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Question: Neural sequence generation is commonly approached by using maximum- likelihood (ML) estimation or reinforcement learning (RL). However, it is known that they have their own shortcomings; ML presents training/testing discrepancy, whereas RL suffers from sample inefficiency. We point out that it is difficult to resolve all of the shortcomings simultaneously because of a tradeoff between ML and RL. In order to counteract these problems, we propose an objective function for sequence generation using α-divergence, which leads to an ML-RL integrated method that exploits better parts of ML and RL. We demonstrate that the proposed objective function generalizes ML and RL objective functions because it includes both as its special cases (ML corresponds to α → 0 and RL to α → 1). We provide a proposition stating that the difference between the RL objective function and the proposed one monotonically decreases with increasing α. Experimental on machine translation tasks show that minimizing the proposed objective function achieves better sequence generation performance than ML-based methods. Neural sequence models have been successfully applied to various types of machine learning tasks, such as neural machine translation;, caption generation (; BID6, conversation , and speech recognition BID8 BID2. Therefore, developing more effective and sophisticated learning algorithms can be beneficial. Popular objective functions for training neural sequence models include the maximum-likelihood (ML) and reinforcement learning (RL) objective functions. However, both have limitations, i.e., training/testing discrepancy and sample inefficiency, respectively. indicated that optimizing the ML objective is not equal to optimizing the evaluation metric. For example, in machine translation, maximizing likelihood is different from optimizing the BLEU score BID19, which is a popular metric for machine translation tasks. In addition, during training, ground-truth tokens are used for the predicting the next token; however, during testing, no ground-truth tokens are available and the tokens predicted by the model are used instead. On the contrary, although the RL-based approach does not suffer from this training/testing discrepancy, it does suffer from sample inefficiency. Samples generated by the model do not necessarily yield high evaluation scores (i.e., rewards), especially in the early stage of training. Consequently, RL-based methods are not self-contained, i.e., they require pre-training via ML-based methods. As discussed in Section 2, since these problems depend on the sampling distributions, it is difficult to resolve them simultaneously. Our solution to these problems is to integrate these two objective functions. We propose a new objective function α-DM (α-divergence minimization) for a neural sequence generation, and we demonstrate that it generalizes ML-and RL-based objective functions, i.e., α-DM can represent both functions as its special cases (α → 0 and α → 1). We also show that, for α ∈, the gradient of the α-DM objective is a combinations of the ML-and RL-based objective gradients. We apply the same optimization strategy as BID18, who useed importance sampling, to optimize this proposed objective function. Consequently, we avoid on-policy RL sampling which suffers from sample inefficiency, and optimize the objective function closer to the desired RL-based objective than the ML-based objective. The experimental for a machine translation task indicate that the proposed α-DM objective outperforms the ML baseline and the reward augmented ML method (RAML; BID18, upon which we build the proposed method. We compare our to those reported by BID3, who proposed an on-policy RL-based method. We also confirm that α-DM can provide a comparable BLEU score without pre-training. The contributions of this paper are summarized as follows.• We propose the α-DM objective function using α-divergence and demonstrate that it can be considered a generalization of the ML-and RL-based objective functions (Section 4).• We prove that the α-DM objective function becomes closer to the desired RL-based objectives as α increases in the sense that the upper bound of the maximum discrepancy between ML-and RL-based objective functions monotonically decreases as α increases.• The of machine translation experiments demonstrate that the proposed α-DM objective outperforms the ML-baseline and RAML (Section 7). In this section, we introduce ML-based and RL-based objective functions and the problems in association with learning neural sequence models using them. We also explain why it is difficult to resolve these problems simulataneously. Maximum-likelihood An ML approach is typically used to train a neural sequence model. Given a context (or input sequence) x ∈ X and a target sequence y = (y 1, . . ., y T) ∈ Y, ML minimizes the negative log-likelihood objective function DISPLAYFORM0 where q(y|x) denotes the true sampling distribution. Here, we assume that x is uniformly sampled from X and omit the distribution of x from Eq. for simplicity. For example, in machine translation, if a corpus contains only a single target sentence y * for each input sentence x, then q(y|x) = δ(y − y *) and the objective becomes L(θ) = − x∈X log p θ (y * |x).ML does not directly optimize the final performance measure; that is, training/testing discrepancy exists. This arises from at leset these two problems:(i) Objective score discrepancy. The reward function is not used while training the model; however, it is the performance measure in the testing (evaluation) phase. For example, in the case of machine translation, the popular evaluation measures such as BLEU or edit rate differ from the negative likelihood function. (ii) Sampling distribution discrepancy. The model is trained with samples from the true sampling distribution q(y|x); however, it is evaluated using samples generated from the learned distribution p θ (y|x).Reinforcement learning In most sequence generation task, the optimization of the final performance measure can be formulated as the minimization of the negative total expected rewards expressed as follows: DISPLAYFORM1 where r(y, y * |x) is a reward function associated with the sequence prediction y, i.e., the BLEU score or the edit rate in machine translation. RL is an approach to solve the above problems. The objective function of RL is L * in Eq., which is a reward-based objective function; thus, there is no objective score discrepancy, thereby resolbing problem (i). Sampling from p θ (y|x) and taking the expectation with p θ (y|x) in Eq. also resolves problem (ii). BID20 and BID3 directly optimized L * using policy gradient methods . A sequence prediction task that selects the next token based on an action trajectory (y 1, . . ., y t−1) can be considered to be an RL problem. Here the next token selection corresponds to the next action selection in RL. In addition, the action trajectory and the context x correspond to the current state in RL.RL can suffer from sample inefficiency; thus, it may not generate samples with high rewards, particularly in the early learning stage. By definition, RL generates training samples from its model distribution. This means that, if model p θ (y|x) has low predictive ability, only a few samples will exist with high rewards.(iii) Sample inefficiency. The RL model may rarely draw samples with high rewards, which hinders to find the true gradient to optimize the objective function. Machine translation suffers from this problem because the action (token) space is vast (typically >10, 000 dimensions) and rewards are sparse, i.e., positive rewards are observed only at the end of a sequence. Therefore, the RL-based approach usually requires good initialization and thus is not self-contained. Previous studies have employed pre-training with ML before performing on-policy RL-based sampling BID20 BID3.Entropy regularized RL To prevent the policy from becoming overly greedy and deterministic, some studies have used the following entropy-regularized version of the policy gradient objective function BID17: DISPLAYFORM2 Reward augmented proposed RAML, which solves problems (i) and (iii) simultaneously. RAML replaces the sampling distribution of ML, i.e., q(y|x) in Eq., with a reward-based distribution q (τ) (y|x) ∝ exp {r(y, y * |x)/τ }. In other words, RAML incorporates the reward information into the ML objective function. The RAML objective function is expressed as follows: DISPLAYFORM3 However, problem (ii) remains. Despite these various attempts, a fundamental technical barrier exists. This barrier prevents solving the three problems using a single method. The barrier originates from a trade-off between sampling distribution discrepancy (ii) and sample inefficiency (iii), because these issues are related to the sampling distribution. Thus, our approach is to control the trade-off of the sampling distributions by combining them. The proposed method utilizes α-divergence DA (p q), which measures the asymmetric distance between two distributions p and q BID0. A prominent feature of α-divergence is that it can behave as D KL (p q) or D KL (q p) depending on the value of α, i.e., D DISPLAYFORM0 where DISPLAYFORM1. Furthermore, α-divergence becomes a Hellinger distance when α equals to 1/2. In this section, we describe the proposed objective function α-DM and its gradient. Furthermore, we demonstrate that it can smoothly bridge both ML-and RL-based objective functions. We define the α-DM objective function as the α-divergence between p θ and q (τ): This DISPLAYFORM0 DISPLAYFORM1 Figure 1 illustrates how the α-DM objective bridges the ML-and RL-based objective functions. Although the objectives L * DISPLAYFORM2, and L (τ) (θ) have the same global minimizer p θ (y|x) = q (τ) (y|x), empirical solutions often differ. To train neural network or other machine learning models via α-divergence minimization, one can use the gradient of α-DM objective function. The gradient of Eq. can be expressed as DISPLAYFORM0 where p DISPLAYFORM1 is a weight that mixes sampling distributions p θ and q (τ). This weight makes it clear that the α-DM objective can be considered as a mixture of ML-and RL-based objective functions. See Appendix A for the derivation of this gradient. It converges to the gradient of entropy regularized RL or RAML by taking α → 1 or α → 0 limits, respectively (up to constant); i.e., DISPLAYFORM2 In Appendix C, we summarize all of the objective functions, gradients, and their connections. In this section, we characterize the difference