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Appendix D Derivation of the Variational Online-Newton method (VON) In this section, we derive the variational online-Newton (VON) method proposed in Section . We will modify the NGVI update in ( 41 ). |
The variational lower-bound in ( 25 ) can be re-expressed as [EQUATION] where [MATH] To derive VON, we use the Bonnet’s and Price’s theorems Opper & Archambeau 2009 Rezende et al. 2014 to express the gradients of the expectation of [MATH] with respect to [MATH] and [MATH] in terms of the gradient and Hessian of [MATH] ... |
[EQUATION] where [MATH] and [MATH] denote the gradient and Hessian of [MATH] , respectively. Using these, we can rewrite the gradients of [MATH] required in the NGVI update in ( 41 ) as |
[EQUATION] By substituting these into the NGVI update of ( 41 ) and then approximating the expectation by one Monte-Carlo sample [MATH] , we get the following update: |
[EQUATION] By defining a matrix [MATH] , we get the following: [EQUATION] where [MATH] with [MATH] We refer to this update as the Variational Online-Newton (VON) method because it resembles a regularized version of online Newton’s method where the scaling matrix is estimated online using the Hessians. |
For the mean-field variant, we can use a diagonal Hessian: [EQUATION] where [MATH] denote the element-wise division operation between vectors [MATH] and [MATH] , and we have defined [MATH] , and [MATH] with [MATH] |
By replacing [MATH] and [MATH] by their stochastic estimates, we obtain the VON update shown in ( ) of the main text. D.1 Hessian Approximation Using the Reparameterization Trick |
In this section we briefly discuss an alternative Hessian approximation approach for mean-field VI beside the generalized Gauss-Newton and gradient magnitude which are discussed in the main paper. This approach is based on the reparameterization trick for the expectation of function over a Gaussian distribution. By usi... |
[EQUATION] where [MATH] and [MATH] By defining [MATH] , we can write the VON update using the reparameterization trick Hessian approximation as |
[EQUATION] where [MATH] and [MATH] One major issue with this approximation is that it might have a high variance and [MATH] may be negative. To make sure that [MATH] for all [MATH] , we can use a simple back-tracking method described below. Denote element [MATH] of [MATH] as [MATH] and simplify notation by denoting [MA... |
[EQUATION] where [MATH] is the maximum learning rate and [MATH] is a constant to keep [MATH] strictly within the feasible set (away from the borders). However, this back-tracking method may be computationally expensive and is not trivial to implement within the RMSProp and Adam optimizers. |
Appendix E Derivation of Vadam E.1 Adam as an Adaptive Heavy-Ball Method Consider the following update of Adam (in the pseudocode in the main text, we used [MATH] ): |
[EQUATION] This update can be expressed as the following adaptive version of the Polyak’s heavy ball method as shown in Wilson et al. 2017 |
[EQUATION] where [MATH] are appropriately defined in terms of [MATH] as shown below: [EQUATION] We will now show that, by using natural gradients in the Polyak’s heavy ball, we get an update that is similar to ( 65 ). This allows us to implement our approximated NGVI methods by using Adam. |
E.2 Natural Momentum for Natural Gradient VI We propose the following update: [EQUATION] We can show that ( 67 ) can be written as the following mirror descent extension of ( 27 ), |
[EQUATION] where [MATH] refers to the variational lower-bound defined in Appendix , and [MATH] and [MATH] are two learning rates defined in terms of [MATH] and [MATH] The last term here is a natural momentum term, which is very similar to the momentum term in the heavy-ball methods. For example, ( 16 ) can be written a... |
[EQUATION] In our natural-momentum method, the Euclidean distance is replaced by a KL divergence, which explains the name natural -momentum. |
Equivalence between ( 68 ) to ( 67 ) can be established by directly taking the derivative, setting it to zero, and simplifying: [EQUATION] |
