54 | 80 | Times in the non-exponential regime | Denoting the eigenvalues and projectors of this
Liouvillian by \tilde{\gamma }_j^{\text{st}} and \tilde{P}_j^{\text{st}}, with j=\text{st},0,\pm , we show in
Appendix that we get in analogy to
(REF -) and (REF -)\tilde{\gamma }_{\text{st}}^{\text{st}} &= 0\quad ,\\
\tilde{\gamma }_0^{\text{st}} &=
-i\Big (\Gamma ^{(1... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.05358843132853508,
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0.03671234846115112,
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-0.... |
92f1f3ecb4b2c01944663e18ab4480c42feee9b0 | subsection | 55 | 80 | Times in the non-exponential regime | This
goes beyond the scope of this paper. However, in Ref. divincenzoloss05, such an
analysis has been performed in bare perturbation theory (i.e. using the unrenormalized tunneling)
with the result (note that we slightly changed the result such that it is valid for a Lorentzian
cutoff function in the bath){\langle \si... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
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0.012152919545769691,
... |
ca8678795d51071c685d65034381b512437b7606 | subsection | 56 | 80 | Times in the non-exponential regime | There is a Z-factor renormalization
Z=\tilde{\Delta }^2/\Delta ^2 for F^c_{x,y} and F^s_{x,y}, and terms \sim \Delta or \sim \Delta ^3
in the Bloch-Redfield solution are replaced by \sqrt{Z}\Delta =\tilde{\Delta }^2/\Delta and
Z^2\Delta ^3=\tilde{\Delta }^4/\Delta , respectively.The most interesting correction in O(\al... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
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0.02... |
a66789570b2ad5511baac4be6fd97aca5e8ce852 | subsection | 57 | 80 | Times in the non-exponential regime | In this case one can use the approximation
H((z_+-\Omega _0)t)\approx -\gamma -\ln (-i(z_+-\Omega _0)t) and
e^{-iz_+t}\approx e^{-i\Omega _0t} in (REF ) leading to&{i\over 2\pi }\int dE e^{-iEt}{1\over E-z_+}\ln {-i(E-\Omega _0)\over \Omega _0}\,\approx \\
& =\,-\Big (\gamma + \ln (\Omega _0 t)\Big )e^{-i\Omega _0 t}\... | {
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"raw": "A.J. Leggett, S. Chakravarty, T.A. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).",
"source_ref_id": "c23ab74cf76ec9d3bef89e71e0bd211e2c5eb7d3",
"start": ... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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... |
5846d1bb93f0609ec648f6800563e7583a087a48 | subsection | 58 | 80 | Times in the non-exponential regime | In this case and for \Omega t\gg 1 our solution reduces to{\langle \sigma _{z}\rangle }(t)\,\approx \,
(1+\alpha )\cos (\tilde{\Delta }t)e^{-{\Gamma \over 2}t}
- 2\alpha {1\over (\tilde{\Delta } t)^2}e^{-\Gamma t}\quad ,which, up to the missing exponential for the second term on the r.h.s., agrees with the
NIBA result ... | {
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"raw": "A.J. Leggett, S. Chakravarty, T.A. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).",
"source_ref_id": "c23ab74cf76ec9d3bef89e71e0bd211e2c5eb7d3",
"start": 0... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
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"cond-mat.mes-hall"
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... |
2d6b04c8ec74b49629a55f2921054e897039abfa | subsection | 59 | 80 | Times in the non-exponential regime | Keeping only the consistent terms falling off
\sim \alpha /(\Omega t) we obtain for large times \Omega t\gg 1:F_x^0(t) \,&=\,
-{\langle \sigma _x\rangle }_{\text{st}}
\,-\, \Big (1+2\alpha {\tilde{\Delta }^2\over \Omega ^2}\Big ){\tilde{\Delta }^2\over \Delta \Omega }
{\langle \tilde{\sigma }_z\rangle }_0
\,+\, \pi \al... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
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0.0... |
38ae904dc8c6551b046e7dcc4fee5890fd0f65e0 | subsection | 60 | 80 | Times in the non-exponential regime | As we discuss later on, the main result is that the
function f_t has to be replaced by the power-lawf_t\,\rightarrow \,\left({1\over \Omega t}\right)^{2\alpha {\epsilon ^2\over \Omega ^2}}
\Big (1-2\alpha \gamma {\epsilon ^2\over \Omega ^2}+\alpha {\tilde{\Delta }^2\over \Omega ^2}\Big )\quad .with a power-law exponent... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
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... |
ab1c03698aa6aedbafc47b1e3058f79d51956629 | subsection | 61 | 80 | Times in the non-exponential regime | The logarithmic terms
are a result of a combination of logarithmic terms arising from the terms
\sim \alpha \ln (\Omega t) appearing explicitly in (-)
and those arising from the functions H_t and \tilde{H}_t, which, for small argument,
can be expanded asH^\prime _t\,&=\,-\gamma \,-\,\ln (\Omega t)\,+\,O(\Omega t)\quad ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
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0.028948115184903145,
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0.031801726669073105,
-0.04022521525621414,
... |
51afb0e864b2ccd38f7614abf92b377350c47ede | subsection | 62 | 80 | Times in the non-exponential regime | In this regime our full solution (REF -)
is needed to calculate all terms one order beyond Bloch-Redfield. In this case the
time dependence of the preexponential functions is governed by a complicated combination of slowly
varying logarithmic terms and terms arising from the functions H_t and \tilde{H}_t containing
the... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0.01958259381353855,
0.0... |
72fc566d2e7bc1e6485632debfe9b3156b05b8fe | subsection | 63 | 80 | Real-time renormalization group | In this section we will present the real-time renormalization group approach to calculate
the Liouvillian L(E) beyond perturbation theory by including the leading logarithmic series at
low and high energies. This provides the basis for the renormalized perturbation theory in the
non-exponential regime together with the... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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27954621a67c1f015fdf3902a57f496fde8f01e5 | subsection | 64 | 80 | RG equations | The leading order RG equations to determine the Liouvillian L(E) for the ohmic spin boson model
have been derived in Ref. kashubaschoeller13. Using the definitions (REF ),
(REF ) and (REF ), they read{d\over dE}\tilde{L}_\Delta (E)\,&=\,2\alpha \sum _i Z^{\prime }(E) G(E) P_i(E) Z^{\prime }(E) G(E)
{\tilde{L}_\Delta (E... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
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8a379ad5ae60cfd315bdf5d5c302ce0332e9feaa | subsection | 65 | 80 | RG equations | For \omega =0 and \Lambda \rightarrow 0,
we obtain the Liouvillian L(0^+) from which we can obtain the stationary state \rho _{\text{st}} via
(REF ). For \omega =\pm \eta (\eta =0^+) and \Lambda < -\Gamma , we obtain a jump
between L(-i\Lambda +\eta ) and L(-i\Lambda -\eta ) indicating the branch cut starting at
z_0=-i... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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8e9c838d8c16a809ea4d8a7499fda3b4228f7046 | subsection | 66 | 80 | RG equations | Therefore, the RG equations provide a well-defined set
of differential equations which can be systematically truncated at order \alpha , including secular terms
and the leading logarithmic series to all orders.We note that the vertex renormalization is very essential. At large energies we show in
Section REF that the v... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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18240f26f08f08f3d48ec00a80eb6ae433f5a2bd | subsection | 67 | 80 | Large energies | For large energies, where \Omega \ll |E| \ll D, we will show in this section that \tilde{L}_\Delta (E) and
Z^{\prime }(E) are given by (REF -). For |E|\gg |\lambda _i(E)|,