between α-DM objective function L (α,τ) and the desired RL-based objective function L * (τ) with respect to sup-norm. Our main claim is that, with respect to sup-norm, the discrepancy between L (α,τ) and L * (τ) decreases linearly as α increases to 1. We utilize this analysis to motivate our α-DM objective function with larger α if there are no concerns about the sampling inefficiency. Proposition 1 Assume that p θ has the same finite support S as that of q (τ), and that for any s ∈ S, there exists δ > 0 such that p θ (s) > δ holds. For any α ∈, the following holds. DISPLAYFORM0 whereL (α,τ):= αL (α,τ). Here, C 1, C 2 is universal constants irrelevant to α. The following proposition immediately proves the theorem above. Proposition 2 Assume that probability distribution p has the same finite support S as that of q, and that for any s ∈ S there exists δ > 0 such that p(s) > δ holds. For any α ∈, the following holds. sup DISPLAYFORM1 Here, C = max sup p p log 2 (q/p), sup p q log 2 (q/p).For the proof of the Proposition 1 and Proposition 2, see Appendix B. In this paper, we employed the optimization strategy which is similar to that of RAML. We sample target sentence y for each x from another data augmentation distribution q 0 (y|x), and then estimate the gradient by importance sampling (IS). For example, we add some noise to the ground truth target sentence y * by insertion, substitution, or deletion, and the distribution p 0 (y|x) assigns some probability to each modified target sentence. Given samples from this proposal ditribution p 0 (y|x), we update the parameter using the following IS estimator DISPLAYFORM0 Here, {(x 1, y 1),..., (x N, y N)} are the N samples from the proposal distribution q 0 (y|x), and w i is the importance weight which is proportional to p (α,τ) θ (y i |x i): DISPLAYFORM1 Note that the difference betweene RAML and α-DM is only this importance weight w i. In RAML, w i depends only on q (τ) (y i |x i) but not on p θ (y i |x i). We normalize w i in each minibatch in order to use same hyperparameter (e.g., learning rate) as ML baseline. Thus, this estimator becomes a weighted IS estimator. A weighted IS estimator is not unbiased, yet but it has smaller variance. Also, we found that normalizing q (τ) (y i |x i) and p θ (y i |x i) in each minibatch leads to good . We evaluate the effectiveness of α-DM experimentally using neural machine translation tasks. We compare the BLEU scores of ML, RAML, and the proposed α-DM on the IWSLT'14 GermanEnglish corpus BID5. In order to evaluate the impact of training objective function, we train the same attention-based encoder-decoder model BID16 for each objective function. Furthermore, we use the same hyperparameter (e.g., learning rate, dropout rate, and temperature τ) between all the objective functions. For RAML and α-DM, we employ a data augmentation procedure similar to that of BID18, and thus we generate samples from a data augmentation distribution q 0 (y|x). Note that the difference between RAML and α-DM is only the weight w i of Eq.. The details of data augmentation distribution are described in Section 7.2. BID3. Specifically, we trained attention-based encoder-decoder model with the encoder of a bidirectional LSTM with 256 units and the LSTM decoder with the same number of layers and units. We exponentially decay the learning rate, and the initial learning rate is chosen using grid search to maximize the BLEU performance of ML baseline on development dataset. The important hyperparameter τ of RAML and α-DM is also determined to maximize the BLEU performance of RAML baseline on development dataset. As a , the initial learning rate of 0.5 and τ of 1.0 were used. Our α-DM used the same hyperparameters as ML and RAML including the initial learning rate, τ, and so on. Details about the models and parameters are discussed in Section 7.2. To investigate the impact of hyperparameter α, we train the neural sequence models using α-DM 5 times for each fixed α ∈ {0.0, 0.1, . . ., 0.9}, and then reported the BLEU score of test dataset. Moreover, assuming that the underfitted model prevents the gradient from being stable in the early stage of training, we train the same models with α being linearly annealed from 0.0 to larger values; we increase the value of α by adding 0.03 at each epoch. Here, the beam width k was set to 1 or 10. All BLEU scores and their averages are plotted in FIG1. The show that for both k = 1, 10, the models performance are better than smaller or larger α when α is around 0.5 (α = 0.5). However, for larger fixed α, the performance was worse than RAML and ML baselines. On the other hand, we can see that the annealed versions of α-DM improve the performance of the corresponding fixed versions in relatively larger α. As a , in the annealed scenario, α-DM with wide range of α ∈ improves on the performance consistently. This implies that the underfitted model makes the performance worse. We summarize the average BLEU scores and their standard deviation of ML, RAML, and α-DM with α ∈ {0.3, 0.4, 0.5} in Table. 1. The shows that the BLEU score (k = 10) of our α-DM outperforms ML and RAML baseline. Furthermore, although the ML baseline performances differ between our and those of BID3, the proposed α-DM performance with α = 0.5 without pre-training is comparable with the on-policy RL-based methods BID3. We believe that these come from the fact that α-DM with α > 0 has smaller bias than that of α = 0 (i.e., RAML). We utilized a stochastic gradient descent with a decaying learning rate. The learning rate decays from the initial learning rate to 0.05 with dev-decay , i.e., after training each epoch, we monitored the perplexity for the development set and reduced the learning rate by multiplying it with δ = 0.5 only when the perplexity for the development set does not update the best perplexity. The mini-batch size is 128. We used the dropout with probability 0.3. Gradients are rescaled when the norms exceed 5. In addition, if an unknown token, i.e., a special token representing a word that is not in the vocabulary, is generated in the predicted sentence, it was replaced by the token with the highest attention in the source sentence BID13. We implemented our models using a fork from the PyTorch 1 version of the OpenNMT toolkit BID14. We calculated the BLEU scores with multi-bleu.perl 2 script for both the development and test sets. We obtained augmented data in the same manner as the RAML framework BID18. For each target sentence, some tokens were replaced by other tokens in the vocabulary and we used the negative Hamming distance as reward. We assumed that Hamming distance e for each sentence is less than [m × 0.25], where m is the length of the sentence and [a] denotes the maximum integer which is less than or equal to a ∈ R. Moreover, the Hamming distance for a sample is uniformly selected from 0 to [m × 0.25]. One can also use BLEU or another machine translation metric for this reward. However, we assumed proposal distribution q 0 (y|x) different from that of RAML. We assumed the simplified proposal distribution q 0 (y|x), which is a discrete uniform distribution over [0, m × 0.25]. This in hyperparameter τ used in this experiment being different from that of RAML. We search the τ, which maximize the BLEU score of RAML on the development set. As a , τ = 1.0 was chosen, and α-DM also uses this fixed τ in all the experiments. From the RL literature, reward-based neural sequence model training can be separated into on-policy and off-policy approaches, which differ in the sampling distributions. The proposed α-DM approach can be considered an off-policy approach with importance sampling. Recently, on-policy RL-based approaches for neural sequence predictions have been proposed. BID20 proposed a method that uses the REINFORCE algorithm . Based on Ranzato et al. FORMULA0, BID3 proposed a method that estimates a critic network and uses it to reduce the variance of the estimated gradient. proposed a method that replaces some ground-truth tokens in an output sequence with generated tokens. Yu et al. FORMULA0, BID15 proposed methods based on GAN (generative adversarial net) approaches BID11. Note that on-policy RL-based approaches can directly optimize the evaluation metric. BID10 proposed off-policy gradient methods using importance sampling, and the proposed α-DM off-policy approach utilizes importance sampling to reduce the difference between the objective function and the evaluation measure when α > 0.As mentioned previously, the proposed α-DM can be considered an off-policy RL-based approach in that the sampling distribution differs from the model itself. Thus, the proposed α-DM approach has the same advantages as off-policy RL methods compared to on-policy RL methods, i.e., computational efficiency during training and learning stability. On-policy RL approaches must generate samples during training, and immediately utilize these samples. This property leads to high computational costs during training and if the model falls into a poor local minimum, it is difficult to recover from this failure. On the other hand, by exploiting data augmentation, the proposed α-DM can collect samples before training. Moreover, because the sampling distribution is a stationary distribution independent of the model, one can expect that the learning process of α-DM is more stable than that of on-policy RL approaches. Several other methods that compute rewards before training can be considered off-policy RL-based approaches, e.g., minimum risk training (MRT; BID21, RANDOMER (, and Google neural machine translation (GNMT; .While the proposed approach is a mixture of ML-and RL-based approaches, this attempt is not unique. The sampling distribution of scheduled sampling is also a mixture of ML-and RL-based sampling distributions. However, the sampling distributions of scheduled sampling can differ even in the same sentence, whereas ours are sampled from a stationary distribution. To bridge the ML-and RL-based approaches, BID12 considered the weights of the gradients of the ML-and RL-based approaches by directly comparing both gradients. In contrast, the weights of the proposed α-DM approach are obtained as the of defining the α-divergence objective function. GNMT considered a mixture of ML-and RL-based objective functions by the weighted arithmetic sum of L and L *. Comparing this weighted mean objective function and α-DM's objective function could be an interesting research direction in future. In this study, we have proposed a new objective function as α-divergence minimization for neural sequence model training that