def kl(self, other):
     a0 = self.logits - tf.reduce_max(self.logits, axis=-1, keepdims=True)
     a1 = other.logits - tf.reduce_max(other.logits, axis=-1, keepdims=True)
     ea0 = tf.exp(a0)
     ea1 = tf.exp(a1)
     z0 = tf.reduce_sum(ea0, axis=-1, keepdims=True)
     z1 = tf.reduce_sum(ea1, axis=-1, keepdims=True)
     p0 = ea0 / z0
     return tf.reduce_sum(p0 * (a0 - tf.log(z0) - a1 + tf.log(z1)), axis=-1)
 def __init__(self, flat):
     self.flat = flat
     mean, logstd = tf.split(axis=len(flat.shape) - 1,
                             num_or_size_splits=2,
                             value=flat)
     self.mean = mean
     self.logstd = logstd
     self.std = tf.exp(logstd)
def learn(*,
          network,
          env,
          total_timesteps,
          timesteps_per_batch=1024,  # what to train on
          max_kl=0.001,
          cg_iters=10,
          gamma=0.99,
          lam=1.0,  # advantage estimation
          seed=None,
          ent_coef=0.0,
          cg_damping=1e-2,
          vf_stepsize=3e-4,
          vf_iters=3,
          max_episodes=0, max_iters=0,  # time constraint
          callback=None,
          load_path=None,
          **network_kwargs
          ):
    '''
    learn a policy function with TRPO algorithm

    Parameters:
    ----------

    network                 neural network to learn. Can be either string ('mlp', 'cnn', 'lstm', 'lnlstm' for basic types)
                            or function that takes input placeholder and returns tuple (output, None) for feedforward nets
                            or (output, (state_placeholder, state_output, mask_placeholder)) for recurrent nets

    env                     environment (one of the gym environments or wrapped via tensorflow_code-pytorch.common.vec_env.VecEnv-type class

    timesteps_per_batch     timesteps per gradient estimation batch

    max_kl                  max KL divergence between old policy and new policy ( KL(pi_old || pi) )

    ent_coef                coefficient of policy entropy term in the optimization objective

    cg_iters                number of iterations of conjugate gradient algorithm

    cg_damping              conjugate gradient damping

    vf_stepsize             learning rate for adam optimizer used to optimie value function loss

    vf_iters                number of iterations of value function optimization iterations per each policy optimization step

    total_timesteps           max number of timesteps

    max_episodes            max number of episodes

    max_iters               maximum number of policy optimization iterations

    callback                function to be called with (locals(), globals()) each policy optimization step

    load_path               str, path to load the model from (default: None, i.e. no model is loaded)

    **network_kwargs        keyword arguments to the policy / network builder. See baselines.common/policies.py/build_policy and arguments to a particular type of network

    Returns:
    -------

    learnt model

    '''

    nworkers = MPI.COMM_WORLD.Get_size()
    rank = MPI.COMM_WORLD.Get_rank()

    cpus_per_worker = 1
    U.get_session(config=tf.ConfigProto(
        allow_soft_placement=True,
        inter_op_parallelism_threads=cpus_per_worker,
        intra_op_parallelism_threads=cpus_per_worker
    ))

    policy = build_policy(env, network, value_network='copy', **network_kwargs)
    set_global_seeds(seed)

    np.set_printoptions(precision=3)
    # Setup losses and stuff
    # ----------------------------------------
    ob_space = env.observation_space
    ac_space = env.action_space

    ob = observation_placeholder(ob_space)
    with tf.variable_scope("pi"):
        pi = policy(observ_placeholder=ob)
    with tf.variable_scope("oldpi"):
        oldpi = policy(observ_placeholder=ob)

    atarg = tf.placeholder(dtype=tf.float32, shape=[None])  # Target advantage function (if applicable)
    ret = tf.placeholder(dtype=tf.float32, shape=[None])  # Empirical return

    ac = pi.pdtype.sample_placeholder([None])

    kloldnew = oldpi.pd.kl(pi.pd)
    ent = pi.pd.entropy()
    meankl = tf.reduce_mean(kloldnew)
    meanent = tf.reduce_mean(ent)
    entbonus = ent_coef * meanent

    vferr = tf.reduce_mean(tf.square(pi.vf - ret))

