Two recently proposed learning algorithms, herding and fast persistent contrastive divergence (FPCD), share the following interesting characteristic: they exploit changes in the model parameters while sampling in order to escape modes and mix better during the sampling process that is part of the learning algorithm. We justify such approaches as ways to escape modes while keeping approximately the same asymptotic distribution of the Markov chain. In that spirit, we extend FPCD using an idea borrowed from Herding in order to obtain a pure sampling algorithm, which we call the rates-FPCD sampler. Interestingly, this sampler can improve the model as we collect more samples, since it optimizes a lower bound on the log likelihood of the training data. We provide empirical evidence that this new algorithm displays substantially better and more robust mixing than Gibbs sampling.
Because the exact gradient of their parameters with respect to log likelihood cannot be computed tractably, training undirected graphical models such as Markov random fields (MRF) and variants of Boltzmann machines requires extensive use of samples taken from a model's underlying distribution. This has led to various sampling schemes for use during training. Here, we study the models structured like the restricted Boltzmann machines (RBM) (Hinton, Osindero, & Teh, 2006)—those with a set of visible (observed) variables, a set of hidden variables, and an energy function that contains only terms for (hidden, visible) pairs.
Unfortunately, sampling from an RBM's distribution is not an easy task. While Gibbs sampling lends itself naturally to sampling in an RBM and is guaranteed to produce samples from the model at equilibrium, it is easy to observe in experiments that the time to equilibrium is considerably larger than what would be practical. And so although the standard sampling method for RBMs is to run a Gibbs chain, it is never actually run to equilibrium except in the most trivial of cases.
Several algorithms have been successful at improving the quality of sampling and have mostly done so by running the sampling process in parallel to learning. For instance, stochastic maximum likelihood (SML) (Younes, 1998), also known as persistent contrastive divergence (PCD) (Tieleman, 2008) keeps the current sample as an initialization to the Gibbs chain in the next training iteration, even though the parameters have changed in between. More recent work has gone even further: the fast persistent contrastive divergence (FPCD) algorithm (Tieleman & Hinton, 2009) keeps a second set of parameters (updated with a stronger gradient but decayed to zero) that are added to the true parameters only for sampling part of the learning algorithm. The herding algorithm (Welling, 2009a, 2009b) outright gives up on getting a point estimate of the parameters and instead simply aims to produce samples like those of the training set through a dynamical system. It can be shown that the system will produce samples with the sought-after sufficient statistics under a condition that may be checked easily (sublinear growth of its parameters) in spite of the fact that its parameters never converge. Interestingly, the distribution of the generated samples may not be expressed at a given time frame as a simple function of the parameters as in a Boltzmann machine. Instead, it is defined procedurally by the dynamics of the generating procedure. However, what matters is that the statistics of the generated samples can be made to match some target statistics (derived from the training data) even though the parameters keep changing.
It is worth noting that while much work has focused on sampling during training, sampling could be legitimately seen as an end in itself—as a creative task, for instance. In this case, what we want are samples that mix quickly and ideally do not require keeping the training set around. By “mixing”, we mean the rate at which the distribution of a stream of samples (at some finite point t in the stream) converges to its asymptotic distribution—in other words, how quickly we can expect it to cover its major modes. Gibbs sampling, which is customarily used for this purpose, mixes very poorly, getting stuck in the same modes for thousands of iterations. Borrowing ideas from FPCD and herding, we obtain a new sampling algorithm, rates-FPCD, which mixes better by a considerable margin. Like herding, it changes the parameters as it is producing samples, but in such a way that they hover around an optimal value corresponding to a lower bound of the likelihood of the training data.
Although rates-FPCD is meant to be a sampling rather than a training algorithm and we initialize its parameters to those of a previously trained RBM, no one should mistake the technique for an unbiased scheme in order to sample already trained RBMs. In fact, the metric we use in order to measure the quality of samples does not allow us to directly measure deviations from the original model. What we attempt to measure is how close the samples we produce are to the samples from a test set. Therefore, any sampling method that modifies the model in order to make it generalize better would score better, even though in the process, it might deviate from the original RBM. While we believe that in practice, provided that the model was trained properly, the deviations are small, we know that they occur. Nonetheless, the bottom line is that the samples score much better than the ones obtained with other methods, which we consider to be an improvement. In section 4, we give insight into to how rates-FPCD typically ends up improving the original model, what kind of deviation one may expect, and its convergence properties.