unifies ML-and RL-based objective functions. In addition, we proved that the gradient of the objective function is the weighted sum of the gradients of negative loglikelihoods, and that the weights are represented as a mixture of the sampling distributions of the ML-and RL-based objective functions. We demonstrated that the proposed approach outperforms the ML baseline and RAML in the IWSLT'14 machine translation task. In this study, we focus our attention on the neural sequence generation problem, but we expect our framework may be useful to broader area of reinforcement learning. The sample inefficiency is one of major problems in reinforcement learning, and people try to mitigiate this problem by using several type of supervised learning frameworks such as imitation learning or apprenticisip learning. This alternative approaches bring another problem similar to the neural sequence generaton problem that is originated from the fact that the objective function for training is different from the one for testing. Since our framework is general and independent from the task, our approach may be useful to combine these approaches. A GRADIENT OF α-DM OBJECTIVEThe gradient of α-DM can be obtained as follows: DISPLAYFORM0 DISPLAYFORM1 DISPLAYFORM2 where DISPLAYFORM3 In Eq. FORMULA0, we used the so-called log-trick: ∇ θ p θ (y|x) = p θ (y|x)∇ θ log p θ (y|x). Proposition 1 Assume that probability distribution p has the same finite support S as that of q, and that for any s ∈ S there exists δ > 0 such that p(s) > δ holds. For any α ∈ the following holds. DISPLAYFORM0 A (p q) ≤ C(1 − α).Here, C = max sup p p log 2 (q/p), sup p q log 2 (q/p).Proof. By Taylor's theorem, there is an α ∈ (α, 1) such that DISPLAYFORM1 DISPLAYFORM2 where C 1 = τ max sup θ x∈X y∈Y p θ log 2 (q (τ) /p θ ), sup θ x∈X y∈Y q (τ) log 2 (q (τ) /p θ ) and C 2 = |Z(τ)|. In this section, we summarize the objective functions of• ML (Maximum Likelihood),• RL (Reinforcement Learning),• RAML (Reward Augmented Maximum Likelihood; BID18,• EnRL (Entropy regularized Reinforcement Learning), and• α-DM (α-Divergence Minimization Training).Objectives. The objective functions of ML, RL, RAML, EnRL, and α-DM are as follows DISPLAYFORM0 DISPLAYFORM1 DISPLAYFORM2 DISPLAYFORM3 DISPLAYFORM4 where q (τ) (y|x) ∝ exp {r(y, y * |x)/τ }. Typically, q(y|x) = δ(y, y * |x) where y * is the target with the highest reward. We can rewrite some of these functions using KL or α-divergences: DISPLAYFORM5 DISPLAYFORM6 DISPLAYFORM7 A (p θ q (τ) ).In the limits, there are the following connections between the objectives. DISPLAYFORM8 DISPLAYFORM9 DISPLAYFORM10 DISPLAYFORM11

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The provided query could be "Tailoring predictions from sequence models (such as LDSs and RNNs) via an explicit latent code." and the positive "Dynamical system models (including RNNs) often lack the ability to adapt the sequence generation or prediction to a given context, limiting their real-world application. In this paper we show that hierarchical multi-task dynamical systems (MTDSs) provide direct user control over sequence generation, via use of a latent code z that specifies the customization to the individual data sequence. This enables style transfer, interpolation and morphing within generated sequences. We show the MTDS can improve predictions via latent code interpolation, and avoid the long-term performance degradation of standard RNN approaches. Time series data often arise as a related'family' of sequences, where certain characteristic differences exist between the sequences in a dataset. Examples include the style of handwritten text , the response of a patient to an anaesthetic , or the style of locomotion in motion capture (mocap) data . In this paper, we will consider how such variation may be modelled, and effectively controlled by an end user. Such related data is often pooled to train a single dynamical system, despite the internal variation. For a simple model, such as a linear dynamical system (LDS), this will in learning only an average effect. In contrast, a recurrent neural network (RNN) may model this variation, but in an implicit and opaque manner. Such a'black-box' approach prohibits end-user control, and may suffer from mode drift, such as in , where a generated mocap sequence performs an unprompted transition from walking to drinking. Some of these problems may be alleviated by appending'context labels' to the inputs (see e.g. , §10.2.4) which describe the required customization. However, such labels are often unavailable, and the approach may fail to model the variation adequately even when they are. To move beyond these approaches, we consider latent variable models, where a latent variable z characterizes each sequence. This may be seen as a form of multi-task learning (MTL, see), from which we derive the name multi-task dynamical system (MTDS), with each sequence treated as a task. A straightforward approach is to append the latent z to the inputs of the model, similarly to the'context label' approach, thereby providing customization of the various bias (or offset) parameters of the model. A number of examples of this have been proposed recently, e.g. in and Miladinović et al.. Nevertheless, this'bias customization' has limited expressiveness and is often unsuitable for customizing simple models. In this paper we investigate a more powerful form of customization which modulates all the system and emission parameters. In this approach, the parameters of each task are constrained to lie on a learned low dimensional manifold, indexed by the latent z. Our experiments show that this approach in improved performance and/or greater data efficiency than existing approaches, as well as greater robustness to unfamiliar test inputs. Further, varying z can generate a continuum of models, allowing interpolation between sequence predictions (see Figure 1b for an example), and potentially morphing of sequence characteristics over time. Contributions In this paper we propose the MTDS, which goes beyond existing work by allowing full adaptation of all parameters of general dynamical systems via use of a learned nonlinear manifold. We show how the approach may be applied to various popular models, and provide general purpose...... learning and inference algorithms. Our experimental studies use synthetic data (sum of two damped harmonic oscillators) and real-world human locomotion mocap data. We illuminate various properties of the MTDS formulation in our experiments, such as data efficiency, user control, and robustness to dataset shift, and show how these go beyond existing approaches to time series modelling. We finally utilize the increased user control in the context of mocap data to demonstrate style morphing. To this end, we introduce the model in Section 2, giving examples and discussing the particular challenges in learning and inference. We discuss the relation to existing work in Section 3. Experimental setup and are given in Section 4 with a in Section 5. Consider a collection of input-output sequences D = {Y Ti}, i = 1,..., N, where T i denotes the length of sequence i. Each sequence i is described by a different dynamical system, whose parameter θ (i) depends on the hierarchical latent variable z (i) ∈ Z: for t = 1,..., T i. The state variables X (i) = {x Ti}, x t ∈ X follow the latent dynamics starting from x 0:= 0 (other choices of initial state are possible). See Figure 1a for a graphical model. In this paper we assume Z = R k which the vector-valued function h φ (·) transforms to conformable model parameters θ ∈ R d, d k. Note that h φ may keep some dimensions of θ constant with respect to z. We call this a Multi-Task Dynamical System, going beyond the usage in. In order to make the framework more concrete we will describe two general choices of the base model. In what follows we will write each parameter with a subscript z to denote dependence on z (e.g. A z := A(z)) to reduce notational clutter. The choice of p(z) and h φ will depend on the application, but a fairly general choice is a deep latent Gaussian model . See section A.1.1 in the supplementary material for further discussion. For a given z, a multi-task linear dynamical system (MTLDS) can be described by: w t ∼ N (0, R z), t ∼ N (0, S z), with θ z = {A z, B z, b z, C z, D z, d z, R z, S z} = h φ (z). The parameterization of θ z must satisfy the constraints of positive definite R z and S z and stable A z (i.e. A z 2 ≤ 1) for all z, hence projection methods such as in are not applicable. We choose an alternative formulation of the LDS, replacing the latent dynamics in eq. by: where Σ z is a diagonal matrix and Q z orthogonal with no loss of generality (proof in supp. mat.). Since Σ z Q z ≤ Σ z Q z = Σ z, stability can be enforced e.g. by Σ = diag{tanh (υ)} for some vector υ. For more details see section A.1.2 in the supplementary material. Due to the nonlinearity of an RNN, enforcing stability of A z is not strictly required (see e.g. , §4.4), although bounding the spectral radius may be useful for learning (e.g.). The dynamics of a multi-task RNN (MT-RNN) are described by: Combined with the emission model of eq., we have If long-term dependencies are important we may consider an orthogonal transition matrix (parameterized as for the MTLDS) to create a multi-task version of the Orthogonal RNN . The parameters φ of an MTDS can be learned from a dataset The first term in the integrand,. For clarity of exposition we only consider models with deterministic state, which extends to the MTRNN and MTLDS above (in the case w t = 0 for all t). This also avoids the interaction effect with the choice of approximate marginalization over X. Equation also cannot be computed in closed form in general, and so we resort to approximate learning. A natural choice is via the ELBO (see an alternative MCO approach for unsupervised tasks in Section A.1.3, supp. mat.). We write the ELBO of eq. as: where D KL is the Kullback-Leibler divergence and q λ (z|Y, U) an approximate posterior for z. We can now optimize a lower bound of the marginal likelihood via arg max φ,λ where low variance unbiased gradients of eq. are available via reparameterization , where µ λ, s λ are inference networks (e.g. Fabius & van). It can be difficult to learn a sensible latent representation if the base model is a powerful RNN. When each output can be identified unambiguously via the inputs preceding it, a larger ELBO can be obtained by the RNN learning the relationship without using the latent variable (see e.g.). A useful heuristic for avoiding such optima is KL annealing (e.g.). In our experiments we perform an initial optimization without the KL penalty (second term in eq. 9), initializing s λ (Y, U) to a small constant value. For an unseen test sequence {Y, U}, the posterior predictive distribution is p(y t+1:T | y 1:t, u 1:T) = Z p(y t+1:T | u 1:T, z) p(z | y 1:t, u 1:t) dz, usually estimated via Monte Carlo. The key quantity is the posterior over z, which may be approximated by the inference networks µ λ, s λ. However, for novel test sequences, the inference networks may perform poorly and standard approximate inference techniques may be preferred. For further discussion and a description of our inference approach, see sections A.1.4, A.1.5 in the supp. mat. We note that z may not require inference, for instance by using the posterior of a sequence in the the training set. This may be useful for artistic control, style transfer, embedding domain knowledge or overriding misleading observations. We also note that the latent code can be varied during the state rollout to simulate a task which varies over time. A number of dynamical models following the'bias customization' approach have been proposed recently. Miladinović et al. and propose models where the biases of an LSTM cell depend on a (hierarchical) latent variable. propose a dynamical system where the latent dynamics are concatenated with a time-constant latent variable. In contrast, our MTDS model performs full parameter customization, and this on both the dynamics and emission distributions. A number of other proposals may be considered specialized applications of the MTDS. use a small deterministic nonlinear dynamical system for the base model whose parameters depend on z using a nonlinear factor analysis structure. use a small RNN as the base model, where the transition matrix depends on z via multilinear decomposition. use a small stochastic nonlinear dynamical system with h φ a set of parameter vectors chosen discretely (or in convex combination) via z. Controlling and customizing sequence prediction has received much attention in the case of video data. As in the MTDS, these approaches learn features that are constant (or slowly varying) within a subsequence. and propose methods for disentangling time-varying and static features, but do not provide a useful density over the latter, nor an obvious way to customize the underlying dynamics. use a GAN architecture where z factorizes into content and motion components. force a parts-based decomposition of the scene before inferring the latent content z. However, as before, the dynamic evolution cannot be easily customized with these methods. Hierarchical approaches for dynamical systems with time-varying parameters are proposed in (corresponding to non-stationary assumptions) and in (for the purposes of local LDS approximation). These models, like the MTDS can adapt all the parameters, but are linear and correspond to single task problems. predict the parameters of simple time-varying LDS models directly via an RNN. While this is a multi-task problem, it is assumed that all necessary variation can be inferred from the inputs U. Multi-task GPs are commonly used for sequence prediction. Examples include those in; Titsias & Lázaro-; Álvarez et al.;. MTGPs however can only be linear combinations of (a small number of) latent functions, further, predictions depend critically upon often unknown mean functions, and inputs are not easily integrated. Note that an MTDS with no inputs, an LDS base model, a linear-Gaussian prior over the emission parameters We investigate the performance of the MTDS on two datasets. The first experiment investigates the performance of the MTDS on synthetic data generated by linear superposition of damped harmonic oscillation (DHO). The second experiment considers real-world mocap data for human locomotion. Data The generative model for J oscillators with constant amplitudes γ and variable frequency and decay factors,.., 80 for tasks i = 1, 2,..., N. The emission noise is distributed iid as Model We model the DHO data using an MTLDS with deterministic state X = R 4 and a k = 4 latent variable z. All LDS parameters were adapted via the latent z except D:= 0 and the emission variance s 2, which was learned. For optimization, we use the MCO algorithm of section A.1.3. This can obtain a tighter bound than the ELBO, and is useful to investigate convergence to the true model over increasing N. We contrast this with a bias customization approach (e.g. Miladinović et al., 2019), implemented similarly, but such that only the parameter b (eq. 6) depends on z. We also train a Pooled LDS, which is the standard approach, using the same parameters for all tasks, and a single-task (STL) LDS which is learned from scratch using Bayesian inference over all parameters for each task. The Pooled-LDS was initialized using spectral methods (see Van Evaluation We assess how quickly and effectively the models can adapt to novel test sequences with a training set size of N = 2 1, 2 2, . . ., 2 7 . (The STL approach effectively uses N = 0.) The test set comprises 20 additional sequences drawn from the generating distribution. For an initial subsequence y 1:t, we estimate the predictive posterior p(y t+1:T |y 1:t) for various t and assess the predictions via root mean squared error (RMSE) and negative log likelihood (NLL). For MTL we use the Monte Carlo inference method described in supp. mat. A.1.5 and for STL we use Hamiltonian Monte Carlo . Each experiment is repeated 10 times to estimate sampling variance. The , shown in Table 1 and supp. mat. section A.2.2, show substantial advantage of using the MTLDS ('MT Full') over single-task or pooled approaches. The MTLDS consistently outperforms the Pooled-LDS for all training sizes N ≥ 4. Merely performing bias customization ('MT Bias') is insufficient to perform much better than a pooled approach. An example of MTLDS test time prediction is shown in Figure 2, with Figures 2c and 2d demonstrating effective generalization from the N = 4 training examples (Figure 2a). Even after 40 observations, the STL approach (which is capable of fitting each sequence exactly) does not significantly outperform the N = 4 MTLDS. Furthermore, the runtime was approx. 1000 times longer since STL inference is higher dimensional and poorly conditioned, and requires a more expensive algorithm. Note that with a larger training set size of N = 128, the MLTDS approaches the likelihood of the true model (Figure 7, supp. mat.). Data The dataset consists of 31 sequences from (ca. 2000 frames average at 30fps) in 8 styles: angry, childlike, depressed, neutral, old, proud, sexy, strutting. In this case the family of possible sequences corresponds to differing walking styles. Each observation represents a 21-joint skeleton in a Lagrangian frame, y t ∈ R 64 where the root movement is represented by a smoothed component and its remainder. we represent joints by their spatial position rather than their rotation. We also provide inputs that an animator may wish to control: the root trajectory over the next second, the gait cycle and a boolean value determining whether the skeleton turns around the inside or outside of a corner. See section A.3.1 in the supplementary materials for more details. Model We use a recurrent 2-layer base model where the first hidden layer is a 1024 unit GRU and the second hidden layer is a 128 unit standard RNN, follwed by a linear decoding layer. The first-layer GRU does not vary with z, i.e. it learns a shared representation of the input sequence across all i. Explicitly, omitting index i, the model for a given z is: for t = 1,..., T. The parameters are θ = {ψ 1, ψ 2, H, C, d} where ψ 1 and H are constant wrt. z. The matrix H ∈ R ×1024 (< 1024) induces a bottleneck between layers, forcing z to explain more of the variance. For our experiments, a small can be used (we use = 24). The first layer GRU uses 1024 units since it was observed experimentally to produce smoother animations than smaller networks. The second layer does not use a gated architecture, as gates appear to learn style inference more easily, and in less use of z. For learning, each sequence was broken into overlapping segments of length 64 (approx. two second intervals), which allows z to vary across a sequence. We learn the model using an open-loop objective, i.e. the y t are not appended to the inputs. This forces the model to recover from its mistakes as in , although unlike these approaches, we do not append predictions to the inputs either. Our rationale is that the state captures the same information as the predictions, and while previous approaches required observations y 1:τ as inputs to seed the state, we can use the latent z. The model was optimized using the variational procedure in section 2.2, where a slower learning rate (by a factor of 10-50) for the first layer parameters (i.e. ψ 1, H) usually ed in a more descriptive z. We also found that standard variational inference for each z (i) worked better in general than using amortized inference. For comparison, we implement a bias customization model ('MTBias') via a deterministic state version of Miladinović et al., 2019, which follows eqs.