Mercurial > pylearn
view pylearn/algorithms/mcRBM.py @ 1000:d4a14c6c36e0
mcRBM - post code-review #1 with Guillaume
author | James Bergstra <bergstrj@iro.umontreal.ca> |
---|---|
date | Tue, 24 Aug 2010 19:24:54 -0400 |
parents | c6d08a760960 |
children | 075c193afd1b |
line wrap: on
line source
""" 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 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 # ########################################### #TODO: Document, move to pylearn's math lib def l1(X): return abs(X).sum() #TODO: Document, move to pylearn's math lib def l2(X): return TT.sqrt((X**2).sum()) #TODO: Document, move to pylearn's math lib def contrastive_cost(free_energy_fn, pos_v, neg_v): return (free_energy_fn(pos_v) - free_energy_fn(neg_v)).sum() #TODO: Typical use of contrastive_cost is to later use tensor.grad, but in that case we want to # block gradient going through neg_v def contrastive_grad(free_energy_fn, pos_v, neg_v, params, other_cost=0): """ :param pos_v: positive-phase sample of visible units :param neg_v: negative-phase sample of visible units """ #block the grad through neg_v cost=contrastive_cost(free_energy_fn, pos_v, neg_v) if other_cost: cost = cost + other_cost return theano.tensor.grad(cost, wrt=params, consider_constant=[neg_v]) ########################################### # # Expressions that are mcRBM-specific # ########################################### #TODO: make global function to initialize parameter def hidden_cov_units_preactivation_given_v(rbm, 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 """ (U,W,a,b,c) = rbm unit_v = v / (TT.sqrt(TT.mean(v**2, axis=1)+small)).dimshuffle(0,'x') # adjust row norm return b - 0.5 * dot(unit_v, U)**2 def free_energy_terms_given_v(rbm, 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` """ U, W, a, b, c = rbm t0 = -TT.sum(TT.nnet.softplus(hidden_cov_units_preactivation_given_v(rbm, v)),axis=1) t1 = -TT.sum(TT.nnet.softplus(c + dot(v,W)), axis=1) t2 = 0.5 * TT.sum(v**2, axis=1) t3 = -TT.dot(v, a) return [t0, t1, t2, t3] def free_energy_given_v(rbm, v): """Returns theano expression for free energy of visible vector `v` in an mcRBM """ return sum(free_energy_terms_given_v(rbm,v)) def expected_h_g_given_v(rbm, 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. """ (U, W, a, b, c) = rbm h = TT.nnet.sigmoid(hidden_cov_units_preactivation_given_v(rbm, v)) g = nnet.sigmoid(c + dot(v,W)) return (h, g) def n_visible_units(rbm): """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. """ W = rbm[1] try: return W.value.shape[0] except AttributeError: return W.shape[0] def sampler(rbm, 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 = n_visible_units(rbm) rval = HMC_sampler( positions = [shared( rng.randn( n_particles, n_visible).astype(floatX), name='particles')], energy_fn = lambda p : free_energy_given_v(rbm, p[0]), seed=int(rng.randint(2**30))) return rval ############################# # # Convenient data container # ############################# class MeanCovRBM(object): """Container for mcRBM parameters It provides parameter lookup by name, as well as a heuristic for initializing the parameters for effective learning. 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) """ params = property(lambda s: [s.U, s.W, s.a, s.b, s.c]) n_visible = property(lambda s: s.W.value.shape[0]) def __init__(self, U, W, a, b, c): self.__dict__.update(locals()) del self.__dict__['self'] def __getitem__(self, idx): # support unpacking of this container as if it were a tuple return self.params[idx] @classmethod def new_from_dims(cls, n_I, n_K, n_J, rng = 8923402190): """ Return a MeanCovRBM instance with randomly-initialized 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 params """ if not hasattr(rng, 'randn'): rng = np.random.RandomState(rng) def shrd(X,name): return shared(X.astype(floatX), name=name) # initialization taken from train_mcRBM.py return cls( U = shrd(0.02 * rng.randn(n_I, n_K),'U'), W = shrd(0.05 * rng.randn(n_I, n_J),'W'), a = shrd(np.ones(n_I)*(0),'a'), b = shrd(np.ones(n_K)*2,'b'), c = shrd(np.ones(n_J)*(-2),'c')) def __getstate__(self): # unpack shared containers, which may have references to Theano stuff and are not a # long-term stable data type. return dict( U = self.U.value, W = self.W.value, b = self.b.value, c = self.c.value) def __setstate__(self, dct): d = dict(dct) for key in ['U', 'W', 'a', 'b', 'c']: d[key] = shared(d[key], name=key) self.__init__(**d) #TODO: put the normalization of U as a global function #TODO: put the learning loop as a global function or class, so that someone could load and *TRAIN* an mcRBM!!! if __name__ == '__main__': import pylearn.algorithms.tests.test_mcRBM pylearn.algorithms.tests.test_mcRBM.test_reproduce_ranzato_hinton_2010(as_unittest=True)