2. RBMs, CD, SML, FPCD, and Herding
The difference between CD (for contrastive divergence), SML (for stochastic maximum likelihood, also known as persistent CD or PCD), FPCD (for fast PCD), and herding (and thus their respective contributions) lies in how the negative samples are obtained.
The CD training algorithm (Hinton, 1999, 2002; Hinton et al., 2006) obtains these samples by iterating one or more steps of a block Gibbs chain initialized on an observed example x. The CD update is a biased estimator of the log-likelihood gradient (Bengio & Delalleau, 2009). In addition, the fact that negative samples often lie in the immediate vicinity of the positive samples can yield spurious modes elsewhere and create energy barriers (Desjardins, Courville, Bengio, Vincent, & Delalleau, 2010) around the training examples. Gibbs sampling typically works very poorly as soon as it ventures outside the wells corresponding to training examples. This is particularly evident when starting the Gibbs chain from a random state rather than from a real data point.
To address this problem and obtain models with better likelihood, the SML training algorithm (Younes, 1998) estimates the negative-phase contribution of the gradient by sampling continuously from a unique Gibbs chain initialized at the very beginning, but whose transition probabilities slowly change as a result of parameter updates. For each training example or minibatch, we run one or more iterations of a block Gibbs chain initialized at h−t−1 to conditionally sample x−t from, and then h−t from, x−t. The interaction between the positive-phase and the negative-phase contributions is only through the weights.
3. Temporarily Unlearning the Current State to Escape Modes
Before going any further and in order to better understand the proposed method, it is useful to lay down some principles through which the mixing of an MCMC may be improved.
Obviously this trick is not directly applicable to an RBM (or any other model of appreciable complexity), since the transition matrix A is never handled directly. It is, however, quite similar to the effect stochastic gradient updates, such as those in equation 2.3, would have on sampling a persistent Gibbs chain: since the effect of the negative phase contribution to the gradient is effectively to remove a factor from the parameters equal to the corresponding sufficient statistics of the negative sample, that sample is less likely to be generated in the next iteration. Eventually, through reinforcement by positive phase samples that come from the training set, negative samples that look like training examples will come back, but not before the chain was diverted away from them. In other words, a sampling procedure that “unlearns” what it yields is likely to have good mixing. Note that because of the intricate relationship between states represented by an RBM's parameterization, penalizing a sample will also penalize other similar samples, forcing the chain to eventually jump from a major mode to another instead of simply cycling through closely related minor modes.
Because SML, FPCD, and herding all use persistent chains, they all benefit from this effect to some extent. In SML's case, it is mitigated by the need to decay the step size in order to satisfy convergence criteria. As the step size is driven to zero, mixing properties of the persistent chain suffer from an insufficient “unlearning” effect, a problem that might even lead the model to worsen: the negative samples may cease moving, always penalizing the same region of the energy surface. FPCD addresses this concern by supplementing the sampling weights with an additional factor, whose step size may be independently adjusted. Herding, because it uses a large, constant step size and explicitly relies on the training dynamic to generate samples, is likely to reap significant benefits.
It is worth noting that FPCD seems to bring something more than herding (and SML with a large step size, for that matter): it acts a bit more conservatively by making the effect of unlearning parameters temporary, with an exponential decay (with factor α in RatesFPCDSample()). This essentially reins in the sampling parameters so that they do not stray too far from the model's true parameters as the latter move at a slower, stable pace—or not at all. As it turns out, however, the decay is most useful during training, presumably early on, but in sampling with an already trained model as a starting point, we will show theoretically and experimentally that it can actually be a hindrance.