- but only the RNN bias in eq. is a function of z. We also implement a 1-layer and 2-layer GRU without the multi-task apparatus, which serves both as an ablation test and a competitor model on the new dataset. Style inference is performed with the same network given an initial seed sequence y 1:τ. We train these in closed-loop (i.e. traditional next step 'teacher forcing' criterion) and open-loop settings. For baselines, we use constant predictions of (i) the training set mean and (ii) the last observed frame of the seed sequence ('zero-velocity' prediction). We test the data efficiency of the MTDS by training the models on subsets of the original dataset. Besides the models described above, 8'single-task' versions of the GRU models are trained which only see data for a single style. We use six training sets of approximate size 2 8, 2 9, 2 10, 2 11, 2 12, 2 13 frames per style, where sampling is stratified carefully across all styles, and major variations thereof. For all experiments, the model fit (MSE) is calculated from the same 32 held out sequences (each of length 64). The are shown in Figure 3a. As expected, the MTDS, MTBias and Pooled models obtain'multi-task' gains over STL approaches for small datasets. However, the MTDS demonstrates much greater data efficiency, achieving close to the minimum error with only 7% of the dataset. The MTBias model requires more than twice this amount to obtain the same performance, and the Pooled model requires more than four times this amount. More details, as with all mocap experiments, can be found in supp. mat. section A.2.3. We investigate how well the MTDS can generalize to novel sequence styles via use of a leave-one-out (LOO) setup, similar to transfer learning. For each test style, a model is trained on the other 7 styles in the training set, and hence encounters novel sequence characteristics at test time. We average the test error over the LOO folds as well as 32 different starting locations on each test sequence. The are given in Figure 3b. We see that while the competitor (pooled) models perform well initially, they usually degrade quickly (worse for closed-loop models). In contrast, the multi-task models finds a better customization which evidences no obvious worsening over the predictive interval. Unlike pooled-RNNs, the MTDS and MTBias models can firstly perform correct inference of their customization, and secondly can'remember' it over long intervals. We note that all models struggle to customize the arms effectively, since their test motions are often entirely novel. Customization to the legs and trunk is easier since less extrapolation is required (see animation videos linked in section A.4.1). We investigate the control available in the latent z by performing style transfer. For various inputs U (s1) from each source style s 1, we generate predictions from the model using target style s 2, encoded by z (s2). We use a classifier with multinomial outputs, trained on the 8 styles of the training set, to test whether the target style s 2 can be recognized from the data generated by the MTDS. Figure 3c gives the classifier'probability' for each target style s 2, averaged over all the inputs {U (s1): s 1 = s 2 }. Successful style transfer should in a the classifier assigning a high probability to the target style. These suggest that the prediction style can be well controlled by z (s2) in the case of the full MTDS, but the MTBias demonstrates reduced control for some (source, target) pairs. See the videos linked in section A.4.1 for examples, and sec. A.2.3 for more details. Qualitative investigation Qualitatively, the MTDS appears to learn a sensible manifold of walking styles, which we assess through visualization of the latent space. A k = 2 latent embedding can be seen in Figure 4 where the z (i) for each training segment i is coloured by the true style label. Some example motions are plotted in the figure. The MTDS embedding broadly respects the style label, but learns a more nuanced representation, splitting some labels into multiple clusters and coalescing others. These appear broadly valid, e.g. the'proud' style contains both marching and arm-waving, with the latter similar to an arm-waving motion in the'childlike' style. This highlights the limitation of relying on task labels. Visualizations such as Fig. 1b indicate that smooth style interpolation is available via interpolation in latent space. We take advantage of this in the animations (linked from sec. A.4.1) by morphing styles dynamically. In this work we have shown how to extend dynamical systems with a general-purpose hierarchical structure for multi-task learning. Our MTDS framework performs customization at the level of all parameters, not just the biases, and adapts all parameters for general classes of dynamical systems. We have seen that the latent code can learn a fine-grained embedding of sequence variation and can be used to modulate predictions. Clearly good predictive performance for sequences requires task inference, whether implicit or explicit. There are three advantages of making this inference explicit. Firstly, it enhances control over predictions. This might be used by animators to control the style of predictions for mocap models, or to express domain knowledge, such as ensuring certain sequences evolve similarly. Secondly, it can improve generalization from small datasets since task interpolation is available out-of-the-box. Thirdly, it can be more robust against changes in distribution at test time than a pooled model: is a unit Gaussian p(z) = N (0, I). This choice allows simple sampling schemes, and straight-forward posterior approximations. It is also a useful choice for interpolation, since it allows continuous deformation of its outputs. An alternative choice might be a uniform distribution over a compact set, however posterior approximation is more challenging, see Svénsen for one approach. Sensible default choices for h φ include affine operators and multilayer perceptrons (MLPs). However, when the parameter space R d is large, it may be infeasible to predict d outputs from an MLP. Consider an RNN with 100k parameters. If an MLP has m L−1 = 300 units in the final hidden layer, the expansion to the RNN parameters in the final layer will require 30×10 6 parameters alone. A practical approach is to use a low rank matrix for this transformation, equivalent to adding an extra linear layer of size m L where we must have m L m L−1 to reduce the parameterization sufficiently. Since we will typically need m L to be O, we are restricting the parameter manifold of θ to lie in a low dimensional subspace. Since MLP approaches with a large base model will then usually have a restricted final layer, are there any advantages over a simple linear-Gaussian model for the prior p(z) and h φ? There may indeed be many situations where this simpler model is reasonable. However, we note some advantages of the MLP approach: 1. The MLP parameterization can shift the density in parameter space to more appropriate regions via nonlinear transformation. 2. A linear space of recurrent model parameters can yield highly non-linear changes even to simple dynamical systems (see e.g. the bifurcations in §8 of). We speculate it might be advantageous to curve the manifold to avoid such phenomena. 3. More expressive choices may help utilization of the latent space (e.g.). This may in fact motivate moving beyond a simple MLP for the h φ. The matrices A, B, R, S of the MTLDS can benefit from specific parameterizations, which we will discuss in turn. Degeneracy of LDS. It will be useful to begin with the well-known over-parameterization of linear dynamical systems. The hidden dynamics of a LDS can be transformed by any invertible matrix G while retaining the same distribution over the emissions Y. This follows essentially because the basis used to represent X is arbitrary. The distribution over Y is unchanged under the following parameter transformations: Parameterization of A. The stability constraint, is equivalent to ensuring that the singular values of A lie within the unit hypercube (since singular values are non-negative). Let A = U ΣV T be the singular value decomposition (SVD) of A. Now we have from the previous that if an LDS has latent dynamics with transition parameter A, we may replace the dynamics under the similarity transform G −1 AG. Choose G = U, i.e. the left singular values of A, and hence A = ΣV T U =: ΣQ for some orthogonal matrix Q. This follows from the closure of the orthogonal group under multiplication, which is easily verified. Note that in choosing this transformation, no additional constraints are placed on the other parameters in the LDS. Orthogonal matrices can be parameterized in a number of ways (see e.g.). A straight-forward choice is the Cayley transform.: "if Q is an orthogonal matrix that does not have the eigenvalue -1, then it may be written in Cayley's form: where S is skew-symmetric". In order to permit negative eigenvalues, we can pre-multiply by a diagonal matrix E with elements in {+1, −1}. Since we then have A = ΣEQ, E can be absorbed into Σ, and so the stability constraint can be satisfied with the parameterization A = ΣQ where Σ is a diagonal matrix with elements in [−1, +1] and Q is a Cayley-transform of a skew-symmetric matrix. This follows from the overparameterization of the LDS, and we emphasise that the system equations and are not equivalent, but any LDS distribution over Y can be written with latent dynamics of the form. Choose G = κ −1 I in eq.. It may be observed that the scale κ of the latent system can be chosen arbitrarily without affecting A. We wish to avoid such degeneracies in a hierachical model, since we may otherwise waste statistical strength and computation on learning equivalent representations. We can remove this by fixing the scale of B. An indirect but straightforward approach is to upper bound the magnitude of each element of B. ForB predicted by h φ (z) we might choose the transformation B = tanh(B) where tanh acts element-wise. If a sparse B is desired, one can use an over-parameterization of two matricesB 1,B 2, and choose B = σ(B 1) • tanh(B 2), where • is element-wise multiplication, and σ a logistic sigmoid. The former parameterization is unlikely to find a sparse representation since the gradient of tanh is greatest at 0. Parameterization of R, S. The covariance matrices R, S must be in the positive definite cone. Where a diagonal covariance will suffice, any parameterization for enforcing positivity can be used, such as exponentiation, squaring or softplus. A number of parameterizations are available for full covariance matrices (see). A simple choice is to decompose the matrix, say R = LL T, where L is a lower triangular Cholseky factor. As before, it is useful to enforce uniqueness, which can be done by ensuring the diagonal is positive. We provide an alternative learning algorithm to the VB approach in section 2.2 which obtains a tighter lower bound. This was important for the DHO experiments in order to monitor convergence to the true model. The below is perhaps a novel approach for learning in unsupervised cases (i.e. where U = ∅), but cannot be performed efficiently for supervised problems without modification. Monte Carlo Objectives construct a lower bound for marginal likelihoods via a transformation of an appropriate Monte Carlo estimator. Specifically we consider the logarithmic transformation of: m = 1,..., M; an importance sampling estimator for p(Y). Using Jensen's inequality, we show that the following is a lower bound on the log marginal likelihood: where p(z 1:M):= p(z 1)...p(z M). The tightness of the bound can be increased by increasing the number of samples M . Assuming p(z) has been re-parameterized to be parameter-free, we can easily calculate the gradient (if not, see). By exchanging integration and differentiation, we can calculate the gradient as: Note that eq. is an importance sampled version of the Fisher identity. We might expect this estimator to suffer from high