the RG equations (REF -) can be approximated by{d\over dE}\tilde{L}_\Delta (E)\,&=\,2\alpha \sum _i Z^{\prime }(E) G(E) P_i(E) Z^{\prime }(E) G(E)
{... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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750effa1610eb2dc854e7be7766a4c65f1930ed9 | subsection | 68 | 80 | Large energies | Furthermore we find{d\over dE} Z(E)^2 g(E)^2\,&=\,0\quad ,\\
{d\over dE} Z(E)\,&=\,2\alpha Z(E) {1\over E}\quad ,with the solutionZ(E)^2 g(E)^2\,&=\, 1\quad ,\\
Z(E)\,&=\,\left({-iE\over D}\right)^{2\alpha }\quad .In conclusion, (REF ), () and
() prove the form (REF -) of
the Liouvillian at large energies which was u... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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871bbb8bbfb646ba4e9a42e21072bdf8a605046c | subsection | 69 | 80 | The non-exponential regime | In the non-exponential regime (REF ), where \alpha \ln (-i(E-\lambda _i(E))/\Omega )\ll 1,
the RG equations can be solved perturbatively around the solution
(REF -) at high energies evaluated at
E=i\Omega , see Ref. schoeller09 for details. Denoting the latter by\tilde{L}_0\,&=\,
\left(\begin{array}{cc} 0 & \Delta \tau... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
] | 2,018 | en | Physics | [
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0914c6ae16169d7b5c7042318e2925cb17bb8418 | subsection | 70 | 80 | The non-exponential regime | For large |E|\gg |\lambda _i(E)| but not
exponentially large the solution (REF -)
is consistent with the result at large energies, given by (REF ),
() and (), when expanded in \alpha \ln (-iE/\Omega ).
As a consequence it is straightforward to see that (REF -) is
indeed the solution of the RG equations in the non-expon... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
"cond-mat.stat-mech",
"cond-mat.mes-hall"
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15dd7ab2532a4bcebe80179ed21e5a36e09d9bd3 | subsection | 71 | 80 | The non-exponential regime | After a straightforward calculation
we obtainA\tilde{L}_\Delta (E)A&=
\Omega \left(\begin{array}{cc} 0 & 0 \\ 0 & \sigma _z \end{array}\right)+
2\alpha {\tilde{\Delta }^2\over \Omega }{\cal {L}}_0(E)
\left(\begin{array}{cc} 0 & 0 \\ 0 & \tau _-\sigma _z \end{array}\right)\\
& \hspace{-28.45274pt}
-\alpha {\tilde{\Delt... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
] | [
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042de0796dafde19ffb0475991ca0fb996f8dcaa | subsection | 72 | 80 | Exponentially large times | For exponentially large times, where higher powers in \alpha \ln (\Omega t) become significant and can
no longer be treated in lowest order to analyze the corrections to Bloch-Redfield, we need a solution
of the RG equations exponentially close to the branching points z_i. Analytically, such an analysis
is very complic... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
"Carsten J. Lindner",
"Herbert Schoeller"
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306b7a627c3d11b05dd263eb9d66f6711b6617c8 | subsection | 73 | 80 | Exponentially large times | As already discussed in
Ref. kashubaschoeller13, we note that there is no change of the power-law exponent of
the 1/t^2 parts, in particular for the time dynamics of {\langle \sigma _z\rangle }(t), see
(REF ), in contrast to the NIBA solution which predicts an incorrect
power-law exponent 2-2\alpha , .As discussed in d... | {
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"start": 0... | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
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6014e70145191c547eac3b986c714347bfc401e2 | subsection | 74 | 80 | Summary | In this work we have presented the solution for the time dynamics of the ohmic spin boson
model at finite bias by systematically expanding one order beyond Boch-Redfield.
Using real-time RG and perturbation theory we have set up a renormalized
perturbation theory to study analytically the whole time regime from exponen... | {
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} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
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3ecb7db322a30909e9a583536031efdbc60222ae | subsection | 75 | 80 | Summary | We solved this problem
by expanding all analytic parts of \Sigma (E) around E=z_i and keeping \Sigma (z_i) in the denominator
whereas all other higher terms of the Taylor expansion are at least of O(\alpha ^2) and can be taken as
a small correction. The non-analytic terms of \Sigma (E) are more subtle and are some func... | {
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weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
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3f01544d1019acda01b32d06c13933c4897df265 | subsection | 76 | 80 | Summary | Therefore, second order terms are needed for the Liouvillian
to calculate the stationary state and all terms of the time evolution of the purely decaying modes
up to first order in \alpha . Again this problem occurs only
for times of the order of the inverse decay rate, since for small times |E-L_0|\sim 1/t is
much lar... | {
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b722b9bd561612595897fb40e0052998254d6e03 | subsection | 77 | 80 | Projectors for | To calculate the projectors of the matrix \tilde{L}_0+\tilde{\Sigma }_a^i up to O(\alpha ),
we first set up the matrix \tilde{\Sigma }_a^i=\tilde{\Sigma }_a(z_i) by setting E=z_i in
() and use& {\cal {F}}_0(0)\,\sim \,O(\alpha )\quad ,\quad {\cal {F}}_0(z_0)\,=\,{\cal {F}}_\sigma (z_\sigma )\,=\,0 \quad ,\\
& {\cal {F}... | {
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} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
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0f7328a9c7400295914466d8a1a40703ba673c59 | subsection | 78 | 80 | Projectors for | Due to the matrix structure of \tilde{\Sigma }_a(E), one
projector is exactly known (in all orders of perturbation theory, see () and
(REF ))A \tilde{P}^i_{\text{st}} A\,&=\,
\left(\begin{array}{cc} \tau _+ & 0 \\ 0 & 0 \end{array}\right)\quad .Using usual perturbation theory it is straightforward to calculate the proj... | {
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} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
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2bac19d99fde9e9e88212716271447f0676fde7d | subsection | 79 | 80 | Projectors for | This is a particular advantage for the
spin boson model.To derive the formula () for the eigenvalue \tilde{\gamma }_0^\text{st} of
\tilde{L}_a(0) up to second order in \alpha , we relate it to the eigenvalue
\tilde{\gamma }_0^0=z_0=-i(\Gamma ^{(1)}+\Gamma ^{(2)}+O(\alpha ^3)) of \tilde{L}_a(z_0). We first note
that, du... | {
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} | 10.1103/PhysRevB.98.115425 | 1802.09846 | Dissipative quantum mechanics beyond Bloch-Redfield: A consistent
weak-coupling expansion of the ohmic spin boson model at arbitrary bias | [
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556c4a6e6a58ea1085bb757993996936ed01f19b | abstract | 0 | 106 | Abstract | We study the computational tractability of PAC reinforcement learning with
rich observations. We present new provably sample-efficient algorithms for
environments with deterministic hidden state dynamics and stochastic rich
observations. These methods operate in an oracle model of computation --
accessing policy and va... | {
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} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
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"John Langford",
"Robert E. Schapire"
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cef249bc1dbbe1fcbff8183e0f920a6b74618d39 | subsection | 1 | 106 | Introduction | We study
episodic reinforcement learning (RL)
in environments with realistically rich observations such as images or text, which we refer to broadly as contextual decision processes.