4. The Rates-FPCD Sampler
Then we train a sampling model using the point estimate θ as the initial parameters, with a constant step size of reasonable magnitude. Each training update consists of the difference (scaled by the step size) between the rates (which are constant) and the sufficient statistics of the next sample, drawn from a persistent Gibbs chain. Each update will thus penalize the occurrence of samples similar to the last one, until the chain has had time to cycle through all of the distribution's major modes. The sampler's output is the sequence of samples from this chain. It is equivalent to training a fully observed Boltzmann machine on (hidden, visible) tuples, where hidden units are computed from the associated visible units using the mapping learned by an existing RBM. Algorithm 1 presents the rates-FPCD sampling algorithm, where the step size is given by εF. Note that the updates are not applied directly to the original parameters. Instead, we have parameters θF containing the difference, which adds some flexibility to the implementation.
In order to achieve the mixing properties we seek, it is necessary to train using a large enough constant step size. It can be shown that when a stochastic approximation algorithm (SML/PCD) is run with a constant step size ε, the expected value of the distance of the current parameters to the local minimum is bounded and proportional to the square root of the step size Borkar (2008); that is, (for some constant K). This gives us confidence that the parameters will hover around the local minimum encoded by the rates as long as the step size is not excessive. This being said, taking inspiration from the FPCD algorithm, we also include an additional parameter, α, which rescales the deviation of the sampling parameters from the original parameters on every update. When α < 1, this serves to explicitly constrain the movement of the parameters about their starting point.
Since the application of the rates-FPCD sampler on a pretrained model will optimize it in order to reach the local maximum encoded by the rates’ parameters (i.e., the sufficient statistics of the visible and hidden units), it is pertinent to ask how that model relates to the original. Note that if the training of the RBM had converged to a local maximum of the log likelihood, then the parameters are already optimal with respect to the hidden variables. That is, whereas the original training aims to maximize the log likelihood of the training set under various possible instantiations of the latent variables, the sampling process maximizes the same quantity under the instantiation defined by the original training (which is a convex optimization problem). Rates-FPCD ends up optimizing a lower bound on the log likelihood and thus will work to improve the model unless it had already converged to a local maximum. Should we want to avoid this improvement and generate samples from the original model, it is also possible to use the decay parameter α that we described previously. Setting α < 1 guarantees that the parameters will not stray too far from their original values. However, preventing the movement of parameters hinders their optimization, meaning that while the distribution we sample from using decay might be closer to the original, it may produce samples that are further away from the training distribution.
The strategy described and justified here was originally suggested for herding: observing that sampling requires keeping around (and using) the training set (unless there are no hidden units). Welling (2009a) proposes an approximation based on a set of “rates,” which are averages of the positive-phase-sufficient statistics through training (in contrast, our rates are computed from applying a point estimate of the parameters on the full training data, but this is not very different). By combining the rates idea with the fast weights of FPCD, we propose a new sampling algorithm that we call the rates-FPCD sampler (see algorithm 2). In practice, this is FPCD with a step size of zero for the slow weights and the rates as the (constant) positive-phase contribution to the update. Unlike for herding, meaningful conditional expectations may be computed for the visible and hidden units. For example, this is very useful to sample grayscale images. Conversely, we can imagine changing the step size for herding (which should control the mixing rate) or using it with fast weights. We thus back-ported our additional parameters to herding in order to make the rates-herding sampling algorithm (see algorithm 2).
Finally, we consider yet another sampling algorithm where the rates are actually set to zero; we call it the sample penalization (SP) sampling algorithm. This draws a closer parallel to the example of section 3 (with ) where we simply penalize the transition of a state to itself. Assuming that a sufficient weight decay is applied to the fast parameters θF, the effective parameters θ− for the negative phase and thus the distribution that we effectively sample from might stay sufficiently close to the original while improving mixing to prove useful. However, note that unlike rates-FPCD, SP will always create a heavy distortion of the parameters (in fact, without a decay of the fast weights, the samples rapidly converge to a uniform distribution on the whole configuration space).
