Yufei Liu Examples: 2. Bayesian statistics Introduction Outline Standard Given a prior distribution p ( θ ) , a likelihood model P ( D | θ ) and measures of performance data D . π is the posterior distribution and their shortcomings π ( θ | D ) ∝ P ( D | θ ) p ( θ ) . An accelerated algorithm: parallel tempering MCMC: Metropolis-Hastings type algorithm. Large Define deviation V ( θ ) . properties and = − log P ( D | θ ) . the infinite swapping limit then π ( θ | D ) ∝ e − V ( θ ) p ( θ ) . For τ ≥ 1 define Implementation issues and partial infinite π τ ( θ | D ) ∝ e − V ( θ ) /τ p ( θ ) = P ( D | θ ) 1 /τ p ( θ ) . swapping Concluding remarks
Yufei Liu Examples: 2. Bayesian statistics Introduction Outline Standard Given a prior distribution p ( θ ) , a likelihood model P ( D | θ ) and measures of performance data D . π is the posterior distribution and their shortcomings π ( θ | D ) ∝ P ( D | θ ) p ( θ ) . An accelerated algorithm: parallel tempering MCMC: Metropolis-Hastings type algorithm. Large Define deviation V ( θ ) . properties and = − log P ( D | θ ) . the infinite swapping limit then π ( θ | D ) ∝ e − V ( θ ) p ( θ ) . For τ ≥ 1 define Implementation issues and partial infinite π τ ( θ | D ) ∝ e − V ( θ ) /τ p ( θ ) = P ( D | θ ) 1 /τ p ( θ ) . swapping Concluding MC for π τ ( τ > 1) results in easier movement among local remarks minima of V ( θ ) .
Yufei Liu Examples: 3. A minimization Introduction problem Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Examples: 3. A minimization Introduction problem Outline Standard measures of Minimize a function f over a set Ω . Construct a Markov chain performance and their using a Metropolis-Hastings type algorithm with π τ as the shortcomings invariant distribution: An accelerated algorithm: parallel 1 Z ( τ ) e − f ( x ) /τ . tempering π τ ( x ) = Large deviation properties and Here τ is chosen such that τ > 0. the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Examples: 3. A minimization Introduction problem Outline Standard measures of Minimize a function f over a set Ω . Construct a Markov chain performance and their using a Metropolis-Hastings type algorithm with π τ as the shortcomings invariant distribution: An accelerated algorithm: parallel 1 Z ( τ ) e − f ( x ) /τ . tempering π τ ( x ) = Large deviation properties and Here τ is chosen such that τ > 0. Markov chain favors better the infinite swapping limit minimization solutions of f . Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Examples: 3. A minimization Introduction problem Outline Standard measures of Minimize a function f over a set Ω . Construct a Markov chain performance and their using a Metropolis-Hastings type algorithm with π τ as the shortcomings invariant distribution: An accelerated algorithm: parallel 1 Z ( τ ) e − f ( x ) /τ . tempering π τ ( x ) = Large deviation properties and Here τ is chosen such that τ > 0. Markov chain favors better the infinite swapping limit minimization solutions of f . As τ → 0, π τ sharply peaked Implementation around global minimum; as τ → ∞ , π τ approximate uniform issues and partial infinite distribution on Ω . swapping Concluding remarks
Yufei Liu Examples: 3. A minimization Introduction problem Outline Standard measures of Minimize a function f over a set Ω . Construct a Markov chain performance and their using a Metropolis-Hastings type algorithm with π τ as the shortcomings invariant distribution: An accelerated algorithm: parallel 1 Z ( τ ) e − f ( x ) /τ . tempering π τ ( x ) = Large deviation properties and Here τ is chosen such that τ > 0. Markov chain favors better the infinite swapping limit minimization solutions of f . As τ → 0, π τ sharply peaked Implementation around global minimum; as τ → ∞ , π τ approximate uniform issues and partial infinite distribution on Ω . swapping Minimization algorithm: sample Markov chain under small τ . Concluding remarks
Yufei Liu Examples: 3. A minimization Introduction problem Outline Standard measures of Minimize a function f over a set Ω . Construct a Markov chain performance and their using a Metropolis-Hastings type algorithm with π τ as the shortcomings invariant distribution: An accelerated algorithm: parallel 1 Z ( τ ) e − f ( x ) /τ . tempering π τ ( x ) = Large deviation properties and Here τ is chosen such that τ > 0. Markov chain favors better the infinite swapping limit minimization solutions of f . As τ → 0, π τ sharply peaked Implementation around global minimum; as τ → ∞ , π τ approximate uniform issues and partial infinite distribution on Ω . swapping Minimization algorithm: sample Markov chain under small τ . However, small τ results in less mobility, the chain more easily Concluding remarks get stuck in local minima of f .
