.. _DLmath: Deep Learning ================ H\ :sub:`2`\ O's Deep Learning is based on a multi-layer feed-forward artificial neural network that is trained with stochastic gradient descent using back-propagation. The network can contain a large number of hidden layers consisting of neurons with tanh, rectifier and maxout activation functions. Advanced features such as adaptive learning rate, rate annealing, momentum training, dropout, L1 or L2 regularization, checkpointing and grid search enable high predictive accuracy. Each compute node trains a copy of the global model parameters on its local data with multi-threading (asynchronously), and contributes periodically to the global model via model averaging across the network. Defining a Deep Learning Model """""""""""""""""""""""""""""""" H\ :sub:`2`\ O Deep Learning models have many input parameters, many of which are only accessible via the expert mode, and their default values should be fine for most use cases. Please read the following instructions before building extensive Deep Learning models. Many of the parameters allow specification of multiple values for grid search (e.g., comma-separated list "10,2,40" or from:to:step range "10:40:5"). The application of grid search and successive continuation of winning models via checkpoint restart is highly recommended as model performance can vary greatly. **Destination Key** Name of the model to be trained. Will be auto-generated if omitted. **Source** A hex key associated with the parsed training data. **Response** The dependent or target variable of interest. Can be numerical or a factor (categorical). **Ignored Columns** This field will auto populate a list of the columns from the data set in use. The user-selected set of columns are the features that will be omitted from the model. Additionally - users can specify whether the model should omit constant columns by selecting expert settings and checking the tic box indicating **Ignore Const Cols**. **Classification** Check box indicating whether the dependent variable is to be treated as a factor or a continuous variable. **Validation** A unique data set with the same shape and features as the training data to be used in model validation (i.e., production of error rates on data not used in model building). **Checkpoint** A model key associated with a previously trained Deep Learning model. This option allows users to build a new model as a continuation of a previously generated model (e.g., by a grid search). **Best Model Key** If given, store the best model so far under this key. Model performance is measured by MSE for regression and overall error rate for classification (at F1-optimal threshold for binary classification). **Expert Mode** Unlock expert mode parameters than can affect model building speed, predictive accuracy and scoring. Leaving expert mode parameters at default values is fine for many problems, but best results on complex datasets are often only attainable via expert mode options. **Activation** The activation function (non-linearity) to be used the neurons in the hidden layers. *Tanh*: Hyperbolic tangent function (same as scaled and shifted sigmoid). *Rectifier*: Chooses the maximum of (0, x) where x is the input value. *Maxout*: Choose the maximum coordinate of the input vector. *With Dropout*: Zero out a random user-given fraction of the incoming weights to each hidden layer during training, for each training row. This effectively trains exponentially many models at once, and can improve generalization. **Hidden** The number and size of each hidden layer in the model. For example, if a user specifies "100,200,100" a model with 3 hidden layers will be produced, and the middle hidden layer will have 200 neurons.To specify a grid search, add parentheses around each model's specification: "(100,100), (50,50,50), (20,20,20,20)". **Epochs** The number of passes over the training dataset to be carried out. It is recommended to start with lower values for initial grid searches. This value can be modified during checkpoint restarts and allows continuation of selected models. **Train Samples per Iteration** The number of training data rows to be processed per iteration. Note that independent of this parameter, each row is used immediately to update the model with (online) stochastic gradient descent. This parameter controls the synchronization period between nodes in a distributed environment and the frequency at which scoring and model cancellation can happen. For example, if it is set to 10,000 on H\ :sub:`2`\ O running on 4 nodes, then each node will process 2,500 rows per iteration, sampling randomly from their local data. Then, model averaging between the nodes takes place, and scoring can happen (dependent on scoring interval and duty factor). Special values are 0 for one epoch per iteration and -1 for processing the maximum amount of data per iteration. If **replicate training data** is enabled, N epochs will be trained per iteration on N nodes, otherwise one epoch. **Seed** The random seed controls sampling and initialization. Reproducible results are only expected with single-threaded operation (i.e., when running on one node, turning off load balancing and providing a small dataset that fits in one chunk). In general, the multi-threaded asynchronous updates to the model parameters will result in (intentional) race conditions and non-reproducible results. Note that deterministic sampling and initialization might still lead to some weak sense of determinism in the model. **Adaptive Rate** The implemented adaptive learning rate algorithm (ADADELTA) automatically combines the benefits of learning rate annealing and momentum training to avoid slow convergence. Specification of only two parameters (rho and epsilon) simplifies hyper-parameter search. In some cases, manually controlled (non-adaptive) learning rate and momentum specifications can lead to better results, but require the specification (and hyper parameter search) of up to 7 parameters. If the model is built on a topology