Mercurial > pylearn
view pylearn/algorithms/mcRBM.py @ 1275:f0129e37a8ef
mcRBM - changed params from lambda to method for pickling
author | James Bergstra <bergstrj@iro.umontreal.ca> |
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date | Wed, 08 Sep 2010 13:18:13 -0400 |
parents | 7bb5dd98e671 |
children | 1817485d586d |
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""" This file implements the Mean & Covariance RBM discussed in Ranzato, M. and Hinton, G. E. (2010) Modeling pixel means and covariances using factored third-order Boltzmann machines. IEEE Conference on Computer Vision and Pattern Recognition. and performs one of the experiments on CIFAR-10 discussed in that paper. There are some minor discrepancies between the paper and the accompanying code (train_mcRBM.py), and the accompanying code has been taken to be correct in those cases because I couldn't get things to work otherwise. Math ==== Energy of "covariance RBM" E = -0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i C_{if} v_i )^2 = -0.5 \sum_f (\sum_k P_{fk} h_k) ( \sum_i C_{if} v_i )^2 "vector element f" "vector element f" In some parts of the paper, the P matrix is chosen to be a diagonal matrix with non-positive diagonal entries, so it is helpful to see this as a simpler equation: E = \sum_f h_f ( \sum_i C_{if} v_i )^2 Version in paper ---------------- Full Energy of the Mean and Covariance RBM, with :math:`h_k = h_k^{(c)}`, :math:`g_j = h_j^{(m)}`, :math:`b_k = b_k^{(c)}`, :math:`c_j = b_j^{(m)}`, :math:`U_{if} = C_{if}`, E (v, h, g) = - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i (U_{if} v_i) / |U_{.f}|*|v| )^2 - \sum_k b_k h_k + 0.5 \sum_i v_i^2 - \sum_j \sum_i W_{ij} g_j v_i - \sum_j c_j g_j For the energy function to correspond to a probability distribution, P must be non-positive. P is initialized to be a diagonal, and in our experience it can be left as such because even in the paper it has a very low learning rate, and is only allowed to be updated after the filters in U are learned (in effect). Version in published train_mcRBM code ------------------------------------- The train_mcRBM file implements learning in a similar but technically different Energy function: E (v, h, g) = - 0.5 \sum_f \sum_k P_{fk} h_k (\sum_i U_{if} v_i / sqrt(\sum_i v_i^2/I + 0.5))^2 - \sum_k b_k h_k + 0.5 \sum_i v_i^2 - \sum_j \sum_i W_{ij} g_j v_i - \sum_j c_j g_j There are two differences with respect to the paper: - 'v' is not normalized by its length, but rather it is normalized to have length close to the square root of the number of its components. The variable called 'small' that "avoids division by zero" is orders larger than machine precision, and is on the order of the normalized sum-of-squares, so I've included it in the Energy function. - 'U' is also not normalized by its length. U is initialized to have columns that are shorter than unit-length (approximately 0.2 with the 105 principle components in the train_mcRBM data). During training, the columns of U are constrained manually to have equal lengths (see the use of normVF), but Euclidean norm is allowed to change. During learning it quickly converges towards 1 and then exceeds 1. It does not seem like this column-wise normalization of U is justified by maximum-likelihood, I have no intuition for why it is used. Version in this code -------------------- This file implements the same algorithm as the train_mcRBM code, except that the P matrix is omitted for clarity, and replaced analytically with a negative identity matrix. E (v, h, g) = + 0.5 \sum_k h_k (\sum_i U_{ik} v_i / sqrt(\sum_i v_i^2/I + 0.5))^2 - \sum_k b_k h_k + 0.5 \sum_i v_i^2 - \sum_j \sum_i W_{ij} g_j v_i - \sum_j c_j g_j Conventions in this file ======================== This file contains some global functions, as well as a class (MeanCovRBM) that makes using them a little more convenient. Global functions like `free_energy` work on an mcRBM as parametrized in a particular way. Suppose we have I input dimensions, F squared filters, J mean variables, and K covariance variables. The mcRBM is parametrized by 5 variables: - `U`, a matrix whose rows are visible covariance directions (I x F) - `W`, a matrix whose rows are visible mean directions (I x J) - `b`, a vector of hidden covariance biases (K) - `c`, a vector of hidden mean biases (J) Matrices are generally layed out and accessed according to a C-order convention. """ # # WORKING NOTES # THIS DERIVATION IS BASED ON THE ** PAPER ** ENERGY FUNCTION # NOT THE ENERGY FUNCTION IN THE CODE!!! # # Free energy