We aim for methods that use function approximation in a provably
effective manner to find the best possible
policy through strategic expl... | {
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3daebfc2ebf3f50c4bc52d53b9d63eaf85a1709d | subsection | 2 | 106 | Introduction | Together, these results
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dfc6272e8e6d193a6b4ef95af85d178b0b9a1653 | subsection | 3 | 106 | Related Work | There is abundant work on strategic exploration in the
tabular setting , , , , , , , . The computation in these algorithms often involves planning in optimistic models and can be solved efficiently via dynamic programming.
To extend
the theory to the more practical settings of large state spaces,
typical approaches inc... | {
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90076560a7c9cfab38e76aef193c708921824193 | subsection | 4 | 106 | Setting and Background | We consider reinforcement learning (RL) in a common special case of contextual decision processes , , sometimes referred to as rich observation MDPs .
We assume an H-step process where in each episode, a
random trajectory
s_1, x_1, a_1, r_1, s_2, x_2, \ldots , s_H, x_H, a_H, r_H is
generated. For each time step (or lev... | {
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713af3062c0e0a0e00d7dc216bf024008bcf2135 | subsection | 5 | 106 | Setting and Background | Let \pi ^\star denote the optimal policy, which maximizes V^\pi , with optimal value function g^\star
defined as g^\star (x) := \mathbf {E}[\sum _{h^{\prime }=h}^H r_{h^{\prime }} | x_h=x, a_{h:H} \sim \pi ^\star ].
As is standard, g^\star satisfies the Bellman equation:
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595c4f5d7237eba9456c43bb0c1c6da66d54f8d0 | subsection | 6 | 106 | Function Classes and Optimization Oracles | As can be rich, the agent must use function approximation to
generalize across observations.
To that end, we assume a given value function class
\subset (\rightarrow [0,1]) and policy class
\Pi \subset (\rightarrow ).
Our algorithm is agnostic to the specific function classes used,
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3879b65db9232bed1fad45d6a83ec51abfa677fd | subsection | 7 | 106 | Function Classes and Optimization Oracles | Formally, for a program of the form\textstyle \max _{g \in } o(g) \textrm {, ~ subject to } h_j(g) \le c_j, ~~ \forall j \in [m],with constants \lbrace c_j\rbrace _{j\in [m]}, an LP oracle with approximation
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02cd41f30e6251711aaca4dec3390ca5acc80bd0 | subsection | 8 | 106 | Body | In this section we propose and analyze a new
algorithm, Valor (Values stored Locally for RL) shown in
Algorithm (with
&
as subroutines). As we will show, this
algorithm is oracle-efficient and enjoys a polynomial
sample-complexity guarantee in the deterministic
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92a5b8d0d5a403cf79afc6007553853fe54e8a27 | subsection | 9 | 106 | Body | \in T
(a_{1:h}) *Alg.
failure
Main Algorithm Valor
[H]
2mm2mm
myfunFunction
\hat{V}^{\star } \leftarrow V of the only dataset in _1
h=1:H
CSC-oracle
\hat{\pi }_h \leftarrow \limits _{\pi \in \Pi _h}\hspace{-3.33328pt}\sum \limits _{(D,V, \lbrace V_{a}\rbrace ) \in _h} \hspace{-10.0pt}V_D(\pi ; \lbrace V_a\rbrace... | {
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91b799b0b6b39b365f0c16277164007e2bd7547b | subsection | 10 | 106 | Body | \forall (D, V,\_) \in _{h+1}:
\vspace*{2.84526pt}
~~|V - \hat{\mathbf {E}}_{D}[g(x_{h+1})]| \le \phi _{h+1}
|V_{opt} - V_{pes}| \le 2\phi _{h+1} + 4 \epsilon _{\textrm {stat}}+ 2\epsilon _{\textrm {feas}}
V_a \leftarrow (V_{opt} + V_{pes})/2
*consensus among remaining functions
V_a \leftarrow (p \circ a)
*no conse... | {
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084050a7ced58075b0c050bc77ca1792ec83f816 | subsection | 11 | 106 | Body | For convenience, we also define _{H+1} to be the singleton \lbrace x\mapsto 0\rbrace . This notation also
allows our algorithms to handle more general non-stationary
function classes.Details of depth-first search exploration. Valor maintains many data sets collected at paths visited by .
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b0afe450420c462128b8079bc35d742fd0d52cec | subsection | 12 | 106 | Body | Assuming P \ne NP, even with algorithm parameter \phi =0 and perfect evaluation of expectations, Olive is not oracle-efficient, that is, it cannot be implemented with polynomially many basic arithmetic operations and calls to CSC, LP, and LS oracles.
The assumptions of perfect evaluation of expectations and \phi =0 ar... | {
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222172858666da84dc0424e426065ae6c3d5be53 | subsection | 13 | 106 | Body | At the same time, oracles are implementable in polynomial time:
For tabular value
functions = (\rightarrow [0,1]) and policies \Pi = (\rightarrow ), the CSC, LP, and LS oracles can be implemented in time polynomial in ||, K = || and the input size.
Both proofs are in Appendix .