Yufei Liu The challenge Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu The challenge Introduction Use Gibbs distribution to illustrate. π ( x ) ∝ e − V ( x ) /τ . When τ Outline is small, the main contribution of Standard measures of � performance and their R d f ( x ) π ( dx ) shortcomings An accelerated algorithm: comes from the global minimum and “important” local minima parallel tempering of V . Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu The challenge Introduction Use Gibbs distribution to illustrate. π ( x ) ∝ e − V ( x ) /τ . When τ Outline is small, the main contribution of Standard measures of � performance and their R d f ( x ) π ( dx ) shortcomings An accelerated algorithm: comes from the global minimum and “important” local minima parallel tempering of V . Large When V has various deep local minima that are separated by deviation properties and steep "barriers", the underlying probability distribution π has the infinite swapping limit multiple isolated parts that communicate poorly with each Implementation other, in which case the scheme can be extremely slow to issues and partial infinite converge (the rare event problem). swapping Concluding remarks
Yufei Liu The challenge Introduction Use Gibbs distribution to illustrate. π ( x ) ∝ e − V ( x ) /τ . When τ Outline is small, the main contribution of Standard measures of � performance and their R d f ( x ) π ( dx ) shortcomings An accelerated algorithm: comes from the global minimum and “important” local minima parallel tempering of V . Large When V has various deep local minima that are separated by deviation properties and steep "barriers", the underlying probability distribution π has the infinite swapping limit multiple isolated parts that communicate poorly with each Implementation other, in which case the scheme can be extremely slow to issues and partial infinite converge (the rare event problem). swapping An example of such is the Lennard-Jones cluster of 38 atoms. This potential has ≈ 10 14 local minima. Concluding remarks
Yufei Liu The challenge Introduction Use Gibbs distribution to illustrate. π ( x ) ∝ e − V ( x ) /τ . When τ Outline is small, the main contribution of Standard measures of � performance and their R d f ( x ) π ( dx ) shortcomings An accelerated algorithm: comes from the global minimum and “important” local minima parallel tempering of V . Large When V has various deep local minima that are separated by deviation properties and steep "barriers", the underlying probability distribution π has the infinite swapping limit multiple isolated parts that communicate poorly with each Implementation other, in which case the scheme can be extremely slow to issues and partial infinite converge (the rare event problem). swapping An example of such is the Lennard-Jones cluster of 38 atoms. This potential has ≈ 10 14 local minima. The lowest 150 and Concluding remarks their “connectivity” graph are as in the figure (taken from Doyle, Miller & Wales, JCP, 1999).
Yufei Liu Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks Global minimum only discovered 10 + years ago.
Yufei Liu Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks Global minimum only discovered 10 + years ago. Focus on overcoming rare-event sampling issues.
Yufei Liu Outline Introduction Outline Standard measures of performance and their shortcomings 1 Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Outline Introduction Outline Standard measures of performance and their shortcomings 1 Standard measures of performance and their shortcomings An accelerated algorithm: 2 An accelerated algorithm: parallel tempering (aka replica parallel tempering exchange) Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Outline Introduction Outline Standard measures of performance and their shortcomings 1 Standard measures of performance and their shortcomings An accelerated algorithm: 2 An accelerated algorithm: parallel tempering (aka replica parallel tempering exchange) Large 3 Large deviation properties and the infinite swapping limit deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Outline Introduction Outline Standard measures of performance and their shortcomings 1 Standard measures of performance and their shortcomings An accelerated algorithm: 2 An accelerated algorithm: parallel tempering (aka replica parallel tempering exchange) Large 3 Large deviation properties and the infinite swapping limit deviation properties and the infinite 4 Implementation issues and partial infinite swapping swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Outline Introduction Outline Standard measures of performance and their shortcomings 1 Standard measures of performance and their shortcomings An accelerated algorithm: 2 An accelerated algorithm: parallel tempering (aka replica parallel tempering exchange) Large 3 Large deviation properties and the infinite swapping limit deviation properties and the infinite 4 Implementation issues and partial infinite swapping swapping limit 5 Concluding remarks Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Standard measures of performance Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Standard measures of performance Introduction How should one describe the rate of convergence Outline � T � Standard 1 measures of f ( X ( t )) dt → R d f ( x ) π ( dx )? performance T 0 and their shortcomings None of the standard descriptions work directly with An accelerated algorithm: convergence of the empirical measure. parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Standard measures of performance Introduction How should one describe the rate of convergence Outline � T � Standard 1 measures of f ( X ( t )) dt → R d f ( x ) π ( dx )? performance T 0 and their shortcomings None of the standard descriptions work directly with An accelerated algorithm: convergence of the empirical measure. parallel tempering 2 nd eigenvalue. Consider the transition kernel Large deviation properties and the infinite p ( dx , T , x 0 ) = P { X ( T ) ∈ dx | X ( 0 ) = x 0 } . swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Standard measures of performance Introduction How should one describe the rate of convergence Outline � T � Standard 1 measures of f ( X ( t )) dt → R d f ( x ) π ( dx )? performance T 0 and their shortcomings None of the standard descriptions work directly with An accelerated algorithm: convergence of the empirical measure. parallel tempering 2 nd eigenvalue. Consider the transition kernel Large deviation properties and the infinite p ( dx , T , x 0 ) = P { X ( T ) ∈ dx | X ( 0 ) = x 0 } . swapping limit Implementation Under mild conditions the exponential rate of convergence issues and partial infinite swapping p ( dx , T , x 0 ) → π ( dx ) Concluding remarks is determined by the sub-dominant eigenvalue of the operator corresponding to X . Used to characterize “efficiency” of the corresponding Monte Carlo.