is the marginal energy of visible units # Recall: # Q(x) = exp(-E(x))/Z ==> -log(Q(x)) - log(Z) = E(x) # # # E (v, h, g) = # - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / |U_{*f}|^2 |v|^2 # - \sum_k b_k h_k # + 0.5 \sum_i v_i^2 # - \sum_j \sum_i W_{ij} g_j v_i # - \sum_j c_j g_j # - \sum_i a_i v_i # # # Derivation, in which partition functions are ignored. # # E(v) = -\log(Q(v)) # = -\log( \sum_{h,g} Q(v,h,g)) # = -\log( \sum_{h,g} exp(-E(v,h,g))) # = -\log( \sum_{h,g} exp(- # - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}| * |v|) # - \sum_k b_k h_k # + 0.5 \sum_i v_i^2 # - \sum_j \sum_i W_{ij} g_j v_i # - \sum_j c_j g_j # - \sum_i a_i v_i )) # # Get rid of double negs in exp # = -\log( \sum_{h} exp( # + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}| * |v|) # + \sum_k b_k h_k # - 0.5 \sum_i v_i^2 # ) * \sum_{g} exp( # + \sum_j \sum_i W_{ij} g_j v_i # + \sum_j c_j g_j)) # - \sum_i a_i v_i # # Break up log # = -\log( \sum_{h} exp( # + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}|*|v|) # + \sum_k b_k h_k # )) # -\log( \sum_{g} exp( # + \sum_j \sum_i W_{ij} g_j v_i # + \sum_j c_j g_j ))) # + 0.5 \sum_i v_i^2 # - \sum_i a_i v_i # # Use domain h is binary to turn log(sum(exp(sum...))) into sum(log(.. # = -\log(\sum_{h} exp( # + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}|* |v|) # + \sum_k b_k h_k # )) # - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j )) # + 0.5 \sum_i v_i^2 # - \sum_i a_i v_i # # = - \sum_{k} \log(1 + exp(b_k + 0.5 \sum_f P_{fk}( \sum_i U_{if} v_i )^2 / (|U_{*f}|*|v|))) # - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j )) # + 0.5 \sum_i v_i^2 # - \sum_i a_i v_i # # For negative-one-diagonal P this gives: # # = - \sum_{k} \log(1 + exp(b_k - 0.5 \sum_i (U_{ik} v_i )^2 / (|U_{*k}|*|v|))) # - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j )) # + 0.5 \sum_i v_i^2 # - \sum_i a_i v_i import sys, os, logging import numpy as np import numpy import theano from theano import function, shared, dot from theano import tensor as TT floatX = theano.config.floatX sharedX = lambda X, name : shared(numpy.asarray(X, dtype=floatX), name=name) import pylearn #TODO: clean up the HMC_sampler code #TODO: think of naming convention for acronyms + suffix? from pylearn.sampling.hmc import HMC_sampler from pylearn.io import image_tiling from pylearn.gd.sgd import sgd_updates import pylearn.dataset_ops.image_patches ########################################### # # Candidates for factoring # ########################################### def l1(X): """ :param X: TensorType variable :rtype: TensorType scalar :returns: the sum of absolute values of the terms in X :math: \sum_i |X_i| Where i is an appropriately dimensioned index. """ return abs(X).sum() def l2(X): """ :param X: TensorType variable :rtype: TensorType scalar :returns: the sum of absolute values of the terms in X :math: \sqrt{ \sum_i X_i^2 } Where i is an appropriately dimensioned index. """ return TT.sqrt((X**2).sum()) def contrastive_cost(free_energy_fn, pos_v, neg_v): """ :param free_energy_fn: lambda (TensorType matrix MxN) -> TensorType vector of M free energies :param pos_v: TensorType matrix MxN of M "positive phase" particles :param neg_v: TensorType matrix MxN of M "negative phase" particles :returns: TensorType scalar that's the sum of the difference of free energies :math: \sum_i free_energy(pos_v[i]) - free_energy(neg_v[i]) """ return (free_energy_fn(pos_v) - free_energy_fn(neg_v)).sum() def contrastive_grad(free_energy_fn, pos_v, neg_v, wrt, other_cost=0): """ :param free_energy_fn: lambda (TensorType matrix MxN) -> TensorType vector of M free energies :param pos_v: positive-phase sample of visible units :param neg_v: negative-phase sample of visible units :param wrt: TensorType variables with respect to which we want gradients (similar to the 'wrt' argument to tensor.grad) :param other_cost: TensorType scalar :returns: TensorType variables for the gradient on each of the 'wrt' arguments :math: Cost = other_cost + \sum_i free_energy(pos_v[i]) - free_energy(neg_v[i]) :math: d Cost / dW for W in `wrt` This function is similar to tensor.grad - it returns the gradient[s] on a cost with respect to one or more parameters. The difference between tensor.grad and this function is that the negative phase term (`neg_v`) is considered constant, i.e. d `Cost` / d `neg_v` = 0. This is desirable because `neg_v` might be the result of a sampling expression involving some of the parameters, but the contrastive divergence algorithm does not call for backpropagating