variance, since the prior is a poor proposal for the posterior. However, the prior should not be a poor proposal for the aggregate posterior, i.e. ). In fact, importance sampling from the prior may serve as a useful bias in this case, attracting the posterior distributions which have a large D KL (p(z | Y (i) )||p(z)) towards the prior. Our observation is that sampling from the prior can be amortized over each sequence Y (i), i = 1,..., N. Specifically, for each particle z m, the dynamics, can be run forward once to calculatê Y m, from which the likelihood Y (i), for all tasks i = 1,..., N can be calculated inexpensively. The amortized cost of taking M samples (e.g. M ∈ O(10 3)) now becomes M/N, which may be relatively small. We can also take advantage of low-discrepancy random variates such as Sobol sequences to reduce variance. We propose that each sequence i resamples a small number M rsmp ≤ 5 of particles from the importance weights for each i to reduce the cost of backpropagation (a similar resampling scheme is suggested in). See Algorithm 1. In the supervised case (i.e. where each observation Y (i) has a different input U (i) ), running the dynamics forward from a particle z m can no longer be amortized over all {Y (i) } since the prediction We can therefore only amortize the parameter generation θ = h φ (z), which is often less expensive than running the dynamics forward. For this reason Algorithm 1 is primarily restricted to unsupervised problems. A hybrid approach would essentially in the importance weighted autoencoder (IWAE) of. Inference at test time can be performed by any number of variational or Monte Carlo approaches. As in the main text, our focus here is on deterministic state dynamical systems. For stochastic state models, additional reasoning similar to Miladinović et al. will be required. A gold standard of inference over z may be the No U-Turn Sampler (NUTS) of (a form of Hamiltonian Monte Carlo), provided k is not too large and efficiency is not a concern. However, given the sequential nature of the model, it is natural to consider exploiting the posterior at time t for calculating the posterior at time t + 1. Bayes' rule suggests an update of the end end Optimize(optimizer, φ, g); end end following form: following the conditional independence assumptions of the MTDS. This update (in principle) incorporates the information learned at time t in an optimal way, and further suggests a constant time update wrt t. However, evaluation of p(y t+1 | u 1:t+1, h φ (z)) usually scales linearly with t, since the state x t+1 must be calculated recursively from x 0 given z and u 1:t+1. Nevertheless, sequential incorporation of previous information will perform a kind of annealing which reduces the difficulty, and hopefully the runtime of inference at each stage. We first provide some of the difficulties of such an approach, looking first at Monte Carlo (MC) methods. Naïve application of Sequential Monte Carlo (SMC) will in severe particle depletion over time. To see this, let the posterior after time t be p(z | y 1:t, u 1:t) = 1 M M m=1 w m δ(z− z m). Then the updated posterior at time t + 1 will be:, simply a re-weighting of existing particles. Over time, the number of particles with significant weights w m will substantially reduce. But since the model is static with respect to z (see), there is no dynamic process to'jitter' the {z m} as in a typical particle filter, and hence a resampling step cannot improve diversity. discusses two related solutions: firstly using'rejuvenation steps' (cf.) which applies a Markov transition kernel to each particle. The downside to this approach is the requirement to run until convergence; and the diagnosis thereof, which can in substantial extra computation. One might instead sample from a fixed proposal distribution (accepting a move with the usual Metropolis-Hastings probability) for which convergence is more easily monitored. A Sequential Monte Carlo sampler approach may be preferred, which permits local moves, and can reduce sample impoverishment via resampling (similar to SMC). However, the approach requires careful choices of both forward and backward Markov kernels which substantially reduces its ease of use. A well-known variational approach to problems with the structure of eq. is assumed density filtering (ADF, see e.g.). For each t, ADF performs the Bayesian update and the projects the posterior into a parametric family Q. The projection is done with respect to the reverse KL Divergence, i.e. q t+1 = arg min q∈Q D KL p(z | y 1:t+1, u 1:t+1) || q. Intuitively, the projection finds an'outer approximation' of the true posterior, avoiding the'mode seeking' behaviour of the forward KL, which is particularly problematic if it attaches to the wrong mode. Clearly the performance of ADF depends crucially on the choice of Q. Unfortunately, where Q is expressive enough to capture a good approximation, the optimization problem will usually be challenging, and must resort to stochastic gradient approaches, ing in an expensive inner loop. Furthermore, when the changes from q t to q t+1 are relatively small, the gradient signal will be weak, ing perhaps in misdiagnosed convergence and hence accumulation of error over increasing t. A recent suggestion of is to improve efficiency via re-use of previous (stale) gradient evaluations. Standard variance reduction techniques may also be considered to improve convergence in the inner loop. In our experiments, we found sampling approaches faster and more reliable for each update, as well as providing diagnostic information, and so we eschew variational approaches. (Our experiments used a fairly small k (≤ 10); variational approaches may be preferred in higher dimensional problems.) Specifically we use iterated importance sampling (IS) to update the posterior at each t. The key quantity for IS is the proposal distribution q prop: we need a proposal that is well-matched to the target distribution. Our observation is that the natural annealing properties of the filtering distributions (eq. 23) allow a slow and reliable adaptation of q prop. In order to capture complex multimodal posteriors, we parameterize q prop by a mixture of Gaussians (MoG). For each t, the proposal distribution is improved over N AIS iterations using adaptive importance sampling (AdaIS), described for mixture models in Cappé et al.. We briefly review the methodology for a target distribution p *. Let the AdaIS procedure at the nth iteration use the proposal: α j ∈ R + s.t. For our experiments, this approach worked robustly and efficiently, and appears superior to the alternatives discussed. Unlike SMC, we obtain a q prop which is a good parameteric approximation of the true posterior. We therefore avoid the sample impoverishment problem discussed above (eq. 25). Due to the small number of iterations of AdaIS required (usually ≤ 5 for our problems), it is substantially faster than MCMC moves, and since stochastic gradients are avoided, convergence is much faster than variational approaches. The scheme benefits from the observed fast initial convergence rates of the EM algorithm (see e.g.), particularly since early stopping can be used for the initial iterates. In practice, one may not wish to calculate a posterior at every t, but instead intervals of length τ. In our DHO experiments (k = 4) we use τ = 5, and usually have ESS > 0.6M after n = 4 inner iterations, with total computation per q t requiring 250-300ms on a laptop. We observe in our experiments that posteriors are often multimodal for t ≤ 20 and sometimes beyond, motivating the MoG parameterization. In these experiments, the MoG appears to capture the salient characteristics of the target distribution well. Note as in section A.1.3, Sobol or other low-discrepancy sequences may be used to reduce sampling variance from q prop. Under review as a conference paper at ICLR 2020 for t = 1,..., T, z ∈ R 4, x t ∈ R 4, suppressing task index i for clarity. Define x 0:= 0, and for all tasks u = [1, 0, 0, 0, . . .]. A is parameterized as discussed in section A.1.2 using a product of a diagonal and orthogonal matrix ΣQ. The diagonal of Σ is constrained to lie in [−1, +1] using the tanh function, and Q is parameterized by the Cayley transform of a skew symmetric matrix S. Using an upper triangular matrix Γ, we have S = Γ − Γ T, and Q = (I − S)(I + S) −1. We parameterize B via the product of logistic sigmoid and tanh functions as in section A.1.2 in order to learn a sparse parameterization. C is unconstrained, and the parameter s is optimized as a constant wrt. z. The STLDS is parameterized in the same way. The prior p(z) is a unit Gaussian distribution, and h φ is a 2 hidden-layer neural network. We use a fixed feature extractor T in the first layer in order to help encode a rectangular support within a spherically symmetric distribution. The second layer is a fully-connected 300 unit layer with sigmoid activations. Learning. The output of an MTLDS is very sensitive to the parameter A, and care must be taken to avoid divergence during optimization. The diagonal-orthogonal parameterization greatly helped to stabilize the optimization over a more naïve orthogonal-diagonal-orthogonal SVD parameterization. We also reduced the learning rate by a factor of 10 for A. It proved useful to artificially elevate the estimate of s during training using a prior log s ∼ N (−1.5, 0.05) (derived from preliminary experiments) since the MTDS can otherwise overfit small datasets (see also discussion in §3, Svénsen, 1998), with associated instability in optimization. The learning rate schedule is given in Table 2, for which the prior over log s ∼ N (m, 0.05) was annealed from m = −1.0 to m = −1.5. The "momentum" parameter β 1 (c.f.) is also reduced at the end of optimization. The latter was motivated by oscillation and moderate deviations observed near optima, apparently caused (upon investigation) by strong curvature of the loss surface. Inference. The latent z are inferred online using the adaptive IS scheme of section A.1.5. We also perform inference over log s since it is held artificially high for optimization, and its true optimal value is not known. An informative prior close to the learned value log s ∼ N −2.0, 0.1 2, was Epoch η β 1 log s mean M 1 8e-4 0.9 -1.0 1 000 200 8e-4 0.9 -1.3 1 000 600 4e-4 0.9 -1.5 2 000 1000 2e-4 0.8 -1.5 4 000 nevertheless used since the posterior was sometimes approximately singular, causing high condition numbers in the estimated covariance matrix of the proposal. The hyperparameters are given in Table 3. These parameters did not require tuning as for optimization, but were sensible defaults. These also seem to work well without tuning for other experiments such as the Mocap data. Each posterior for a given time t took on average approx. 