5. Experimental Results
We used three data sets for which it made sense to binarize the data and for which visualizing samples helped us understand qualitatively the properties and quality of the learned model and the sampling procedure: the USPS digits images, the MNIST digit images, and a bag-of-words representation of the OpenTable data set. The USPS data set has 16×16 pixel digit images: 6291 are used for training, 1000 for hyperparameter selection, and 2007 for testing. The MNIST data set has 28×28 pixel images: 50,000 for training, 10,000 for hyperparameter selection, and 10,000 for testing (only 2000 were used to generate the likelihood curves shown here, to reduce computation time). The OpenTable data set consists of 435,786 reviews of restaurants, from which we picked a subset of 80,000 train, 10,000 validation, and 10,000 test examples. Each example is a vector of 1000 binary values, the ith of which encoding the presence or absence of the ith most common word stem in the data set.
To estimate the quality of the samples generated by a sampling procedure, we use the indirect sampling likelihood (ISL) procedure (Breuleux, Bengio, & Vincent, 2010; Desjardins et al., 2010). The main idea of this procedure is to compare the generated samples to the test set and exploit a nonparametric density estimator to evaluate how close each test example is from any of the generated examples (i.e., what likelihood is given to the test set by the nonparametric density estimator trained on the generated samples). We plot the ISL likelihood proxy against the number of samples generated (see Figure 2) to evaluate how quickly the generated samples cover the main modes of the distribution of interest (i.e., cover the test set examples). When the ISL curve rises quickly, the model is mixing quickly. The asymptotic value of ISL (as the number of examples becomes large) is more indicative of the quality of the generative model in general but tells less about the quality of the mixing process (how quickly it covers the main modes).
5.1. Comparing the Mixing Achieved with Different Samplers
Four classes of sampling methods were evaluated: rates-FPCD, rates-herding, sample penalization, and Gibbs sampling, the last of which is the standard sampling procedure for RBMs. When applicable, the methods are parameterized by a step size εF, a weight decay rate α for the difference parameters θF, and the number k of Gibbs iterations between each sample. These hyperparameters were selected to maximize ISL so as to compare each sampling method in its best light. To focus on the effect of the sampling method, all are compared on the same model, which is chosen here as the best (according to ISL) among all training methods and hyperparameters. For the MNIST and USPS data sets, this turned out to be a model trained by FPCD. In general, the results of the comparison hold regardless of the trained model we start with.
Figure 1 shows sequences of samples obtained from the same model on MNIST but using different sampling methods. Visually speaking, the log likelihood of the test set under a density defined by these samples seems to be a fair assessment of the mixing capabilities of the models shown, with rates-herding and rates-FPCD leading and Gibbs a distant last. Note that the increase in the mixing rate is sometimes mitigated by a higher proportion of spurious samples, that is, samples that lie on the path from a major mode to another but are not valid by themselves. This indicates that sampling through rates-FPCD might indeed have an effect similar to an increase in temperature (where energy differences are smoothed). Despite this, the samples look faithful to the training set, unlike samples from a Gibbs chain with a temperature high enough to mix well.
Figure 2 plots the ISL with generated samples of only the USPS test set for these sampling methods, in function of the number of generated samples. It is readily apparent that Gibbs sampling performs very poorly (note that it often performs much worse). By correlating the curve's bumps with visual inspection of samples, we find that they indicate the times at which the chain suddenly switches from one mode to the next. In contrast, the curves for rates-FPCD and rates-herding are quite smooth and well mixing. Furthermore, the performance of the training set itself as a Parzen classifier is eventually surpassed by a slight margin (we contend that the margin would become wider if more samples were drawn): −78.5 (rates-FPCD) versus -80.0 (train set). Rates-herding typically starts out with the fastest rise in ISL (faster mixing) but is caught up with by rates-FPCD. As discussed below, this may be related to a difference in optimal εF.