through the sampling procedure. Warning - if other_cost depends on pos_v or neg_v and you *do* want to backpropagate from the `other_cost` through those terms, then this function is inappropriate. In that case, you should call tensor.grad separately for the other_cost and add the gradient expressions you get from ``contrastive_grad(..., other_cost=0)`` """ cost=contrastive_cost(free_energy_fn, pos_v, neg_v) if other_cost: cost = cost + other_cost return theano.tensor.grad(cost, wrt=wrt, consider_constant=[neg_v]) ########################################### # # Expressions that are mcRBM-specific # ########################################### class mcRBM(object): """Light-weight class that provides the math related to inference Attributes: - U - the covariance filters (theano shared variable) - W - the mean filters (theano shared variable) - a - the visible bias (theano shared variable) - b - the covariance bias (theano shared variable) - c - the mean bias (theano shared variable) """ def __init__(self, U, W, a, b, c): self.U = U self.W = W self.a = a self.b = b self.c = c def hidden_cov_units_preactivation_given_v(self, v, small=0.5): """Return argument to the sigmoid that would give mean of covariance hid units See the math at the top of this file for what 'adjusted' means. return b - 0.5 * dot(adjusted(v), U)**2 """ unit_v = v / (TT.sqrt(TT.mean(v**2, axis=1)+small)).dimshuffle(0,'x') # adjust row norm return self.b - 0.5 * dot(unit_v, self.U)**2 def free_energy_terms_given_v(self, v): """Returns theano expression for the terms that are added to form the free energy of visible vector `v` in an mcRBM. 1. Free energy related to covariance hiddens 2. Free energy related to mean hiddens 3. Free energy related to L2-Norm of `v` 4. Free energy related to projection of `v` onto biases `a` """ t0 = -TT.sum(TT.nnet.softplus(self.hidden_cov_units_preactivation_given_v(v)),axis=1) t1 = -TT.sum(TT.nnet.softplus(self.c + dot(v,self.W)), axis=1) t2 = 0.5 * TT.sum(v**2, axis=1) t3 = -TT.dot(v, self.a) return [t0, t1, t2, t3] def free_energy_given_v(self, v): """Returns theano expression for free energy of visible vector `v` in an mcRBM """ return TT.add(*self.free_energy_terms_given_v(v)) def expected_h_g_given_v(self, v): """Returns tuple (`h`, `g`) of theano expression conditional expectations in an mcRBM. `h` is the conditional on the covariance units. `g` is the conditional on the mean units. """ h = TT.nnet.sigmoid(self.hidden_cov_units_preactivation_given_v(v)) g = nnet.sigmoid(self.c + dot(v,self.W)) return (h, g) def n_visible_units(self): """Return the number of visible units of this RBM For an RBM made from shared variables, this will return an integer, for a purely symbolic RBM this will return a theano expression. """ try: return self.W.value.shape[0] except AttributeError: return self.W.shape[0] def sampler(self, n_particles, n_visible=None, rng=7823748): """Return an `HMC_sampler` that will draw samples from the distribution over visible units specified by this RBM. :param n_particles: this many parallel chains will be simulated. :param rng: seed or numpy RandomState object to initialize particles, and to drive the simulation. """ if not hasattr(rng, 'randn'): rng = np.random.RandomState(rng) if n_visible is None: n_visible = self.n_visible_units() rval = HMC_sampler.new_from_shared_positions( shared_positions = sharedX( rng.randn( n_particles, n_visible), name='particles'), energy_fn=self.free_energy_given_v, seed=int(rng.randint(2**30))) return rval def as_feedforward_layer(self, v): """Return a dictionary with keys: inputs, outputs and params The inputs is [v] The outputs is :math:`[E[h|v], E[g|v]]` where `h` is the covariance hidden units and `g` is the mean hidden units. The params are ``[U, W, b, c]``, the model parameters that enter into the conditional expectations. :TODO: add an optional parameter to return only one of the expections. """ return dict( inputs = [v], outputs = list(self.expected_h_g_given_v(v)), params = [self.U, self.W, self.b, self.c], ) def params(self): """Return the elements of [U,W,a,b,c] that are shared variables WRITEME : a *prescriptive* definition of this method suitable for mention in the API doc. """ return list(self._params) @classmethod def alloc(cls, n_I, n_K, n_J, rng = 8923402190, U_range=0.02, W_range=0.05, a_ival=0, b_ival=2, c_ival=-2): """ Return a MeanCovRBM instance with randomly-initialized shared variable parameters. :param n_I: input dimensionality :param n_K: number of covariance hidden units :param n_J: number