0.3 seconds. We used the No U-Turn Sampler for the STL experiments due to poor conditioning and the higher complexity and dimensionality of the posterior (19 dimensions). Tuning is performed using ideas from Hoffman & Gelman (2014, Algorithm 4, 5), and the mass matrix is estimated from the warmup phase. 2 The warmup stage lasted 1000 samples and the subsequent 600 samples were used for inference. Each sampler was initialized from a MAP value, obtained via optimization with 10 random restarts. For both MAP optimization and sampling, we found it essential to enforce a low standard deviation (we used log s = −2 and log s ∼ N −2, 0.2 2 respectively) similarly to the MTL experiments. The autocorrelation-based effective sample size (, ch. 11.5) typically exceeds 100 for each parameter. Each posterior for a given time t took on average approx. 300 seconds. Note that as discussed in section A.1.4, unlike our AdaIS procedure, we cannot make much re-use of previous computation here. The average (over the 10 repetitions) are given in Table 4, which extends Table 1 in the main text with the NLL . The distribution of these can be seen in the violin plots of Figure 8. The RMSE of the MTLDS are all significantly better than both the pooled and single-task models according to a Welch's t-test and Mann-Whitney U-test, except for MTLDS-4 at t = 40. The latter is significantly better than the pooled model, but is indistinguishable from the STLDS at the level α = 0.05. We also consider the convergence of the MTLDS to the true model with increasing N. For each experiment, we average the log marginal likelihood of the test sequences estimated via 10 000 (Sobol) samples from the prior. As before, the prior should be a good proposal for the aggregate posterior, and we amortize the same samples over all test sequences. In order to interpret the difference to the true distribution log p * (Y test) − log p(Y test | φ), we use the Bayes Factor interpretations given by. For instance a difference of 1.0 is'barely worth mentioning', but a difference of 4.0 is'strong evidence' that the distributions are different. We average over 10 000 test examples to avoid sampling variation of the test set. Figure 7 show boxplots of the log marginal likelihood for each model over increasing N, where the boxes show the interquartile range (IQR) over the 10 repetitions. We see convergence towards the true value with increasing N, with the difference of the MTLDS-128'barely worth mentioning'. Figure 9a, which is a subset of the CMU skeleton. Representation in observation space. We choose a Lagrangian representation (Figure 9c) where the coordinate frame is centered at the root joint of the skeleton (joint 1 in Fig. 9a, the pelvis), projected onto the ground. The frame is rotated such that the z-axis points in the "forward" direction, roughly normal to the body. This is in contrast to the Eulerian frame (Figure 9b) which has an absolute fixed position for all t. In the Lagrangian frame, the joint positions are always relative to the root joint, which avoids confusing the overall trajectory of the skeleton (typified by the root joint), and the overall rotation of the skeleton, with the local motions of the joints. The relative joint positions can be represented by spatial position or by joint angle. For the latter, the spatial positions of all joints can be recovered from the angle made with their parent joint via use of forward kinematics (FK). This construction ensures the constant bone length of the skeleton over time, which is a desirable property. However, it also substantially increases the sensitivity of internal joints. For instance, the rotation of the trunk will disproportionately affect the error of the joints in both arms. For this reason, we have chosen to model the spatial position of joints, which may in violations of bone length, but avoids these sensitivity issues. See also §2.1. One can further encode the joint positions via velocity (i.e. differencing) which may in smoother predictions. We avoid this encoding for the local joint motion (joints 2 to 21) since it can suffer from accumulated errors, but we do use it to predict the co-ordinate frame as is standard in mocap models. Hence our per-frame representation consists of the velocityẋ,ż,ω of the co-ordinate frame, the relative vertical position of the root joint, and 3-d position of the remaining 20 joints, which gives y t ∈ R 64. Choice of inputs. Our choice of inputs will reflect controls that an animator may wish to manipulate. The first input will be the trajectory that the skeleton is to follow. As in Holden et al. (2017b), we provide the trajectory over the next second (30 frames), sampled uniformly every 5 frames. Unlike previous work, there is no trajectory history in the inputs since this can be kept in the recurrent state. The (2-d) trajectory co-ordinates are given wrt. the current co-ordinate frame, and hence can rotate rapidly during a tight corner. In order to provide some continuity in the inputs, we also provide a first difference of the trajectory in Eulerian co-ordinates. Table 5: Hyper-parameters of mocap models. η denotes the learning rate. The velocity implied by the differenced trajectory does not disambiguate the gait frequency vs. stride length. The same motion might be achieved with fast short steps, or slower long strides. We therefore provide the gait frequency via a phasor (as in b), whose frequency may be externally controlled. This is provided by sine and cosine components to avoid the discontinuity at 2π. A final ambiguity exists from the trajectory at tight corners: the skeleton can rotate either towards the focus of the corner, or towards the outside. Figure 9d demonstrates the latter, which appears not infrequently in the data. We provide a boolean indicator alongside the trajectory which identifies corners for which this happens. Altogether we have u t ∈ R 32: 12 inputs for the Lagrangian trajectory, 12 inputs for the differenced Eulerian trajectory, 2 inputs for the gait phase and 6 inputs for the turning indicators. Extracting the root trajectory. The root trajectory is computed by projecting the root joint onto the ground. However, this projection may still contain information about the style of locomotion, for instance via swaying. We wish to remove all such information, since a model can otherwise learn the style without reference to a latent z. Our goal is to find an appropriately smoothed version of the extracted trajectory T. We use a cubic B-spline fit to control points fitted to the'corners' of the trajectory. These control points are selected using a polygonal approximation to T using the Ramer-Douglas-Peucker algorithm (RDP, e.g.). Briefly, the RDP algorithm uses a divide-and-conquer approach which greedily chooses points that minimize the Hausdorff distance of T to the polygonal approximation. Some per-style tuning of the RDP parameter, and a small number of manually added control points rendered this a semi-automatic process. Extracting the gait phase. Foot contacts are calculated via the code used by Holden et al. (2017b), which is based on thresholding of vertical position and velocity of each foot. As in we check visually for outliers and correct misclassified foot contacts manually. The leading edge of each foot contact is taken to represent 0 (left) and π (right), and the gait phase is calculated by interpolation. In this section we discuss elements of the experimental setup, learning and inference common to all experiments. Details particular to each experiment can be found in the following section. Further Model Details The MTDS architecture is described in section A.3, aside from the choice of prior. We tested both linear and nonlinear h φ in preliminary experiments and the performance was often similar. The nonlinear version used a one hidden layer MLP with 300 hidden units with tanh activations. For the final affine layer, we used a rank 30 matrix which, chosen pragmatically as a trade-off between flexibility and parameter count (see discussion in section A.1.1). Both choices often performed similarly, however the linear approach was chosen, since optimization of the latent z on new data was faster, and apparently more robust to choice of initialization. A nonlinear h φ may be more important when the base model is simpler. The benchmark models use an encoding length of τ = 64 frames. The encoder shares parameters with the decoder, i.e. the RNN is simply'warm started' for 64 frames before prediction. The benchmark models, unlike the MTDS, predict the difference from the previous frame (or 'velocity') via a residual architecture, as this performs better in. Further Learning Details Our primary goal was qualitative: to obtain good style-content separation, high quality animations and smooth interpolation between sequences. Therefore hyperparameter selection for the MTDS proceeded via quantitative means (via the ELBO) and visual inspection of the qualitative criteria. The qualitative desiderata motivated split learning rates between shared and multi-task networks (cf. section 4.2), and the amount of L2 regularization. See Table 5 for the chosen values. The main learning rate η applies to the fixed parameters wrt. z (i.e. ψ 1, H), and the multi-task learning rate applies to the parameter generation parameters φ and inference parameters λ. Standard variational inference proved more reliable than amortized inference: we used a Gaussian with diagonal covariance (parameterized using softplus) for the variational posterior over each z. L2 regularization was applied to φ, ψ 1, H. Unless otherwise specified, we optimized each model using a batch size N batch = 16 for 20 000 iterations. The ELBO had often reached a plateau by this time, and training even longer ed in a worse latent representation at times (as evidenced through poor style transfer). As noted in the main text, we remove the KL penalty of eq. for the initial 2 000 iterations, and enforce a small posterior standard deviation (s λ = 10 −3) for the same duration. This is similar to finding a MAP estimate for the {z}. For the remaining iterations, the original ELBO criterion is used, and the constraint on s λ is removed. The model is implemented in PyTorch and trained on GPUs. Since we use a fairly small max. sequence length L = 64, truncated backpropagation through time was not necessary. The hyper-parameters for the benchmark models were found (Table 5) using a grid search over learning rate and regularization, as well as the optimizers {Adam, (vanilla) SGD}. We performed the search over the pooled data for all 8 styles, with a stratified sample of 12.5% held out for a validation set. Once the hyperparameters were chosen, benchmark models were also trained for 20 000 iterations, recording the validation error every 1 000 iterations on a stratified 12.5% held out sample. The model with the lowest validation error during optimization is chosen. We standardize the data so that when pooled, each dimension has zero mean and unit variance. Finally, note that as discussed in section A.3.1, the data are represented in Lagrangian form, therefore drifts in the predicted trajectory from the true one are not necessarily heavily penalized. This can be altered by changing the weights on the root velocities, but we did not do this. Inference. At test time, especially for experiment 2, we cannot expect amortized inference to perform optimally, and we consider standard inference techniques. We want to understand the nature of the posterior distributions, and so we again used the AdaIS approach of section A.1.5. In practice, each posterior was unimodal and approximately Gaussian. Furthermore, the variation in sequence space for different z in the posterior was usually fairly small, and the posterior predictive mean performed similarly to using a point estimate. Each observation from which z is inferred is of size 64 × 64 and hence the posterior is fairly concentrated. Unlike the DHO model, this is a more expensive procedure. Our k = 3 experiments took approx. 24 seconds per observation for inference. An optimization approach using standard techniques may be expected to perform similarly at a reduced computational cost. Hence unless otherwise specified, inference was done via optimization. Experiment 1 -MTL The training data for each style uses 4 subsequences chosen carefully to represent the inter-style variation. Obviously it is important that frames are consecutive rather than randomly sampled. Over the increasing size training sets, each of these subsequences is a superset of the previous one. The 6 training set sizes (2 8, 2 9, 2 10, 2 11, 2 12, 2 13 frames per style) are not exact since short subsequences are discarded (e.g. at file boundaries), and the largest set contains all the training data except the test set 3, where data are not evenly distributed over styles. The test set comprises 4 sequences from each style, each of length 64, and is the same for all experiments. A length-64 seed sequence immediately preceding each test sequence was used for inference for all models. The models are trained as described above, except for the single task (STL) models. The STL models use an identical architecture to the pooled 1-layer GRU models, except they are trained only on the data for their style. Since there is less data for these models, we train them for a Table 6: Mocap Experiment 1 (MTL): predictive MSE for length-64 predictions where training sets are a given fraction of the original dataset. maximum of 5 000 iterations. We do not train 2-layer GRUs, since the amount of data is small for most experiments. The full are given in Table 6. We use fractions of the dataset instead of absolute training set sizes to aid understanding. The performance of the MTDS appears to increase with larger k, and suggests that we need k > 3 to achieve optimal performance on unseen training data. The demonstrate substantial benefit of the MTDS over a pooled RNN model in terms of sample efficiency, but not in asymptotic performance, as might be expected. According to a paired t-test, the improvements of the k = 7 MTDS over the (1-layer, open loop) pooled GRU are significant for training set sizes 3%, 7%, 13% and 27%. 4 At a style level, the k = 7 MTDS performs at least as well as the pooled GRUs for the first four training set sizes. See Figure 10. Note that the'angry' and'childlike' styles appear to be harder than the others, most likely due to their relatively high speed. For example animations of the MTL experiments, see the linked video in section A.4.1. Table 7 provides the aggregate of experiment 2 for each of the mocap models. A visualization is given in Figure 11. The 2-layer competitors are shown here for completeness, but they achieve similar performance to the 1-layer models on aggregate. Figure 12 provides a breakdown of these on a per-style basis. Styles 5-8 appear to be easier from the point of view of the benchmarks, but the MTDS shows equal or better performance on all styles except style 5. The competitor achieve better short-term performance than the MTDS. However, note that the zero-velocity baseline performs similarly to the open-loop GRUs for the first 5 predictions. This suggests that the MTDS may be improved for these early predictions simply by interpolating from the zero-velocity baseline for small values of t. We are unable to conclude from these experiments that the benchmark models can represent the style better initially, but simply that they can smooth the transition from the seed sequence better. The classifier is learned on the original observations to distinguish between the 8 styles. We use a 512-unit GRU to encode an observation sequence (usually of 64 frames), and transform the final state via a 300-unit hidden layer MLP with sigmoid activations into multinomial emissions. The model is trained via cross-entropy with 20% of the training data held out as a validation set; training was stopped as the validation error approached 0. We perform a standardization of the gait frequency across all styles, since some styles can be identified purely by calculating the frequency. The mean frequency across all styles (1 cycle per 33 frames) is applied to all sequences via linear interpolation. In this we make use of the instantaneous phase given in the inputs. We use a k = 8 latent code for the MTDS as the model is trained on all styles. for each source style s 1, with examples j = 1,..., 4. We next seek the'archetypal' latent code z associated with each target style s 2. For each s 2, we optimize z over 20 candidate values, obtained from the posterior mean of the style s 2 in the training set. Data are generated from all {U (s1) j } and the z which provides the greatest success in style transfer is chosen. The 32 highly varied input sequences guard against overfitting -the'archetypal' codes for each style must perform well across much of the variety of the original dataset. We provide a scalar measurement of the'success' of style transfer for each pair (s 1, s 2) by using the ing'probability' that the classifier assigns the target style s 2, averaged across the four input sequences for the source s 1. The of these experiments are shown in Figure 13a for the model with multi-task bias, and Figure 13b shows the for the full MTDS. Table 3c in the main text gives the marginal of these wrt. the target style. The cells in Figure 13 give the classifier probability for the target style for each (source, target) combination, averaged over the four source inputs. Successful style transfer should in a the classifier assigning a high score in every cell of the table. For most (source, target) pairs, the full MTDS model substantially outperforms the MTBias model: it appears that MTDS can control the prediction well in the majority of cases, and the MTBias model offers reduced control in general. However, we observe for both models that it is more difficult when styles are associated with extremes of the input distribution. Specifically, both the'childlike' and'angry' styles have unusually high speed inputs, and the'old' style has unusually low speeds. Note that in order to provide style transfer, the models are mostly ignoring these correlations, even though they are very useful for prediction. Further improvements may be available, perhaps by using an adversarial loss, or applying domain knowledge to the model. This is orthogonal to our contribution, and we leave this to future work. Providing style transfer from all varieties of source style is a challenging task. For instance, some styles include sources with widely varying speeds and actions, which may be mismatched to the target style. To understand what may be more typical use of the model, we provide an easier variant of this experiment where only one example of each source style is provided, rather than four. Note nevertheless that the same z (s2) is still used across all sources s 1. The of this secondary experiment are provided in Figure 14. In this case, style transfer is successful for almost all (source, target) pairs in the case of the MTDS, except for the angry style. The MTBias model still has many notable failures. 1. In-sample predictions https://vimeo.com/362069486. The goal is to showcase the best possible performance of the models by predicting from inputs in the training set. 2. MTL examples https://vimeo.com/362122944. Examples from Experiment 1. We compare the quality of animations and fit to the ground truth for two limited training set sizes (6.7% and 13.3% of the full data). For both models, MSE to the ground truth is given, averaged over the entire predictive window (length 256). This is different to the experimental setup which uses only the first 64 frames. 3. Novel test examples https://vimeo.com/362068342. Examples from Experiment 2. We show the adaptions obtained by each model to novel sequences, in particular showcasing examples of the pooled GRU models inferring suboptimal styles wrt. MSE. Again, MSE to the ground truth is given averaged over the predictive window (length 256). 4. Style morphing https://vimeo.com/361910646. This animation demonstrates the effect of changing the latent code over time. This also demonstrates style transfer and style interpolation from experiment 3. For style morphing, we found it useful to fix the dynamical bias of the second layer (parameter b in eq. 7) wrt. z since it otherwise ed in'jumps' while interpolating between sequences. We speculate that shifting the bias induces bifurcations in the state space, whereas adapting the transition matrix allows for smooth interpolation."

Propose new objective function for neural sequence generation which integrates ML-based and RL-based objective functions.

Represent this text to find a much much longer article about it with say hundreds-thousands of words

A backward model of previous (state, action) given the next state, i.e. P(s_t, a_t | s_{t+1}), can be used to simulate additional trajectories terminating at states of interest! Improves RL learning efficiency.