Yufei Liu Standard measures of performance Introduction and their shortcomings Outline Standard measures of performance Problem: Only indirectly related to problem of interest. and their shortcomings Information on density, but not on empirical measure which An accelerated depends on sample path; neglects potentially significant effect algorithm: parallel of time averaging in empirical measure (Rosenthal, tempering Gubernatis). Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Standard measures of performance Introduction and their shortcomings Outline Standard measures of performance Problem: Only indirectly related to problem of interest. and their shortcomings Information on density, but not on empirical measure which An accelerated depends on sample path; neglects potentially significant effect algorithm: parallel of time averaging in empirical measure (Rosenthal, tempering Gubernatis). Large deviation properties and Asymptotic variance. Also a popular quantity for comparing the infinite swapping limit efficiency of algorithms, but is a property of the algorithm once Implementation one is already at equilibrium. Also does not properly reflect the issues and partial infinite time averaging. swapping Concluding remarks
Yufei Liu Standard measures of performance Introduction and their shortcomings Outline Standard measures of performance Problem: Only indirectly related to problem of interest. and their shortcomings Information on density, but not on empirical measure which An accelerated depends on sample path; neglects potentially significant effect algorithm: parallel of time averaging in empirical measure (Rosenthal, tempering Gubernatis). Large deviation properties and Asymptotic variance. Also a popular quantity for comparing the infinite swapping limit efficiency of algorithms, but is a property of the algorithm once Implementation one is already at equilibrium. Also does not properly reflect the issues and partial infinite time averaging. swapping Large deviation rate. We will use the LD rate I , where a Concluding larger rate implies faster convergence. remarks
Yufei Liu A representative example Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu A representative example Introduction Outline Standard Compute the average potential energy and other functionals measures of performance with respect to a Gibbs measure of the form and their shortcomings 1 An accelerated Z ( τ ) e − V ( x ) /τ π τ ( x ) = algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu A representative example Introduction Outline Standard Compute the average potential energy and other functionals measures of performance with respect to a Gibbs measure of the form and their shortcomings 1 An accelerated Z ( τ ) e − V ( x ) /τ π τ ( x ) = algorithm: parallel tempering Large A corresponding continuous time model is deviation properties and √ the infinite swapping limit dX = −∇ V ( X ) dt + 2 τ dW , X ( 0 ) = x 0 , Implementation issues and where τ is a fixed temperature (properly scaled). partial infinite swapping Concluding remarks
Yufei Liu A representative example Introduction Outline Standard Compute the average potential energy and other functionals measures of performance with respect to a Gibbs measure of the form and their shortcomings 1 An accelerated Z ( τ ) e − V ( x ) /τ π τ ( x ) = algorithm: parallel tempering Large A corresponding continuous time model is deviation properties and √ the infinite swapping limit dX = −∇ V ( X ) dt + 2 τ dW , X ( 0 ) = x 0 , Implementation issues and where τ is a fixed temperature (properly scaled). partial infinite swapping • Simulations are done using a discrete time model. Concluding remarks
Yufei Liu An accelerated algorithm: parallel Introduction tempering Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu An accelerated algorithm: parallel Introduction tempering Outline Standard measures of Besides τ 1 = τ , introduce higher temperature τ 2 > τ 1 . performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu An accelerated algorithm: parallel Introduction tempering Outline Standard measures of Besides τ 1 = τ , introduce higher temperature τ 2 > τ 1 . Thus performance and their shortcomings √ dX a 1 = −∇ V ( X a 1 ) dt + 2 τ 1 dW 1 An accelerated √ algorithm: dX a 2 = −∇ V ( X a parallel 2 ) dt + 2 τ 2 dW 2 , tempering Large with W 1 and W 2 independent. deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu An accelerated algorithm: parallel Introduction tempering Outline Standard measures of Besides τ 1 = τ , introduce higher temperature τ 2 > τ 1 . Thus performance and their shortcomings √ dX a 1 = −∇ V ( X a 1 ) dt + 2 τ 1 dW 1 An accelerated √ algorithm: dX a 2 = −∇ V ( X a parallel 2 ) dt + 2 τ 2 dW 2 , tempering Large with W 1 and W 2 independent. Now introduce swaps deviation properties and (Swendsen, Geyer), i.e., X a 1 and X a 2 exchange locations with the infinite swapping limit state dependent intensity Implementation � � issues and partial infinite 1 ∧ π τ 1 ( x 2 ) π τ 2 ( x 1 ) ag ( x 1 , x 2 ) . swapping = a , π τ 1 ( x 1 ) π τ 2 ( x 2 ) Concluding with a > 0, as the “swap rate.” remarks