where we use the fact that gradient of the KL divergence with respect to [MATH] is qual to the difference between the natural parameters of the distributions Raskutti & Mukherjee 2015 Khan & Lin 2017 Noting that the gradient with respect to [MATH] is the natural-gradient with respect to [MATH] . Therefore, defining [MA... |
E.3 NGVI with Natural Momentum for Gaussian Approximations We will now derive the update for a Gaussian approximation [MATH] Recalling that the mean parameters of a Gaussian [MATH] are [MATH] and [MATH] Similarly to Appendix , By using the chain rule, we can express the gradient [MATH] in terms of [MATH] and [MATH] as |
[EQUATION] Using the natural parameters of a Gaussian defined as [MATH] and [MATH] , we can rewrite the update ( 72 ) in terms of the update for [MATH] and [MATH] First, the update for [MATH] is obtained by plugging [MATH] into ( 72 ): |
[EQUATION] Now, for [MATH] , we first plugging [MATH] into ( 72 ) and then rearrange the update to express some of the terms as [MATH] |
[EQUATION] where in the final step, we substitute the definition of [MATH] from ( 75 ). To express these updates similar to VON, we make an approximation where we replace the instances of [MATH] by [MATH] in both ( 75 ) and ( 79 ). With this approximation, we get the following: |
[EQUATION] We build upon this update to express it as VON update with momentum. E.4 Variational Online Newton with Natural Momentum |
Now, we derive VON with natural momentum. To do so, we follow the same procedure used to derive the VON update in Section . That is, we first use Bonnet’s and Price’s theorem to express the gradients with respect to [MATH] and [MATH] in terms of the expectations of gradients and Hessian of [MATH] . Then, we substitute ... |
[EQUATION] where [MATH] with [MATH] To get a momentum version of Vprop, we follow a similar method to Section That is, we first employ a mean-field approximation, and then replace the Hessian by the gradient-magnitude approximation. Doing so gives us |
[EQUATION] where [MATH] with [MATH] Finally, we use an unbiased gradient estimate [MATH] , introduce the square-root for the scaling vector in the mean update, and define step-sizes [MATH] and [MATH] The result is a Vprop with momentum update: |
[EQUATION] where [MATH] with [MATH] This is very similar to the update ( 65 ) of Adam expressed in the momentum form. By introducing the bias correction term for [MATH] and [MATH] , we can implement this update by using Adam’s update shown in Fig. The final update of Vadam is shown below, where we highlight the differe... |
[EQUATION] Note that we do not use the same step-size [MATH] for [MATH] and [MATH] , but rather choose the step-sizes according to the Adam update. In the pseudocode, we define [MATH] |
Appendix F The VadaGrad Update By setting [MATH] in ( 23 ), we get the following update: [EQUATION] where [MATH] with [MATH] By replacing the Hessian by a GM approximation, and taking the square-root as in Vprop, we get the following update we call VadaGrad: |
[EQUATION] where [MATH] with [MATH] Appendix G Proof of Theorem 1 Let [MATH] denote the gradient for an individual data point, [MATH] denote the average gradient over a minibatch of size [MATH] and [MATH] denote the average full-batch gradient. Let [MATH] denote a uniform distribution over the data samples [MATH] and [... |
[EQUATION] Using the following two results, [EQUATION] along with Theorem 2.2 of Cochran 1977 which states that [EQUATION] we get the following: |
[EQUATION] where [MATH] Denoting dimension [MATH] of the full-batch gradient by [MATH] , dimension [MATH] of the average gradient over a minibatch by [MATH] and dimension [MATH] of the diagonal of the average GGN, we get the stated result. |
Appendix H Proof to Show That Fixed-Point of Vprop Do Not Change with Square-root We now show that the fixed-points do not change when we take the square root of [MATH] Denote the variational distribution at iteration [MATH] by [MATH] . Assume no stochasticity, i.e., we compute the full-batch gradients and also can exa... |
A fixed point [MATH] of the variational objective satisfies the following: [EQUATION] If we replace the Hessian by the GM approximation, we get the following fixed-point: |