with many local minima or long plateaus, it is possible for a constant learning rate to produce sub-optimal results. Learning rate annealing allows digging deeper into local minima, while rate decay allows specification of different learning rates per layer. When the gradient is being estimated in a long valley in the optimization landscape, a large learning rate can cause the gradient to oscillate and move in the wrong direction. When the gradient is computed on a relatively flat surface with small learning rates, the model can converge far slower than necessary. **Rho** The first of two hyper parameters for ADADELTA. It is similar to momentum and relates to the memory to prior weight updates. Typical values are between 0.9 and 0.999. This parameter is only active if adaptive learning rate is enabled. **Epsilon** The second of two hyper parameters for ADADELTA. It is similar to learning rate annealing during initial training and momentum at later stages where it allows forward progress. Typical values are between 1e-10 and 1e-4. This parameter is only active if adaptive learning rate is enabled. **Rate** When adaptive learning rate is disabled, the magnitude of the weight updates are determined by the user specified learning rate (potentially annealed), and are a function of the difference between the predicted value and the target value. That difference, generally called delta, is only available at the output layer. To correct the output at each hidden layer, back propagation is used. Momentum modifies back propagation by allowing prior iterations to influence the current update. Using the momentum parameter can aid in avoiding local minima and the associated instability. Too much momentum can lead to instabilities, that's why the momentum is best ramped up slowly. This parameter is only active if adaptive learning rate is disabled. **Rate Annealing** Learning rate annealing reduces the learning rate to "freeze" into local minima in the optimization landscape. The annealing rate is the inverse of the number of training samples it takes to cut the learning rate in half (e.g., 1e-6 means that it takes 1e6 training samples to halve the learning rate). This parameter is only active if adaptive learning rate is disabled. **Rate Decay** The learning rate decay parameter controls the change of learning rate across layers. For example, assume the rate parameter is set to 0.01, and the rate decay parameter is set to 0.5. Then the learning rate for the weights connecting the input and first hidden layer will be 0.01, the learning rate for the weights connecting the first and the second hidden layer will be 0.005, and the learning rate for the weights connecting the second and third hidden layer will be 0.0025, etc. This parameter is only active if adaptive learning rate is disabled. **Momentum Start** The momentum_start parameter controls the amount of momentum at the beginning of training. This parameter is only active if adaptive learning rate is disabled. **Momentum Ramp** The momentum_ramp parameter controls the amount of learning for which momentum increases (assuming momentum_stable is larger than momentum_start). The ramp is measured in the number of training samples. This parameter is only active if adaptive learning rate is disabled. **Momentum Stable** The momentum_stable parameter controls the final momentum value reached after momentum_ramp training samples. The momentum used for training will remain the same for training beyond reaching that point. This parameter is only active if adaptive learning rate is disabled. **Nesterov Accelerated Gradient** The Nesterov accelerated gradient descent method is a modification to traditional gradient descent for convex functions. The method relies on gradient information at various points to build a polynomial approximation that minimizes the residuals in fewer iterations of the descent. This parameter is only active if adaptive learning rate is disabled. **Input Dropout Ratio** A fraction of the features for each training row to be omitted from training in order to improve generalization (dimension sampling). **Hidden Dropout Ratios** A fraction of the inputs for each hidden layer to be omitted from training in order to improve generalization. Defaults to 0.5 for each hidden layer if omitted. **L1 Regularization** A regularization method that constrains the absolute value of the weights and has the net effect of dropping some weights (setting them to zero) from a model to reduce complexity and avoid overfitting. **L2 Regularization** A regularization method that constrains the sum of the squared weights. This method introduces bias into parameter estimates, but frequently produces substantial gains in modeling as estimate variance is reduced. **Max w2** A maximum on the sum of the squared incoming weights into any one neuron. This tuning parameter is especially useful for unbound activation functions such as Maxout or Rectifier. **Initial Weight Distribution** The distribution from which initial weights are to be drawn. The default option is an optimized initialization that considers the size of the network. The "uniform" option uses a uniform distribution with a mean of 0 and a given interval. The "normal" option draws weights from the standard normal distribution with a mean of 0 and given standard deviation. **Initial Weight Scale** The scale of the distribution function for Uniform or Normal distributions. For Uniform, the values are drawn uniformly from initial weight scale. For Normal, the values are drawn from a Normal distribution with the standard deviation of the initial weight scale. **Loss Function** The loss (error) function to be optimized by the model. *Cross Entropy* Used when the model output consists of independent hypotheses, and the outputs can be interpreted as the probability that each hypothesis is true. Cross entropy is the recommended loss function when the target values are class labels, and especially for imbalanced data. It strongly penalizes error in the prediction of the