    ratio = tf.exp(pi.pd.logp(ac) - oldpi.pd.logp(ac))  # advantage * pnew / pold
    surrgain = tf.reduce_mean(ratio * atarg)

    optimgain = surrgain + entbonus
    losses = [optimgain, meankl, entbonus, surrgain, meanent]
    loss_names = ["optimgain", "meankl", "entloss", "surrgain", "entropy"]

    dist = meankl

    all_var_list = get_trainable_variables("pi")
    # var_list = [v for v in all_var_list if v.name.split("/")[1].startswith("pol")]
    # vf_var_list = [v for v in all_var_list if v.name.split("/")[1].startswith("vf")]
    var_list = get_pi_trainable_variables("pi")
    vf_var_list = get_vf_trainable_variables("pi")

    vfadam = MpiAdam(vf_var_list)

    get_flat = U.GetFlat(var_list)
    set_from_flat = U.SetFromFlat(var_list)
    klgrads = tf.gradients(dist, var_list)
    flat_tangent = tf.placeholder(dtype=tf.float32, shape=[None], name="flat_tan")
    shapes = [var.get_shape().as_list() for var in var_list]
    start = 0
    tangents = []
    for shape in shapes:
        sz = U.intprod(shape)
        tangents.append(tf.reshape(flat_tangent[start:start + sz], shape))
        start += sz
    gvp = tf.add_n([tf.reduce_sum(g * tangent) for (g, tangent) in zipsame(klgrads, tangents)])  # pylint: disable=E1111
    fvp = U.flatgrad(gvp, var_list)

    assign_old_eq_new = U.function([], [], updates=[tf.assign(oldv, newv)
                                                    for (oldv, newv) in
                                                    zipsame(get_variables("oldpi"), get_variables("pi"))])

    compute_losses = U.function([ob, ac, atarg], losses)
    compute_lossandgrad = U.function([ob, ac, atarg], losses + [U.flatgrad(optimgain, var_list)])
    compute_fvp = U.function([flat_tangent, ob, ac, atarg], fvp)
    compute_vflossandgrad = U.function([ob, ret], U.flatgrad(vferr, vf_var_list))

    @contextmanager
    def timed(msg):
        if rank == 0:
            print(colorize(msg, color='magenta'))
            tstart = time.time()
            yield
            print(colorize("done in %.3f seconds" % (time.time() - tstart), color='magenta'))
        else:
            yield

    def allmean(x):
        assert isinstance(x, np.ndarray)
        out = np.empty_like(x)
        MPI.COMM_WORLD.Allreduce(x, out, op=MPI.SUM)
        out /= nworkers
        return out

    U.initialize()
    if load_path is not None:
        pi.load(load_path)

    th_init = get_flat()
    MPI.COMM_WORLD.Bcast(th_init, root=0)
    set_from_flat(th_init)
    vfadam.sync()
    print("Init param sum", th_init.sum(), flush=True)

    # Prepare for rollouts
    # ----------------------------------------
    seg_gen = traj_segment_generator(pi, env, timesteps_per_batch, stochastic=True)

    episodes_so_far = 0
    timesteps_so_far = 0
    iters_so_far = 0
    tstart = time.time()
    lenbuffer = deque(maxlen=40)  # rolling buffer for episode lengths
    rewbuffer = deque(maxlen=40)  # rolling buffer for episode rewards

    if sum([max_iters > 0, total_timesteps > 0, max_episodes > 0]) == 0:
        # noththing to be done
        return pi

    assert sum([max_iters > 0, total_timesteps > 0, max_episodes > 0]) < 2, \
        'out of max_iters, total_timesteps, and max_episodes only one should be specified'

    while True:
        if callback: callback(locals(), globals())
        if total_timesteps and timesteps_so_far >= total_timesteps:
            break
        elif max_episodes and episodes_so_far >= max_episodes:
            break
        elif max_iters and iters_so_far >= max_iters:
            break
        logger.log("********** Iteration %i ************" % iters_so_far)

        with timed("sampling"):
            seg = seg_gen.__next__()
        add_vtarg_and_adv(seg, gamma, lam)