Proposition REF implies that if Olive c... | {
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bef2834b85bbbac7d1dc950208fc5224c8de542b | subsection | 14 | 106 | Body | We construct a
family of MDPs as shown in Figure REF that
encodes the 3-SAT problem for this formula as follows: For each
variable x_i there are two terminal states x_i^1 and x_i^0
corresponding to the Boolean assignment to the variable. For each
variable, the reward in either x_i^1 or x_i^0 is 1 and 0 in
the other. Th... | {
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0b4ab865ef8f4a723e20baf469fde760aeb2603c | subsection | 15 | 106 | Body | We list these constraints in the following writing out the constraints for each optimal action that are implied by the indicator of the original constraints in Problem (REF ):
From initial state:f(s_0, \texttt {[try c_j]}) &= \max _b f(c_1, b) - 1/m
& \textrm {if } \pi _{f}(s_0) =& \texttt {[try c_j]}\\
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50f80f5846c59678735459daa029972def3d2797 | subsection | 16 | 106 | What is new compared to | The overall structure of Valor is similar to Lsvee . The main differences are
in the pruning mechanism, where we use a novel state-identity
test, and the policy optimization step in
Algorithm .Lsvee uses a Q-value function
class \subset (\times \rightarrow [0,1]) and a state
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803889b8f0b2186431cea827b4dbe6c7f8417a56 | subsection | 17 | 106 | Computational and Sample Complexity of | Valor requires two types of nontrivial computations over the
function classes. We show that they can be reduced to CSC on \Pi and LP on (recall Section REF ), respectively, and hence Valor is oracle-efficient.First, Lines in and in involve optimizing V_D(\pi ; \lbrace V_a\rbrace ) (Eq. (REF )) over \Pi ,
which can be... | {
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a9f9ec35108ee826bcc5a8e44416a8facc28a4c0 | subsection | 18 | 106 | Computational and Sample Complexity of | Then
for any \epsilon , \delta \in (0, 1), with probability at least 1- \delta ,
Valor makes O\left( \frac{MH^2}{\epsilon }\log \frac{MH}{\delta }\right) CSC oracle calls and at most O\left( \frac{MKH^2}{\epsilon }\log \frac{MH}{\delta }\right) LP oracle calls with required accuracy \epsilon _{\textrm {feas}}= \epsilon... | {
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c84ff247083bf44f9cfcb3e216932c8299540335 | subsection | 19 | 106 | Toward Oracle-Efficient PAC-RL with Stochastic Hidden State Dynamics | Valor demonstrates that provably sample- and oracle-efficient RL with rich
stochastic observations is possible and, as such, makes progress toward reliable and
practical RL in many applications. In this section, we discuss the natural
next step of allowing stochastic hidden-state transitions. | {
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} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
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5b5ccd5bb3952a2475c38c86a90e934b11114f7b | subsection | 20 | 106 | Computational Barriers with Decoupled Learning Rules. | One factor contributing to the computational intractability of Olive is that (REF ) involves optimizing over policies and values jointly.
It is therefore promising to look for
algorithms that separate optimizations over policies and values, as in Valor. In Appendix , we provide a
series of examples that illustrate some... | {
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3222078c983cea35d1cc5073f54c04947b673aed | subsection | 21 | 106 | Alternative Algorithms. | An important element of Valor is that it explicitly stores value
estimates of the hidden states, which we call “local values." Local
values lead to statistical and computational efficiency under weak
realizability conditions, but this approach is unlikely to generalize
to the stochastic setting where the agent may not ... | {
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39374ebd0a540aa1bf61b040631e53f7b86c71d0 | subsection | 22 | 106 | Conclusion | This paper describes new RL algorithms for environments with rich
stochastic observations and deterministic hidden state dynamics. Unlike other
existing approaches, these algorithms are computationally efficient in
an oracle model, and we emphasize that the oracle-based approach has
led to practical algorithms for many... | {
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f70e24fab61cc42a6d0e3fb7a546e714e4fa3380 | subsection | 23 | 106 | Additional Notation and Definitions | In the next few sections we analyze the new algorithms for the deterministic setting. We will adopt the following conventions:In the deterministic setting (which we focus on here), a path p always deterministically leads to some state s, so we use them interchangeably, e.g., V^\star (p) \equiv V^\star (s), x \sim p \Le... | {
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cfe43b323ff147982828442d155452c9ec6e8ce2 | subsection | 24 | 106 | Least-Squares (LS) Oracle | The least-squares oracle takes as inputs a parameter \epsilon _{\textrm {sub}} and a sequence
\lbrace (x^{(i)}, v^{(i)})\rbrace _{i\in [n]} of observations x^{(i)}\in and values v^{(i)}\in . It outputs a value function \hat{g} \in whose squared
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7b380a2424809d8101dfe6850e6cceb9a5d3b2dd | subsection | 25 | 106 | Multi Data Set Classification Oracle | The multi data set classification oracle receives as inputs a parameter \epsilon _{\textrm {feas}}, m scalars that are upper bounds on the allowed cost \lbrace U_j\rbrace _{j \in [m]} \in ^{m}, and m cost-sensitive classification data sets D_1, \dots D_m, each of which
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-0.023793619126081467,
0.014315852895379066,
0.033851344138383865,
... | |
537d0975ee270bfc58990c8310de677ca7db6968 | subsection | 26 | 106 | Assumptions on the Function Classes | While Valor only requires realizability of the policy and the value function
classes, our other algorithms require stronger assumptions which we introduce
below.[Policy realizability]
\pi ^\star \in \Pi .[Value realizability]
g^\star \in .[Policy-value completeness]
At each level h, \forall g^{\prime } \in _{h+1}, t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1551,
"openalex_id": "https://openalex.org/W2117355432",
"raw": "Rémi Munos and Csaba Szepesvári. Finite-time bounds for fitted value iteration. Journal of Machine Learning Research, 2008.",
"source_ref_id": "6f51d77deaa045f... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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65b8765db527d09b733181e97f11b244a5abea56 | subsection | 27 | 106 | Analysis of | A state s \in _h is called learned if there is a
data set in _h that is sampled from a path leading to that
state. The set of all learned states at level h is ^{\textrm {learned}}_h
and
^{\textrm {learned}}:= \bigcup _{h \in [H]} ^{\textrm {learned}}_h. | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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0... | |
e14532426725864514c0f577b2e34ab7f95fc4bf | subsection | 28 | 106 | Concentration Results | We now define an event that holds with high probability and will be the main concentration argument in the proof. This event uses a parameter \epsilon _{\textrm {stat}} whose value we will set later.[Deviation Bounds]
Let denote the event that for all h \in [H] the total number of calls to (p) at level h is at most T_... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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c1aff27ce2e86ad8510d94a201af90c64cda30c1 | subsection | 29 | 106 | Concentration Results | The current path is denoted by p_j at level h_j and data sets D^{\prime }_{a}, \tilde{D} collected are denoted by D^{\prime }_{j,a} and \tilde{D}_{j} respectively.