Yufei Liu An accelerated algorithm: parallel Introduction tempering Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu An accelerated algorithm: parallel Introduction tempering Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit One can check (detailed balance condition): with this swapping Implementation issues and intensity, invariant distribution of the joint process partial infinite swapping � = π τ 1 ( x 1 ) π τ 2 ( x 2 ) = e − V ( x 1 ) τ 1 e − V ( x 2 ) π ( x 1 , x 2 ) . Z ( τ 1 ) Z ( τ 2 ) . τ 2 Concluding remarks
Yufei Liu An accelerated algorithm: parallel Introduction tempering Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit One can check (detailed balance condition): with this swapping Implementation issues and intensity, invariant distribution of the joint process partial infinite swapping � = π τ 1 ( x 1 ) π τ 2 ( x 2 ) = e − V ( x 1 ) τ 1 e − V ( x 2 ) π ( x 1 , x 2 ) . Z ( τ 1 ) Z ( τ 2 ) . τ 2 Concluding remarks Use the first marginal of the empirical measure.
Yufei Liu Parallel tempering analysis Introduction Outline Standard measures of performance In practice, much more temperatures (30 − 50) are used. and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Parallel tempering analysis Introduction Outline Standard measures of performance In practice, much more temperatures (30 − 50) are used. Bring and their shortcomings in higher temperatures An accelerated algorithm: 1 Higher temperature simulations correspond to higher parallel tempering volatility. Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Parallel tempering analysis Introduction Outline Standard measures of performance In practice, much more temperatures (30 − 50) are used. Bring and their shortcomings in higher temperatures An accelerated algorithm: 1 Higher temperature simulations correspond to higher parallel tempering volatility. Large deviation 2 High-energy barriers are more easily crossed for properties and the infinite simulations carried out in higher temperatures. swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Parallel tempering analysis Introduction Outline Standard measures of performance In practice, much more temperatures (30 − 50) are used. Bring and their shortcomings in higher temperatures An accelerated algorithm: 1 Higher temperature simulations correspond to higher parallel tempering volatility. Large deviation 2 High-energy barriers are more easily crossed for properties and the infinite simulations carried out in higher temperatures. swapping limit 3 Swapping enables information flow from high temperatures Implementation issues and to low temperatures. partial infinite swapping Concluding remarks
Yufei Liu Donsker-Varadhan rate of decay Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Donsker-Varadhan rate of decay Introduction Outline Standard measures of performance How does convergence depend on swap rate a ? and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Donsker-Varadhan rate of decay Introduction Outline Standard measures of performance How does convergence depend on swap rate a ? and their shortcomings Donsker-Varadhan theory for empirical measure. Let I denote An accelerated the large deviations rate function. algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Donsker-Varadhan rate of decay Introduction Outline Standard measures of performance How does convergence depend on swap rate a ? and their shortcomings Donsker-Varadhan theory for empirical measure. Let I denote An accelerated the large deviations rate function. algorithm: parallel Let S denote the state space, for any µ ∈ P ( S ) , if N δ ( µ ) is a tempering δ -neighborhood of µ under weak topology Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Donsker-Varadhan rate of decay Introduction Outline Standard measures of performance How does convergence depend on swap rate a ? and their shortcomings Donsker-Varadhan theory for empirical measure. Let I denote