        # ob, ac, atarg, ret, td1ret = map(np.concatenate, (obs, acs, atargs, rets, td1rets))
        ob, ac, atarg, tdlamret = seg["ob"], seg["ac"], seg["adv"], seg["tdlamret"]
        vpredbefore = seg["vpred"]  # predicted value function before udpate
        atarg = (atarg - atarg.mean()) / atarg.std()  # standardized advantage function estimate

        if hasattr(pi, "ret_rms"): pi.ret_rms.update(tdlamret)
        if hasattr(pi, "ob_rms"): pi.ob_rms.update(ob)  # update running mean/std for policy

        args = seg["ob"], seg["ac"], atarg
        fvpargs = [arr[::5] for arr in args]

        def fisher_vector_product(p):
            return allmean(compute_fvp(p, *fvpargs)) + cg_damping * p

        assign_old_eq_new()  # set old parameter values to new parameter values
        with timed("computegrad"):
            *lossbefore, g = compute_lossandgrad(*args)
        lossbefore = allmean(np.array(lossbefore))
        g = allmean(g)
        if np.allclose(g, 0):
            logger.log("Got zero gradient. not updating")
        else:
            with timed("cg"):
                stepdir = cg(fisher_vector_product, g, cg_iters=cg_iters, verbose=rank == 0)
            assert np.isfinite(stepdir).all()
            shs = .5 * stepdir.dot(fisher_vector_product(stepdir))
            lm = np.sqrt(shs / max_kl)

            # logger.log("lagrange multiplier:", lm, "gnorm:", np.linalg.norm(g))
            fullstep = stepdir / lm
            expectedimprove = g.dot(fullstep)
            surrbefore = lossbefore[0]
            stepsize = 1.0
            thbefore = get_flat()
            for _ in range(10):
                thnew = thbefore + fullstep * stepsize
                set_from_flat(thnew)
                meanlosses = surr, kl, *_ = allmean(np.array(compute_losses(*args)))
                improve = surr - surrbefore
                logger.log("Expected: %.3f Actual: %.3f" % (expectedimprove, improve))
                if not np.isfinite(meanlosses).all():
                    logger.log("Got non-finite value of losses -- bad!")
                elif kl > max_kl * 1.5:
                    logger.log("violated KL constraint. shrinking step.")
                elif improve < 0:
                    logger.log("surrogate didn't improve. shrinking step.")
                else:
                    logger.log("Stepsize OK!")
                    break
                stepsize *= .5
            else:
                logger.log("couldn't compute a good step")
                set_from_flat(thbefore)
            if nworkers > 1 and iters_so_far % 20 == 0:
                paramsums = MPI.COMM_WORLD.allgather((thnew.sum(), vfadam.getflat().sum()))  # list of tuples
                assert all(np.allclose(ps, paramsums[0]) for ps in paramsums[1:])

        for (lossname, lossval) in zip(loss_names, meanlosses):
            logger.record_tabular(lossname, lossval)

        with timed("vf"):

            for _ in range(vf_iters):
                for (mbob, mbret) in dataset.iterbatches((seg["ob"], seg["tdlamret"]),
                                                         include_final_partial_batch=False, batch_size=64):
                    g = allmean(compute_vflossandgrad(mbob, mbret))
                    vfadam.update(g, vf_stepsize)

        logger.record_tabular("ev_tdlam_before", explained_variance(vpredbefore, tdlamret))

        lrlocal = (seg["ep_lens"], seg["ep_rets"])  # local values
        listoflrpairs = MPI.COMM_WORLD.allgather(lrlocal)  # list of tuples
        lens, rews = map(flatten_lists, zip(*listoflrpairs))
        lenbuffer.extend(lens)
        rewbuffer.extend(rews)

        logger.record_tabular("EpLenMean", np.mean(lenbuffer))
        logger.record_tabular("EpRewMean", np.mean(rewbuffer))
        logger.record_tabular("EpThisIter", len(lens))
        episodes_so_far += len(lens)
        timesteps_so_far += sum(lens)
        iters_so_far += 1

        logger.record_tabular("EpisodesSoFar", episodes_so_far)
        logger.record_tabular("TimestepsSoFar", timesteps_so_far)
        logger.record_tabular("TimeElapsed", time.time() - tstart)

        if rank == 0:
            logger.dump_tabular()