[EQUATION] This fixed-point does not depend on the fact whether we scale by using the square-root or not. However, the iterations do depend on it and the scaling is expected to affect the convergence and also the path that we take to approach the solution. |
Appendix I Details for the Logistic Regression Experiments I.1 Toy Example We used the toy example given in Murphy 2012 (see Fig. 8.6 in the book). The data is [MATH] examples. For all methods, a prior precision of [MATH] and [MATH] MC sample is used. The initial settings of all methods are [MATH] and [MATH] . For ever... |
[EQUATION] Vadam and VOGN are run for 83,333 epochs using a minibatch size of [MATH] (corresponding to 500,000 iterations). For Vadam, [MATH] is set to [MATH] VOGN-1 is run for 8000 epochs with a minibatch size of [MATH] (also corresponding to 500,000 iterations). |
I.2 Real-Data Experiments Datasets for logistic regression are available at . For the Breast Cancer dataset, we use the hyper-parameters found by Khan & Lin 2017 . For USPS, we used the procedure of Khan & Lin 2017 to find the hyperparameter. All details are given in Table . For all datasets we use 20 random splits. |
Performance comparison of MF-Exact, VOGN-1, Vadam: We used 20 random 50-50 splits of the USPS 3vs5 dataset. For all methods, a prior precision of [MATH] is used. MF-Exact and Vadam are run for [MATH] epochs with a minibatch size of [MATH] . The learning rates for both methods are decayed according to ( 104 ) with initi... |
Minibatch experiment comparing VOGN-1 and Vadam: We use the Breast-Cancer dataset with 20 random initializations. For both VOGN-1 and Vadam, a prior precision of [MATH] is used. For VOGN-1, the learning rates are set to [MATH] and [MATH] . It is run for [MATH] epochs using a minibatch size of [MATH] and 1 MC sample. Fo... |
Appendix J Details for the Bayesian Neural Network Experiment J.1 UCI Regression Experiments The 8 datasets together with their sizes [MATH] and number of features [MATH] are listed in Table . For each of the datasets, we use the 20 random train-test splits provided by Gal & Ghahramani 2016 Following earlier work, we u... |
Following earlier work, we use neural networks with one hidden layer and 50 hidden units with ReLU activation functions. All networks were trained for 40 epochs. For the 4 smallest datasets, we use a minibatch size of 32, 10 MC samples for Vadam and 20 MC samples for BBVI. For the 4 larger datasets, we use a minibatch ... |
For BBVI, we optimize the variational objective using the Adam optimizer. For both BBVI and Vadam we use a learning rate of [MATH] and set [MATH] and [MATH] to encourage convergence within 40 epochs. For both BBVI and Vadam, the initial precision of the variational distribution [MATH] was set to 10. |
J.2 VOGN Convergence Experiments We apply BBVI, Vadam, and VOGN to train a neural network with a single-hidden layer of 64 units and ReLU activations on a random train-test split of the Australian-Scale dataset ( [MATH] [MATH] ). For VOGN, we do not use the natural-momentum term. The prior precision [MATH] is set to [M... |
Appendix K Details for the Exploration for Deep Reinforcement Learning Experiment Reinforcement learning (RL) aims to solve the sequential decision making problem where at each discrete time step [MATH] an agent observes a state [MATH] and selects an action [MATH] using a policy [MATH] , i.e., [MATH] The agent then rec... |
A central component of RL algorithms is the Q-function, [MATH] , which denotes the expected return after executing an action [MATH] in a state [MATH] and following the policy [MATH] afterwards. Formally, the Q-function is defined as |
[MATH] The Q-function also satisfies a recursive relation also known as the Bellman equation: [MATH] Using the Q-function and a parameterized policy [MATH] , the goal of reinforcement learning can be simply stated as finding a policy parameter [MATH] which maximizes the expected Q-function |
[EQUATION] In practice, the Q-function is unknown and is commonly approximated by a parameterized function [MATH] with parameter [MATH] learned such that it satisfies the Bellman equation on average: |