An accelerated the large deviations rate function. algorithm: parallel Let S denote the state space, for any µ ∈ P ( S ) , if N δ ( µ ) is a tempering δ -neighborhood of µ under weak topology Large deviation properties and P ( µ T ∈ N δ ( µ )) ≈ e − T ( I ( µ )+ ε ( δ )) the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Donsker-Varadhan rate of decay Introduction Outline Standard measures of performance How does convergence depend on swap rate a ? and their shortcomings Donsker-Varadhan theory for empirical measure. Let I denote An accelerated the large deviations rate function. algorithm: parallel Let S denote the state space, for any µ ∈ P ( S ) , if N δ ( µ ) is a tempering δ -neighborhood of µ under weak topology Large deviation properties and P ( µ T ∈ N δ ( µ )) ≈ e − T ( I ( µ )+ ε ( δ )) the infinite swapping limit Implementation To achieve maximum rate of convergence, we choose a such issues and partial infinite that I a is the largest possible. swapping Concluding remarks
Yufei Liu Large deviation rate function Introduction Outline Under mild conditions on V , one can calculate I explicitly Standard measures of (Donser-Varadhan). performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Large deviation rate function Introduction Outline Under mild conditions on V , one can calculate I explicitly Standard measures of (Donser-Varadhan). Suppose ν ∈ P ( S ) is given by performance and their shortcomings θ ( x 1 , x 2 ) = d ν An accelerated d π ( x 1 , x 2 ) . algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Large deviation rate function Introduction Outline Under mild conditions on V , one can calculate I explicitly Standard measures of (Donser-Varadhan). Suppose ν ∈ P ( S ) is given by performance and their shortcomings θ ( x 1 , x 2 ) = d ν An accelerated d π ( x 1 , x 2 ) . algorithm: parallel tempering Then we have monotonic form Large deviation properties and I a ( ν ) = J 0 ( ν ) + aJ 1 ( ν ) the infinite swapping limit Implementation where J 0 is the rate for “no swap” dynamics; J 1 is nonnegative issues and partial infinite and swapping Concluding remarks
Yufei Liu Large deviation rate function Introduction Outline Under mild conditions on V , one can calculate I explicitly Standard measures of (Donser-Varadhan). Suppose ν ∈ P ( S ) is given by performance and their shortcomings θ ( x 1 , x 2 ) = d ν An accelerated d π ( x 1 , x 2 ) . algorithm: parallel tempering Then we have monotonic form Large deviation properties and I a ( ν ) = J 0 ( ν ) + aJ 1 ( ν ) the infinite swapping limit Implementation where J 0 is the rate for “no swap” dynamics; J 1 is nonnegative issues and partial infinite and swapping J 1 ( ν ) = 0 iff θ ( x 2 , x 1 ) = θ ( x 1 , x 2 ) ν -a.s. Concluding remarks
Yufei Liu Limit of rate function Introduction Thus for I a ( ν ) ↑ ∞ as a ↑ ∞ ( ν is very unlikely) unless Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function Introduction Thus for I a ( ν ) ↑ ∞ as a ↑ ∞ ( ν is very unlikely) unless Outline Standard θ ( x 2 , x 1 ) = θ ( x 1 , x 2 ) ν -a.s. measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function Introduction Thus for I a ( ν ) ↑ ∞ as a ↑ ∞ ( ν is very unlikely) unless Outline Standard θ ( x 2 , x 1 ) = θ ( x 1 , x 2 ) ν -a.s. measures of performance and their shortcomings If we call measures that place precisely same relative weight on An accelerated permutations ( x 1 , x 2 ) and ( x 2 , x 1 ) as π symmetrized measures , algorithm: parallel then tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function Introduction Thus for I a ( ν ) ↑ ∞ as a ↑ ∞ ( ν is very unlikely) unless Outline Standard θ ( x 2 , x 1 ) = θ ( x 1 , x 2 ) ν -a.s. measures of performance and their shortcomings If we call measures that place precisely same relative weight on An accelerated permutations ( x 1 , x 2 ) and ( x 2 , x 1 ) as π symmetrized measures , algorithm: parallel then tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function (cont’d) Introduction Outline Standard measures of By contraction principle, for probability measure γ performance and their shortcomings I a 1 ( γ ) = inf { I a ( ν ) : first marginal of ν is γ } . An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function (cont’d) Introduction Outline Standard measures of By contraction principle, for probability measure γ performance and their shortcomings I a 1 ( γ ) = inf { I a ( ν ) : first marginal of ν is γ } . An accelerated algorithm: parallel tempering I a Large 1 ( γ ) ↑ as a ↑ . deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function (cont’d) Introduction Outline Standard measures of By contraction principle, for probability measure γ performance and their shortcomings I a 1 ( γ ) = inf { I a ( ν ) : first marginal of ν is γ } . An accelerated algorithm: parallel tempering I a Large 1 ( γ ) ↑ as a ↑ . deviation properties and the infinite This suggests one consider the infinite swapping limit a ↑ ∞ . swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function (cont’d) Introduction Outline Standard measures of By contraction principle, for probability measure γ performance and their shortcomings I a 1 ( γ ) = inf { I a ( ν ) : first marginal of ν is γ } . An accelerated algorithm: parallel tempering I a Large 1 ( γ ) ↑ as a ↑ . deviation properties and the infinite This suggests one consider the infinite swapping limit a ↑ ∞ . swapping limit Unfortunately, limit process is not well defined (no tightness). Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Limit of rate function (cont’d) Introduction Outline Standard measures of By contraction principle, for probability measure γ performance and their shortcomings I a 1 ( γ ) = inf { I a ( ν ) : first marginal of ν is γ } . An accelerated algorithm: parallel tempering I a Large 1 ( γ ) ↑ as a ↑ . deviation properties and the infinite This suggests one consider the infinite swapping limit a ↑ ∞ . swapping limit Unfortunately, limit process is not well defined (no tightness). Implementation issues and An alternative perspective: rather than swap particles, swap partial infinite temperatures, and use “weighted” empirical measure. swapping Concluding remarks
Yufei Liu "Temperature swapping "process Introduction Temperature swapping process: Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu "Temperature swapping "process Introduction Temperature swapping process: Outline Standard � measures of 2 τ 1 1 ( Z a = 1 ) + 2 τ 2 1 ( Z a = 2 ) dW 1 dY a 1 = −∇ V ( Y a 1 ) dt + performance � and their 2 τ 2 1 ( Z a = 1 ) + 2 τ 1 1 ( Z a = 2 ) dW 2 , dY a 2 = −∇ V ( Y a shortcomings 2 ) dt + An accelerated algorithm: where Z a ( t ) jumps from 1 → 2 with intensity ag ( Y a 1 ( t ) , Y a 2 ( t )) parallel tempering and from 2 → 1 with intensity ag ( Y a 2 ( t ) , Y a 1 ( t )) . Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu "Temperature swapping "process Introduction Temperature swapping process: Outline Standard � measures of 2 τ 1 1 ( Z a = 1 ) + 2 τ 2 1 ( Z a = 2 ) dW 1 dY a 1 = −∇ V ( Y a 1 ) dt + performance � and their 2 τ 2 1 ( Z a = 1 ) + 2 τ 1 1 ( Z a = 2 ) dW 2 , dY a 2 = −∇ V ( Y a shortcomings 2 ) dt + An accelerated algorithm: where Z a ( t ) jumps from 1 → 2 with intensity ag ( Y a 1 ( t ) , Y a 2 ( t )) parallel tempering and from 2 → 1 with intensity ag ( Y a 2 ( t ) , Y a 1 ( t )) . Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Infinite swapping limit Introduction Instead of using ordinary empirical measure Outline � T Standard measures of T ( · ) = 1 µ a performance 2 )( · ) dt , δ ( X a 1 , X a and their T 0 shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Infinite swapping limit Introduction Instead of using ordinary empirical measure Outline � T Standard measures of T ( · ) = 1 µ a performance 2 )( · ) dt , δ ( X a 1 , X a and their T 0 shortcomings use weighted empirical measure η a An accelerated T : algorithm: parallel � T � � tempering 1 1 ( Z a = 1 ) δ ( Y a 2 ) ( · ) + 1 ( Z a = 2 ) δ ( Y a 1 ) ( · ) dt . 1 , Y a 2 , Y a Large T deviation 0 properties and the infinite Ergodic theory η a T → π . ( Y a 1 , Y a 2 , η a T ) admits a well defined swapping limit weak limit a → ∞ . Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Infinite swapping limit Introduction Instead of using ordinary empirical measure Outline � T Standard measures of T ( · ) = 1 µ a performance 2 )( · ) dt , δ ( X a 1 , X a and their T 0 shortcomings use weighted empirical measure η a An accelerated T : algorithm: parallel � T � � tempering 1 1 ( Z a = 1 ) δ ( Y a 2 ) ( · ) + 1 ( Z a = 2 ) δ ( Y a 1 ) ( · ) dt . 1 , Y a 2 , Y a Large T deviation 0 properties and the infinite Ergodic theory η a T → π . ( Y a 1 , Y a 2 , η a T ) admits a well defined swapping limit weak limit a → ∞ . Define state dependent weight Implementation issues and partial infinite swapping π τ 1 ( x 1 ) π τ 2 ( x 2 ) . ρ 1 ( x 1 , x 2 ) = π τ 1 ( x 1 ) π τ 2 ( x 2 ) + π τ 1 ( x 2 ) π τ 2 ( x 1 ) , Concluding remarks π τ 1 ( x 2 ) π τ 2 ( x 1 ) . ρ 2 ( x 1 , x 2 ) = π τ 1 ( x 1 ) π τ 2 ( x 2 ) + π τ 1 ( x 1 ) π τ 2 ( x 2 ) .