Consider a fix a \in and g \in and defineY_{i,j} = {\left\lbrace \begin{array}{ll}
0 & \textrm {if } j > N_{dfs}\\
g(x_{h+1}^{(i,j)}) - \mathbf {E}_{p_j \ci... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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a5879782468396b2d07437c1ccea127f5952ab6e | subsection | 30 | 106 | Concentration Results | With a union bound over and , the following statement holds: Given a fix j \in , with probability at least 1-\delta ^{\prime }, if j \le N_{dfs} then for all g \in _{h+1} and a \in\left| \hat{\mathbf {E}}_{D^{\prime }_{j,a}}[g(x_{h+1})] - \mathbf {E}_{p_j\circ a}[g(x_{h+1})]\right|
\le \sqrt{\frac{\log (2K||/\delta ^{\... | {
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{
"arxiv_id": "",
"doi": "10.48550/arxiv.1610.09512",
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"raw": "Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E. Schapire. Contextual decision processes with low Bellman rank a... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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cd97ea1d91caea52915244e5832893801962b0ee | subsection | 31 | 106 | Concentration Results | As
such, Bernstein's inequality with a union bound over \pi \in \Pi gives that with probability 1-\delta ^{\prime },\left| (\hat{\mathbf {E}}_{\tilde{D}_j}- \mathbf {E}_{p_j})[K \lbrace \pi (x_h)=a_h\rbrace (r_h + V_{a_h})]\right| \le \sqrt{\frac{4 K\log (2 |\Pi |/\delta ^{\prime })}{n_{\textrm {train}}}} + \frac{4K}{3... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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1590afcf94a5878e760bf2d07fab11ec043c761e | subsection | 32 | 106 | Concentration Results | This
means that D^{\prime }_a and D are two data sets sampled from the same
distribution, and as such, we haveV_{opt} - V_{pes}
& = \hat{\mathbf {E}}_{D_a^{\prime }}[g_{opt}(x_{h+1}) - g_{pes}(x_{h+1})]
\le \mathbf {E}_s[g_{opt}(x_{h+1}) - g_{pes}(x_{h+1})] + 2\epsilon _{\textrm {stat}}\\
& \le \hat{\mathbf {E}}_D[g_{o... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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... | |
562d1e50ff8eb18a6b6bc1df1d613690f4ecba02 | subsection | 33 | 106 | Bound on Oracle Calls | [Proof of Theorem REF ]
Consider event from Definition REF which by Lemma REF has probability at least 1 - \delta /2.
Valor requires two types of nontrivial computations over the
function classes. We show that they can be reduced to CSC on \Pi and LP on (recall Sec. REF ), respectively, and hence Valor is oracle-effici... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0... | |
e02a5f2052678b788a25510ec663b0c9e6c871f4 | subsection | 34 | 106 | Bound on Oracle Calls | In addition, there at at most MH calls to the CSC oracle in .
The statement follows with realizing that T_{\max }= MH n_{\textrm {exp}}+ M = O\left(\frac{MH}{\epsilon } \ln \left( \frac{MH}{\delta }\right) \right). | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.0... | |
3bb4f376d69930acce25e6c105b87464f8c38577 | subsection | 35 | 106 | Depth First Search and Estimated Values | In this section, we show that in the high-probability event
(Definition REF ), produces good estimates of optimal values on learned states.
The next lemma first quantifies the error in the value estimate at
level h in terms of the estimation error of the values of the next
time step \lbrace V_a\rbrace _a.
[Error propa... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.019471550360322,
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b6704986cdf8f91a2a24cfde4c9d76d85320a726 | subsection | 36 | 106 | Depth First Search and Estimated Values | The third inequality uses
that \pi ^\star is the global and point-wise maximizer of the
long-term expected reward, which is precisely r_h+g^\star .Similarly, we can lower bound \tilde{V} by\tilde{V}
& = \hat{\mathbf {E}}_{\tilde{D}} [ K \lbrace \tilde{\pi }(x_h) = a_h\rbrace (r_h + V_{a_h})] - \epsilon _{\textrm {sub}}... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.... | |
81bc99c1945e2b33598fd96c0295359f4c260812 | subsection | 37 | 106 | Depth First Search and Estimated Values | Since g^\star _{h+1} is feasible for both V_{opt}
and V_{pes}, we haveV_{pes} - \epsilon _{\textrm {sub}}=& \hat{\mathbf {E}}_{D_a^{\prime }}[g_{pes}(x_{h+1})] - \epsilon _{\textrm {sub}}\le \hat{\mathbf {E}}_{D_a^{\prime }}[g^\star (x_{h+1})]
\le \hat{\mathbf {E}}_{D_a^{\prime }}[g_{opt}(x_{h+1})] + \epsilon _{\textrm... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0... | |
308e20420d9df5b1dd506dbd72bc10428f2dcb17 | subsection | 38 | 106 | Depth First Search and Estimated Values | Therefore,\hat{\mathbf {E}}_{D^{\prime }}[g^\star (x_{h+1})] - V_a & \le V_{opt} - V_a + \epsilon _{\textrm {sub}}= \frac{V_{opt} - V_{pes}}{2} + \epsilon _{\textrm {sub}}\le \phi _{h+1} + 2\epsilon _{\textrm {stat}}+ \epsilon _{\textrm {feas}}+ \epsilon _{\textrm {sub}}.\\
V_a - \hat{\mathbf {E}}_{D^{\prime }}[g^\star... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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... | |
65ac6a6383fb834012574f6798f9e895b91bf908 | subsection | 39 | 106 | Depth First Search and Estimated Values | For h=H+1 the
statement holds trivially since _{H+1} = \lbrace g^\star _{h+1} \rbrace
the constant 0 function is the only function in _{H+1} and
therefore the algorithm always returns on Line and
never calls level H+1 recursively.Consider now some data set
(\tilde{D},\tilde{V}, \lbrace V_a\rbrace ) \in _h at level h ... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.02... | |
3b69289c8eeec3aa48e9026e875098e4b7beed62 | subsection | 40 | 106 | Policy Performance | In this section, we bound the quality of the policy returned by
in the good event by using the fact that produces accurate estimates of the optimal values (previous section).
Before we state the main result of this section in
Proposition REF , we prove the following
helpful lemma. This Lemma is essentially Lemma 4.3 in... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 323,
"openalex_id": "https://openalex.org/W112666333",
"raw": "Stephane Ross and J Andrew Bagnell. Reinforcement and imitation learning via interactive no-regret learning. arXiv:1406.5979, 2014.",
"source_ref_id": "97e1333f7... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.028... | |
f9070632f3f794cacc932573543c4b436325289e | subsection | 41 | 106 | Policy Performance | Then the policy \hat{\pi } = \hat{\pi }_{1:H}
returned by satisfiesV^{\hat{\pi }} \ge V^\star - p_{ul}^{\hat{\pi }} - 2 H^2 T_{\max }(7\epsilon _{\textrm {stat}}+ 3\epsilon _{\textrm {sub}}+ 2\epsilon _{\textrm {feas}})where
p_{ul}^{\hat{\pi }}= \mathbf {P}( \exists h \in [H] ~:~ s_h \notin ^{\textrm {learned}}~|~ a_{1... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0... | |
187940d900b822a0ee61f0aabf52267027b02b1e | subsection | 42 | 106 | Policy Performance | For a state
s \in ^{\textrm {learned}} at level h, we have& V^\star (s) - Q^\star (s, \hat{\pi }_h)\\
& = \mathbf {E}_s[K(\lbrace \pi ^\star _h(x_h) = a_h\rbrace - \lbrace \hat{\pi }_h(x_h) = a_h\rbrace )(r_h + g^\star _{h+1}(x_{h+1})]\\
& \le \sum _{s \in ^{\textrm {learned}}_h} \mathbf {E}_s[K(\lbrace \pi ^\star _h(x... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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689c7e526079649670b062e592f35a3400698e6f | subsection | 43 | 106 | Policy Performance | The third inequality uses the deviation bound that
holds in event .Since per call, only one data set can be added to _{h}, the magnitude |_{h}| \le T_{\max } is bounded by the total number of calls to at each level. Using
Lemma REF , the suboptimality of \hat{\pi } is therefore at mostV^\star - V^{\hat{\pi }} & \le p_{... | {
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"raw": "Akshay Krishnamurthy, Alekh Agarwal, and John Langford. PAC reinforcement learning with rich observations. In Advances in Neural Information Processing Systems, 2016... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
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0.01629495620727539,
-0.011076908558607101,
0.004550534766167402,
0.010542898438870907,
0.02374059334397316,
-0.08409906923770905,
0.04998340830206871,
0.012595024891197681,... | |
ab3be4ed087ad09aa7703929fdac50ce4b09c36c | subsection | 44 | 106 | Meta-Algorithm Analysis | Now that we have the main guarantees for and , we
may turn to the analysis of .