[MATH] where [MATH] is a behavior policy used to collect samples and [MATH] is either a copy or a slowly updated value of [MATH] whose [MATH] By using an approximated Q-function, the goal of RL is to find a policy parameter maximizing the expected value of [MATH] |
[EQUATION] In the remainder, we consider the minimization problem [MATH] to be consistent with the variational optimization problem setting in the main text. |
K.1 Stochastic Policy Gradient and Deterministic Policy Gradient The RL objective in ( 106 ) is often minimized by gradient descent. The gradient computation depends on stochasticity of [MATH] For a stochastic policy, [MATH] policy gradient or REINFORCE can be computed using the likelihood ratio trick: |
[EQUATION] For a deterministic policy [MATH] deterministic policy gradient (DPG) Silver et al. 2014 can be computed using the chain-rule: |
[EQUATION] As discussed by Silver et al. 2014 , the deterministic policy gradient is more advantageous than the stochastic counter part due to its lower variance. However, the issue of a deterministic policy is that it does not perform exploration by itself. In practice, exploration is done by injecting a noise to the ... |
K.2 Parameter-based Exploration Policy Gradient Parameter-based exploration policy gradient Rückstieß et al. 2010 relaxes the RL objective in ( 106 ) by assuming that the parameter [MATH] is sampled from a Gaussian distribution [MATH] with a diagonal covariance. Formally, it solves an optimization problem |
[EQUATION] where [MATH] is either the objective function for the stochastic policy in ( 107 ) or the deterministic policy in ( 108 ). In each time step, the agent samples a policy parameter [MATH] and uses it to determine an action This exploration strategy is advantageous since the stochasticity of [MATH] allows the a... |
Notice that ( 109 ) is exactly the variational optimization problem discussed in the main text. As explained in the main text, this problem can be solved by our methods. In the next section, we apply VadaGrad and Vadam to the deep deterministic policy gradient and show that a parameter-exploration strategy induced by o... |
K.3 Parameter-based Exploration Deep Deterministic Policy Gradient via VadaGrad and Vadam While parameter-based exploration strategy can be applied to both stochastic and deterministic policies, it is commonly applied to a deterministic policy. In our experiment, we adopt a variant of deterministic policy gradient call... |
[EQUATION] Gradient descent on ( 110 ) yields an update: [MATH] , where [MATH] is the step-size. DDPG also uses a replay buffer which is a first-in-first-out queue that store past collected samples. In each update iteration, DDPG uniformly draws [MATH] minibatch training samples from the replay buffer to approximate th... |
To apply VadaGrad to solve ( 109 ), in each update iteration we sample [MATH] and then updates the mean and variance using the VadaGrad update in ( 96 ). The deterministic policy gradient [MATH] can be computed using the chain-rule as shown in ( 108 ). The computational complexity of DDPG with VadaGrad is almost identi... |
In VadaGrad, the precision matrix always increases overtime and it guarantees that the policy eventually becomes deterministic. This is beneficial since it is known that there always exists a deterministic optimal policy for MDP. However, this behavior may not be desirable in practice since the policy may become determ... |
We perform experiment using the Half-Cheetah task from the OpenAI gym platform. We compare DDPG with VadaGrad and Vadam against four baseline methods. |
SGD-Plain: the original DDPG without any noise injection optimized by SGD, Adam-Plain: the original DDPG without any noise injection optimized by Adam, |
SGD-Explore: a naive parameter exploration DDPG based on VO optimized by SGD, and Adam-Explore: a naive parameter exploration DDPG based on VO optimized by Adam. |
In SGD-Explore and Adam-Explore, we separately optimizes the mean and variance of the Gaussian distribution: [EQUATION] where [MATH] is the mean step-size and [MATH] is the variance step-size. The gradients of [MATH] are computed by chain-rule and automatic-differentiation. For Adam-Explore, two Adam optimizers with di... |