Yufei Liu Infinite swapping limit (cont’d) Introduction Outline The triple has following weak limit Standard measures of � performance and their dY 1 = −∇ V ( Y 1 ) dt + 2 τ 1 ρ 1 ( Y 1 , Y 2 ) + 2 τ 2 ρ 2 ( Y 1 , Y 2 ) dW 1 shortcomings � dY 2 = −∇ V ( Y 2 ) dt + 2 τ 2 ρ 1 ( Y 1 , Y 2 ) + 2 τ 1 ρ 2 ( Y 1 , Y 2 ) dW 2 , An accelerated algorithm: parallel � T tempering � � η T ( dx ) = 1 ρ 1 ( Y 1 , Y 2 ) δ ( Y 1 , Y 2 ) + ρ 2 ( Y 1 , Y 2 ) δ ( Y 2 , Y 1 ) dt , Large deviation T 0 properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Infinite swapping limit (cont’d) Introduction Outline The triple has following weak limit Standard measures of � performance and their dY 1 = −∇ V ( Y 1 ) dt + 2 τ 1 ρ 1 ( Y 1 , Y 2 ) + 2 τ 2 ρ 2 ( Y 1 , Y 2 ) dW 1 shortcomings � dY 2 = −∇ V ( Y 2 ) dt + 2 τ 2 ρ 1 ( Y 1 , Y 2 ) + 2 τ 1 ρ 2 ( Y 1 , Y 2 ) dW 2 , An accelerated algorithm: parallel � T tempering � � η T ( dx ) = 1 ρ 1 ( Y 1 , Y 2 ) δ ( Y 1 , Y 2 ) + ρ 2 ( Y 1 , Y 2 ) δ ( Y 2 , Y 1 ) dt , Large deviation T 0 properties and � � the infinite η a T Theorem: for any sequence a T ↑ ∞ , satisfies the swapping limit T uniform large deviations principle (in T ) with rate I ∞ Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Infinite swapping limit (cont’d) Introduction Outline The triple has following weak limit Standard measures of � performance and their dY 1 = −∇ V ( Y 1 ) dt + 2 τ 1 ρ 1 ( Y 1 , Y 2 ) + 2 τ 2 ρ 2 ( Y 1 , Y 2 ) dW 1 shortcomings � dY 2 = −∇ V ( Y 2 ) dt + 2 τ 2 ρ 1 ( Y 1 , Y 2 ) + 2 τ 1 ρ 2 ( Y 1 , Y 2 ) dW 2 , An accelerated algorithm: parallel � T tempering � � η T ( dx ) = 1 ρ 1 ( Y 1 , Y 2 ) δ ( Y 1 , Y 2 ) + ρ 2 ( Y 1 , Y 2 ) δ ( Y 2 , Y 1 ) dt , Large deviation T 0 properties and � � the infinite η a T Theorem: for any sequence a T ↑ ∞ , satisfies the swapping limit T uniform large deviations principle (in T ) with rate I ∞ Implementation issues and partial infinite � J 0 ( ν ) swapping a →∞ I a ( ν ) = I ∞ ( ν ) . if θ ( x 1 , x 2 ) = θ ( x 2 , x 1 ) lim = ∞ otherwise Concluding remarks
Yufei Liu Implementation issues Introduction Outline Standard measures of • Applications of parallel tempering use many temperatures performance and their (e.g., K = 30 to 50) when V is complicated to overcome shortcomings barriers of all different heights. An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Implementation issues Introduction Outline Standard measures of • Applications of parallel tempering use many temperatures performance and their (e.g., K = 30 to 50) when V is complicated to overcome shortcomings barriers of all different heights. An accelerated algorithm: parallel • Straightforward extension of infinite swapping to K tempering temperatures τ 1 < τ 2 < · · · < τ K . Benefits of Large deviation symmetrization even greater. properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Implementation issues Introduction Outline Standard measures of • Applications of parallel tempering use many temperatures performance and their (e.g., K = 30 to 50) when V is complicated to overcome shortcomings barriers of all different heights. An accelerated algorithm: parallel • Straightforward extension of infinite swapping to K tempering temperatures τ 1 < τ 2 < · · · < τ K . Benefits of Large deviation symmetrization even greater. properties and the infinite • But, coefficients become complex, e.g., K ! weights, and swapping limit each involves many calculations. Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Implementation issues Introduction Outline Standard measures of • Applications of parallel tempering use many temperatures performance and their (e.g., K = 30 to 50) when V is complicated to overcome shortcomings barriers of all different heights. An accelerated algorithm: parallel • Straightforward extension of infinite swapping to K tempering temperatures τ 1 < τ 2 < · · · < τ K . Benefits of Large deviation symmetrization even greater. properties and the infinite • But, coefficients become complex, e.g., K ! weights, and swapping limit each involves many calculations. Implementation issues and partial infinite • Need for computational feasibility leads to partial infinite swapping swapping . Concluding remarks