actual class label. *Mean Square* Used when the model output are continuous real values, but can be used for classification as well (where it emphasizes the error on all output classes, not just for the actual class). **Score Interval** The minimum time (in seconds) to elapse between model scoring. The actual interval is determined by the number of training samples per iteration and the scoring duty cycle. **Score Training Samples** The number of training dataset points to be used for scoring. Will be randomly sampled. Use 0 for selecting the entire training dataset. **Score Validation Samples** The number of validation dataset points to be used for scoring. Can be randomly sampled or stratified (if "balance classes" is set and "score validation sampling" is set to stratify). Use 0 for selecting the entire training dataset. **Score Duty Cycle** Maximum fraction of wall clock time spent on model scoring on training and validation samples, and on diagnostics such as computation of feature importances (i.e., not on training). **Classification Stop** The stopping criteria in terms of classification error (1-accuracy) on the training data scoring dataset. When the error is at or below this threshold, training stops. **Regression Stop** The stopping criteria in terms of regression error (MSE) on the training data scoring dataset. When the error is at or below this threshold, training stops. **Quiet Mode** Enable quiet mode for less output to standard output. **Max Confusion Matrix** For classification models, the maximum size (in terms of classes) of the confusion matrix for it to be printed. This option is meant to avoid printing extremely large confusion matrices. **Max Hit Ratio K** The maximum number (top K) of predictions to use for hit ratio computation (for multi-class only, 0 to disable) **Balance Classes** For imbalanced data, balance training data class counts via over/under-sampling. This can result in improved predictive accuracy. **Max After Balance Size** When classes are balanced, limit the resulting dataset size to the specified multiple of the original dataset size. **Score Validation Sampling** Method used to sample the validation dataset for scoring, see Score Validation Samples above. **Diagnostics** Gather diagnostics for hidden layers, such as mean and RMS values of learning rate, momentum, weights and biases. **Variable Importance** Whether to compute variable importances for input features. The implemented method (by Gedeon) considers the weights connecting the input features to the first two hidden layers. **Fast Mode** Enable fast mode (minor approximation in back-propagation), should not affect results significantly. **Ignore Const Cols** Ignore constant training columns (no information can be gained anyway). **Force Load Balance** Increase training speed on small datasets by splitting it into many chunks to allow utilization of all cores. **Replicate Training Data** Replicate the entire training dataset onto every node for faster training on small datasets. **Single Node Mode** Run on a single node for fine-tuning of model parameters. Can be useful for checkpoint resumes after training on multiple nodes for fast initial convergence. **Shuffle Training Data** Enable shuffling of training data (on each node). This option is recommended if training data is replicated on N nodes, and the number of training samples per iteration is close to N times the dataset size, where all nodes train will (almost) all the data. It is automatically enabled if the number of training samples per iteration is set to -1 (or to N times the dataset size or larger). Interpreting A Model """""""""""""""""""""""" The model view page displays information about the Deep Learning model being trained. **Diagnostics Table** If diagnostics is enabled, information for each layer is displayed. *Units* The number of units (or artificial neurons) in the layer *Type* The type of layer (used activation function). Each model will have one input and one output layer. Hidden layers are identified by the activation function specified. *Dropout* For input layer, the percentage of dropped features for each training row. For hidden layers, the percentage of incoming weights dropped from training at that layer. Note that dropout is randomized for each training row. *L1, L2* The L1 and L2 regularization penalty applied by layer. *Rate, Weight and Bias* The per-layer learning rate, weight and bias statistics are displayed. **Scoring** If a validation set was given, the scoring results are displayed for the validation set (or a sample thereof). Otherwise, scoring is performed on the training dataset (or a sample thereof). **Confusion Matrix** For classification models, a table showing the number of actual observations in a particular class relative to the number of predicted observations in a class. **Hit Ratio Table** A table displaying the percentage of instances where the actual class label assigned to an observation is in the top K classes predicted by the model. For instance, in a four class classifier on values A, B, C, D, a particular observation is predicted to be class A with a probability of .6 of being A, .2 probability of being B, a .1 probability of being C, and a .1 probability of being D. If the true class is B, the observation will be counted in the hit rate for K=2, but not in the hit rate of K=1. **Variable Importance** A table listing the importance of variables listed from greatest importance, to least importance. Note that variable importances are notoriously difficult to compute for Neural Net models. Gedeon's method is implemented here. References """""""""""""""""""""""""""""""" Deep Learning http://en.wikipedia.org/wiki/Deep_learning Artificial Neural Network http://en.wikipedia.org/wiki/Artificial_neural_network ADADELTA http://arxiv.org/abs/1212.5701 Momentum http://www.cs.toronto.edu/~fritz/absps/momentum.pdf Dropout http://arxiv.org/pdf/1207.0580.pdf and http://arxiv.org/abs/1307.1493 Feature Importance http://www.ncbi.nlm.nih.gov/pubmed/9327276