All methods use the DDPG network architectures as described by Lillicrap et al. 2015 two-layer neural networks with [MATH] and [MATH] ReLU hidden units. The output of the policy network is scaled by a hyperbolic tangent to bound the actions. The minibatch size is [MATH] All methods optimize the Q-network by Adam with s... |
For Adam we use [MATH] . We also use the same value of [MATH] for Vadam. The initial precision for SGD-Explore, Adam-Explore, and VadaGrad is [MATH] |
For Vadam we use [MATH] for the initial second-order moment estimate and add a constant value of [MATH] to the precision matrix [MATH] for sampling (see line 3 of Algorithm ). We do this for two reasons. First, we set [MATH] so that the initial conditions and hyper-parameters of Vadam and Adam-Plain are exactly the sam... |
We perform experiment using the Half-Cheetah task from the OpenAI gym platform Brockman et al. 2016 We measure the performance of each method by computing cumulative rewards along 20 test episodes without exploration. The early learning performance of Vadam, Adam-Plain, and Adam-Explore in Figure shows that Vadam learn... |
Figure shows the learning performance for a longer period of training for all methods. We can see that VadaGrad learns faster than SGD and Adam-based methods initially, but it suffers from a premature convergence and are outperformed by Adam-based methods. In contrast, Vadam does not suffer from the premature convergen... |
1: Initialize: Variational distribution [MATH] with random initial mean and initial precision [MATH] 2: for Time step [MATH] do 3: Sample policy parameter [MATH] |
4: Observe [MATH] , execute [MATH] observe [MATH] and transit to [MATH] . Then add [MATH] to a replay buffer [MATH] 5: Drawn M minibatch samples [MATH] from [MATH] |
6: Update the Q-network weight [MATH] by stochastic gradient descent or Adam: [EQUATION] 7: Compute deterministic policy gradient using the sampled policy parameter: |
[EQUATION] 8: Update the mean [MATH] and variance [MATH] by VadaGrad: [EQUATION] 9: Update target network parameters [MATH] and [MATH] by moving average: |
[EQUATION] 10: end for Algorithm 1 Parameter-based exploration DDPG via VadaGrad 1: Initialize: Initial mean [MATH] , 1st-order moment [MATH] , 2nd-order moment [MATH] , prior [MATH] , constant [MATH] |
2: for Time step [MATH] do 3: Sample policy parameter [MATH] 4: Observe [MATH] , execute [MATH] observe [MATH] and transit to [MATH] . Then add [MATH] to a replay buffer [MATH] |
5: Drawn M minibatch samples [MATH] from [MATH] 6: Update the Q-network weight [MATH] by stochastic gradient descent or Adam: [EQUATION] |
7: Compute deterministic policy gradient using the sampled policy parameter: [EQUATION] 8: Update and correct the bias of the 1st-order moment [MATH] and the 2nd-order moment [MATH] by Vadam: |
[EQUATION] 9: Update the mean [MATH] using the moment estimates by Vadam: [EQUATION] 10: Update target network parameters [MATH] and [MATH] by moving average: |
[EQUATION] 11: end for Algorithm 2 Parameter-based exploration DDPG via Vadam Appendix L Toy Example on Local-Minima Avoidance using Vadam |
Fig. shows an illustration of variational optimization on a two-dimensional objective function. The objective function [MATH] is taken from Fig. 5.2 in Robert & Casella 2005 . Variational optimization is performed by gradually turning off the KL-term for Vadam, thus annealing Vadam towards VadaGrad. This is referred to... |
Appendix M Experiment on Improving “Marginal Value of Adaptive-Gradient Methods” Recently, Wilson et al. 2017 show some examples where adaptive gradient methods, namely Adam and AdaGrad, generalize worse than SGD. We repeated their experiments to see whether weight-perturbation in VadaGrad improves the generalization p... |
We use the following hyper-parameter setting in this experiment. For all methods, we divide the step-size [MATH] by 10 once every [MATH] epochs, as described by Wilson et al. 2017 For VadaGrad, AdaGrad, and Adam, we fixed the value of scaling vector step-size [MATH] and do not decay. For AdaGrad and Adam, the initial v... |