Yufei Liu Partial infinite swapping Introduction Outline Standard measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Partial infinite swapping Introduction Partial infinite swapping. Instead of instantly symmetrizing Outline all permutations, pick subgroups of the set of permutations Standard measures of (that can generate the whole permutation set) and construct performance and their corresponding partial infinite swapping dynamics within each shortcomings group. Then alternate among each dynamics (need certain An accelerated algorithm: handoff rule, use proper weight). parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Partial infinite swapping Introduction Partial infinite swapping. Instead of instantly symmetrizing Outline all permutations, pick subgroups of the set of permutations Standard measures of (that can generate the whole permutation set) and construct performance and their corresponding partial infinite swapping dynamics within each shortcomings group. Then alternate among each dynamics (need certain An accelerated algorithm: handoff rule, use proper weight). parallel tempering Examples are Dynamics A and B in figure: Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Comparison of PINS and PT Introduction Relaxation study of convergence to equilibrium for LJ-38: Outline parallel tempering versus partial infinite swapping, only lowest Standard temperature illustrated. measures of performance and their shortcomings An accelerated algorithm: parallel tempering Large deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Concluding remarks Introduction References: Outline • Mathematical paper: “On the infinite swapping limit for Standard measures of parallel tempering”, Dupuis, Liu, Plattner and Doll, to be performance appeared in SIAM J. on MMS and their shortcomings • Applications paper (lots of numerical data): “An infinite An accelerated algorithm: swapping approach to the rare-event sampling problem”, parallel tempering Plattner, Doll, Dupuis, Wang, Liu and Gubernatis, J. of Large Chem. Phy. 135, 134111 (2011) deviation properties and the infinite swapping limit Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Concluding remarks Introduction References: Outline • Mathematical paper: “On the infinite swapping limit for Standard measures of parallel tempering”, Dupuis, Liu, Plattner and Doll, to be performance appeared in SIAM J. on MMS and their shortcomings • Applications paper (lots of numerical data): “An infinite An accelerated algorithm: swapping approach to the rare-event sampling problem”, parallel tempering Plattner, Doll, Dupuis, Wang, Liu and Gubernatis, J. of Large Chem. Phy. 135, 134111 (2011) deviation properties and Many open questions. the infinite swapping limit • Selection of set of temperatures. Implementation issues and partial infinite swapping Concluding remarks
Yufei Liu Concluding remarks Introduction References: Outline • Mathematical paper: “On the infinite swapping limit for Standard measures of parallel tempering”, Dupuis, Liu, Plattner and Doll, to be performance appeared in SIAM J. on MMS and their shortcomings • Applications paper (lots of numerical data): “An infinite An accelerated algorithm: swapping approach to the rare-event sampling problem”, parallel tempering Plattner, Doll, Dupuis, Wang, Liu and Gubernatis, J. of Large Chem. Phy. 135, 134111 (2011) deviation properties and Many open questions. the infinite swapping limit • Selection of set of temperatures. Implementation issues and • Selection of “best” subgroups for partial infinite swapping partial infinite swapping approximations. Concluding remarks
Yufei Liu Concluding remarks Introduction References: Outline • Mathematical paper: “On the infinite swapping limit for Standard measures of parallel tempering”, Dupuis, Liu, Plattner and Doll, to be performance appeared in SIAM J. on MMS and their shortcomings • Applications paper (lots of numerical data): “An infinite An accelerated algorithm: swapping approach to the rare-event sampling problem”, parallel tempering Plattner, Doll, Dupuis, Wang, Liu and Gubernatis, J. of Large Chem. Phy. 135, 134111 (2011) deviation properties and Many open questions. the infinite swapping limit • Selection of set of temperatures. Implementation issues and • Selection of “best” subgroups for partial infinite swapping partial infinite swapping approximations. • Better quantitative understanding of rate of marginals Concluding such as I ∞ 1 ( γ ) remarks
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