Consider running with and (Algorithm + + ) with parametersn_{\textrm {exp}}\ge \frac{8}{\epsilon }\ln \left(\frac{4MH}{\delta }\right), \quad n_{eval} \ge \frac{32}{\epsilon ^2}\ln \left( \frac{8MH}{\delta }\right),
\quad \epsilon _{\tex... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0... | |
633e42973816762d1ccfbef0161aefdc71c2be38 | subsection | 45 | 106 | Meta-Algorithm Analysis | With these two bounds, if terminates, the termination condition implies thatV^\star - V^{\hat{\pi }^{(k)}} \le \hat{V}^{(k)} - \hat{V}^{\hat{\pi }^{(k)}} + \frac{\epsilon }{4} \le \frac{3}{4}\epsilon \le \epsilonand hence the returned policy is \epsilon -optimal.On the other hand, if the algorithm does not terminate in... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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b271830553824f55ae8a6b5a495427e8373a2b8e | subsection | 46 | 106 | Meta-Algorithm Analysis | With this fact, since there
are at most MH states, the algorithm must terminate and return a
near-optimal policy after at most MH iterations.In a non-terminal iteration k, the probability that we do not hit an
unlearned state in Line is(1 - p_{ul}^{\hat{\pi }^{(k)}})^{n_{\textrm {exp}}} \le (1 - \epsilon /8)^{n_{\text... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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0... | |
05a0ce94544f4711cea2f0782b4712789debd3c9 | subsection | 47 | 106 | Proof of Sample Complexity: Theorem | We now have all parts to complete the proof of
Theorem REF . For the calculation, we instantiate all the parameters as\epsilon _{\textrm {stat}}&= \epsilon _{\textrm {sub}}= \epsilon _{\textrm {feas}}= \frac{\epsilon }{2^6 7 H^2 T_{\max }},\\
\phi _h &= (H-h+1)(6\epsilon _{\textrm {stat}}+ 2\epsilon _{\textrm {sub}}+ \... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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... | |
8743027387f24cce75a3815141b15038f7d1e5ee | subsection | 48 | 106 | Extension: | We note that Theorem REF suffers relatively high sample complexity compared to the original Lsvee. The issue is that Valor pools all the data sets together for policy optimization (Algorithm ). This implicitly weights all data sets uniformly, and allows some undesired trade-off: the policy that maximizes the objective ... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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0.046... | |
9b95b72665a56439f6ce4bf956c5eb65d9365e78 | subsection | 49 | 106 | Extension: | Using Proposition REF , we can bound the deviation of the optimal policy for each constraint as& V - \hat{\mathbf {E}}_{D}[K \lbrace \pi ^\star (x_h) = a_h\rbrace (r_h + V_{a_h})] \\
& \le V^\star (s) + \phi _h - 2\epsilon _{\textrm {stat}}- \hat{\mathbf {E}}_{D}[K \lbrace \pi ^\star (x_h) = a_h\rbrace (r_h + V_{a_h})]... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.0... | |
749664fd29fa850277c21154626178f07e7a7f04 | subsection | 50 | 106 | Extension: | Then the policy \hat{\pi } = \hat{\pi }_{1:H}
returned by in Algorithm REF satisfiesV^{\hat{\pi }} \ge V^\star - p_{ul}^{\hat{\pi }} - 32H^2(\epsilon _{\textrm {stat}}+\epsilon _{\textrm {feas}}+ \epsilon _{\textrm {sub}})where
p_{ul}^{\hat{\pi }}= \mathbf {P}( \exists h \in [H] ~:~ s_h \notin ^{\textrm {learned}}~|~ a... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.01... | |
6c828cedf43e22fdfd2f4f3d46169601ddc7f372 | subsection | 51 | 106 | Extension: | For a state s \in ^{\textrm {learned}}_h, we have& V^\star (s) - Q^\star (s, \hat{\pi }_h)\\
& = \mathbf {E}_s[K(\lbrace \pi ^\star _h(x_h) = a_h\rbrace - \lbrace \hat{\pi }_h(x_h) = a_h\rbrace ) (r_h + g^\star _{h+1}(x_{h+1})]\\
& \le \mathbf {E}_s[K(\lbrace \pi ^\star _h(x_h) = a_h\rbrace - \lbrace \hat{\pi }_h(x_h) ... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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0.02... | |
3467297fc178473da0884130d46a7cfd333b5649 | subsection | 52 | 106 | Extension: | Using Lemma REF , the suboptimality of
\hat{\pi } is therefore at mostV^\star - V^{\hat{\pi }} & \le p_{ul}^{\hat{\pi }} + (1 -p_{ul}^{\hat{\pi }}) \sum _{h=1}^H
(4 \phi _{h+1} + 16 \epsilon _{\textrm {stat}}+ 5 \epsilon _{\textrm {feas}}+ 6 \epsilon _{\textrm {sub}})\\
&\le p_{ul}^{\hat{\pi }} + 16H\epsilon _{\textrm ... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0... | |
95764aeee4163b98413f1e83c2c02def4bda1d1a | subsection | 53 | 106 | Extension: | Finally, we are ready to assemble all statements to the following sample-complexity bound:
Consider a Markovian CDP with deterministic dynamics over M hidden states, as described in Section . When \pi ^\star \in \Pi and g^\star \in (Assumptions REF and REF hold), for any \epsilon , \delta \in (0, 1), the local value a... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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... | |
a56154fead4b3f8bdddbbb8192425fac7da4f913 | subsection | 54 | 106 | Extension: | For the
sample complexity, since T_{\max } is an upper bound on the number of
data sets we collect (because T_{\max } is an upper bound on the number of execution of
at any level), and we also n_{\textrm {eval}} trajectories for each of the MH
iterations of , the total sample complexity is& HT_{\max }n_{\textrm {train}... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.01196250412613153... | |
621d541f73387545fafeaeb101a308791993386e | subsection | 55 | 106 | Alternative Algorithms | [Informal statement]
Under Assumption REF or
Assumptions REF +REF , there
exist oracle-efficient algorithms with polynomial sample complexity
in CDPs (contextual decision processes) with deterministic dynamics over small hidden states. These algorithms do not store or
use local values. | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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a736dd5cda9e8a101213f66edee4a870cf75727c | subsection | 56 | 106 | Algorithm with Two-Sample State-Identity Test | See Algorithm + REF . The algorithm uses
a novel state identity test which compares two distributions using a
two-sample test in Line REF (recall that _h = for h \in [H] and _{H+1} = \lbrace x \mapsto 0 \rbrace ). Such
an identity test mechanism is very different from the one used in the Valor algorithm, and the two ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.5555/2188385.2188410",