of mean filters (linear) :param rng: seed or numpy RandomState object to initialize parameters :note: Constants for initial ranges and values taken from train_mcRBM.py. """ if not hasattr(rng, 'randn'): rng = np.random.RandomState(rng) rval = cls( U = sharedX(U_range * rng.randn(n_I, n_K),'U'), W = sharedX(W_range * rng.randn(n_I, n_J),'W'), a = sharedX(np.ones(n_I)*a_ival,'a'), b = sharedX(np.ones(n_K)*b_ival,'b'), c = sharedX(np.ones(n_J)*c_ival,'c'),) rval._params = [rval.U, rval.W, rval.a, rval.b, rval.c] return rval class mcRBMTrainer(object): """Light-weight class encapsulating math for mcRBM training Attributes: - rbm - an mcRBM instance - sampler - an HMC_sampler instance - normVF - geometrically updated norm of U matrix columns (shared var) - learn_rate - SGD learning rate [un-annealed] - learn_rate_multipliers - the learning rates for each of the parameters of the rbm (in order corresponding to what's returned by ``rbm.params()``) - l1_penalty - float or TensorType scalar to modulate l1 penalty of rbm.U and rbm.W - iter - number of cd_updates (shared var) - used to anneal the effective learn_rate - lr_anneal_start - scalar or TensorType scalar - iter at which time to start decreasing the learning rate proportional to 1/iter """ # TODO: accept a GD algo as an argument? @classmethod def alloc(cls, rbm, visible_batch, batchsize, initial_lr=0.075, rng=234, l1_penalty=0, learn_rate_multipliers=[2, .2, .02, .1, .02], lr_anneal_start=2000, ): """ :param rbm: mcRBM instance to train :param visible_batch: TensorType variable for training data :param batchsize: the number of rows in visible_batch :param initial_lr: the learning rate (may be annealed) :param rng: seed or RandomState to initialze PCD sampler :param l1_penalty: see class doc :param learn_rate_multipliers: see class doc :param lr_anneal_start: see class doc """ #TODO: :param lr_anneal_iter: the iteration at which 1/t annealing will begin #TODO: get batchsize from visible_batch?? # allocates shared var for negative phase particles # TODO: should normVF be initialized to match the size of rbm.U ? return cls( rbm=rbm, visible_batch=visible_batch, sampler=rbm.sampler(batchsize, rng=rng), normVF=sharedX(1.0, 'normVF'), learn_rate=sharedX(initial_lr/batchsize, 'learn_rate'), iter=sharedX(0, 'iter'), l1_penalty=l1_penalty, learn_rate_multipliers=learn_rate_multipliers, lr_anneal_start=lr_anneal_start) def __init__(self, **kwargs): self.__dict__.update(kwargs) def normalize_U(self, new_U): """ :param new_U: a proposed new value for rbm.U :returns: a pair of TensorType variables: a corrected new value for U, and a new value for self.normVF This is a weird normalization procedure, but the sample code for the paper has it, and it seems to be important. """ U_norms = TT.sqrt((new_U**2).sum(axis=0)) new_normVF = .95 * self.normVF + .05 * TT.mean(U_norms) return (new_U * new_normVF / U_norms), new_normVF def contrastive_grads(self): """Return the contrastive divergence gradients on the parameters of self.rbm """ return contrastive_grad( free_energy_fn=self.rbm.free_energy_given_v, pos_v=self.visible_batch, neg_v=self.sampler.positions, wrt = self.rbm.params(), other_cost=(l1(self.rbm.U)+l1(self.rbm.W)) * self.l1_penalty) def cd_updates(self): """ Return a dictionary of shared variable updates that implements contrastive divergence learning by stochastic gradient descent with an annealed learning rate. """ grads = self.contrastive_grads() # contrastive divergence updates # TODO: sgd_updates is a particular optization algo (others are possible) # parametrize so that algo is plugin # the normalization normVF might be sgd-specific though... # TODO: when sgd has an annealing schedule, this should # go through that mechanism. lr = TT.clip( self.learn_rate * TT.cast(self.lr_anneal_start / (self.iter+1), floatX), 0.0, #min self.learn_rate) #max ups = dict(sgd_updates( self.rbm.params(), grads, stepsizes=[a*lr for a in self.learn_rate_multipliers])) ups[self.iter] = self.iter + 1 # sampler updates ups.update(dict(self.sampler.updates())) # add trainer updates (replace CD update of U) ups[self.rbm.U], ups[self.normVF] = self.normalize_U(ups[self.rbm.U]) return ups if __name__ == '__main__': import pylearn.algorithms.tests.test_mcRBM rbm,smplr = pylearn.algorithms.tests.test_mcRBM.test_reproduce_ranzato_hinton_2010( as_unittest=False, n_train_iters=10) import cPickle print '' print 'Saving rbm...' cPickle.dump(rbm, open('mcRBM.rbm.pkl', 'w'), -1) print 'Saving sampler...' cPickle.dump(smplr, open('mcRBM.smplr.pkl', 'w'), -1)