    return pi
Пример #4
0
def learn(
    env,
    policy_fn,
    *,
    timesteps_per_actorbatch,  # timesteps per actor per update
    clip_param,
    entcoeff,  # clipping parameter epsilon, entropy coeff
    optim_epochs,
    optim_stepsize,
    optim_batchsize,  # optimization hypers
    gamma,
    lam,  # advantage estimation
    max_timesteps=0,
    max_episodes=0,
    max_iters=0,
    max_seconds=0,  # time constraint
    callback=None,  # you can do anything in the callback, since it takes locals(), globals()
    adam_epsilon=1e-5,
    schedule='constant'  # annealing for stepsize parameters (epsilon and adam)
):
    # Setup losses and stuff
    # ----------------------------------------
    ob_space = env.observation_space
    ac_space = env.action_space
    pi = policy_fn("pi", ob_space,
                   ac_space)  # Construct network for new policy
    oldpi = policy_fn("oldpi", ob_space, ac_space)  # Network for old policy
    atarg = tf.placeholder(
        dtype=tf.float32,
        shape=[None])  # Target advantage function (if applicable)
    ret = tf.placeholder(dtype=tf.float32, shape=[None])  # Empirical return

    lrmult = tf.placeholder(
        name='lrmult', dtype=tf.float32,
        shape=[])  # learning rate multiplier, updated with schedule
    clip_param = clip_param * lrmult  # Annealed cliping parameter epislon

    ob = U.get_placeholder_cached(name="ob")
    ac = pi.pdtype.sample_placeholder([None])

    kloldnew = oldpi.pd.kl(pi.pd)
    ent = pi.pd.entropy()
    meankl = tf.reduce_mean(kloldnew)
    meanent = tf.reduce_mean(ent)
    pol_entpen = (-entcoeff) * meanent

    ratio = tf.exp(pi.pd.logp(ac) - oldpi.pd.logp(ac))  # pnew / pold
    surr1 = ratio * atarg  # surrogate from conservative policy iteration
    surr2 = tf.clip_by_value(ratio, 1.0 - clip_param,
                             1.0 + clip_param) * atarg  #
    pol_surr = -tf.reduce_mean(tf.minimum(
        surr1, surr2))  # PPO's pessimistic surrogate (L^CLIP)
    vf_loss = tf.reduce_mean(tf.square(pi.vpred - ret))
    total_loss = pol_surr + pol_entpen + vf_loss
    losses = [pol_surr, pol_entpen, vf_loss, meankl, meanent]
    loss_names = ["pol_surr", "pol_entpen", "vf_loss", "kl", "ent"]

    var_list = pi.get_trainable_variables()
    lossandgrad = U.function([ob, ac, atarg, ret, lrmult],
                             losses + [U.flatgrad(total_loss, var_list)])
    adam = MpiAdam(var_list, epsilon=adam_epsilon)

    assign_old_eq_new = U.function(
        [], [],
        updates=[
            tf.assign(oldv, newv)
            for (oldv,
                 newv) in zipsame(oldpi.get_variables(), pi.get_variables())
        ])
    compute_losses = U.function([ob, ac, atarg, ret, lrmult], losses)

    U.initialize()
    adam.sync()

    # Prepare for rollouts
    # ----------------------------------------
    seg_gen = traj_segment_generator(pi,
                                     env,
                                     timesteps_per_actorbatch,
                                     stochastic=True)

    episodes_so_far = 0
    timesteps_so_far = 0
    iters_so_far = 0
    tstart = time.time()
    lenbuffer = deque(maxlen=100)  # rolling buffer for episode lengths
    rewbuffer = deque(maxlen=100)  # rolling buffer for episode rewards

    assert sum(
        [max_iters > 0, max_timesteps > 0, max_episodes > 0,
         max_seconds > 0]) == 1, "Only one time constraint permitted"

    while True:
        if callback: callback(locals(), globals())
        if max_timesteps and timesteps_so_far >= max_timesteps:
            break
        elif max_episodes and episodes_so_far >= max_episodes:
            break
        elif max_iters and iters_so_far >= max_iters:
            break
        elif max_seconds and time.time() - tstart >= max_seconds:
            break

        if schedule == 'constant':
            cur_lrmult = 1.0
        elif schedule == 'linear':
            cur_lrmult = max(1.0 - float(timesteps_so_far) / max_timesteps, 0)
        else:
            raise NotImplementedError

        logger.log("********** Iteration %i ************" % iters_so_far)

        seg = seg_gen.__next__()
        add_vtarg_and_adv(seg, gamma, lam)