"end": 218,
"openalex_id": "https://openalex.org/W2212660284",
"raw": "Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander J. Smola. A kernel two-sample test. Journal of Machine ... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0... | |
77c436755ea1d06f35258b7bea510f4d43c791d3 | subsection | 57 | 106 | Algorithm with Two-Sample State-Identity Test | Given the novelty of the mechanism, we believe analyzing the
two-sample test algorithm and understanding its computational and
statistical properties enriches our toolkit for dealing with the
challenges addressed in this paper.myfunFunction
\hat{g}_{H+1} \leftarrow 0
h=H:1
\hat{\pi }_h \leftarrow _{\pi \in \Pi _h} \s... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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-... | |
dfe9c7dde5a0c4bda05243bef27245ea738416ae | subsection | 58 | 106 | Computational considerations | The two-sample test algorithm requires three types nontrivial
computation. Line REF requires importance weighted
policy optimization, which is simply a call to the CSC oracles. Line REF performs squared-loss
regression on _h, which is a call to a LS oracle.The slightly unusual computation occurs on Line REF : we
comput... | {
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{
"arxiv_id": "",
"doi": "10.1198/jasa.2003.s269",
"end": 922,
"openalex_id": "https://openalex.org/W3144619878",
"raw": "Bernhard Schölkopf and Alexander J. Smola. Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press, ... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
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] | 2,018 | en | Computer Science | [
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182d22c4681ed02fd06151bd5f4bcd61d4a6ed21 | subsection | 59 | 106 | Sample complexity | Consider the same Markovian CDP setting as in Theorem REF but we explicitly require here that the process is an MDP over . Under Assumption REF , for any \epsilon , \delta \in (0, 1), the two-sample state-identity test algorithm (Algorithm +REF ) returns a policy \pi such that V^\star - V^{\pi } \le \epsilon with proba... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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96cdb82c500e9de59cd7caa691c8fa617069dbdf | subsection | 60 | 106 | Sample complexity | \\
L_{^{\textrm {val}}_h}(g; \pi , g^{\prime }) :=& \sum _{D \in ^{\textrm {val}}_{h}} L_D(g; \pi , g^{\prime }).Also define V_s, V_{^{\textrm {learned}}_h}, L_s, L_{^{\textrm {val}}_h} as the population version of V_D, V_{^{\textrm {learned}}_h}, L_D, L_{^{\textrm {val}}_h}, respectively.Consider a Markovian contextua... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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4f3718f344609d03897a166cfff4f26ac449ae42 | subsection | 61 | 106 | Sample complexity | We say the deviation
bound holds for a data set of
n_{\textrm {test}} observations sampled in
Line REF during a call to if for all
g \in _h:|\hat{\mathbf {E}}_{D} [g(x_h)] - \mathbf {E}_{p}[g(x_h)]| & \le \tau _{val}, \qquad |\hat{\mathbf {E}}_{D} [\bar{r}] - V^{\hat{\pi }_{h+1:H}}(p)| \le \tau _{val}.We say the deviat... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
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94a377ee947a6143636e05d3ecb85d1afac578b7 | subsection | 62 | 106 | Concentration Results. | For our analysis we rely on the following concentration bounds that
define the good event . This definition involves
parameters \tau , \tau _L, \tau _V whose values we will set later.
Let denote the event that for all h \in [H] the total number of calls to (p) at level h is at most T_{\max }= M(K+1)(1 + Hn_{\textrm {... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.0085448632016778,
-0.047668129205703735,
-0.050811417400836945,
-0.009963920339941978,
-0.019851546734571457,
0.013847255147993565,
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0.013290313072502613,
0.004157991148531437,
0.05245935544371605,
-0.03884860873222351,
0.016158945858478546,
0.008575379848480225,
0.... | |
06d011be116855f01dcb8993a2ca5ac3cd2a4caf | subsection | 63 | 106 | Concentration Results. | By Hoeffding's
inequality and a union bound, with probability 1-\delta ^{\prime }, for all g
\in _{h}\left| \hat{\mathbf {E}}_{\tilde{D}}[g(x_h)] - \mathbf {E}_{s}[g(x_{h})]\right|
\le \sqrt{\frac{\log (2||/\delta ^{\prime })}{2n_{\textrm {train}}}}.With \delta ^{\prime } = \frac{\delta }{6HT_{\max }} the choice for n_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1610.09512",
"end": 600,
"openalex_id": "https://openalex.org/W2545659366",
"raw": "Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E. Schapire. Contextual decision processes with low Bellman rank ar... | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.029337143525481224,
-0.0422808974981308,
0.02207154408097267,
0.02718493901193142,
0.004... | |
ed89a19ab76164547a2f591939e40af0593c479a | subsection | 64 | 106 | Concentration Results. | Using
a union bound, the deviation bounds
(REF )–() hold for a single call to
with probability 1 - 3 \delta ^{\prime }.Consider now the event ^{\prime } that these bounds hold for the first
T_{\max } calls at each level h. Applying a union bound let us bound
\mathbf {P}(^{\prime }) \ge 1 - 3 H T_{\max }\delta ^{\prime ... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
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0.030232038348913193,
-0.059456851333379745,
0.016268223524093628,
-0.0050933584570884705... | |
6abbfc13e07fb013d6f73317b62e6f55a4a5d1f4 | subsection | 65 | 106 | Concentration Results. | The bound |^{\textrm {val}}_h| \le M follows
from the fact that in no state can be added twice to
^{\textrm {val}}_h since as soon as it is in ^{\textrm {val}}_h once,
d_{MMD} \le 2 \tau holds (see Eq.(REF )) and the
current data set is not added to ^{\textrm {val}}_h. | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.010190718807280064,
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-0.03255537897348404,
0.005438609514385462,
... | |
202147a7ae8de8b55dc8340bdb9e6f675ba5d3f1 | subsection | 66 | 106 | Depth-first search and learning optimal values. | We now prove that and produce good value function estimates.