# Source: arxiv 1806.05019 # Title: Explicit solutions for nonlocal NLS: GBDT and algebro-geometric approaches # Sections: all # Downloaded: 2026-03-02T08:54:04.934919+00:00 |
\newsymbol \blackbox 1004 Explicit solutions for nonlocal NLS: GBDT and algebro-geometric approaches Abstract We apply the GBDT version of the Bäcklund-Darboux transformation to the nonlocal NLS (focusing and defocusing cases). The matrix case is included and solutions in the form of rectangular [MATH] matrix functions... |
MSC(2010): 35B06, 35Q55, 14H70, 37K40 Keywords: Nonlocal nonlinear Schrödinger equation, explicit solution, wave function, Bäcklund-Darboux transformation, algebro-geometric solution. |
Introduction Nonlocal nonlinear integrable equations and, in particular, the nonlocal nonlinear Schrödinger equation (nonlocal NLS) have actively been studied during the last years (see the important papers |
and references therein), starting from the seminal article by M.J. Ablowitz and Z.H. Musslimani. The nonlocal NLS is a special case of the coupled NLS: |
[EQUATION] where [MATH] is the [MATH] identity matrix. Indeed, setting in ( 1.5 [EQUATION] we transform the first equation in ( 1.5 ) into the nonlocal matrix NLS: |
[EQUATION] Here [MATH] is an [MATH] matrix function. Although the scalar case of the nonlocal NLS where [MATH] is usually considered in the literature, the matrix case is of interest as well (see, e.g., |
), and we deal in the present paper with this more general situation. We note that under the assumptions ( 1.6 ) the first and second equations in ( 1.5 ) are equivalent, and so ( 1.7 ) is equivalent to ( 1.1 ). Although the cases [MATH] and [MATH] differ in some important aspects, we often formulate the results for th... |
The nonlocal NLS is closely related to the PT -symmetric theory. In this paper, we use some ideas from , where the generalized Bäcklund-Darboux transformation (GBDT) was applied to the linear PT -symmetric Schrödinger equation, in order to apply GBDT to the nonlocal NLS ( 1.7 ). We note that interesting new symmetries ... |
. Bäcklund-Darboux transformations and commutation methods (see, e.g., are well-known tools for explicitly solving integrable equations and spectral and scattering problems. In particular, GBDT (where generalized eigenvalues are [MATH] matrices with an arbitrary Jordan structure) allows to construct wide classes of exp... |
and references cited there). In this note, we apply GBDT to construct a large class of explicit solutions of the nonlocal matrix NLS and corresponding wave functions. The construction of the wave functions is of interest in itself and for possible further applications to spectral and scattering results. |
The more complicated class of algebro-geometric solutions is both interesting and important (see, e.g. for algebro-geometric solutions in the context of several different nonlinear evolution equations). Its elements can still be regarded as explicit solutions, even though their complexity increases due to the underlyin... |
Section contains some necessary preliminary results on the GBDT approach. In Section , we apply GBDT to the nonlocal NLS. Section is dedicated to the construction of explicit solutions of the nonlocal NLS, examples are considered in Section . A nonlocal analog of the important algebro-geometric Theorem 3.11 from |
is presented in Section . The necessary results on algebro-geometric solutions are given in Appendix Preliminaries The zero curvature representation |
[EQUATION] is an important modification of the famous Lax pairs (see and more historical remarks in ). System ( 2.1 ) is the compatibility condition for the auxiliary linear systems |
[EQUATION] This fact is easily proved in one direction and in a more complicated way (see 41 , Ch. 12] and ) in the opposite direction. The coupled NLS ( 1.1 ) admits representation 2.1 ) where [MATH] and [MATH] are matrix polynomials of the first and second orders (with respect to [MATH] ): |
[EQUATION] Here [MATH] and [MATH] have the form ( 1.4 ). From here on in the text we consider [MATH] [MATH] [MATH] and [MATH] as given by ( 2.3 ) and ( 2.4 |
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