        # ob, ac, atarg, ret, td1ret = map(np.concatenate, (obs, acs, atargs, rets, td1rets))
        ob, ac, atarg, tdlamret = seg["ob"], seg["ac"], seg["adv"], seg[
            "tdlamret"]
        vpredbefore = seg["vpred"]  # predicted value function before udpate
        atarg = (atarg - atarg.mean()
                 ) / atarg.std()  # standardized advantage function estimate
        d = Dataset(dict(ob=ob, ac=ac, atarg=atarg, vtarg=tdlamret),
                    shuffle=not pi.recurrent)
        optim_batchsize = optim_batchsize or ob.shape[0]

        if hasattr(pi, "ob_rms"):
            pi.ob_rms.update(ob)  # update running mean/std for policy

        assign_old_eq_new()  # set old parameter values to new parameter values
        logger.log("Optimizing...")
        logger.log(fmt_row(13, loss_names))
        # Here we do a bunch of optimization epochs over the data
        for _ in range(optim_epochs):
            losses = [
            ]  # list of tuples, each of which gives the loss for a minibatch
            for batch in d.iterate_once(optim_batchsize):
                *newlosses, g = lossandgrad(batch["ob"], batch["ac"],
                                            batch["atarg"], batch["vtarg"],
                                            cur_lrmult)
                adam.update(g, optim_stepsize * cur_lrmult)
                losses.append(newlosses)
            logger.log(fmt_row(13, np.mean(losses, axis=0)))

        logger.log("Evaluating losses...")
        losses = []
        for batch in d.iterate_once(optim_batchsize):
            newlosses = compute_losses(batch["ob"], batch["ac"],
                                       batch["atarg"], batch["vtarg"],
                                       cur_lrmult)
            losses.append(newlosses)
        meanlosses, _, _ = mpi_moments(losses, axis=0)
        logger.log(fmt_row(13, meanlosses))
        for (lossval, name) in zipsame(meanlosses, loss_names):
            logger.record_tabular("loss_" + name, lossval)
        logger.record_tabular("ev_tdlam_before",
                              explained_variance(vpredbefore, tdlamret))
        lrlocal = (seg["ep_lens"], seg["ep_rets"])  # local values
        listoflrpairs = MPI.COMM_WORLD.allgather(lrlocal)  # list of tuples
        lens, rews = map(flatten_lists, zip(*listoflrpairs))
        lenbuffer.extend(lens)
        rewbuffer.extend(rews)
        logger.record_tabular("EpLenMean", np.mean(lenbuffer))
        logger.record_tabular("EpRewMean", np.mean(rewbuffer))
        logger.record_tabular("EpThisIter", len(lens))
        episodes_so_far += len(lens)
        timesteps_so_far += sum(lens)
        iters_so_far += 1
        logger.record_tabular("EpisodesSoFar", episodes_so_far)
        logger.record_tabular("TimestepsSoFar", timesteps_so_far)
        logger.record_tabular("TimeElapsed", time.time() - tstart)
        if MPI.COMM_WORLD.Get_rank() == 0:
            logger.dump_tabular()

    return pi
Пример #5
0
    def __init__(self, *, policy, ob_space, ac_space, nbatch_act, nbatch_train,
                nsteps, ent_coef, vf_coef, max_grad_norm):
        sess = get_session()

        with tf.variable_scope('ppo2_model', reuse=tf.AUTO_REUSE):
            # CREATE OUR TWO MODELS
            # act_model that is used for sampling
            act_model = policy(nbatch_act, 1, sess)

            # Train model for training
            train_model = policy(nbatch_train, nsteps, sess)

        # CREATE THE PLACEHOLDERS
        A = train_model.pdtype.sample_placeholder([None])
        ADV = tf.placeholder(tf.float32, [None])
        R = tf.placeholder(tf.float32, [None])
        # Keep track of old actor
        OLDNEGLOGPAC = tf.placeholder(tf.float32, [None])
        # Keep track of old critic
        OLDVPRED = tf.placeholder(tf.float32, [None])
        LR = tf.placeholder(tf.float32, [])
        # Cliprange
        CLIPRANGE = tf.placeholder(tf.float32, [])

        neglogpac = train_model.pd.neglogp(A)