Note that for equal k, our implementation of Gibbs sampling is about two (on USPS) to seven (on MNIST; this may be due to cache or paging issues) times faster than that of rates-FPCD. This is mostly due to the use of slow and fast weights, the presence of parameter updates, and some additional bookkeeping, so it is an overestimate. Even taking this into account, a look at Figure 2 shows that the latter covers the state-space at least 20 times faster than the former on USPS. On MNIST, that would be at least 250 times faster (100 samples from rates-FPCD being better than 200,000 Gibbs samples). A fair comparison of rates-FPCD with rates-herding is more difficult to make since they give very similar advantages at a similar cost and the latter can produce only discrete samples. Depending on the data set we used, rates-FPCD was either equivalent or superior to rates-herding. We do not believe that our experiments, as they stand, provide compelling reasons to prefer one over the other when both are applicable (on binary inputs).
The optimal step size for the rates-FPCD method in the long run was observed to be lower than the optimal step size for the rates-herding method by a factor of 10 to 100 in most cases. The step size seems to control a trade-off between mixing rate and sampling noise, meaning that the mixing rate increases as the step-size εF increases, but so does the frequency of spurious samples. It seems that rates-herding is more robust to larger step sizes, but that this advantage subsides as more samples are drawn and rapid mixing is less of a concern. A tentative explanation for this robustness is that herding provides a halting criterion for the number of iterations k between each produced sample: the chain is run until a fixed point is reached. Presumably this allows herding to loop exactly as many times as needed in order to guarantee that only low-energy samples will be produced. In the long run, however, rates-FPCD tends to perform better. This is most observable on MNIST where rates-FPCD, for a comparable k, scores significantly higher. On other data sets, there is no clear difference, which prompts some caution in comparing both approaches.
An interesting observation is that the optimal value for the decay rate α for sampling purposes is 1, meaning that no decay is applied at all and the fast parameters may roam freely. Applying a decay of 0.95 (see Figure 3, left) yielded worse results. This can be explained by the fact that the models are not fully optimized and that α = 1 allows the sampler to optimize them further, thereby improving the quality of samples according to our metric. It is worth noting that such is not the case during training (α < 1 works better in that case). We believe this may be because of a need to keep the fast and slow weights evolving in sync while the relationship between the visible and hidden units is learned.
The SP method systemically performed better than Gibbs but still significantly worse than the other methods. It can be seen on Figure 1 that its good mixing is mitigated by a skewing in the distribution: the samples produced by this method, while recognizable, appear different from the true distribution (they seem larger and bolder in general and not only on the figure shown), as expected according to the earlier discussion.
Experiments on the OpenTable data set showed a different but just as interesting picture. On this data set, the ISL performance of Gibbs sampling was on par with rates-FPCD for some models (models trained with SML or FPCD), but Gibbs sampling failed badly on other models, in particular those trained using CD. As expected, we find rates-FPCD to work more consistently. As shown in Figure 3 on the right, when applied on models trained using CD, Gibbs sampling quickly and systemically gets stuck in a very narrow mode (usually, this mode is a vector of zeros). This causes the ISL measure to decrease, since an increasingly disproportionate weight is given to a single sample. Rates-FPCD does not encounter any obvious mixing problems. We surmise that CD training creates very deep spurious energy wells for Gibbs samplers to get stuck in, but that rates-FPCD optimizes them away during sampling (thereby improving the model, as explained in section 4). To a lesser extent, we also observe better consistency from rates-FPCD on models trained with SML or FPCD, the mean and variance of the ISL performance being, respectively, higher and lower.
In this letter, we set out to evaluate several sampling methods over RBM models whose objective is to mix well: rates-herding, the essence of which had previously been proposed (Welling, 2009a) and which we extended with variable step size; sample penalization, which directly tries to move away from just visited configurations; and rates-FPCD, which is inspired from rates-herding, FPCD, and sample penalization. Interestingly, the rates-FPCD sampling procedure can generate samples that get better as we generate more of them, as it optimizes a lower bound on the log likelihood of the training data. We showed that in cases where Gibbs sampling does not mix well, rates-herding and rates-FPCD produced samples that much more quickly visited modes and covered the test examples, both from visual inspection and quantitatively.