In event , consider an execution of and let
\lbrace \hat{g}_h,\hat{\pi }_h\rbrace _{h \in [H]} denote the learned value
functions and policies. Then every state s in ^{\textrm {check}}_h
satisfies\left|\mathbf {E}_{s} [\hat{g}_h(x_h)] - \mathbf {E}_{s}[g^\s... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.004051321651786566,
-0.009285232052206993,
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0.00014782363723497838,
0.006912424229085445,
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0.03573708236217499,
0.00867486372590065,
-0.005676427856087685,
0.015045586042106152,... | |
862d95850555d0eea5af5f877815fbcfc4d4e1de | subsection | 67 | 106 | Base case: | Both statement holds trivially for h=H+1
since the LHS is 0 and the RHS is non-negative. In particular
there are no actions, so Eq. (REF ) is trivial. | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.03155405819416046,
0.0274801068007946,
-0.023589253425598145,
-0.005462453234940767,
-0.01128347497433424,
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0.04443202167749405,
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0.023589253425598145,
-0.0069386884570121765,
-0.027144424617290497,
-0.02868550829589367,
0.003379702102392912,
0.... | |
ad029db95604b5539877a0f88710965653b055a1 | subsection | 68 | 106 | Inductive case: | Assume that Eq. (REF ) holds on level h+1. For any learned s \in ^{\textrm {learned}}_h, we first show that \hat{\pi }_h achieves high value compared to \pi _{\hat{g}_{h+1}}^\star (recall its definition from Assumption REF ) under V_s(\cdot ; \hat{g}_{h+1}):V_s(\pi _{\hat{g}_{h+1}}^\star ; \hat{g}_{h+1}) - V_s(\hat{\pi... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.004497884307056665,
-0.006397753953933716,
-0.01593143679201603,
-0.0019399271113798022,
-0.0077902283519506454,
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0.00935437809675932,
0.02818521484732628,
0.0398285910487175,
-0.009239927865564823,
-0.010453097522258759,
0.0020677295979112387,
-0.030855713412165642,... | |
94be810965f84944fe5d20e1868872e0f8338934 | subsection | 69 | 106 | Inductive case: | First we introduce and recall the definitions:&~ g_{\hat{\pi }_h,\hat{g}_{h+1}}(x) = \mathbf {E}[r + \hat{g}_{h+1}(x_{h+1}) \mid x_h = x, a_h = \hat{\pi }_h(x)], \\
&~ g_{\star ,\hat{g}_{h+1}}(x) = \mathbf {E}[r + \hat{g}_{h+1}(x_{h+1}) \mid x_h = x, a_h = \pi ^\star _{\hat{g}_{h+1}}(x)].Note that g_{\hat{\pi }_h,\hat{... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02099880576133728,
-0.023104790598154068,
-0.040227364748716354,
0.0038113747723400593,
0.014558765105903149,
-0.04437829181551933,
0.024493519216775894,
0.025042908266186714,
0.012658800929784775,
0.019976334646344185,
-0.05200866982340813,
0.007233600132167339,
-0.016435839235782623,
... | |
a2aacd566bac3b879ee17a990c1773168bf967be | subsection | 70 | 106 | Inductive case: | On the other
hand, Assumption REF guarantees that
g_{\star ,\hat{g}_{h+1}} \in _h, for any \hat{g}_{h+1}.The LHS of Eq.(REF ) can be bounded as\left|\mathbf {E}_{s} [g^\star (x_h)] - \mathbf {E}_{s}[\hat{g}_h(x_h)] \right|
\le \left|\mathbf {E}_{s} [g^\star (x_h)] - \mathbf {E}_{s}[g_{\hat{\pi }_h, \hat{g}_{h+1}}(x_h)]... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.02178892306983471,
0.00043629342690110207,
-0.007205755449831486,
-0.010375677607953548,
-0.011436132714152336,
-0.00929233431816101,
-0.015197315253317356,
0.02197202295064926,
0.06463436782360077,
-0.0010633162455633283,
-0.038054320961236954,
0.01747080869972706,
-0.01222956646233797,
... | |
d8e26071c73619dda9ed4db09d428f6185e34a77 | subsection | 71 | 106 | Inductive case: | The last inequality follows from the fact that if s \in ^{\textrm {learned}}_h \Rightarrow s \circ a \in ^{\textrm {check}}_{h+1} and we can therefore apply the induction hypothesis.
We can use the same argument to lower bound the above quantity. | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.023724131286144257,
0.019482774659991264,
0.010427937842905521,
0.00743000116199255,
-0.013708123005926609,
-0.03219158574938774,
-0.012891891412436962,
0.02482261136174202,
0.0030284500680863857,
-0.0037550493143498898,
-0.044152818620204926,
-0.013746264390647411,
-0.011488276533782482,... | |
a27e23501e9a5480c14edbc108637719b6efbfc2 | subsection | 72 | 106 | Inductive case: | This gives\left|\mathbf {E}_{s} [g^\star (x_h)] - \mathbf {E}_{s}[g_{\hat{\pi }_h, \hat{g}_{h+1}}(x_h)] \right|
\le &~ (H-h)(2\epsilon _V + \sqrt{4M\epsilon _V + 2\epsilon _L} + 8\tau ) + 2\epsilon _V.Next, we work with the second term in Equation (REF ):&~ \left|\mathbf {E}_{s} [\hat{g}_h(x_h)] - \mathbf {E}_{s}[g_{\h... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.00961332768201828,
0.0048791454173624516,
0.0053331078961491585,
-0.03341775760054588,
0.00210005440749228,
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0.014336065389215946,
-0.02081361785531044,
0.009704883210361004,
0.013183992356061935,
-0.... | |
a980f6d3621afef19a908c7b0483f1026f5b39ef | subsection | 73 | 106 | Inductive case: | According to the algorithm, this only happens when the MMD test suggests that the data set \tilde{D} drawn from s looks very similar to a previous data set D \in ^{\textrm {val}}_h, which corresponds to some s^{\prime } \in ^{\textrm {val}}_h. So,&~ |\mathbf {E}_{s}[\hat{g}_h(x_h)] - \mathbf {E}_{s}[g^\star (x_h)]| \\
... | {
"cite_spans": []
} | 1803.00606 | On Oracle-Efficient PAC RL with Rich Observations | [
"Christoph Dann",
"Nan Jiang",
"Akshay Krishnamurthy",
"Alekh Agarwal",
"John Langford",
"Robert E. Schapire"
] | [
"cs.LG",
"stat.ML"
] | 2,018 | en | Computer Science | [
-0.005875302944332361,
0.0023081547114998102,
-0.03845652937889099,
-0.018602583557367325,
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0.0017368387198075652,
0.037785064429044724,
-0.027774158865213394,
-0.02638545259833336,
0.007573036476969719... |
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