        # Calculate the entropy
        # Entropy is used to improve exploration by limiting the premature convergence to suboptimal policy.
        entropy = tf.reduce_mean(train_model.pd.entropy())

        # CALCULATE THE LOSS
        # Total loss = Policy gradient loss - entropy * entropy coefficient + Value coefficient * value loss

        # Clip the value to reduce variability during Critic training
        # Get the predicted value
        vpred = train_model.vf
        vpredclipped = OLDVPRED + tf.clip_by_value(train_model.vf - OLDVPRED, - CLIPRANGE, CLIPRANGE)
        # Unclipped value
        vf_losses1 = tf.square(vpred - R)
        # Clipped value
        vf_losses2 = tf.square(vpredclipped - R)

        vf_loss = .5 * tf.reduce_mean(tf.maximum(vf_losses1, vf_losses2))

        # Calculate ratio (pi current policy / pi old policy)
        ratio = tf.exp(OLDNEGLOGPAC - neglogpac)

        # Defining Loss = - J is equivalent to max J
        pg_losses = -ADV * ratio

        pg_losses2 = -ADV * tf.clip_by_value(ratio, 1.0 - CLIPRANGE, 1.0 + CLIPRANGE)

        # Final PG loss
        pg_loss = tf.reduce_mean(tf.maximum(pg_losses, pg_losses2))
        approxkl = .5 * tf.reduce_mean(tf.square(neglogpac - OLDNEGLOGPAC))
        clipfrac = tf.reduce_mean(tf.to_float(tf.greater(tf.abs(ratio - 1.0), CLIPRANGE)))

        # Total loss
        loss = pg_loss - entropy * ent_coef + vf_loss * vf_coef

        # UPDATE THE PARAMETERS USING LOSS
        # 1. Get the model parameters
        params = tf.trainable_variables('ppo2_model')
        # 2. Build our trainer
        trainer = MpiAdamOptimizer(MPI.COMM_WORLD, learning_rate=LR, epsilon=1e-5)
        # 3. Calculate the gradients
        grads_and_var = trainer.compute_gradients(loss, params)
        grads, var = zip(*grads_and_var)

        if max_grad_norm is not None:
            # Clip the gradients (normalize)
            grads, _grad_norm = tf.clip_by_global_norm(grads, max_grad_norm)
        grads_and_var = list(zip(grads, var))
        # zip aggregate each gradient with parameters associated
        # For instance zip(ABCD, xyza) => Ax, By, Cz, Da

        _train = trainer.apply_gradients(grads_and_var)

        def train(lr, cliprange, obs, returns, masks, actions, values, neglogpacs, states=None):
            # Here we calculate advantage A(s,a) = R + yV(s') - V(s)
            # Returns = R + yV(s')
            advs = returns - values

            # Normalize the advantages
            advs = (advs - advs.mean()) / (advs.std() + 1e-8)
            td_map = {train_model.X:obs, A:actions, ADV:advs, R:returns, LR:lr,
                    CLIPRANGE:cliprange, OLDNEGLOGPAC:neglogpacs, OLDVPRED:values}
            if states is not None:
                td_map[train_model.S] = states
                td_map[train_model.M] = masks
            return sess.run(
                [pg_loss, vf_loss, entropy, approxkl, clipfrac, _train],
                td_map
            )[:-1]
        self.loss_names = ['policy_loss', 'value_loss', 'policy_entropy', 'approxkl', 'clipfrac']


        self.train = train
        self.train_model = train_model
        self.act_model = act_model
        self.step = act_model.step
        self.value = act_model.value
        self.initial_state = act_model.initial_state

        self.save = functools.partial(save_variables, sess=sess)
        self.load = functools.partial(load_variables, sess=sess)

        if MPI.COMM_WORLD.Get_rank() == 0:
            initialize()
        global_variables = tf.get_collection(tf.GraphKeys.GLOBAL_VARIABLES, scope="")
        sync_from_root(sess, global_variables) #pylint: disable=E1101
 def entropy(self):
     a0 = self.logits - tf.reduce_max(self.logits, axis=-1, keepdims=True)
     ea0 = tf.exp(a0)
     z0 = tf.reduce_sum(ea0, axis=-1, keepdims=True)
     p0 = ea0 / z0
     return tf.reduce_sum(p0 * (tf.log(z0) - a0), axis=-1)